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+{"text":"\\section{Introduction}\nIn the composite fermion (CF) picture of the fractional quantum Hall\n(FQH) effect, fundamental interactions are taken into account at the\nmean field level by mapping the system of strongly interacting 2D\nelectrons in magnetic field into a system of weakly interacting\nComposite Fermions (CF) moving in a reduced effective magnetic field\n\\cite{Jain}. The reduction in magnetic field follows from the\nbinding of $\\phi$ flux quanta to electrons, so that effective\nmagnetic field experienced by CF quasiparticles is $B^*=\\pm B\/(\\phi\np\\pm 1)$, where $p$ is an integer that enumerates members of a\nparticular sequence and $\\phi$ is a even integer that labels\ndifferent sequences. In this picture, the FQH effect can be\nunderstood by the emergence of CF Landau levels  with cyclotron\nfrequency: $\\omega_{CF}=\\frac{eB^*}{cm^{*}}$, where $m^{*}$ is an\neffective CF mass. Evidence for a spin split Landau level structure\nof CF for the $\\phi=2$ sequence has been provided by\nmagnetotransport experiments in tilted magnetic fields at filling\nfactors near $\\nu=3\/2$ \\cite{Du} and by inelastic light scattering\nstudies of spin excitations in the range $1\/3<\\nu<2\/5$ \\cite{Irene}.\nFor the $\\phi=4$ sequence  (i.e. $\\nu\\lesssim1\/3$), however, direct\nevidence for such CF Landau level structure is lacking. Studies of\nthe $\\phi=4$ sequence are more difficult because of the higher\nmagnetic fields that are required and, compared to the $\\phi=2$\nsequence, the smaller energy scales in the excitations. Insight on\nthe energy scales for excitations were revealed by activated\ntransport measurements \\cite{Pan} and by the recent observations of\n$\\phi=4$ quasiparticle excitations in light scattering experiments\n\\cite{Cyrus}.\n\nIn this work, we present a resonant inelastic light scattering study\nof spin excitations for $\\nu\\lesssim1\/3$. The excitations are spin\nwaves (SW) and spin-flip (SF) modes. The SF excitations involve a\nchange in both the spin orientation and CF Landau level quantum\nnumber. We monitor the evolution of these spin excitations below and\naway from $\\nu=1\/3$ when the population of the excited CF Landau\nlevel increases. Our results reveal the existence of spin split CF\nLandau levels in the $\\phi=4$ sequence. The SW-SF splitting is\nlinear in magnetic field. This determination suggests an effective\nmass significantly larger than the activation mass of CF with\n$\\phi=4$.\n\\section{Sample and Experiment}\nThe 2D electron (2DES) system studied here is a GaAs single quantum\nwell of width $w=330~\\AA$. The electron density at small magnetic\nfields is n=5.5$\\times$10$^{10}$~cm$^{-2}$ and the low temperature\nmobility is $\\mu$=7.2$\\times$10$^6$\/Vs. The sample is mounted in a\nbackscattering geometry, making an angle $\\theta$ between the\nincident photons and the normal to the sample surface. The magnetic\nfield perpendicular to the sample is B=B$_T$cos$\\theta$, where B$_T$ is\nthe total applied field. The results reported here have been\nobtained at $\\theta$=50$\\pm$2 degrees. Similar results have been seen at 30 degrees \\cite{Cyrus-thesis}.The ingoing and outgoing\npolarizations were chosen to be orthogonal (depolarized spectra)\nsince excitations which involve a change in the spin degrees of\nfreedom are stronger in this configuration. The sample was cooled in\na dilution refrigerator with windows for optical access. All the\nmeasurements were performed at the base temperature T=23~mK and the\npower density was kept below 10$^{-5}$W\/cm$^2$ to avoid heating the\nelectron system. The energy of the incident photons was tuned to be\nin resonance with the excitonic optical transitions of the 2DES in\nthe FQH regime \\cite{Goldberg,Bar-joseph,Cyrus2}.\n\\section{Results and discussion}\n\\begin{figure}\n\\centering \\epsfig{figure=spectres.eps, width=0.79\\linewidth, clip=}\n\\caption{Low energy spectra of spin excitations in the filling\nfactor range 0.31$<\\nu<$0.33. The most intense peak is the long\nwavelength spin-wave at the Zeeman energy E$_z$ while the peak its\nlow energy side is assigned to a spin-flip transition (see text and\nfigure \\ref{levels}). The left inset shows the result of a\ntwo-gaussian fitting procedure for the two peaks to extract their\nrespective energies. The right inset shows the backscattering\nconfiguration} \\label{spectres}\n\\end{figure}\nFigure \\ref{spectres} shows the evolution of the low energy spectrum\nfor $\\nu\\lesssim1\/3$. $\\nu=1\/3$ corresponds to a perpendicular field\nof 6.5T and the filling factor range studied corresponds to the\nrange 0.31$<\\nu<$0.33. Close to $\\nu=1\/3$, the spectra are dominated\nby the long wavelength SW at the 'bare' Zeeman energy\nE$_z=g\\mu_BB_T$, where g=0.44 is the Lande factor for electrons in\nGaAs and $\\mu_B$ is the Bohr magneton. For filling factors away from\n$\\nu=1\/3$ an excitation emerges on the low energy side of E$_z$. We\nassign this excitation to a SF mode linked to transitions in the CF\nframework that involve the first excited CF Landau level as depicted\nin figure \\ref{levels} . At $\\nu=1\/3$ the first CF Landau\n($\\ket{0,\\uparrow}$) level is fully occupied while for $\\nu=2\/7$ the\nfirst two CF Landau levels ($\\ket{0,\\uparrow}$ and\n$\\ket{1,\\uparrow}$) are occupied. In between the two incompressible\nstates, the first excited Landau level is partially populated and SF\ntransitions between $\\ket{1,\\uparrow}$ and $\\ket{0,\\downarrow}$\nstarting from the partially filled level become possible. Thus the\nstudy of the SF excitations in the filling factor range\n2\/7$<\\nu<$1\/3 probe directly the CF level structure for $\\phi=4$.\n\\begin{figure}\n\\centering \\epsfig{figure=levels.eps, width=0.7\\linewidth, clip=}\n\\caption{Structure of spin split CF Landau levels for 2\/7$<\\nu<$1\/3 ($\\phi=4$).\nTwo $^4$CF spin transitions are possible. The large $q$ spin wave is\nat E$^*_z$=E$_z$+E$^{\\uparrow\\downarrow}$ where\nE$^{\\uparrow\\downarrow}$ is the spin reversal energy. The spin-flip\nexcitation at E$_{SF}$. The spin-flip excitation emerges when the\n$\\ket{1,\\uparrow}$ level is populated.} \\label{levels}\n\\end{figure}\nFor small occupation of the $\\ket{1,\\uparrow}$ excited level and\nwhen the coupling between the excited quasiparticle and its\nquasihole is negligible, the SF transition energy can be written as\nin the $\\phi=2$ case:\n\\begin{equation}\nE_{SF}=E_z+E^{\\uparrow\\downarrow}-\\hbar\\omega_c\n\\label{ESF}\n\\end{equation}\nwhere E$^{\\uparrow\\downarrow}$ is the spin reversal energy which is\na measure of the residual interactions between $\\phi=4$ CF\nquasiparticles \\cite{Pinczuk,Longo,Aoki,Mandal,Irene}.\n\\begin{figure}\n\\centering \\epsfig{figure=energies.eps, width=0.65\\linewidth, clip=}\n\\caption{Magnetic field dependence of the Zeeman (E$_z$) and\nspin-flip excitation energies for 0.31$<\\nu<$0.33. Also shown is the\nevolution of the splitting E$_z$-E$_{SF}$.} \\label{energies}\n\\end{figure}\nThe energy E$_{SF}$ was extracted for each filling factor by\nperforming a simple analysis of the low energy spectra using a\ntwo-gaussian fitting procedure as shown in the inset of figure\n\\ref{spectres}. Figure \\ref{energies} displays the corresponding\nenergies, E$_z$ and E$_{SF}$ as a function of filling factor. The\nstrong dependence of E$_{SF}$ confirms our assignment of the peak as\nexcitation involving spin degrees of freedom. More importantly, the\nspacing between E$_z$ and E$_{SF}$ is not constant and decreases\nwith the filling factor. From equation \\ref{ESF}, we easily see that\nthis spacing is directly related to the CF cyclotron energy so that\nthe splitting between the two spin excitations is\nE$_z$-E$_{SF}$=$\\hbar\\omega_c$-E$^{\\uparrow\\downarrow}$.\nThe magnetic field dependence of the E$_z$-E$_{SF}$ spacing is set\nby the effective field B$^*$. For the $\\phi=2$ sequence, $^2$CF\nemanate from the $\\nu=1\/2$ state and the effective field has its\norigin at B$_{1\/2}$. For the $\\phi=4$ sequence however, $^4$CF\nemanate from the $\\nu$=1\/4 state and the origin is at B$_{1\/4}$. and\nthe effective magnetic field should then decrease when going from\n0.33 to 0.31. This is indeed consistent with our data and to the\nexistence of $^4$CF or $\\phi=4$ spin-flip excitations below\n$\\nu$=1\/3. Our results support the CF Landau level picture shown figure\n\\ref{levels} for the $\\phi=4$ sequence.\n\nThe linear decrease of E$_z$-E$_{SF}$ with the effective magnetic\nfield makes very tempting the evaluation of an effective mass by using\na slope that is simply given by $\\frac{\\hbar e}{m^{*}c}$ in the\nframework of equation \\ref{ESF} for E$_{SF}$. Our data between\nfilling factors 0.33 and 0.31 give m$^{*}$=1.5($\\pm 0.1$)~m$_e$\nwhere m$_e$ is the bare electron mass. Previous determinations using\nthe activation gap values at $\\nu$=2\/7, 3\/11 and 4\/15 on samples\nwith similar densities yield values around 0.9~m$_e$ \\cite{Pan}. We\nnote that while our determination of m$^{*}$ is performed between\n$\\nu$=0.33 and $\\nu$=0.31, i.e. in between the incompressible states\nat 1\/3 and 2\/7, the effective mass determined via transport\nmeasurements comes from a linear scaling of the activation gap\nvalues at the incompressible states. The high\neffective mass obtained in our analysis may be linked to the onset\nof significant CF interactions in the partially populated CF Landau\nlevel.\n\\begin{figure}\n\\centering\n\\epsfig{figure=intensities.eps, width=0.65\\linewidth, clip=}\n\\caption{Evolution of the SF integrated intensity for 0.31$<\\nu<$0.33.}\n\\label{intensities}\n\\end{figure}\nAdditional insights can be obtained by tracking the evolution of the\nintensity of the SF excitation when the population of the first\nexcited CF level increases. This is done in Fig. \\ref{intensities}\nwhere the integrated spectral weight of the SF excitation is plotted\nas a function of filling factor. As expected, the SF intensity\nincreases when the population of $\\ket{1,\\uparrow}$ increases but\ndisplays an intriguing saturation around below $\\nu$=0.32. As\nalready mentioned for the effective mass, the saturation may result\nfrom increasing impact of CF residual interactions. These\ninteractions could possibly lead to further condensation into higher\norder CF in the partially populated level. Recent transport\nmeasurements have indeed shown the possible existence of such higher\norder states even for the $\\phi$=4 sequence \\cite{Pan2}.\n\\section{Conclusion}\nIn this study, we have shown the existence of spin-flip excitations\nof $^4$CF quasiparticle below $\\nu$=1\/3. The results indicate the\nexistence of a spin-split CF Landau level structure for the $\\phi=4$\nsequence of the fractional quantum Hall effect that is similar to\nthe one found for the $\\phi=2$ sequence. The evolution of the SF\nenergy with effective magnetic field yields an effective mass\nof 1.5m$_e$. The evolution of the SF intensity with filling factor\nmight signal the onset of significant CF-CF interactions that could\npossibly lead to further CF condensation.\n\nThis work is supported by the National Science Foundation under Grant No. NMR-0352738, by the Department of Energy under Grant No. DE-AIO2--04ER46133, and by a research grant from the W.M. Keck Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRare flavour-changing neutral-current (FCNC) $B^0_{(s)} \\rightarrow \\mu^+ \\mu^-$  and $b \\rightarrow s \\mu^+ \\mu^-$ decays\nare considered among the most promising  probes\nof the standard model (SM) and its extensions.\nThe precise measurement of several observables (total and differential\nbranching ratios, angular distributions of the decay products) of these decays\nmight provide interesting clues for new physics (NP) phenomena,\nif any sizable deviation  from the SM predictions is observed.\nIn this paper we review the current status of these indirect searches at the Collider \nDetector at Fermilab (CDF II), which\nreached a sensitivity very close to SM predictions after a \ndecade of Tevatron leadership in the exploration of $B^0_s$ dynamics.\nThe Tevatron $p \\bar{p}$ collider, whose operations ended in October 2011 after 20 years of operation, \nprovided excellent opportunities to study $B$ physics.\nCDF II is a multipurpose detector, consisting of a central charged particle tracking system, \nsurrounded by calorimeters and muon chambers. It collected a final data set corresponding\nto about 10 fb$^{-1}$ of integrated luminosity. \n\nIn the search for rare $B$ decays,\nthe  experimental challenge is to reject a huge background while keeping the signal efficiency high.\nA dedicated dimuon trigger has been used to select \nevents with a pair of muons in the final state in the pseudorapidity region $|\\eta|<$1.1.\nIn the reconstruction and analysis of $B$ hadron decays, CDF takes advantage of\nan excellent transverse momentum resolution, \n$\\frac{\\sigma_{p_T}}{p_T} = 0.07\\%\\, p_T$ (GeV\/c),\nwhich implies a resolution on the invariant\ndimuon mass of 24 MeV in the $B^0_{(s)} \\rightarrow \\mu^+ \\mu^-$ decay, \na vertex resolution  of about 30 $\\mu$m in the transverse plane, and \nparticle identification (PID) capability, \nbased on multiple measurements of the ionization per unit of path length ($dE\/dx$) in the drift chambers.  \n\\section{Search for $B^0_{(s)} \\rightarrow \\mu^{+} \\mu^{-}$}\n$B_{(s)}^0 \\rightarrow \\mu^{+} \\mu^{-}$ decays are mediated by FCNC and thus forbidden at first order in the SM. \nMoreover they are further suppressed by helicity factors $(m_{\\mu}\/m_B)^2$ in the final dimuon state.\nThey can only occur at second order through penguin and box diagrams. \nThe SM predicts very low rates for these processes: $\\ensuremath{\\mathcal B}(B_{s}^0 \\rightarrow \\mu^{+} \\mu^{-}) = \\left(3.2 \\pm 0.2\\right)\\times 10^{-9}$ \nand  $\\ensuremath{\\mathcal B}(B^0 \\rightarrow \\mu^{+} \\mu^{-}) = \\left(1.0 \\pm 0.1\\right)\\times 10^{-10}$ \\cite{SM1,SM2}.\nHowever, a wide variety of beyond the standard model (BSM) theories predict enhancement of their branching ratios\n by several order of magnitudes, \nmaking these decays one of the most sensitive probes in indirect searches for NP. \n\nIn 2011, CDF observed an intriguing $\\sim$2.5$\\sigma$ excess over background in\n$B_{s}^0 \\rightarrow \\mu^{+} \\mu^{-}$ using  7 fb$^{-1}$ of data \\cite{CDF}.\nThough it was compatible with other experimental results (LHCb \\cite{LHCb}, CMS \\cite{CMS})\nand the SM prediction, it could be interpreted as the first indication of a signal\nand allowed CDF to set a two-sided bound on the rate \n$\\ensuremath{\\mathcal B}(B_{s}^0 \\rightarrow \\mu^{+} \\mu^{-}) = \\left(1.8^{+1.1}_{-0.9}\\right)\\times 10^{-8}$ .\nTo further investigate the nature of the excess, we repeated the analysis unchanged using the whole\n Run II data set, corresponding to 9.7 fb$^{-1}$ of integrated\nluminosity, about 30\\% more data with respect to the 2011 analysis. Here we report on \nthe final results of this search \\cite{CDF2}.\n\\begin{figure}\n  \\centering\n\\subfigure[]{\\label{fig:bmumu1}\n  \\includegraphics[width=.85\\textwidth]{bd-8NNbins_poiE_bgE_v12.eps}\n}\n\\subfigure[]{\\label{fig:bmumu2}\n  \\includegraphics[width=.85\\textwidth]{bs-8NNbins_poiE_bgE_v12.eps}\n}\n\\caption{\nDimuon mass distributions for the (a) $B^0 \\rightarrow \\mu^+ \\mu^-$ and (b) \n$B_s^0 \\rightarrow \\mu^+ \\mu^-$ signal region in the eight NN bins. \nThe observed data (points) are compared to the total background expectation (light gray\nhistogram). The hatched region is the total uncertainty on the background expectation.\nIn (b) the dark gray histogram represents the SM signal expectation enhanced by a factor 4.1. \n}\n\\label{fig:bmumu}\n\\end{figure}\n\nThe baseline selection  requires high quality muon candidates with opposite charge,\n transverse momentum $p_T >$ 2 GeV\/c, and a dimuon invariant mass $m_{\\mu\\mu}$ in the range 4.669-5.969 GeV\/c$^2$. \nThe muon pairs are constrained to originate from a common, well-measured reconstructed decay point. \nA likelihood-based muon identification method, \nis used to suppress contributions \nfrom hadrons misidentified as muons. \nThe branching ratios of $B_{(s)}^0 \\rightarrow \\mu^{+} \\mu^{-}$ are measured\nby normalizing to a sample of 40225$\\pm$ 267\n $B^+ \\rightarrow J\/\\psi (\\rightarrow \\mu^+ \\mu^-)\\,K^+$ candidates, \nselected with the same baseline requirements.\nA Neural Network (NN) classifier is used to improve signal over background  separation. \nFourteen variables are used to construct the NN discriminant that ranges between 0 and 1. \nThe six most discriminating variables \ninclude the 3D opening angle between the dimuon momentum and the displacement vector between the\nprimary and secondary vertex; \nthe isolation $I$ \\footnote{$I = |\\vec{p}_T^{\\mu\\mu}|\/(\\sum_i p_T^i + |\\vec{p}_T^{\\mu\\mu}|)$, where $\\vec{p}^{\\mu\\mu}$ is the momentum\nof the dimuon pair; the sum is over all tracks with $\\sqrt{ (\\Delta\\phi)^2 + (\\Delta\\eta)^2} \\le 1$; $\\Delta\\eta$ and $\\Delta\\phi$ are the relative azimuthal angle and pseudorapidity of track $i$ with respect to $\\vec{p}^{\\mu\\mu}$.}\nof the candidate $B^0_{s}$ ;\nthe muon and $B^0_{(s)}$ impact parameters; the $B^0_{(s)}$ decay length significance; the vertex-fit $\\chi^2$.\nThe NN from the 2011 analysis was used with the same training.\nThe final search region in the dimuon invariant mass has a half width of about 60 MeV corresponding to 2.5 times the dimuon \nmass resolution. \nThe NN was validated with signal and background events.\nCareful checks for possible mass-biases  of the NN output and overtraining show no anomalies. \n\nExtensive and detailed background estimates and checks have been performed.\nBackground is due to  both combinatorial and peaking contributions in the signal region. \nCombinatorial background is estimated by fitting the sidebands  to linear functions,\nafter blinding the signal region in the dimuon mass distribution.\nThe peaking background is due to $B \\rightarrow h^+ h^{'-}$ decays where the hadrons \n($h,h'$ stand for $\\pi$ or $K$) are misidentified as  muons. \nThis has been estimated from both MC and data.  \nThe misidentification probability is parametrized as a function of the track transverse momentum\n using $D^*$-tagged $D^0 \\rightarrow \\pi^+ K^-$ events. \nIt turned out that the peaking background is  about 10\\% of the combinatorial background \nin $B_s^0 \\rightarrow \\mu^+ \\mu^-$ and about 50\\% of the total background in \nthe $B^0 \\rightarrow \\mu^+ \\mu^-$ channel.\nBackground estimates have been cross-checked using independent background-dominated control samples, \nin which the muons have the same measured charge or the reconstructed\ndimuon candidate lifetime is negative. \nNo significant discrepancies between the expected and observed number of events in the control samples have been found.\\\\\nThe data are divided into 8 bins of the NN discriminant to exploit the improved background suppression at high NN values, \nand five bins of mass in the search region.\nIn the $B^0$ search region  data are consistent with the background \nprediction (Fig.~\\ref{fig:bmumu1}) and yield the limit of\n $\\mathcal{B}(B^0 \\rightarrow \\mu^+ \\mu^-) < 3.8 (4.6)\\times 10^{-9}$ at 90\\% (95\\%) C.L.. \nThe significance of the background-only hypothesis expressed as a p-value, estimated from an \nensemble of background-only pseudo-experiments, is 41\\%.\n\nIn the $B^0_s$ search region, a moderate excess in the highest NN bins ($>$0.97)\nis observed (Fig.~\\ref{fig:bmumu2}).\nThe p-value for the\nbackground-only hypothesis is 0.94\\%. We also\nproduce an ensemble of simulated experiments that includes\na  $B^0_s \\rightarrow \\mu^+ \\mu^-$ contribution at the expected SM\nbranching fraction which yields a p-value of 7.1\\%.\nWith respect to the previous CDF result with 7 fb$^{-1}$ of data \\cite{CDF}, the excess in the third significant NN bin\n(0.97-0.987)  softened, as expected for a statistical fluctuation.\nThough the 2011 hint of signal is not reinforced by the new data,\nit is still present and  remains $>$2$\\sigma$ significant over background.\nAssuming the observed\nexcess in the $B^0_s$ region is due to signal, CDF finds \n$\\mathcal{B}(B_s^0 \\rightarrow \\mu^+ \\mu^-) = \\left(1.3^{+0.9}_{-0.7}\\right)\\times 10^{-8}$, \nwhich is still compatible with both\n the SM expectation and the latest  constraints from LHC experiments \\cite{LHCbmumu, LHCb2}.\n\n\\begin{figure}[!h]\n  \\centering\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_kmm_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_kstmm_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_phimm_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_kstmm_kspi_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_ksmm_data_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{fit_bmass_lmmm_data_prl.eps}\n}\n\\caption{Invariant mass of\n(a) $B^+ \\rightarrow K^+ \\mu^+ \\mu^- $, \n(b) $B^0 \\rightarrow K^{*0} \\mu^+ \\mu^-$,\n(c) $B^0_s \\rightarrow \\phi \\mu^+ \\mu^-$,\n(d) $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$,\n(e) $B^0 \\rightarrow K_S \\mu^+ \\mu^-$, \n(f) $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$ with fit results overlaid. }\n\\label{fig:bsmumuyield}\n\\end{figure}\n\\section{$b \\rightarrow  s \\mu^{+} \\mu^{-}$  decays}\nRare decays of bottom hadrons mediated by the FCNC process $b \\rightarrow s \\mu^+ \\mu^-$ \nare suppressed at tree level in the SM and must\noccur through higher-order loop amplitudes. Their expected branching ratios are of the order of 10$^{-6}$.\nBecause of their clean experimental signature and the reliable theoretical predictions for their rates, \nthese are excellent channels for NP searches.\n\n\\begin{table}[h]\n\\begin{center} \n  \\begin{tabular}{l l}\n    \\hline\\hline Decay mode & $\\mathcal{B}(10^{-6})$  \\\\\n    \\hline\n    $B^+ \\rightarrow K^+ \\mu^+ \\mu^- $ & $0.46 \\pm 0.04 \\pm 0.02 $\\\\\n    $B^0 \\rightarrow K^{*0} \\mu^+ \\mu^-$ &  $1.02 \\pm 0.10 \\pm 0.06$\\\\\n    $B^0_s \\rightarrow \\phi \\mu^+ \\mu^-$ &  $1.47 \\pm 0.24 \\pm 0.46$\\\\\n    $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$ & $0.95 \\pm 0.32 \\pm 0.08$\\\\\n    $B^0 \\rightarrow K_S \\mu^+ \\mu^-$ & $0.32 \\pm 0.10 \\pm 0.02$\\\\\n    $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$ & $1.73 \\pm 0.42 \\pm 0.55$\\\\\n    \\hline\\hline\n  \\end{tabular}\n\\caption{Branching ratio of the $H_b\\rightarrow h \\mu^+ \\mu^-$ decays measured by CDF.\nThe first quoted uncertainty is statistical, the second is systematic.}\n  \\label{tabBR}\n\\end{center}\n\\end{table}\nCDF has studied the  FCNC $H_b\\rightarrow h \\mu^+ \\mu^-$ decays (where $H_b$ and $h$ indicate hadrons \ncontaining a $b$ and $s$ quark, respectively) listed in Table~\\ref{tabBR}, \nusing 6.8 fb$^{-1}$ of data collected with the dimuon trigger \\cite{CDFbsmumuBR}. \nCandidates for each decay have been selected by standard kinematics cuts and a NN optimized for best sensitivity.\nSignal yields are obtained by an unbinned maximum log-likelihood fit to the $b$-hadron mass distributions \n(Fig.~\\ref{fig:bsmumuyield}), modelling \nthe signal peak with a Gaussian, and the combinatorial background with a linear function.\nTo cancel dominant systematic uncertainties, the branching ratio of each rare decay  $H_b\\rightarrow h \\mu^+ \\mu^-$ \nis measured relative to the corresponding resonant channel $H_b\\rightarrow J\/\\psi\\, h$,\nused as a normalization and a cross-check of the whole analysis.\nThe results of the total branching ratios are reported in Table~\\ref{tabBR} and include\nthe first observation of the baryonic FCNC decay $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$, \nand the first measurement  of the $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$ and  $B^0 \\rightarrow K_S\\,\\mu^+ \\mu^-$ decays\nat a hadron collider.\n\\begin{figure}[h] \n  \\centering\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_kmm.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_kstmm.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_phimm.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_kstmm_kspi.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_ksmm.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=4.5cm]{dbr_lmmm.eps}\n}\n\\caption{\nDifferential branching ratios of \n(a) $B^+ \\rightarrow K^+ \\mu^+ \\mu^- $, \n(b) $B^0 \\rightarrow K^{*0} \\mu^+ \\mu^-$,\n(c) $B^0_s \\rightarrow \\phi \\mu^+ \\mu^-$,\n(d) $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$,\n(e) $B^0 \\rightarrow K_S\\,\\mu^+ \\mu^-$, \n(f) $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$.\nThe points are the fit result from data. The solid curves are the SM\nexpectation \\cite{SMexp, SMexp2, SMexp3, SMexp4}. The dashed line in (f) is the SM prediction normalized to our total branching\nratio measurement. The hatched regions are the charmonium veto regions.}\n\\label{fig:bsmumudiff}\n\\end{figure}\n\nRich information about the $b \\rightarrow s \\mu^+ \\mu^-$\ndynamics can be obtained by precise measurements of the differential branching ratio as a function of $q^2 = m_{\\mu\\mu}^2 c^2$\nand the angular distributions of the decay products.\n\\begin{figure}[h] \n  \\centering\n\\subfigure[]\n  \\includegraphics[width=5cm]{summary_afb_6bin_all_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=5cm]{summary_fl_6bin_all_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=5cm]{summary_at2_6bin_all_prl.eps}\n}\n\\subfigure[]\n  \\includegraphics[width=5cm]{summary_aim_6bin_all_prl.eps}\n}\n\n\\caption{Measurements of angular observables (a) $A_{FB}$, (b) $F_L$, (c) $A_T^{(2)}$, and (d) $A_{im}$ \n as a function of dimuon mass squared $q^2$\nin the combined decay mode $B \\rightarrow K^{*} \\mu^+ \\mu^-$. \nThe points are the fit results from data.\nThe solid and dotted curves represent expectations from the SM \nand a particular BSM scenario, respectively.}\n\\label{fig:angular}\n\\end{figure}\nThe differential branching ratios with respect to $q^2$ have been measured by dividing \nthe signal region  into six bins in $q^2$ \nand fitting the signal yield in each bin.\nIn each fit, the mean of the $H_b$ mass and the background slope were fixed to the\nvalue from the global fit, so that only the signal fraction\nwas allowed to vary in the fit.\nThe results are shown in Fig.~\\ref{fig:bsmumudiff}. \nFor $B^0_s \\rightarrow \\phi \\mu^+ \\mu^-$ and $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$ \nthese are the first such measurements. At present no significant discrepancy from SM prediction is found.\n\nThe angular distributions \nof the combined  $B^0 \\rightarrow K^{*0} \\mu^+ \\mu^-$\nand $B^+ \\rightarrow K^{*+} \\mu^+ \\mu^-$ decays have been measured and  \nparametrized to four angular observables:\nthe muon forward-backward asymmetry $A_{FB}$, \nthe $K^*$ longitudinal polarization fraction $F_L$,  \nthe transverse polarization asymmetry $A_T^{(2)}$,\nthe time-reversal-odd charge-and-parity asymmetry $A_{im}$, defined in \\cite{ANG,ANG2}. \n$A_T^{(2)}$ and $A_{im}$  have been measured for the first time by CDF \\cite{CDFbsmumuAfb}.\nThe results for these observables, shown in Fig.~\\ref{fig:angular}, are among the most precise to date and consistent \nwith SM predictions and other experiments, but\nstill statistically limited in providing stringent tests on\nvarious BSM models.\n\\section{Conclusion}\nWe have summarized the recent updates on the searches of rare $b$-hadron decays at CDF.\nThe intriguing excess in $B_s^0 \\rightarrow \\mu^+ \\mu^-$ reported in 2011\nis confirmed with the full data set, though its significance is softened to the level of 2$\\sigma$ over background. \nThe measured $B_s^0 \\rightarrow \\mu^+ \\mu^-$ branching ratio   is still compatible with \nthe SM expectation and recent combined results from LHC experiments.\n\nThe dynamics of several rare decays mediated by the FCNC process $b \\rightarrow s \\mu^- \\mu^-$ \nhas been studied in detail extending the reach to new angular observables. \nThe $\\Lambda_b^0 \\rightarrow \\Lambda \\mu^+ \\mu^-$ decay has been observed for the first time. \nAnalyses of the  $b \\rightarrow s \\mu^- \\mu^-$  decays \nare still in progress and may yield interesting results in the\nnear future. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background on \\texorpdfstring{$p$}{p}-local compact groups}\\label{Background}\n\nLet $p$ a prime, to remain fixed for the rest of the paper unless otherwise stated. In this section we review all the definitions and results about $p$-local compact groups that we need in this paper. The main references for this section are the works of C. Broto, R. Levi and B. Oliver \\cite{BLO2, BLO3, BLO6}.\n\nRoughly speaking, $p$-local compact groups are abstractions of the fusion data obtained from finite and compact Lie groups. This idea already implies the existence of some sort of \\emph{Sylow $p$-subgroup}. In the finite case this role was played by finite $p$-groups, while in this more general setup we use discrete $p$-toral groups. Let $\\Z\/p^{\\infty} = \\bigcup_{n \\geq 1} \\Z\/p^n$ be the union of the cyclic $p$-groups $\\Z\/p^n$ under the obvious inclusions.\n\n\\begin{defi}\\label{defidiscptor}\n\nA \\emph{discrete $p$-toral group} $P$ is a group that contains a normal subgroup $P_0$ of finite index and which is isomorphic to $(\\Z\/p^{\\infty})^{\\times r}$ for some finite $r \\geq 0$.\n\n\\end{defi}\n\nIn other words, a discrete $p$-toral group is a group $P$ fitting in an exact sequence\n$$\n\\{1\\} \\to P_0 \\Right4{} P \\Right4{} \\pi \\to \\{1\\},\n$$\nwhere $\\pi$ is a finite $p$-group and $P_0 \\cong (\\Z\/p^{\\infty})^{\\times r}$. The \\emph{rank} of $P$, denoted by $\\mathrm{rk}(P)$, is $r$, and the \\emph{order} of $P$ is then defined as the pair $|P| \\stackrel{def} = (\\mathrm{rk}(P), |\\pi(P)|)$, considered as an element of $\\mathbb{N}^2$. This way we can compare the order of two discrete $p$-toral groups, by writing $|Q|\\leq |P|$ if either $\\mathrm{rk}(Q) < \\mathrm{rk}(P)$, or $\\mathrm{rk}(Q) = \\mathrm{rk}(P)$ and $|\\pi(Q)|\\leq |\\pi(P)|$. Given a discrete $p$-toral group $S$ and subgroups $P, Q\\leq S$, define\n$$\n\\mathrm{Hom}_S(P, Q) = \\{f = c_x \\in \\mathrm{Hom}(P, Q) \\,\\, | \\,\\, \\exists x \\in S \\mbox{ such that } x P x^{-1}\\leq Q\\}.\n$$\n\n\\begin{defi}\n\nGiven a discrete $p$-toral group $S$, a \\emph{fusion system} over $S$ is a category $\\mathcal{F}$ with $\\mathrm{Ob}(\\mathcal{F}) = \\{P\\leq S\\}$, and whose morphisms are actual homomorphisms satisfying the following:\n\\begin{enumerate}[(i)]\n\n\\item $\\mathrm{Hom}_S(P,Q) \\subseteq \\mathrm{Hom}_{\\mathcal{F}}(P,Q) \\subseteq \\operatorname{Inj}\\nolimits(P,Q)$  for all $P, Q \\in \\mathrm{Ob}(\\mathcal{F})$; and\n\n\\item every morphism in $\\mathcal{F}$ is the composition of an isomorphism in $\\mathcal{F}$, followed by an inclusion.\n\n\\end{enumerate}\nThe \\emph{rank of $\\mathcal{F}$} is the rank of $S$.\n\n\\end{defi}\n\nThe following notation will be used tacitly throughout the rest of the paper. Let $\\mathcal{F}$ be a fusion system over $S$, and let $P, Q, X\\leq S$. As objects in $\\mathcal{F}$, we say that $P$ and $Q$ are \\emph{$\\mathcal{F}$-conjugate} if they are isomorphic as objects in $\\mathcal{F}$. The notation $P^X$ and $P^{\\mathcal{F}}$, respectively, stands for the $X$-conjugacy and $\\mathcal{F}$-conjugacy classes of $P$. Note also that $\\mathrm{Aut}_{\\mathcal{F}}(P)$ is a group, by definition of fusion system, and that $\\mathrm{Inn}(P)\\leq \\mathrm{Aut}_{\\mathcal{F}}(P)$. Thus, it is reasonable to define\n$$\n\\mathrm{Out}_S(P) \\stackrel{def} = \\mathrm{Aut}_S(P)\/\\mathrm{Inn}(P) \\qquad \\mbox{and} \\qquad \\mathrm{Out}_{\\mathcal{F}}(P) \\stackrel{def} = \\mathrm{Aut}_{\\mathcal{F}}(P)\/\\mathrm{Inn}(P).\n$$\nFinally, we say that $P$ is \\emph{fully $\\mathcal{F}$-centralized}, respectively \\emph{fully $\\mathcal{F}$-normalized}, if $|C_S(P)| \\geq |C_S(Q)|$ for all $Q \\in P^{\\mathcal{F}}$, respectively  if $|N_S(P)| \\geq |N_S(Q)|$ for all $Q \\in P^{\\mathcal{F}}$.\n\n\\begin{defi}\\label{defisat}\n\nLet $S$ be a discrete $p$-toral group, and let $\\mathcal{F}$ be a fusion system over $S$. We say that $\\mathcal{F}$ is a \\emph{saturated fusion system} if the following conditions are satisfied.\n\\begin{enumerate}[(I)]\n\n\\item If $P\\leq S$ is a fully $\\mathcal{F}$-normalized subgroup, then it is also fully $\\mathcal{F}$-centralized. Moreover, in this case $\\mathrm{Out}_{\\mathcal{F}}(P)$ is a finite group, and $\\mathrm{Out}_S(P) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Out}_{\\mathcal{F}}(P))$.\n\n\\item Suppose $P\\leq S$ and $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,S)$ are such that $f(P)$ is fully $\\mathcal{F}$-centralized, and set\n$$\nN_f = \\{g \\in N_S(P) \\,\\, | \\,\\, f \\circ c_g \\circ f^{-1} \\in \\mathrm{Aut}_S(f(P))\\}.\n$$\nThen, there exists $\\4{f} \\in \\mathrm{Hom}_{\\mathcal{F}}(N_f, S)$ such that $\\4{f}|_P = f$.\n\n\\item Let $P_1\\leq P_2\\leq P_3\\leq \\ldots$ be a sequence of subgroups of $S$, and set $P = \\bigcup_{n = 1}^{\\infty} P_n$. If $f \\in \\mathrm{Hom}(P,S)$ is a homomorphism such that $f|_{P_n} \\in \\mathrm{Hom}_{\\mathcal{F}}(P_n,S)$ for all $n$, then $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,S)$.\n\n\\end{enumerate}\n\n\\end{defi}\n\nWe also recall the definition of centric and radical subgroups, which are crucial concepts in the $p$-local group theory.\n\n\\begin{defi}\n\nLet $\\mathcal{F}$ be a saturated fusion system over a discrete $p$-toral group $S$.\n\\begin{itemize}\n\n\\item A subgroup $P\\leq S$ is \\emph{$\\mathcal{F}$-centric} if $C_S(Q) = Z(Q)$ for all $Q \\in P^{\\mathcal{F}}$.\n\n\\item A subgroup $P\\leq S$ is \\emph{$\\mathcal{F}$-radical} if $\\mathrm{Out}_{\\mathcal{F}}(P)$ contains no nontrivial normal $p$-subgroup.\n\n\\end{itemize}\n\n\\end{defi}\n\nGiven a saturated fusion system $\\mathcal{F}$ over a discrete $p$-toral group $S$, we denote by $\\mathcal{F}^c$ and $\\mathcal{F}^r$ the full subcategories of $\\mathcal{F}$ with object sets the collections of $\\mathcal{F}$-centric and $\\mathcal{F}$-radical subgroups, respectively. We also write $\\mathcal{F}^{cr} \\subseteq \\mathcal{F}$ for the full subcategory of $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups.\n\nProving that a given fusion system is saturated is a rather difficult task, even when the fusion system is finite, but there are some techniques that may be helpful. One of these techniques, which we will use in later sections, is \\cite[Theorem A]{BCGLO1}, restated as Theorem \\ref{5A} below.\n\n\\begin{defi}\n\nLet $\\mathcal{F}$ be a fusion system over a finite $p$-group $S$, and let $\\mathcal{H} \\subseteq \\mathrm{Ob}(\\mathcal{F})$ be a subset of objects.\n\\begin{itemize}\n\n\\item $\\mathcal{F}$ is \\emph{$\\mathcal{H}$-generated} if every morphism in $\\mathcal{F}$ can be described as a composite of restrictions of morphisms in $\\mathcal{F}$ between subgroups in $\\mathcal{H}$.\n\n\\item $\\mathcal{F}$ is \\emph{$\\mathcal{H}$-saturated} if the saturation axioms hold for all subgroups in the set $\\mathcal{H}$.\n\n\\end{itemize}\n\n\\end{defi}\n\n\\begin{thm}\\label{5A}\n\nLet $\\mathcal{F}$ be a fusion system over a finite $p$-group $S$, and let $\\mathcal{H}$ be a subset of objects of $\\mathcal{F}$ closed under $\\mathcal{F}$-conjugacy and such that $\\mathcal{F}$ is $\\mathcal{H}$-generated and $\\mathcal{H}$-saturated. Suppose further that, for each $\\mathcal{F}$-centric subgroup $P \\notin \\mathcal{H}$, $P$ is $\\mathcal{F}$-conjugate to some $Q$ such that\n$$\n\\mathrm{Out}_S(Q) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}}(Q)) \\neq \\{1\\}.\n$$\nThen $\\mathcal{F}$ is saturated.\n\n\\end{thm}\n\nThe following result will be useful in later sections when checking the condition displayed in the previous theorem. We state it in full generality since in fact it will apply in different situations throughout this paper.\n\n\\begin{lmm}\\label{Kpgp}\n\nLet $\\mathcal{F}$ be a fusion system over a discrete $p$-toral group $S$. Let also $P\\leq S$ be a subgroup, and let $P_0 \\lhd P$ be a normal subgroup such that $f|_{P_0} \\in \\mathrm{Aut}_{\\mathcal{F}}(P_0)$ for all $f \\in \\mathrm{Aut}_{\\mathcal{F}}(P)$. Set\n$$\nK_P \\stackrel{def} = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}}(P) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}}(P_0) \\times \\mathrm{Aut}(P\/P_0)).\n$$\nThen, $K_P\\leq O_p(\\mathrm{Aut}_{\\mathcal{F}}(P))$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nBy definition $K_P$ is normal in $\\mathrm{Aut}_{\\mathcal{F}}(P)$, and thus we only have to show that $K_P$ is a discrete $p$-toral subgroup of $\\mathrm{Aut}_{\\mathcal{F}}(P)$ (that is, every element of $K_P$ has order a power of $p$). This in turn follows by \\cite[Theorem 3.2]{Gor}: although the result in \\cite{Gor} is stated for finite groups, the arguments in its proof apply here without modification, since $\\mathrm{Aut}_{\\mathcal{F}}(P)$ is a locally finite group.\n\\end{proof}\n\nThe concept of transporter system associated to a fusion system was first introduced in \\cite{OV} for fusion systems over finite $p$-groups, and then extended to discrete $p$-toral groups in \\cite{BLO6}, with centric linking systems as a particular case. We refer the reader to the aforementioned sources for further details.\n\nLet $G$ be a group and let $\\mathcal{H}$ be a set of subgroups of $G$ that is closed by overgroups, i.e. if $H \\in \\mathcal{H}$ and $K \\geq H$, then $K \\in \\mathcal{H}$, and closed by conjugation in $G$, i.e. if $H \\in \\mathcal{H}$ and $g \\in G$, then $gHg^{-1} \\in \\mathcal{H}$. The transporter category of $G$ with respect to $\\mathcal{H}$ is the category $\\mathcal{T}_{\\mathcal{H}}(G)$ whose object set is $\\mathcal{H}$, and with morphism sets\n$$\n\\mathrm{Mor}_{\\mathcal{T}_{\\mathcal{H}}(G)}(P,Q) = \\{x \\in G \\,\\, | \\,\\, x \\cdot P \\cdot x^{-1}\\leq Q\\}\n$$\nfor each $P,Q \\in \\mathcal{H}$.\n\n\\begin{defi}\\label{defitransporter}\n\nLet $S$ be a discrete $p$-toral group, and let $\\mathcal{F}$ be a fusion system over $S$. A \\emph{transporter system} associated to $\\mathcal{F}$ is a nonempty category $\\mathcal{T}$ whose object set $\\mathrm{Ob}(\\mathcal{T})$ is a subset of $\\mathrm{Ob}(\\mathcal{F})$ that is closed by overgroups and conjugation in $\\mathcal{F}$, together with functors\n$$\n\\mathcal{T}_{\\mathrm{Ob}(\\mathcal{T})}(S) \\Right4{\\varepsilon} \\mathcal{T} \\qquad \\mbox{and} \\qquad \\mathcal{T} \\Right4{\\rho} \\mathcal{F}\n$$\nsatisfying the following conditions.\n\\begin{itemize}\n\n\\item[(A1)] The functor $\\varepsilon$ is the identity on objects and an inclusion on morphism sets, and the functor $\\rho$ is the inclusion on objects and a surjection on morphism sets.\n\n\\item[(A2)] For each $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$, the set $\\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ has a free action of\n$$\nE(P) \\stackrel{def} = \\mathrm{Ker} \\big[\\rho_P \\colon \\mathrm{Aut}_{\\mathcal{T}}(P) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}}(P) \\big]\n$$\nby right composition, and $\\rho_{P,Q}$ is the orbit map of this action. Also, $E(Q)$ acts freely on $\\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ by left composition.\n\n\\item[(B)] Let $P,Q \\in \\mathrm{Ob}(\\mathcal{T})$. Then, the map $\\varepsilon_{P,Q} \\colon N_S(P,Q) \\to \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ is injective, and\n$$\n(\\rho_{P,Q} \\circ \\varepsilon_{P,Q})(g) = c_g \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)\n$$\nfor all $g \\in \\mathrm{Mor}_{\\mathcal{T}_{\\mathrm{Ob}(\\mathcal{T})}(S)}(P, Q) = N_S(P, Q)$.\n\n\\item[(C)] For all $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$, for all $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$, and for all $g \\in P$, the following is a commutative diagram in $\\mathcal{T}$.\n$$\n\\xymatrix{\nP \\ar[r]^{\\varphi} \\ar[d]_{\\varepsilon_P(g)} & Q \\ar[d]^{\\varepsilon_Q(\\rho(\\varphi)(g))} \\\\\nP \\ar[r]_{\\varphi} & Q\n}\n$$\n\n\\item[(I)] Each isomorphism class of objects in $\\mathrm{Ob}(\\mathcal{T})$ contains an element $P$ such that\n$$\n\\varepsilon_P(N_S(P)) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{T}}(P));\n$$\nor, in other words, such that $\\varepsilon(N_S(P))$ has finite index prime to $p$ in $\\mathrm{Aut}_{\\mathcal{T}}(P)$.\n\n\\item[(II)] Let $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$ be isomorphic objects, and let $\\varphi \\in \\mathrm{Iso}_{\\mathcal{T}}(P,Q)$. Let also $\\4{P}\\leq N_S(P)$ and $\\4{Q}\\leq N_S(Q)$ be such that $\\varphi \\circ \\varepsilon_P(\\4{P}) \\circ \\varphi^{-1}\\leq \\varepsilon_Q(\\4{Q})$. Then there is some morphism $\\4{\\varphi} \\in \\mathrm{Mor}_{\\mathcal{T}}(\\4{P}, \\4{Q})$ such that\n$$\n\\4{\\varphi} \\circ \\varepsilon_{P, \\4{P}}(1) = \\varepsilon_{Q, \\4{Q}}(1) \\circ \\varphi.\n$$\n\n\\item[(III)] Let $P_1\\leq P_2\\leq P_3\\leq \\ldots$ be a sequence in $\\mathrm{Ob}(\\mathcal{T})$, and let $\\varphi_n \\in \\mathrm{Mor}_{\\mathcal{T}}(P_n,S)$ be such that $\\varphi_n = \\varphi_{n+1} \\circ \\varepsilon_{P_n, P_{n+1}}(1)$ for all $n \\geq 1$. Then, upon setting $P = \\bigcup_{n \\geq 1} P_n$, there is a morphism $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,S)$ such that $\\varphi_n = \\varphi \\circ \\varepsilon_{P_n,P}(1)$ for all $n \\geq 1$.\n\n\\end{itemize}\nThe \\emph{rank of $\\mathcal{T}$} is the rank of $S$. A \\emph{centric linking system} associated to a saturated fusion system $\\mathcal{F}$ is a transporter system $\\mathcal{L}$ such that $\\mathrm{Ob}(\\mathcal{L})$ is the collection of all $\\mathcal{F}$-centric subgroups of $S$ and $E(P) = \\varepsilon(Z(P))$ for all $P \\in \\mathrm{Ob}(\\mathcal{L})$.\n\n\\end{defi}\n\n\\begin{rmk}\\label{rmktransp}\n\nThe above definition of centric linking system is taken from \\cite{BLO6}, and it is seen in \\cite[Corollary A.5]{BLO6} to coincide with the original \\cite[Definition 4.1]{BLO3}. Notice also that axiom (I) above differs from the corresponding axiom for the finite case, see \\cite[Definition 3.1]{OV}, in that condition (I) above seems to be more restrictive than the corresponding condition in \\cite{OV}:\n\\begin{itemize}\n\n\\item[(I')] $\\varepsilon_{S,S}(S) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{T}}(S))$.\n\n\\end{itemize}\nHowever, \\cite[Proposition 3.4]{OV} implies that both definitions, \\cite[Definition 3.1]{OV} and the above, agree in the finite case.\n\n\\end{rmk}\n\n\\begin{lmm}\\label{epimono}\n\nIn a transporter system, all morphisms are monomorphisms and epimorphisms in the categorical sense.\n\n\\end{lmm}\n\n\\begin{proof}\n\nThis is \\cite[Proposition A.2 (d)]{BLO6}.\n\\end{proof}\n\n\\begin{defi}\n\nA \\emph{$p$-local compact group} is a triple $\\mathcal{G} = (S, \\FF, \\LL)$ formed by a discrete $p$-toral group $S$, a saturated fusion system $\\mathcal{F}$ over $S$, and a centric linking system $\\mathcal{L}$ associated to $\\mathcal{F}$. The \\emph{classifying space} of a $p$-local compact group $\\mathcal{G}$ is the $p$-completed nerve of $\\mathcal{L}$, denoted by $B\\mathcal{G} = |\\mathcal{L}|^{\\wedge}_p$. The \\emph{rank of $\\mathcal{G}$} is the rank of $S$.\n\n\\end{defi}\n\nGeneralizing work of \\cite{Chermak} and \\cite{Oliver}, it is proved in \\cite{Levi-Libman} that every saturated fusion system over a discrete $p$-toral group has an associated centric linking system which is unique up to isomorphism. Thus, from now on we speak of \\emph{the} associated centric linking system for a given saturated fusion system. \n\nFinally, we recall the ``bullet construction'' on a $p$-local compact group.\n\n\\begin{defi}\\label{defibullet}\n\nLet $\\mathcal{F}$ be a saturated fusion system over a discrete $p$-toral group $S$. Let also $T\\leq S$ be the maximal torus, and let $W = \\mathrm{Aut}_{\\mathcal{F}}(T)$. Set the following\n\n\\begin{enumerate}[(i)]\n\n\\item The exponent of $S\/T$ is $e = \\exp(S\/T) = \\min\\{k \\in \\mathbb{N} \\, | \\, x^{p^k} \\in T \\mbox{ for all } x \\in S\\}$.\n\n\\item For each $P\\leq T$, let $I(P) = \\{t \\in T \\, | \\, \\omega(t) = t \\mbox{ for all } \\omega \\in W \\mbox{ such that } \\omega|_P = \\mathrm{Id}_P\\}$, and let $I(P)_0$ denote its maximal torus.\n\n\\item For each $P\\leq S$, set $P^{[e]} = \\{x^{p^e} \\, | \\, x \\in P\\}\\leq T$, and set\n$$\nP^{\\bullet} = P \\cdot I(P^{[e]})_0 = \\{xt \\, | \\, x \\in P, \\, t \\in I(P^{[e]})_0\\}.\n$$\n\n\\item Let $\\mathcal{F}^{\\bullet}$ be the full subcategory of $\\mathcal{F}$ with object set $\\mathrm{Ob}(\\mathcal{F}^{\\bullet}) = \\{P^{\\bullet} \\, | \\, P\\leq S\\}$.\n\n\\end{enumerate}\n\n\\end{defi}\n\nThe following summarizes the main properties of the ``bullet construction''.\n\n\\begin{prop}\\label{3.2BLO3}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group. Then, for each $P, Q \\in \\mathrm{Ob}(\\mathcal{F})$ and each $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$ there is a unique $f^{\\bullet} \\in \\mathrm{Hom}_{\\mathcal{F}}(P^{\\bullet}, Q^{\\bullet})$ whose restriction to $P$ is $f$. This way, the ``bullet construction'' makes $P \\mapsto P^{\\bullet}$ into a functor $(-)^{\\bullet} \\colon \\mathcal{F} \\to \\mathcal{F}$ that satisfies the following properties.\n\\begin{enumerate}[(i)]\n\n\\item The set $\\mathrm{Ob}(\\mathcal{F}^{\\bullet}) = \\{P^{\\bullet} \\,  | \\, P\\leq S\\}$ contains finitely many $S$-conjugacy classes of subgroups of $S$.\n\n\\item For all $P\\leq S$, $(P^{\\bullet})^{\\bullet} = P^{\\bullet}$.\n\n\\item If $P\\leq Q\\leq S$, then $P^{\\bullet}\\leq Q^{\\bullet}$.\n\n\\item For all $P, Q\\leq S$, $N_S(P, Q) \\subseteq N_S(P^{\\bullet}, Q^{\\bullet})$.\n\n\\item For all $P\\leq S$, $C_S(P) = C_S(P^{\\bullet})$.\n\n\\item The functor $(-)^{\\bullet}$ is a left adjoint to the inclusion of $\\mathcal{F}^{\\bullet}$ as a full subcategory of $\\mathcal{F}$.\n\n\\item All $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups of $S$ are in $\\mathcal{F}^{\\bullet}$. In particular, there are only finitely many $\\mathcal{F}$-conjugacy classes of such subgroups.\n\n\\end{enumerate}\nMoreover, if we denote by $\\mathcal{L}^{\\bullet} \\subseteq \\mathcal{L}$ the full subcategory with $\\mathrm{Ob}(\\mathcal{L}^{\\bullet}) = \\{P^{\\bullet} \\, | \\, P \\in \\mathrm{Ob}(\\mathcal{L})\\}$, then there is a unique functor $(-)^{\\bullet} \\colon \\mathcal{L} \\to \\mathcal{L}^{\\bullet}$ such that the following holds.\n\\begin{enumerate}[(a)]\n\n\\item $(-)^{\\bullet} \\circ \\rho = \\rho \\circ (-)^{\\bullet} \\colon \\mathcal{L} \\to \\mathcal{F}$.\n\n\\item For all $P, Q \\in \\mathrm{Ob}(\\mathcal{L})$ and all $\\varphi \\in \\mathrm{Mor}_{\\mathcal{L}}(P,Q)$, we have $\\varepsilon_{Q, Q^{\\bullet}}(1) \\circ \\varphi = \\varphi^{\\bullet} \\circ \\varepsilon_{P, P^{\\bullet}}(1)$.\n\n\\item For all $P, Q \\in \\mathrm{Ob}(\\mathcal{L})$ and all $g \\in N_S(P,Q)$, we have $\\varepsilon_{P,Q}(g)^{\\bullet} = \\varepsilon_{P^{\\bullet}, Q^{\\bullet}}(g)$.\n\n\\item The functor $(-)^{\\bullet} \\colon \\mathcal{L} \\to \\mathcal{L}$ is left adjoint to the inclusion of $\\mathcal{L}^{\\bullet}$ as a full subcategory of $\\mathcal{L}$. In particular, the inclusion $\\mathcal{L}^{\\bullet} \\subseteq \\mathcal{L}$ induces an equivalence $|\\mathcal{L}^{\\bullet}|\\simeq |\\mathcal{L}|$.\n\n\\end{enumerate}\n\n\\end{prop}\n\n\\begin{proof}\n\nThe first part of the statement corresponds to \\cite[Proposition 3.3]{BLO3}. Parts (i), (ii) and (iii)  correspond to \\cite[Lemma 3.2 (a), (b) and (c)]{BLO3} respectively. Part (iv) is an easy variation of \\cite[Lemma 3.2 (b)]{BLO3} (details are left to the reader). For part (v), let $P\\leq S$. Since $P\\leq P^{\\bullet}$, we have $C_S(P) \\geq C_S(P^{\\bullet})$. Let $x \\in C_S(P)$. By (iv), we have $x \\in N_S(P^{\\bullet})$. Since $c_x = \\mathrm{Id} \\in \\mathrm{Aut}_{\\mathcal{F}}(P)$ extends uniquely to $c_x = \\mathrm{Id} \\in \\mathrm{Aut}_{\\mathcal{F}}(P^{\\bullet})$, it follows that $x \\in C_S(P^{\\bullet})$. Part (vi) corresponds to \\cite[Corollary 3.4]{BLO3}, and part (vii) corresponds to \\cite[Corollary 3.5]{BLO3}. The last part of the statement, including parts (a), (b) and (c) corresponds to \\cite[Proposition 1.12]{JLL}. Part (d) corresponds to \\cite[Proposition 4.5 (a)]{BLO3}.\n\\end{proof}\n\n\n\\subsection{Isotypical equivalences and unstable Adams operations}\\label{Ssisotyp}\n\nIn this subsection we review the concept of isotypical equivalence, with particular interest on the unstable Adams operations for $p$-local compact groups originally introduced in \\cite{JLL}.\n\n\\begin{defi}\n\nLet $(\\mathcal{T}, \\varepsilon, \\rho)$ be a transporter system associated to a fusion system $\\mathcal{F}$. An automorphism $\\Psi \\colon \\mathcal{T} \\to \\mathcal{T}$ is \\emph{isotypical} if $\\Psi(\\varepsilon_P(P)) = \\varepsilon_{\\Psi(P)}(\\Psi(P))$ for each $P \\in \\mathrm{Ob}(\\mathcal{T})$.\n\n\\end{defi}\n\nWe denote by $\\mathrm{Aut}_{\\mathrm{typ}}^I(\\mathcal{T})$ the group of isotypical automorphisms $\\Psi$ of $\\mathcal{T}$ which in addition satisfy $\\Psi(\\varepsilon_{P,Q}(1)) = \\varepsilon_{\\Psi(P), \\Psi(Q)}(1)$ whenever $P\\leq Q$. Notice that if $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{T})$, then $\\Psi$ induces an automorphism of $S$ by restricting to the object $S \\in \\mathrm{Ob}(\\mathcal{T})$. By abuse of notation, we will denote the induced automorphism by $\\Psi \\in \\mathrm{Aut}(S)$.\n\nNext we review the concept of unstable Adams operations for $p$-local compact groups. Our definition corresponds to the definition of \\emph{normal Adams operation} in \\cite[Definition 3.3]{JLL}, conveniently adapted to our notation. By $(\\Z^\\wedge_p)^{\\times}$ we denote the subgroup of multiplicative units in the ring of $p$-adic integers $\\Z^\\wedge_p$.\n\n\\begin{defi}\\label{uAo}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group. An \\emph{unstable Adams operation of degree $\\zeta \\in (\\Z^\\wedge_p)^{\\times}$} on $\\mathcal{G}$ is an isotypical equivalence $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{L})$ such that the induced automorphism $\\Psi \\in \\mathrm{Aut}(S)$ satisfies\n\\begin{enumerate}[(i)]\n\n\\item the restriction of $\\Psi$ to the maximal torus $T\\leq S$ is the $\\zeta$-power automorphism; and\n\n\\item $\\Psi$ induces the identity on $S\/T$.\n\n\\end{enumerate}\nAn unstable Adams operation is \\emph{fine} if its degree is $\\zeta \\neq 1$, with $\\zeta$ congruent to $1$ modulo $p$.\n\n\\end{defi}\n\nAs proved in \\cite[Theorem 4.1]{JLL}, unstable Adams operations exist for all $p$-local compact groups, and in particular this applies to the existence of fine unstable Adams operations.\n\n\\begin{thm}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group. Then, for some large enough $m \\in \\mathbb{N}$, $\\mathcal{G}$ has unstable Adams operations of degree $\\zeta$, for each $\\zeta \\in 1 + p^m\\Z^\\wedge_p$.\n\n\\end{thm}\n\n\\begin{rmk}\\label{uAo1}\n\nRoughly speaking, the construction of unstable Adams operations in \\cite{JLL} is done by defining $\\Psi$ to fix enough objects and morphisms in $\\mathcal{L}$. More specifically, $\\Psi$ fixes\n\\begin{enumerate}[(a)]\n\n\\item a set $\\mathcal{H}$ of representatives of the $S$-conjugacy classes $\\mathrm{Ob}(\\mathcal{L}^{\\bullet})$; and\n\n\\item for each $P \\in \\mathcal{H}$, a set of representatives $\\mathcal{M}_P \\subseteq \\mathrm{Aut}_{\\mathcal{L}}(P)$ of the classes in $\\mathrm{Aut}_{\\mathcal{L}}(P)\/P \\cong \\mathrm{Out}_{\\mathcal{F}}(P)$.\n\n\\end{enumerate}\nThis properties will be crucial in our constructions in Section \\ref{Sfam}.\n\n\\end{rmk}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a fusion system over $S$ (not necessarily saturated), and let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$. Let also $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{T})$ be an isotypical automorphism. Set also\n\\begin{equation}\\label{fixS}\nC_S(\\Psi) = \\{g \\in S \\, | \\, \\Psi(\\varepsilon_S(g)) = \\varepsilon_S(g)\\}\\leq S,\n\\end{equation}\nthe subgroup of fixed points of $S$ by $\\Psi$. The following result is the main tool in detecting objects and morphisms in $\\mathcal{T}$ that are invariant by $\\Psi$.\n\n\\begin{lmm}\\label{invar1}\n\nThe following holds.\n\\begin{enumerate}[(i)]\n\n\\item Let $P\\leq C_S(\\Psi)$, and let $Q \\in P^S$. Then, $Q\\leq C_S(\\Psi)$ if and only if, for some $x \\in N_S(Q,P)$,\n$$\nx^{-1} \\cdot \\Psi(x) \\in C_S(Q).\n$$\n\n\\item Let $P, P' ,Q, Q'\\leq C_S(\\Psi)$ be such that $P' \\in P^S$ and $Q' \\in Q^S$, and suppose $P, P', Q, Q' \\in \\mathrm{Ob}(\\mathcal{L})$. Let also $x \\in N_S(P',P)$ and $y \\in N_S(Q', Q)$, and let $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ be such that $\\Psi(\\varphi) = \\varphi$. Set $\\varphi' = \\varepsilon(y^{-1}) \\circ \\varphi \\circ \\varepsilon(x) \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P', Q')$. Then, $\\Psi(\\varphi') = \\varphi'$ if and only if\n$$\n\\varepsilon(y^{-1} \\cdot \\Psi(y)) \\circ \\varphi' = \\varphi' \\circ \\varepsilon(x^{-1} \\cdot \\Psi(x)).\n$$\n\n\\end{enumerate}\n\n\\end{lmm}\n\n\\begin{proof}\n\nFor part (i), let $P\\leq C_S(\\Psi)$, and let $Q \\in P^S$. Let also $g \\in Q$ and $x \\in N_S(Q,P)$, and set $h = x \\cdot g \\cdot x^{-1} \\in P$. Since $P\\leq C_S(\\Psi)$, we get\n$$\nx \\cdot g \\cdot x^{-1} = h = \\Psi(h) = \\Psi(x \\cdot g \\cdot x^{-1}) = \\Psi(x) \\cdot \\Psi(g) \\cdot \\Psi(x)^{-1}.\n$$\nThus, if $Q\\leq C_S(\\Psi)$ then clearly $x^{-1} \\cdot \\Psi(x) \\in C_S(Q)$, and conversely if $x^{-1} \\cdot \\Psi(x) \\in C_S(Q)$ then $h \\in C_S(\\Psi)$. Since the argument works for any $g \\in Q$ and any $x \\in N_S(Q, P)$, part (i) follows.\n\nFor part (ii), let $P, P' ,Q, Q'\\leq C_S(\\Psi)$, with $P' \\in P^S$ and $Q' \\in Q^S$. Let also $x \\in N_S(P',P)$ and $y \\in N_S(Q', Q)$, and let $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$ be such that $\\Psi(\\varphi) = \\varphi$, with $\\varphi' = \\varepsilon(y^{-1}) \\circ \\varphi \\circ \\varepsilon(x) \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P', Q')$. We have\n$$\n\\varepsilon(y) \\circ \\varphi' \\circ \\varepsilon(x^{-1}) = \\varphi = \\Psi(\\varphi) = \\Psi(\\varepsilon(y) \\circ \\varphi' \\circ \\varepsilon(x^{-1})) = \\varepsilon(\\Psi(y)) \\circ \\Psi(\\varphi') \\circ \\varepsilon(\\Psi(x)^{-1}),\n$$\nand (ii) follows easily.\n\\end{proof}\n\n\n\\subsection{Normalizers, centralizers, and related constructions}\\label{Squotient}\n\nIn this subsection we review the construction of the centralizer and normalizer $p$-local compact subgroups for a given $p$-local compact group. The main references here are \\cite[Appendix A]{BLO2} and \\cite[Section 2]{BLO6}. For the rest of this subsection, fix a $p$-local compact group $\\mathcal{G} = (S, \\FF, \\LL)$, a subgroup $A\\leq S$, and a subgroup $K\\leq \\mathrm{Aut}(A)$, and define the following:\n\\begin{itemize}\n\n\\item $\\mathrm{Aut}_{\\mathcal{F}}^K(A) = K \\cap \\mathrm{Aut}_{\\mathcal{F}}(A)$;\n\n\\item $\\mathrm{Aut}_S^K(A) = K \\cap \\mathrm{Aut}_S(A)$; and\n\n\\item $N_S^K(A) = \\{x \\in N_S(A) \\, | \\, c_x \\in K\\}$.\n\n\\end{itemize}\nThe subgroup $A$ is \\emph{fully $K$-normalized in $\\mathcal{F}$} if we have $|N_S^K(A)| \\geq |N_S^{^{f}K}(f(A))|$ for each $f \\in \\mathrm{Hom}_{\\mathcal{F}}(A, S)$, where $^{f}K = \\{f \\gamma f^{-1} \\, | \\, \\gamma \\in K\\}\\leq \\mathrm{Aut}(f(A))$.\n\n\\begin{defi}\\label{definorm}\n\nThe \\emph{$K$-normalizer fusion system of $A$ in $\\mathcal{F}$}, is the fusion system $N_{\\mathcal{F}}^K(A)$ over $N_S^K(A)$ with morphism sets\n$$\n\\begin{aligned}\n\\mathrm{Hom}_{N_{\\mathcal{F}}^K(A)}&(P,Q) = \\\\\n & = \\{f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q) \\,\\, | \\,\\, \\exists \\4{f} \\in \\mathrm{Hom}_{\\mathcal{F}}(PA, QA) \\mbox{ with } \\4{f}|_P = f \\mbox{ and } \\4{f}|_A \\in K\\}\n\\end{aligned}\n$$\nfor each $P, Q\\leq N_S^K(A)$.\n\n\\end{defi}\n\nBy \\cite[Theorem 2.3]{BLO6} we know that $N_{\\mathcal{F}}^K(A)$ is a saturated fusion system whenever $A$ is fully $K$-normalized in $\\mathcal{F}$. For this reason, for the rest of this subsection we assume that $A$ satisfies this property.\n\n\\begin{lmm}\\label{centricNFKA}\n\nIf $P\\leq N_S^K(A)$ is $N_{\\mathcal{F}}^K(A)$-centric, then $P \\cdot A$ is $\\mathcal{F}$-centric.\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $P\\leq N_S^K(A)$ be $N_{\\mathcal{F}}^K(A)$-centric. We have to check that, for each $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(P \\cdot A, S)$, there is an inclusion $C_S(\\gamma(P \\cdot A))\\leq \\gamma(P \\cdot A)$. We can apply \\cite[Proposition A.2]{BLO2}, since the proof in \\cite{BLO2} works without modifications in the compact setup, and it follows that the subgroup $A$ is fully centralized in $\\mathcal{F}$, and there is some $f \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S^{^{\\gamma}K}(\\gamma(A)) \\cdot \\gamma(A),S)$ such that $(f \\circ \\gamma)|_A \\in K$. Thus $f \\circ \\gamma$ is a morphism in $N_{\\mathcal{F}}^K(A)$.\n\nNote that $C_S(\\gamma(P \\cdot A))\\leq C_S(\\gamma(A))\\leq N_S^{^{\\gamma} K}(\\gamma(A))$, and we have inclusions\n$$\n\\begin{aligned}\nf(C_S(\\gamma(P \\cdot A))) &\\leq C_S((f \\circ \\gamma)(P \\cdot A)) = C_S((f \\circ \\gamma)(P) \\cdot A)\\leq \\\\\n &\\leq C_S((f \\circ \\gamma)(P)) \\cap C_S(A)\\leq C_S((f \\circ \\gamma)(P)) \\cap N_S^K(A)\\leq (f \\circ \\gamma)(P),\n\\end{aligned}\n$$\nwhere the last inequality holds since $P \\in N_{\\mathcal{F}}^K(A)^c$. Thus,\n$$\nC_S(\\gamma(P \\cdot A))\\leq \\gamma(P)\\leq \\gamma(P) \\cdot \\gamma(A) = \\gamma(P \\cdot A),\n$$\nand this proves that $P \\cdot A \\in \\mathcal{F}^c$.\n\\end{proof}\n\nIn view of the above, we can now define $N_{\\mathcal{L}}^K(A)$ as the category with objects the set of $N_{\\mathcal{F}}^K(A)$-centric subgroups of $N_S^K(A)$ and with morphism sets\n$$\n\\mathrm{Mor}_{N_{\\mathcal{L}}^K(A)}(P,Q) = \\{\\varphi \\in \\mathrm{Mor}_{\\mathcal{L}}(PA, QA) \\,\\, | \\,\\, \\rho(\\varphi)|_P \\in \\mathrm{Hom}_{N_{\\mathcal{F}}^K(A)}(P,Q) \\mbox{ and } \\rho(\\varphi)|_A \\in K\\}.\n$$\nIn general, $N_{\\mathcal{L}}^K(A)$ need not be a transporter system associated to $N_{\\mathcal{F}}^K(A)$, but there are two particular situations where this is indeed the case.\n\n\\begin{lmm}\n\nIf either $K = \\{\\mathrm{Id}\\}$ or $K = \\mathrm{Aut}(A)$, then the category $N_{\\mathcal{L}}^K(A)$ is a centric linking system associated to $N_{\\mathcal{F}}^K(A)$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nThe case $K = \\{\\mathrm{Id}\\}$ corresponds to \\cite[Proposition 2.5]{BLO2} in the finite case, while the case $K = \\mathrm{Aut}(A)$ corresponds to \\cite[Lemma 6.2]{BLO2} for $p$-local finite groups. In both situations, the proof for $p$-local finite groups applies here without modification to show that $N_{\\mathcal{L}}^{\\{\\mathrm{Id}\\}}(A)$ satisfies all the condition of a centric linking system, except perhaps axiom (III), which is easily checked.\n\\end{proof}\n\n\\begin{defi}\\label{rmknorm}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $A\\leq S$.\n\\begin{enumerate}[(a)]\n\n\\item If $A$ is fully $\\mathcal{F}$-centralized, the \\emph{centralizer $p$-local compact group of $A$ in $\\mathcal{G}$} is the triple\n$$\nC_{\\mathcal{G}}(A) = (C_S(A), C_{\\mathcal{F}}(A), C_{\\mathcal{L}}(A)) \\stackrel{def} = (N_S^{\\{\\mathrm{Id}\\}}(A), N_{\\mathcal{F}}^{\\{\\mathrm{Id}\\}}(A), N_{\\mathcal{L}}^{\\{\\mathrm{Id}\\}}(A)).\n$$\n\n\\item If $A$ is fully $\\mathcal{F}$-normalized, the \\emph{normalizer $p$-local compact group of $A$ in $\\mathcal{G}$} is the triple\n$$\nN_{\\mathcal{G}}(A) = (N_S(A), N_{\\mathcal{F}}(A), N_{\\mathcal{L}}(A)) \\stackrel{def} = (N_S^{\\mathrm{Aut}(A)}(A), N_{\\mathcal{F}}^{\\mathrm{Aut}(A)}(A), N_{\\mathcal{L}}^{\\mathrm{Aut}(A)}(A)).\n$$\n\n\\end{enumerate}\nA subgroup $A\\leq S$ is called \\emph{central in $\\mathcal{F}$} if $C_{\\mathcal{G}}(A) = \\mathcal{G}$. Similarly, $A\\leq S$ is called \\emph{normal in $\\mathcal{F}$} if $N_{\\mathcal{G}}(A) = \\mathcal{G}$. Clearly, if $A\\leq S$ is central in $\\mathcal{F}$ then in particular it is normal in $\\mathcal{F}$.\n\n\\end{defi}\n\n\\begin{lmm}\\label{central1}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $P\\leq S$. Then $P$ is fully $\\mathcal{F}$-centralized if and only if $P^{\\bullet}$ is fully $\\mathcal{F}$-centralized. Furthermore, if this is the case then $C_{\\mathcal{G}}(P) = C_{\\mathcal{G}}(P^{\\bullet})$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nSuppose first that $P^{\\bullet}$ is fully $\\mathcal{F}$-centralized. By Proposition \\ref{3.2BLO3} (v), we have $C_S(P) = C_S(P^{\\bullet})$. If $Q \\in P^{\\mathcal{F}}$, then $Q^{\\bullet} \\in (P^{\\bullet})^{\\mathcal{F}}$ by Proposition \\ref{3.2BLO3}, and we have\n$$\n|C_S(Q)| = |C_S(Q^{\\bullet})|\\leq |C_S(P^{\\bullet})| = |C_S(P)|,\n$$\nwhich implies that $P$ is fully $\\mathcal{F}$-centralized.\n\nConversely, suppose that $P$ is fully $\\mathcal{F}$-centralized, and let $R \\in (P^{\\bullet})^{\\mathcal{F}}$ be fully $\\mathcal{F}$-centralized. Choose some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(P^{\\bullet} C_S(P^{\\bullet}), S)$ such that $\\gamma(P^{\\bullet}) = R$, and set $Q = \\gamma(P)$, with $Q^{\\bullet} = \\gamma(P^{\\bullet}) = R$. By Proposition \\ref{3.2BLO3} (v), we have $C_S(P) = C_S(P^{\\bullet})$, and thus\n$$\n\\gamma(C_S(P^{\\bullet})) = \\gamma(C_S(P)) = C_S(Q) = C_S(Q^{\\bullet}) = C_S(R),\n$$\nwhere the leftmost and rightmost equalities hold by Proposition \\ref{3.2BLO3} (v), and the equality in the middle holds since $P$ is fully $\\mathcal{F}$-centralized. It follows that $P^{\\bullet}$ is fully $\\mathcal{F}$-centralized.\n\nTo finish the proof, suppose that $P$ and $P^{\\bullet}$ are fully $\\mathcal{F}$-centralized, and consider $C_{\\mathcal{G}}(P)$ and $C_{\\mathcal{G}}(P^{\\bullet})$, which are $p$-local compact groups with Sylow $C_S(P) = C_S(P^{\\bullet})$. By definition, it is enough to show that $C_{\\mathcal{F}}(P) = C_{\\mathcal{F}}(P^{\\bullet})$. Notice that there is an obvious inclusion $C_{\\mathcal{F}}(P^{\\bullet}) \\subseteq C_{\\mathcal{F}}(P)$. Let $Q, R\\leq C_S(P)$, and let $f \\in \\mathrm{Hom}_{C_{\\mathcal{F}}(P)}(Q,R)$. By definition of $C_{\\mathcal{F}}(P)$, there is some $\\4{f} \\in \\mathrm{Hom}_{\\mathcal{F}}(QP, RP)$ such that $\\4{f}|_Q = f$ and $\\4{f}|_P = \\mathrm{Id}$. Let $\\gamma = \\4{f}$, and consider $\\gamma^{\\bullet} \\in \\mathrm{Hom}_{\\mathcal{F}}((QP)^{\\bullet}, (RP)^{\\bullet})$. Then $\\gamma^{\\bullet}$ restricts to a morphism $\\omega \\in \\mathrm{Hom}_{\\mathcal{F}}(QP^{\\bullet}, RP^{\\bullet})$. Furthermore, by definition of $\\omega$, we have\n$$\n\\omega|_Q = \\gamma^{\\bullet}|Q = \\gamma|Q = f \\qquad \\mbox{and} \\qquad \\omega|_{P^{\\bullet}} = \\gamma^{\\bullet}|_{P^{\\bullet}} = (f|_P)^{\\bullet} = \\mathrm{Id}.\n$$\nThus $f$ is a morphism in $C_{\\mathcal{F}}(P^{\\bullet})$, and $C_{\\mathcal{F}}(P) = C_{\\mathcal{F}}(P^{\\bullet})$.\n\\end{proof}\n\n\\begin{cor}\\label{central2}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group. Then, for each $P\\leq S$ which is fully $\\mathcal{F}$-centralized, there is a sequence of finite subgroups $P_0\\leq P_1\\leq \\ldots$ such that $P = \\bigcup_{n \\geq 0} P_n$ and such that the following conditions hold for all $n \\geq 0$.\n\\begin{enumerate}[(i)]\n\n\\item $P_n$ is fully $\\mathcal{F}$-centralized and $P_n^{\\bullet} = P^{\\bullet}$.\n\n\\item $C_{\\mathcal{G}}(P_n) = C_{\\mathcal{G}}(P)$.\n\n\\end{enumerate}\n\n\\end{cor}\n\n\\begin{proof}\n\nLet $P\\leq S$. Since $S$ is locally finite, so is $P$, and we can find some sequence of finite subgroups $P_0\\leq P_1\\leq \\ldots$ such that $P = \\bigcup_{n \\geq 0} P_n$. Furthermore, there is some $M \\in \\mathbb{N}$ such that $P_n^{\\bullet} = P^{\\bullet}$ for all $n \\geq M$, and we may assume for simplicity that $M = 0$. By Lemma \\ref{central1}, $P^{\\bullet}$ is fully $\\mathcal{F}$-centralized, and then so is $P_n$, for all $n \\geq 0$. Furthermore,\n$$\nC_{\\mathcal{F}}(P_n) = C_{\\mathcal{F}}(P_n^{\\bullet}) = C_{\\mathcal{F}}(P^{\\bullet}) = C_{\\mathcal{F}}(P),\n$$\nand this finishes the proof.\n\\end{proof}\n\nTo finish this section, we recall the construction of the quotient of a transporter system by a $p$-group. This quotient was already explored in \\cite[Appendix A]{Gonza2}, and here we only recall the necessary definitions. Let $(\\mathcal{T}, \\varepsilon, \\rho)$ be a transporter system associated to a fusion system $\\mathcal{F}$, and let $A\\leq S$ be a normal subgroup in $\\mathcal{F}$. If $P, Q\\leq S$ are such that $A\\leq P,Q$, then each morphism $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$ restricts to an automorphism of $A$, and hence it also induces a homomorphism $\\mathrm{ind}(f) \\colon P\/A \\to Q\/A$. For a subgroup $P\/A\\leq S\/A$, we will denote by $P\\leq S$ the unique subgroup of $S$ that contains $A$ with image $P\/A$ through the projection $S \\to S\/A$.\n\n\\begin{defi}\\label{quotient1}\n\nLet $A$ is a normal subgroup in $\\mathcal{F}$. The \\emph{quotient} of $\\mathcal{T}$ by $A$ is the transporter system $(\\mathcal{T}\/A, \\3{\\varepsilon}, \\3{\\rho})$ associated to the fusion system $\\mathcal{F}\/A$, where\n\\begin{itemize}\n\n\\item $\\mathcal{F}\/A$ is the fusion system over $S\/A$ with morphism sets\n$$\n\\begin{aligned}\n\\mathrm{Hom}_{\\mathcal{F}\/A}&(P\/A, Q\/A) = \\\\\n & = \\{\\overline{f} \\in \\mathrm{Hom}(P\/A,Q\/A) \\,\\, | \\,\\, \\exists f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q) \\mbox{ such that } \\overline{f} = \\mathrm{ind}(f)\\}.\n\\end{aligned}\n$$\n\n\\item $\\mathcal{T}\/A$ is the category with object set $\\{P\/A\\leq S\/A \\, | \\, A\\leq P \\in \\mathrm{Ob}(\\mathcal{T})\\}$ and morphism sets\n$$\n\\mathrm{Mor}_{\\mathcal{T}\/A}(P\/A,Q\/A) = \\mathrm{Mor}_{\\mathcal{T}}(P,Q)\/\\varepsilon_P(A).\n$$\nThe structural functors $\\3{\\varepsilon}$ and $\\3{\\rho}$ are induced, respectively, by the structural functors $\\varepsilon$ and $\\rho$ of $\\mathcal{T}$.\n\n\\end{itemize}\n\n\\end{defi}\n\n\\begin{rmk}\\label{quotient21}\n\nBy \\cite[Proposition A.2]{Gonza2}, $(\\mathcal{T}\/A, \\3{\\varepsilon}, \\3{\\rho})$ is a transporter system associated to $\\mathcal{F}\/A$. \n\n\\end{rmk}\n\n\\begin{lmm}\\label{quotient22}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a saturated fusion system over $S$, and let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$, such that $\\mathrm{Ob}(\\mathcal{T})$ contains all the centric subgroups of $\\mathcal{F}$. Let also $A\\leq S$ be normal in $\\mathcal{F}$. Then the following holds:\n\\begin{enumerate}[(i)]\n\n\\item the fusion system $\\mathcal{F}\/A$ is saturated; and\n\n\\item the transporter system $\\mathcal{T}\/A$ contains all the $\\mathcal{F}\/A$-centric subgroups of $S\/A$.\n\n\\end{enumerate}\n\n\\end{lmm}\n\n\\begin{proof}\n\nPart (i) corresponds to \\cite[Proposition A.3]{Gonza2}, and part (ii) is easily checked: let $P\/A\\leq S\/A$ be $\\mathcal{F}\/A$-centric, and let $Q\/A$ be $\\mathcal{F}\/A$-conjugate to $P\/A$. If $P, Q\\leq S$ denote the preimages of $P\/A$ and $Q\/A$ in $S$ respectively, then $Q$ is $\\mathcal{F}$-conjugate to $P$ by definition of $\\mathcal{F}\/A$. Moreover,\n$$\nC_S(Q)A\/A\\leq C_{S\/A}(Q\/A)\\leq Q\/A,\n$$\nand thus $C_S(Q)\\leq Q$ (since $A\\leq Q$). It follows that $P$ is $\\mathcal{F}$-centric, and hence an object in $\\mathcal{T}$. This implies that $P\/A \\in \\mathrm{Ob}(\\mathcal{T}\/A)$.\n\\end{proof}\n\n\n\\section{Telescopic transporter systems}\\label{Sbig}\n\nIn this section we describe a general procedure to add new objects to a given transporter system. The constructions in this section play a crucial role in the next section.\n\n\\begin{defi}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a fusion system over $S$ (not necessarily saturated), and let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$. The transporter system $\\mathcal{T}$ is \\emph{telescopic} if it satisfies the following condition.\n\\begin{itemize}\n\n\\item[(T)] For each $P \\in \\mathrm{Ob}(\\mathcal{T})$ there is a sequence $P_0\\leq P_1\\leq \\ldots$ of objects in $\\mathcal{T}$ such that $P = \\bigcup_{i \\geq 0} P_i$, and such that $P_i$ is a finite subgroup of $S$ for all $i \\geq 0$.\n\n\\end{itemize}\n\n\\end{defi}\n\n\\begin{lmm}\\label{equinerv}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\widetilde{\\LL}$ be a telescopic transporter system associated to $\\mathcal{F}$. Suppose in addition that $\\widetilde{\\LL}$ contains $\\mathcal{L}$ as a full subcategory. Then, the inclusion $\\mathcal{L} \\subseteq \\widetilde{\\LL}$ induces an equivalence between the corresponding nerves.\n\n\\end{lmm}\n\n\\begin{proof}\n\nThis is an immediate consequence of \\cite[Proposition A.9]{BLO6}.\n\\end{proof}\n\nIn terms of the above definition, in this section we study some situations where we can add objects to a given transporter system to produce a telescopic transporter system, without changing the homotopy type of the nerve of the original transporter system.\n\n\\begin{defi}\\label{compsyst}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a saturated fusion system over $S$, and let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$. Let also $(-)^{\\star}_{\\mathcal{F}} \\colon \\mathcal{F} \\to \\mathcal{F}$ and $(-)^{\\star}_{\\mathcal{T}} \\colon \\mathcal{T} \\to \\mathcal{T}$ be a pair of idempotent functors, and let $\\mathcal{C}^{\\star} \\subseteq \\mathcal{C}$, with $\\mathcal{C} = \\mathcal{F}, \\mathcal{T}$, be the full subcategory with $\\mathrm{Ob}(\\mathcal{C}^{\\star}) = \\{P^{\\star} \\, | \\, P \\in \\mathrm{Ob}(\\mathcal{C})\\}$. The pair $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$ is a \\emph{finite retraction pair} if the following conditions are satisfied.\n\\begin{itemize}\n\n\\item[(1)] For each $P\\leq S$, $P\\leq (P)^{\\star}_{\\mathcal{F}}$. Moreover, if $P \\in \\mathrm{Ob}(\\mathcal{T})$, then $(P)^{\\star}_{\\mathcal{F}} = (P)^{\\star}_{\\mathcal{T}} = P^{\\star}$.\n\n\\item[(2)] For each $P, Q\\leq S$ and each $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$, $(f)^{\\star}_{\\mathcal{F}} \\in \\mathrm{Hom}_{\\mathcal{F}}(P^{\\star}, Q^{\\star})$ extends $f$, and it is the unique extension.\n\n\\item[(i)] $\\mathrm{Ob}(\\mathcal{F}^{\\star})$ contains finitely many $S$-conjugacy classes of subgroups of $S$.\n\n\\item[(ii)] For all $P\\leq S$, $P^{\\star} = (P^{\\star})^{\\star}$.\n\n\\item[(iii)] If $P\\leq Q\\leq S$, then $P^{\\star}\\leq Q^{\\star}$.\n\n\\item[(iv)] For all $P, Q\\leq S$, $N_S(P,Q) \\subseteq N_S(P^{\\star}, Q^{\\star})$.\n\n\\item[(v)] For all $P\\leq S$, $C_S(P) = C_S(P^{\\star})$.\n\n\\item[(a)] $(-)^{\\star}_{\\mathcal{F}} \\circ \\rho = \\rho \\circ (-)^{\\star}_{\\mathcal{T}}$.\n\n\\item[(b)] For all $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$ and all $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$, we have $\\varepsilon_{Q, Q^{\\star}}(1) \\circ \\varphi = (\\varphi)^{\\star}_{\\mathcal{T}} \\circ \\varepsilon_{P, P^{\\star}}(1)$.\n\n\\end{itemize}\n\n\\end{defi}\n\nThe above definition is inspired in the pair of ``bullet'' functors described in Proposition \\ref{3.2BLO3}. In particular, conditions (i)-(v) and (a)-(b) are labelled to emphasize the relation with the motivating example. Properties (vi), (vii), (c) and (d) in \\ref{3.2BLO3} are actually consequences of the definition, as we prove below.\n\n\\begin{lmm}\\label{extraprop}\n\nLet $\\mathcal{F}$ be a saturated fusion system over a discrete $p$-toral group $S$, let $\\mathcal{T}$ be a transporter system associated to $\\mathcal{F}$, and let $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$ be a finite retraction pair. Then the following properties hold (where we label the properties according to \\ref{3.2BLO3} to emphasize the correspondence).\n\\begin{itemize}\n\n\\item[(vi)] The functor $(-)^{\\star}_{\\mathcal{F}}$ is left adjoint to the inclusion of $\\mathcal{F}^{\\star}$ as a full subcategory of $\\mathcal{F}$.\n\n\\item[(vii)] All $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups of $S$ are in $\\mathcal{F}^{\\star}$.\n\n\\item[(c)] For all $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$ and all $g \\in N_S(P,Q)$, we have $(\\varepsilon_{P,Q}(g))^{\\star}_{\\mathcal{T}} = \\varepsilon_{P^{\\star}, Q^{\\star}}(g)$.\n\n\\item[(d)] The functor $(-)^{\\star}_{\\mathcal{T}}$ is left adjoint to the inclusion of $\\mathcal{T}^{\\star}$ as a full subcategory of $\\mathcal{T}$. In particular, the inclusion $\\mathcal{T}^{\\star} \\subseteq \\mathcal{T}$ induces an equivalence $|\\mathcal{T}^{\\star}| \\simeq |\\mathcal{T}|$.\n\n\\end{itemize}\n\n\\end{lmm}\n\n\\begin{proof}\n\nProperty (vi) follows from condition (2) \\ref{compsyst}, since it implies that, for all $P, Q\\leq S$, the restriction map $\\mathrm{Hom}_{\\mathcal{F}}(P^{\\star}, Q^{\\star}) \\to \\mathrm{Hom}_{\\mathcal{F}}(P, Q^{\\star})$ is a bijection. To prove property (vii), let $P\\leq S$ be $\\mathcal{F}$-centric, and suppose that $P \\notin \\mathrm{Ob}(\\mathcal{F}^{\\star})$. Then, $P \\lneqq P^{\\star}$, which implies that $P \\lneqq N_{P^{\\star}}(P)$. Since every element of $\\mathrm{Aut}_{\\mathcal{F}}(P)$ extends uniquely to an element of $\\mathrm{Aut}_{\\mathcal{F}}(P^{\\star})$, it is not hard to see that $1 \\neq N_{P^{\\star}}(P)\/\\mathrm{Inn}(P)$ is normalized by $\\mathrm{Aut}_{\\mathcal{F}}(P)$, and hence $P$ cannot be $\\mathcal{F}$-radical. Property (c) is an immediate consequence of property (b) in \\ref{compsyst}, applied to $\\varphi = \\varepsilon_{P,Q}(g)$, together with Lemma \\ref{epimono}.\n\nFinally, we prove property (d). For each $P \\in \\mathrm{Ob}(\\mathcal{T})$, set as usual\n$$\nE(P) = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{T}}(P) \\Right2{}\\mathrm{Aut}_{\\mathcal{F}}(P)).\n$$\nWe claim that $E(P) = E(P^{\\star})$ for all $P \\in \\mathrm{Ob}(\\mathcal{T})$. Clearly, restriction from $P^{\\star}$ to $P$ maps $E(P^{\\star})$ to $E(P)$, and this restriction is injective by \\ref{epimono}. Let now $\\varphi \\in E(P)$, and consider $(\\varphi)^{\\star}_{\\mathcal{T}} \\in \\mathrm{Aut}_{\\mathcal{T}}(P^{\\star})$. By assumption, $f = \\rho(\\varphi) = \\mathrm{Id}$, and thus $(f)^{\\star}_{\\mathcal{F}} = \\mathrm{Id} \\in \\mathrm{Aut}_{\\mathcal{F}}(P^{\\star})$ by property (2) in \\ref{compsyst}. By property (a) in \\ref{compsyst} we get\n$$\n\\rho((\\varphi)^{\\star}_{\\mathcal{T}}) = (\\rho(\\varphi))^{\\star}_{\\mathcal{F}} = (\\mathrm{Id})^{\\star}_{\\mathcal{F}} = \\mathrm{Id},\n$$\nand $(\\varphi)^{\\star}_{\\mathcal{T}} \\in E(P^{\\star})$. Using axiom (A2) of transporter systems, together with property (vi) above, it is easy to deduce now that the functor $(-)^{\\star}_{\\mathcal{T}}$ is left adjoint to the inclusion of $\\mathcal{T}^{\\star}$ as a full subcategory of $\\mathcal{T}$, and property (d) follows.\n\\end{proof}\n\n\\begin{defi}\\label{wT}\n\nLet $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$ be a finite retraction pair. Define $\\4{\\mathcal{T}}$ to be the category with object set $\\mathrm{Ob}(\\4{\\mathcal{T}}) = \\{P\\leq S \\, | \\, P^{\\star} \\in \\mathrm{Ob}(\\mathcal{T})\\}$, and with morphism sets\n$$\n\\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q) = \\{\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P^{\\star}, Q^{\\star}) \\, | \\, \\varphi \\circ \\varepsilon_{P^{\\star}}(g) \\circ \\varphi^{-1} \\in \\varepsilon_{Q^{\\star}}(Q) \\mbox{, for all } g \\in P\\},\n$$\nfor all $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$. Composition in $\\4{\\mathcal{T}}$ is given by composition in $\\mathcal{T}$. Define also functors\n$$\n\\mathcal{T}_{\\mathrm{Ob}(\\4{\\mathcal{T}})}(S) \\Right3{\\4{\\varepsilon}} \\4{\\mathcal{T}} \\qquad \\mbox{and} \\qquad \\4{\\mathcal{T}} \\Right3{\\4{\\rho}} \\mathcal{F}\n$$\nas follows. The functor $\\4{\\varepsilon}$ is the identity on objects, and the functor $\\4{\\rho}$ is injective on objects. For all $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$, all $g \\in N_S(P,Q)$, and all $\\varphi \\in \\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q)$, define\n$$\n\\4{\\varepsilon}_{P,Q}(g) = \\varepsilon_{P^{\\bullet}, Q^{\\bullet}}(g) \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q) \\qquad \\mbox{and} \\qquad \\4{\\rho}(\\varphi \\colon P \\to Q) = \\rho(\\varphi \\colon P^{\\bullet} \\to Q^{\\bullet})|_P.\n$$\nThe properties of the functors $(-)^{\\star}_{\\mathcal{F}}$ and $(-)^{\\star}_{\\mathcal{T}}$ imply that both $\\4{\\mathcal{T}}$ and the above functors are well-defined, and that $\\mathcal{T}$ is a full subcategory of $\\4{\\mathcal{T}}$.\n\n\\end{defi}\n\n\\begin{prop}\\label{extendL}\n\nFor each finite retraction pair $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$, the category $\\4{\\mathcal{T}}$, with the functors $\\4{\\varepsilon}$ and $\\4{\\rho}$ defined above, is a telescopic transporter system associated to $\\mathcal{F}$. Furthermore, the functor $(-)^{\\star}_{\\mathcal{T}} \\colon \\mathcal{T} \\to \\mathcal{T}$ extends to a functor $(-)^{\\star}_{\\4{\\mathcal{T}}} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}$, which is unique satisfying the following properties\n\\begin{enumerate}[(a)]\n\n\\item There is an equality $(-)^{\\star}_{\\mathcal{F}} \\circ \\4{\\rho} = \\rho \\circ (-)^{\\star}_{\\4{\\mathcal{T}}} \\colon \\4{\\mathcal{T}} \\to \\mathcal{F}$.\n\n\\item For all $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$ and all $\\varphi \\in \\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q)$, we have $\\4{\\varepsilon}_{Q,Q^{\\star}}(1) \\circ \\varphi = (\\varphi)^{\\star}_{\\4{\\mathcal{T}}} \\circ \\4{\\varepsilon}_{P,P^{\\star}}(1)$.\n\n\\end{enumerate}\nIn particular, the inclusion of $\\mathcal{T}$ in $\\4{\\mathcal{T}}$ as a full subcategory induces an equivalence $|\\mathcal{T}| \\simeq |\\4{\\mathcal{T}}|$.\n\n\\end{prop}\n\n\\begin{proof}\n\nBy definition, $\\4{\\mathcal{T}}$ contains $\\mathcal{T}$ as a full subcategory, and the functor $(-)_{\\mathcal{T}}^{\\star} \\colon \\mathcal{T} \\to \\mathcal{T}^{\\star}$ can be extended to a functor $(-)_{\\4{\\mathcal{T}}}^{\\star} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}^{\\star}$ as follows. On objects, $(P)^{\\star}_{\\4{\\mathcal{T}}} = (P)^{\\star}_{\\mathcal{F}} = P^{\\star}$. On morphisms, $(-)^{\\star}_{\\4{\\mathcal{T}}}$ is defined by the inclusion\n$$\n\\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q) \\subseteq \\mathrm{Mor}_{\\mathcal{T}}(P^{\\star}, Q^{\\star})\n$$\ngiven by definition of $\\4{\\mathcal{T}}$. The proof of (a) and (b), as well as the uniqueness of $(-)^{\\star}_{\\4{\\mathcal{T}}}$ satisfying these conditions, is left to the reader as an easy exercise.\n\nNext we show that $\\4{\\mathcal{T}}$ is indeed a transporter system. Conditions (A1), (B) and (C) are clear. Condition (A2) follows from the properties of the functor $(-)^{\\star}_{\\mathcal{F}} \\colon \\mathcal{F} \\to \\mathcal{F}^{\\star}$. Indeed, for each $P \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$ set\n$$\n\\begin{array}{c}\n\\4{E}(P) = \\mathrm{Ker}(\\4{\\rho}_P \\colon \\mathrm{Aut}_{\\4{\\mathcal{T}}}(P) \\to \\mathrm{Aut}_{\\mathcal{F}}(P)) \\\\\nE(P^{\\star}) = \\mathrm{Ker}(\\rho_{P^{\\star}} \\colon \\mathrm{Aut}_{\\mathcal{T}}(P^{\\star}) \\to \\mathrm{Aut}_{\\mathcal{F}}(P^{\\star})).\n\\end{array}\n$$\nIf $\\varphi \\in E(P^{\\star})$, then, by definition of $\\4{\\mathcal{T}}$, together with axiom (C) on $\\mathcal{T}$, it follows that $\\varphi \\in \\4{E}(P)$. Conversely, if $\\varphi \\in \\4{E}(P)$, then by definition $\\varphi \\in \\mathrm{Aut}_{\\mathcal{T}}(P^{\\star})$ is such that $\\rho(\\varphi)|_P = \\mathrm{Id}$. By property (a) on $(-)^{\\star}_{\\4{\\mathcal{T}}}$, we have\n$$\n\\rho(\\varphi) = \\rho((\\varphi)^{\\star}_{\\4{\\mathcal{T}}}) = (\\rho(\\varphi)|_P)^{\\star}_{\\mathcal{F}} = (\\mathrm{Id})^{\\star}_{\\mathcal{F}} = \\mathrm{Id},\n$$\nand thus $\\varphi \\in E(P^{\\star})$. Hence $\\4{E}(P) = E(P^{\\star})$, and the freeness of the action of $\\4{E}(P)$ on $\\mathrm{Mor}_{\\4{\\mathcal{T}}}(P,Q)$ follows from property (A2) in $\\mathcal{T}$. That $\\4{\\rho}_{P,Q}$ is the orbit map of this action follows easily.\n\nNext we check condition (I) for $\\4{\\mathcal{T}}$. Fix $Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$. If $Q \\in \\mathrm{Ob}(\\mathcal{T})$ then there is nothing to show, since $\\mathcal{T}$ is a full subcategory of $\\4{\\mathcal{T}}$. Thus, assume that $Q \\notin \\mathrm{Ob}(\\mathcal{T})$. We can choose $Q$ such that $Q^{\\star}$ is fully $\\mathcal{F}$-normalized, so that\n$$\n\\varepsilon_{Q^{\\star}}(N_S(Q^{\\star})) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{T}}(Q^{\\star})).\n$$\nSet for short $G = \\mathrm{Aut}_{\\mathcal{T}}(Q^{\\star})$ and $K = \\varepsilon_{Q^{\\star}}(N_S(Q^{\\star}))$.\n\nWe claim first that every subgroup $H$ of $G$ has Sylow $p$-subgroups. Notice that $G\/Q^{\\star} \\cong \\mathrm{Out}_{\\mathcal{F}}(Q^{\\star})$, which is a finite group. Thus, $H\/(H \\cap Q^{\\star}) \\cong HQ^{\\star}\/Q^{\\star}\\leq G\/Q^{\\star}$, and thus $H \\cap Q^{\\star}$ is a discrete $p$-toral normal subgroup of $H$ with finite index. The claim follows by \\cite[Lemma 8.1]{BLO3}.\n\nNow, by definition we can consider $H = \\mathrm{Aut}_{\\4{\\mathcal{T}}}(Q)$ as a subgroup of $G$, and in particular the above discussion implies that $H$ has Sylow $p$-subgroups. Fix $R \\in \\operatorname{Syl}\\nolimits_p(H)$ such that $\\varepsilon_Q(N_S(Q))\\leq R$. Since $K \\in \\operatorname{Syl}\\nolimits_p(G)$, there is some $\\varphi \\in G$ such that $\\varphi \\circ R \\circ \\varphi^{-1}\\leq K$. Set $P = \\rho(\\varphi)(Q)\\leq Q^{\\star}$. Note that $P^{\\star}\\leq Q^{\\star}$ by definition of $P$, and $P$ is $\\mathcal{F}$-conjugate to $Q$, which implies that $P^{\\star}$ is $\\mathcal{F}$-conjugate to $Q^{\\star}$ This implies that $P^{\\star} = Q^{\\star}$. Thus,\n$$\n\\mathrm{Aut}_{\\4{\\mathcal{T}}}(P) = \\varphi \\circ H \\circ \\varphi^{-1} = \\varphi \\circ \\mathrm{Aut}_{\\4{\\mathcal{T}}}(Q) \\circ \\varphi^{-1}\\leq G,\n$$\nand $\\varepsilon_P(N_S(P)) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\4{\\mathcal{T}}}(P))$. Condition (I) follows.\n\nCondition (II) for $\\4{\\mathcal{T}}$ follows easily from condition (II) for $\\mathcal{T}$. Indeed, let $\\varphi \\in \\mathrm{Iso}_{\\4{\\mathcal{T}}}(P,Q)$, $P \\lhd \\4{P}\\leq S$ and $Q \\lhd \\4{Q}\\leq S$ be such that $\\varphi \\circ \\4{\\varepsilon}_P(\\4{P}) \\circ \\varphi^{-1}\\leq \\4{\\varepsilon}_Q(\\4{Q})$. By applying the functor $(-)_{\\4{\\mathcal{T}}}^{\\star} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}^{\\star}$, we get $\\varphi^{\\star} \\in \\mathrm{Iso}_{\\mathcal{T}}(P^{\\star}, Q^{\\star})$, and\n$$\nP^{\\star} \\lhd \\widehat{P} \\stackrel{def} = N_{(\\4{P})^{\\star}}(P^{\\star})\\leq S \\qquad \\mbox{and} \\qquad Q^{\\star} \\lhd \\widehat{Q} \\stackrel{def} = N_{(\\4{Q})^{\\star}}(Q^{\\star})\\leq S,\n$$\nsuch that $\\varphi^{\\star} \\circ \\varepsilon_{P^{\\star}}(\\widehat{P}) \\circ (\\varphi^{\\star})^{-1}\\leq \\varepsilon_{Q^{\\star}}(\\widehat{Q})$. Axiom (II) in $\\mathcal{T}$ implies that there exists some $\\widehat{\\varphi} \\in \\mathrm{Mor}_{\\mathcal{T}}(\\widehat{P}, \\widehat{Q})$ such that\n$$\n\\widehat{\\varphi} \\circ \\varepsilon_{P^{\\star}, \\widehat{P}}(1) = \\varepsilon_{Q^{\\star}, \\widehat{Q}}(1) \\circ \\varphi^{\\star}.\n$$\nNote that $\\4{P}\\leq \\widehat{P}$ and $\\4{Q}\\leq \\widehat{Q}$ by property (iv) in \\ref{compsyst}. Thus, we may restrict the morphism $\\widehat{\\varphi}$ to $\\4{P}$, and condition (II) follows.\n\nCondition (III) for $\\4{\\mathcal{T}}$ follows easily by condition (III) for $\\mathcal{T}$, together with the properties of the functor $(-)^{\\star}_{\\4{\\mathcal{T}}} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}^{\\star}$, since $\\mathcal{T}^{\\star}$ contains finitely many isomorphism classes of objects by property (i) in \\ref{compsyst}.\n\nLet us now prove that that $\\4{\\mathcal{T}}$ is a telescopic transporter system. Let $P \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$. By \\cite[Lemma 1.9]{BLO3}, there is a sequence $P_0\\leq P_1\\leq \\ldots$ of finite subgroups of $P$ such that $P = \\bigcup_{i \\geq 0} P_i$. Since $\\mathcal{F}^{\\star}$ contains finitely many $S$-conjugacy classes of subgroups by property (i) in \\ref{compsyst}, it follows that there is some $M \\in \\mathbb{N}$ such that $(P_i)^{\\star} = P^{\\star}$ for all $i \\geq M$, and we may assume that $M = 0$ for simplicity. This way, since $P \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$, it follows that $P_i \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$ for all $i \\geq 0$. Thus $\\4{\\mathcal{T}}$ is a telescopic transporter system.\n\nFinally, we check that the inclusion of $\\mathcal{T}$ in $\\4{\\mathcal{T}}$ as a full subcategory induces an equivalence between the corresponding nerves. Recall from property (d) in \\ref{extraprop} that the inclusion $\\mathcal{T}^{\\star} \\subseteq \\mathcal{T}$ induces an equivalence $|\\mathcal{T}^{\\star}| \\simeq |\\mathcal{T}|$. Thus, we only need to show that $(-)^{\\star}_{\\4{\\mathcal{T}}} \\colon \\4{\\mathcal{T}} \\to \\mathcal{T}^{\\star}$ is (left) adjoint to the inclusion of $\\mathcal{T}^{\\star}$ as a full subcategory of $\\4{\\mathcal{T}}$. That is, given $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$, we have to show that the restriction map\n$$\n\\mathrm{Mor}_{\\4{\\mathcal{T}}}(P^{\\star}, Q^{\\star}) \\Right2{} \\mathrm{Mor}_{\\4{\\mathcal{T}}}(P, Q^{\\star})\n$$\nis a bijection. Let $\\4{E}(P) = E(P^{\\star})$ as above, and recall that there is a bijection between the sets $\\mathrm{Hom}_{\\mathcal{F}}(P^{\\star}, Q^{\\star})$ and $\\mathrm{Hom}_{\\mathcal{F}}(P, Q^{\\star})$, given by the restriction map, by (vi) in \\ref{extraprop}. Thus, by axiom (A2) of transporter systems,\n$$\n\\mathrm{Mor}_{\\mathcal{T}}(P^{\\star}, Q^{\\star})\/E(P^{\\star}) = \\mathrm{Hom}_{\\mathcal{F}}(P^{\\star}, Q^{\\star}) \\cong \\mathrm{Hom}_{\\mathcal{F}}(P,Q) = \\mathrm{Mor}_{\\4{\\mathcal{T}}}(P, Q^{\\star})\/\\4{E}(P),\n$$\nand the claim follows.\n\\end{proof}\n\nWe call $\\4{\\mathcal{T}}$ the \\emph{telescopic transporter system associated to $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$}, or simply the telescopic transporter system associated to $\\mathcal{T}$ if there is no need to specify $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{T}})$.\n\n\\begin{prop}\\label{extendL2}\n\nEach $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^I(\\mathcal{T})$ extends uniquely to some $\\4{\\Psi} \\in \\mathrm{Aut}_{\\mathrm{typ}}^I(\\4{\\mathcal{T}})$.\n\n\\end{prop}\n\n\\begin{proof}\n\nLet $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^I(\\mathcal{T})$ and let $\\psi \\in \\mathrm{Aut}(S)$ be the automorphism induced by $\\Psi$. Then $\\Psi$ extends to $\\4{\\mathcal{T}}$ by the formulas\n$$\n\\4{\\Psi}(P) = \\psi(P) \\qquad \\mbox{and} \\qquad \\4{\\Psi}(\\varphi \\colon P \\to Q) = \\Psi(\\varphi^{\\star}_{\\4{\\mathcal{T}}} \\colon P^{\\star} \\to Q^{\\star}).\n$$\nClearly, this determines an isotypical equivalence $\\4{\\Psi}$ of $\\4{\\mathcal{T}}$. Moreover, since $\\4{\\Psi}$  is isotypical, that is $\\4{\\Psi}(\\4{\\varepsilon}_{P,Q}(1)) = \\4{\\varepsilon}_{\\4{\\Psi}(P), \\4{\\Psi}(Q)}(1)$ for all $P, Q \\in \\mathrm{Ob}(\\4{\\mathcal{T}})$ with $P\\leq Q$, and since morphisms in $\\4{\\mathcal{T}}$ are monomorphisms and epimorphisms in the categorical sense by Lemma \\ref{epimono}, it follows that $\\4{\\Psi}$ is the unique extension of $\\Psi$ to $\\4{\\mathcal{T}}$.\n\\end{proof}\n\nBelow we analyze some examples which will be of interest in later sections.\n\n\\begin{expl}\\label{expl1}\n\nWe start with the most obvious example. Let $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $(-)^{\\bullet}_{\\mathcal{F}} \\colon \\mathcal{F} \\to \\mathcal{F}$ and $(-)^{\\bullet}_{\\mathcal{L}} \\colon \\mathcal{L} \\to \\mathcal{L}$ be the usual ``bullet'' functors. Then, clearly $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ is a finite retraction pair (by \\ref{3.2BLO3}), and this way we obtain a telescopic transporter system $\\widetilde{\\LL}$ which in addition satisfies the following properties.\n\\begin{itemize}\n\n\\item[(1)] For each $P \\in \\mathrm{Ob}(\\widetilde{\\LL})$, we have \n$$\n\\mathrm{Ker}(\\mathrm{Aut}_{\\widetilde{\\LL}}(P) \\to \\mathrm{Aut}_{\\mathcal{F}}(P)) = \\varepsilon_P(C_S(P)) = \\varepsilon_P(Z(P^{\\bullet})).\n$$\n\n\\end{itemize}\nLet $P \\in \\mathrm{Ob}(\\widetilde{\\LL})$, and set $E(P) = \\mathrm{Ker}(\\mathrm{Aut}_{\\widetilde{\\LL}}(P) \\to \\mathrm{Aut}_{\\mathcal{F}}(P))$. If $P \\in \\mathrm{Ob}(\\mathcal{L})$ then $\\mathrm{Aut}_{\\widetilde{\\LL}}(P) = \\mathrm{Aut}_{\\mathcal{L}}(P)$, and there is nothing to prove. Suppose that $P \\notin \\mathrm{Ob}(\\mathcal{L})$. By definition $P^{\\bullet} \\in \\mathrm{Ob}(\\mathcal{L})$, and there is a commutative diagram of group extensions\n$$\n\\xymatrix{\nE(P^{\\bullet}) \\ar[r] & \\mathrm{Aut}_{\\widetilde{\\LL}}(P^{\\bullet}) \\ar[r] & \\mathrm{Aut}_{\\mathcal{F}}(P^{\\bullet}) \\\\\nE(P) \\ar[r] \\ar[u] & \\mathrm{Aut}_{\\mathcal{L}}(P) \\ar[r] \\ar[u] & \\mathrm{Aut}_{\\mathcal{F}}(P) \\ar[u]\n}\n$$\nwhere all the vertical arrows are inclusions. Thus, we have\n$$\nE(P)\\leq E(P^{\\bullet}) = \\varepsilon_{P^{\\bullet}}(Z(P^{\\bullet})) = \\varepsilon_{P^{\\bullet}}(C_S(P^{\\bullet})) = \\varepsilon_P(C_S(P)),\n$$\nwhere $C_S(P^{\\bullet}) = Z(P^{\\bullet})$ since $P^{\\bullet}$ is $\\mathcal{F}$-centric, and $C_S(P^{\\bullet}) = C_S(P)$ by property (v) in \\ref{compsyst}. The inclusion $\\varepsilon_P(C_S(P))\\leq E(P)$ is clear. This proves (1). In particular, every object in $\\widetilde{\\LL}$ is \\emph{quasicentric} (that is $C_{\\mathcal{F}}(P)$ is the fusion system of $C_S(P)$ for all $P \\in \\mathrm{Ob}(\\widetilde{\\LL})$), and in this sense $\\widetilde{\\LL}$ is a \\emph{quasicentric linking system}.\n\\begin{itemize}\n \n\\item[(2)] There is an isomorphism $\\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{L}) \\cong \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\widetilde{\\LL})$.\n\n\\end{itemize}\nThis follows by Proposition \\ref{extendL2}, together with the observation that every isotypical automorphism of $\\widetilde{\\LL}$ must restrict to an isotypical automorphism of $\\mathcal{L}$, since $\\mathrm{Ob}(\\mathcal{L})$ is the set of all $\\mathcal{F}$-centric subgroups of $S$. Moreover, this restriction is injective as a consequence of Lemma \\ref{epimono},  and since every morphism in $\\widetilde{\\LL}$ is the restriction of some morphism in $\\mathcal{L}$.\n\n\\end{expl}\n\n\\begin{expl}\\label{expl3}\n\nThe following is a less obvious example. Again, let $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, let $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ be the finite retraction pair in \\ref{expl1}, and let $\\widetilde{\\LL}$ be the telescopic transporter system associated to $\\mathcal{L}$. Let also $A\\leq S$ be a fully $\\mathcal{F}$-normalized subgroup such that $N_S(A)$ has finite index in $S$ (for example $A\\leq T$ a subgroup of the maximal torus of $S$).\n\nLet $N_{\\mathcal{G}}(A) = (N_S(A), N_{\\mathcal{F}}(A), N_{\\mathcal{L}}(A))$ be the normalizer $p$-local compact group of $A$, as defined in \\ref{rmknorm}. Again, in general $N_{\\mathcal{L}}(A)$ is not a telescopic linking system. In this case, since $N_{\\mathcal{G}}(A)$ is a $p$-local compact group, one could apply Example \\ref{expl1} to produce a telescopic transporter system associated to $N_{\\mathcal{L}}(A)$. However, this way there is no obvious correspondence between the telescopic transporter systems for $N_{\\mathcal{L}}(A)$ and $\\mathcal{L}$, mainly because usually the ``bullet'' functors in $N_{\\mathcal{F}}(A)$ and in $\\mathcal{F}$ do not agree, that is $(P)^{\\bullet}_{\\mathcal{F}} \\neq (P)^{\\bullet}_{N_{\\mathcal{F}}(A)}$ in general.\n\nInstead, we propose a different construction. Set for short $N = N_S(A)$, $\\mathcal{E} = N_{\\mathcal{F}}(A)$ and $\\mathcal{T} = N_{\\mathcal{L}}(A)$. We define a finite retraction pair $((-)^{\\star}_{\\mathcal{E}}, (-)^{\\star}_{\\mathcal{T}})$ as follows. For each $P\\leq N$, notice that $(P)^{\\bullet}_{\\mathcal{F}}\\leq N_S(A)$, since $N_S(A)$ already contains the maximal torus of $S$. Thus, we can define\n$$\nP^{\\star} = P^{\\bullet}.\n$$\nOn morphisms, let $P, Q\\leq N$, and let $f \\in \\mathrm{Hom}_{\\mathcal{E}}(P,Q)$. By definition of $\\mathcal{E}$, the morphism $f$ extends to some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(PA, QA)$ such that $\\gamma|_A \\in \\mathrm{Aut}_{\\mathcal{F}}(A)$. Applying the functor $(-)^{\\bullet}_{\\mathcal{F}}$ to the commutative square\n$$\n\\xymatrix{\nPA \\ar[r]^{\\gamma} & QA\\\\\nP \\ar[u]^{\\mathrm{incl}} \\ar[r]_{f} & Q \\ar[u]_{\\mathrm{incl}}\n}\n$$\nwe see that $(f)^{\\bullet}_{\\mathcal{F}}$ extends to $(\\gamma)^{\\bullet}_{\\mathcal{F}}$, and the latter restricts in turn to a morphism $\\5{\\gamma} \\in \\mathrm{Hom}_{\\mathcal{F}}((P)^{\\bullet}_{\\mathcal{F}}A, (Q)^{\\bullet}_{\\mathcal{F}}A)$ such that $\\5{\\gamma}|_A = \\gamma|_A \\in \\mathrm{Aut}_{\\mathcal{F}}(A)$. We define\n$$\n(f)^{\\star}_{\\mathcal{E}} = (f)^{\\bullet}_{\\mathcal{F}}.\n$$\nProperties (1)-(2) and (i)-(v) in \\ref{compsyst} for $(-)^{\\bullet}_{\\mathcal{F}}$ imply that $(-)^{\\star}_{\\mathcal{E}}$ also satisfies these conditions.\n\nOn $\\mathcal{T}$, define $(-)^{\\star}_{\\mathcal{T}}$ as follows. Let $P, Q \\in \\mathrm{Ob}(\\mathcal{T})$, and let $\\varphi \\in \\mathrm{Mor}_{\\mathcal{T}}(P,Q)$. By definition, $\\varphi$ is a morphism in $\\mathrm{Mor}_{\\mathcal{L}}(PA, QA)$ such that $\\rho(\\varphi)|_P \\in \\mathrm{Mor}_{\\mathcal{E}}(P,Q)$, and $\\rho(\\varphi)|_A \\in \\mathrm{Aut}_{\\mathcal{F}}(A)$. Clearly,\n$$\nP^{\\bullet}A\\leq (PA)^{\\bullet} \\qquad \\mbox{and} \\qquad Q^{\\bullet}A\\leq (QA)^{\\bullet},\n$$\nand thus $(\\varphi)^{\\bullet}_{\\mathcal{L}}$ restricts to a morphism $\\4{\\varphi} \\in \\mathrm{Mor}_{\\mathcal{L}}(P^{\\bullet}A, Q^{\\bullet}A)$ such that $\\rho(\\4{\\varphi})|_{P^{\\bullet}} \\in \\mathrm{Hom}_{\\mathcal{E}}(P^{\\bullet}, Q^{\\bullet})$ and $\\rho(\\4{\\varphi})|_A \\in \\mathrm{Aut}_{\\mathcal{F}}(A)$. Define $(\\varphi)^{\\star}_{\\mathcal{T}} = \\4{\\varphi}$. It is not difficult to check that $(-)^{\\bullet}_{\\mathcal{T}}$ satisfies properties (a)-(b) in \\ref{compsyst}.\n\nLet $\\widetilde{\\LL}$ and $\\4{\\mathcal{T}}$ be the associated telescopic transporter systems for $\\mathcal{L}$ and $\\mathcal{T}$, respectively. In general, $\\mathcal{T}$ is not a subcategory of $\\mathcal{L}$, and neither is $\\4{\\mathcal{T}}$ a subcategory of $\\widetilde{\\LL}$. Let $\\mathcal{T}_{\\geq A} \\subseteq \\mathcal{T}$ be the full subcategory of subgroups that contain $A$, and let $\\4{\\mathcal{T}}_{\\geq A} \\subseteq \\4{\\mathcal{T}}$ be the full subcategory of subgroups $P$ such that $A\\leq P^{\\bullet}$. Then the following is easily checked.\n\\begin{enumerate}[(a)]\n\n\\item $\\mathcal{T}_{\\geq A}$ is a subcategory of $\\mathcal{L}$, and $\\4{\\mathcal{T}}_{\\geq A}$ is a subcategory of $\\widetilde{\\LL}$.\n\n\\item $\\mathcal{T}_{\\geq A}$ contains all the centric radical subgroups of $\\mathcal{E}$.\n\n\\item The functor $(-)^{\\star}_{\\4{\\mathcal{T}}}$ coincides with the functor $(-)^{\\bullet}_{\\widetilde{\\LL}}$ on the subcategory $\\4{\\mathcal{T}}_{\\geq A}$.\n\n\\end{enumerate}\nThis example, including the above remarks, will be very useful in the next section, when we have to compare certain constructions on a $p$-local compact group $\\mathcal{G} = (S, \\FF, \\LL)$ and on the normalizer $N_{\\mathcal{G}}(A) = (N_S(A), N_{\\mathcal{F}}(A), N_{\\mathcal{L}}(A))$ of a certain subtorus $A\\leq S$.\n\n\\end{expl}\n\n\n\\section{Families of unstable Adams operations}\\label{Sfam}\n\nIn this section we prove Theorem \\ref{thmA}, restated as Theorem \\ref{fix6} below: every $p$-local compact group can be approximated by $p$-local finite groups. Roughly speaking, given a $p$-local compact group we produce an \\emph{approximation of $\\mathcal{G}$ by $p$-local finite groups} (defined below) by considering \\emph{fixed point subcategories} of a telescopic transporter system associated to $\\mathcal{L}$ by iterations of a given unstable Adams operation on $\\mathcal{G}$. \n\nEssentially, we follow the same lines as \\cite{Gonza1}. However, the introduction of telescopic transporter systems means a great deal of simplification, and it is actually thanks to this that we are finally able to prove Proposition \\ref{fix5}, basically the missing step in \\cite{Gonza1} in proving Theorem \\ref{thmA}. We have opted for reproving here every property that we need from \\cite{Gonza1} for the sake of completeness as well as for correcting mistakes: while working on \\ref{fix5} below, the author realized that the statement of \\cite[Lemma 2.11]{Gonza1} is false. Nevertheless, this does not affect neither the main results of \\cite{Gonza1} nor the results that we present in this paper, and this comment is just intended as a warning to the interested reader.\n\n\\begin{defi}\\label{defiapprox}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\widetilde{\\LL}$ be a telescopic transporter system associated to $\\mathcal{F}$ and containing $\\mathcal{L}$ as a full subcategory. An \\emph{approximation of $\\mathcal{G}$ by $p$-local finite groups} is a family $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ satisfying the following conditions.\n\\begin{enumerate}[(i)]\n\n\\item $S = \\bigcup_{i \\geq 0} S_i$.\n\n\\item For each $i \\geq 0$, $S_i$ is a finite $p$-group, $\\mathcal{F}_i$ is a saturated fusion system over $S_i$, and $\\mathcal{L}_i$ is a linking system associated to $\\mathcal{F}_i$. Furthermore, $\\mathrm{Ob}(\\mathcal{F}_i^{cr}) \\subseteq \\mathrm{Ob}(\\mathcal{L}_i)$, and there are inclusions $\\mathcal{L}_i \\subseteq \\mathcal{L}_{i+1}$ and $\\mathcal{L}_i \\subseteq \\widetilde{\\LL}$.\n\n\\item For each $P, Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$ and each $\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$ there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, there are objects $P_i, Q_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ and morphisms $\\varphi_i \\in \\mathrm{Mor}_{\\mathcal{L}_i}(P_i, Q_i)$, such that $P = \\bigcup_{i \\geq M} P_i$ and $Q = \\bigcup_{i \\geq M} Q_i$, and $\\4{\\varepsilon}_{Q_i, Q}(1) \\circ \\varphi_i = \\varphi \\circ \\4{\\varepsilon}_{P_i, P}(1)$.\n\n\\end{enumerate}\n\n\\end{defi}\n\nAlthough condition (i), or at least a weaker version of it, can be deduced from condition (iii) applied to $P = Q = S$ and to any $\\varphi \\in \\mathrm{Aut}_{\\mathcal{L}}(S)$, we prefer to include (i)  in the definition for the sake of clarity. We now show some basic properties of approximations of $p$-local compact groups by $p$-local finite groups.\n\n\\begin{lmm}\\label{finmorph}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be a finite approximation of $\\mathcal{G}$ by $p$-local finite groups with respect to some telescopic transporter system $\\widetilde{\\LL}$ satisfying the conditions in \\ref{defiapprox}. Then, for every finite subgroup $P\\leq S$ and every $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,S)$, there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, $P\\leq S_i$, and $f$ is the composition of a morphism $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P, S_i)$ with the inclusion $S_i\\leq S$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $Q = f(P)$, which is also a finite subgroup of $S$, and let $\\gamma \\in \\mathrm{Iso}_{\\mathcal{F}}(P,Q)$ be the restriction of $f$ to its image. By property (i) in \\ref{defiapprox}, it is clear that there is some $M_0 \\in \\mathbb{N}$ such that $P, Q\\leq S_i$ for all $i \\geq M_0$. By Alperin's Fusion Theorem \\cite[Theorem 3.6]{BLO3}, there exist $W_0 = P, W_1, \\ldots, W_n = Q\\leq S$, $U_1, \\ldots, U_n \\in \\mathrm{Ob}(\\mathcal{L}) \\subseteq \\mathrm{Ob}(\\widetilde{\\LL})$, and morphisms $\\varphi_j \\in \\mathrm{Aut}_{\\widetilde{\\LL}}(U_j)$, for $j = 1, \\ldots, n$, such that, for each $j$\n$$\nW_{j-1}, W_j\\leq U_j \\qquad \\mbox{and} \\qquad \\rho(\\varphi_j)(W_{j-1}) = W_j,\n$$\nand $\\gamma = \\rho(\\varphi_n) \\circ \\ldots \\circ \\rho(\\varphi_1)$. Combining properties (i) and (iii) in \\ref{defiapprox}, we see that for each $j = 1, \\ldots, n$ there exists some $M_j \\in \\mathbb{N}$ such that, for all $i \\geq M_j$, there exist $U_{j,i}, V_{j, i} \\in \\mathrm{Ob}(\\mathcal{L}_i)$, together with an isomorphism $\\varphi_{j.i} \\in \\mathrm{Iso}_{\\mathcal{L}_i}(U_{j,i}, V_{j,i})$, such that\n$$\nU_j = \\bigcup_{i \\geq M_j} U_{j,i} = \\bigcup_{i \\geq M_j} V_{j,i} \\qquad \\mbox{and} \\qquad \\4{\\varepsilon}_{V_{j,i}, U_j}(1) \\circ \\varphi_{j,i} = \\varphi_j \\circ \\4{\\varepsilon}_{U_{j,i}, U_j}(1).\n$$\nMoreover, since $W_{j-1}, W_j$ are finite subgroups, we may assume without loss of generality that $W_{j-1}\\leq U_{j,i}$ and $W_j\\leq V_{j, i}$ for all $i \\geq M_j$. Let $M = \\max\\{M_0, \\ldots, M_n\\}$. Then, for all $i \\geq 0$, it follows that\n$$\n\\gamma = \\rho_i(\\varphi_{n,i}) \\circ \\ldots \\circ \\rho_i(\\varphi_{1,i}) \\in \\mathrm{Mor}(\\mathcal{F}_i),\n$$\nand this finishes the proof.\n\\end{proof}\n\n\\begin{lmm}\\label{Quill}\n\nLet $\\mathcal{C}$ be a nonempty category that satisfies the following conditions:\n\\begin{enumerate}[(i)]\n\n\\item given objects $a_1, a_2$, there is an object $b$ and morphisms $a_1 \\Right1{g_1} b \\Left1{g_2} a_2$; and\n\n\\item given morphisms $g_1, g_2 \\colon a \\to b$, there is $h \\colon b \\to c$ such that $h \\circ g_1 = h \\circ g_2$.\n\n\\end{enumerate}\nThen, the nerve of $\\mathcal{C}$ is contractible.\n\n\\end{lmm}\n\n\\begin{proof}\n\nThis is \\cite[Corollary 2]{Quillen}.\n\\end{proof}\n\n\\begin{lmm}\\label{approx-2}\n\nLet $\\mathcal{C}$ be a small category all of whose morphisms are epimorphisms and monomorphisms in the categorical sense, and let $\\mathcal{C}_0 \\subseteq \\mathcal{C}_1 \\subseteq \\ldots $ be a sequence of subcategories such that $\\mathcal{C} = \\bigcup_{i \\geq 0} \\mathcal{C}_i$. Then, $|\\mathcal{C}| \\simeq \\mathrm{hocolim \\,} |\\mathcal{C}_i|$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $I$ be the category of natural ordinals, with objects $\\mathrm{Ob}(I) = \\{i \\in \\mathbb{N}\\}$, and where the morphism set $\\mathrm{Mor}_I(i,j)$ is $\\{\\sigma_{i,j}\\}$ if $i\\leq j$, or empty otherwise. Define a functor\n$$\n\\Theta \\colon I \\Right3{} \\curs{\\mathbf{Cat}}\n$$\nby $\\Theta(i) = \\mathcal{C}_i$ and $\\Theta(\\sigma_{i,j}) = \\mathrm{incl} \\colon \\mathcal{C}_i \\to \\mathcal{C}_j$. The Grothendieck construction on $\\Theta$, namely $G(\\Theta)$, is the category with object set\n$$\n\\{(i, X) \\, | \\, i \\in \\mathrm{Ob}(I) \\mbox{ and } X \\in \\mathrm{Ob}(\\mathcal{C}_i)\\}.\n$$\nThe morphism sets $\\mathrm{Mor}_{G(\\Theta)}((i,X), (j, Y))$ are empty whenever $j < i$. Otherwise they consist of the pairs $(\\sigma_{i,j}, \\varphi)$, with $\\varphi \\in \\mathrm{Mor}_{\\mathcal{C}_j}(X,Y)$. By \\cite[Theorem 1.2]{Thomason}, we have an equivalence $\\mathrm{hocolim \\,} |\\mathcal{C}_i| \\simeq |G(\\Theta)|$.\n\nConsider now the projection functor $\\tau \\colon G(\\Theta) \\to \\mathcal{C}$ that sends an object $(i, X)$ to $\\tau(i, X) = X$, and a morphism $(\\sigma_{i,j}, \\varphi)$ to $\\tau(\\sigma_{i,j}, \\varphi) = \\varphi$. We claim that this functor induces an equivalence between the corresponding nerves. For each $X \\in \\mathcal{C}$, let $\\tau\/X$ be the category with object set\n$$\n\\mathrm{Ob}(\\tau\/X) = \\{((j, Y), \\varphi) \\,\\, | \\,\\, (j, Y) \\in \\mathrm{Ob}(G(\\Theta)) \\mbox{ and } \\varphi \\in \\mathrm{Mor}_{\\mathcal{C}}(Y, X)\\}.\n$$\nA morphism in $\\tau\/X$ from $((j,Y), \\varphi)$ to $((k, Z), \\psi)$ is $(\\sigma_{i,j}, \\gamma) \\in \\mathrm{Mor}_{\\mathcal{C}}((j,Y), (k, Z))$ such that $\\varphi = \\psi \\circ \\gamma$. By \\cite[Theorem A and Corollary 2]{Quillen}, it is enough to check that $\\tau\/X$ satisfies the conditions of Lemma \\ref{Quill}. Clearly, $\\tau\/X$ is nonempty, since $((i, X), 1_X) \\in \\tau\/X$ for some $i \\in \\mathbb{N}$ big enough. Let $((j,Y), \\varphi), ((k,Z), \\psi) \\in \\mathrm{Ob}(\\tau\/X)$, and let $m = \\max\\{i,j,k\\}$. Then condition (i) in Lemma \\ref{Quill} holds with\n$$\n((j,Y), \\varphi) \\Right3{(\\sigma_{j,m},\\varphi)} ((m, X), 1_X) \\Left3{(\\sigma_{k,m}, \\psi)} ((j,Y), \\psi).\n$$\nRegarding condition (ii), notice that $\\mathrm{Mor}_{\\tau\/X}(((j,Y), \\varphi), ((k, Z), \\psi))$ is either empty, or contains a single morphisms, since morphisms in $\\mathcal{C}$ are all epimorphisms and monomorphisms in the categorical sense. Thus, $|\\tau\/X|$ is contractible for all $X \\in \\mathcal{C}$, and the claim follows.\n\\end{proof}\n\n\\begin{lmm}\\label{approx0}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and suppose $\\mathcal{G}$ admits an approximation by $p$-local finite groups $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ with respect to some telescopic transporter system $\\widetilde{\\LL}$ satisfying the conditions in \\ref{defiapprox}. Then, there is an equivalence $B\\mathcal{G} \\simeq (\\mathrm{hocolim \\,} |\\mathcal{L}_i|)^{\\wedge}_p$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nBy \\ref{defiapprox}, $\\widetilde{\\LL}$ contains $\\mathcal{L}$ as a full subcategory, and thus by \\ref{equinerv} the inclusion $\\mathcal{L} \\subseteq \\widetilde{\\LL}$ induces an equivalence $|\\mathcal{L}| \\simeq |\\widetilde{\\LL}|$. It is enough to show that $\\mathrm{hocolim \\,} |\\mathcal{L}_i| \\simeq |\\widetilde{\\LL}|$.\n\nSet $\\mathcal{L}^{\\circ} \\stackrel{def} = \\bigcup_{i \\geq 0} \\mathcal{L}_i \\subseteq \\widetilde{\\LL}$. By Lemma \\ref{epimono}, all morphisms in $\\widetilde{\\LL}$ are epimorphisms and monomorphisms in the categorical sense, and thus the same applies to $\\mathcal{L}^{\\circ}$. By Lemma \\ref{approx-2} it follows that\n$$\n\\mathrm{hocolim \\,} |\\mathcal{L}_i| \\simeq |\\mathcal{L}^{\\circ}|.\n$$\nThus, to finish the proof it is enough to show that the inclusion functor $\\iota \\colon \\mathcal{L}^{\\circ} \\to \\widetilde{\\LL}$ induces an equivalence of nerves. For each $P \\in \\mathrm{Ob}(\\widetilde{\\LL})$, the undercategory $\\iota\/P$ has object set\n$$\n\\mathrm{Ob}(\\iota\/P) = \\{(Q, \\varphi) \\, | \\, Q \\in \\mathrm{Ob}(\\mathcal{L}^{\\circ}) \\mbox{ and } \\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(Q,P)\\}.\n$$\nA morphism in $\\iota\/P$ from $(Q, \\varphi)$ to $(R, \\psi)$ is a morphism $\\gamma \\in \\mathrm{Mor}_{\\mathcal{L}^{\\circ}}(Q,R)$ such that $\\varphi = \\psi \\circ \\gamma$.\n\nWe show that $\\iota\/P$ satisfies the conditions of Lemma \\ref{Quill}, which implies that $|\\iota\/P|$ is contractible. Clearly, $\\iota\/P$ is nonempty. Let $(Q, \\varphi), (R, \\psi) \\in \\mathrm{Ob}(\\iota\/P)$. By property (iii) in Definition \\ref{defiapprox}, there is some $X \\in \\mathrm{Ob}(\\mathcal{L}^{\\circ})$, with $X\\leq P$, such that $\\varphi$ and $\\psi$ restrict to morphisms $\\varphi \\colon Q \\to X$ and $\\psi \\colon R \\to X$ in $\\mathcal{L}^{\\circ}$. Thus, condition (i) of \\ref{Quill} is satisfied with\n$$\n(Q, \\varphi) \\Right3{\\varphi} (X, \\varepsilon(1)) \\Left3{\\psi} (R, \\psi).\n$$\nRegarding condition (ii) in \\ref{Quill}, the set $\\mathrm{Mor}_{\\iota\/P}((Q, \\varphi), (R, \\psi))$ is either empty or contains a single morphism. Since the argument works for all $P \\in \\mathrm{Ob}(\\mathcal{L})$, it follows that $|\\mathcal{L}^{\\circ}| \\simeq |\\widetilde{\\LL}|$.\n\\end{proof}\n\n\\begin{rmk}\\label{approx-1}\n\nSuppose the $p$-local compact group $\\mathcal{G} = (S, \\FF, \\LL)$ admits an approximation by $p$-local finite groups $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$. For each $i$, the fusion system $\\mathcal{F}_i$ is saturated, and we may consider its associated centric linking system $\\mathcal{T}_i$. Let also $\\mathcal{H}_i = \\mathrm{Ob}(\\mathcal{L}_i) \\cap \\mathrm{Ob}(\\mathcal{T}_i)$, and let $\\mathcal{L}_{\\mathcal{H}_i} \\subseteq \\mathcal{L}_i$ be the full subcategory with object set $\\mathcal{H}_i$. Since both $\\mathrm{Ob}(\\mathcal{L}_i)$ and $\\mathrm{Ob}(\\mathcal{T}_i)$ contain $\\mathrm{Ob}(\\mathcal{F}_i^{cr})$, it follows that $\\mathrm{Ob}(\\mathcal{F}_i^{cr}) \\subseteq \\mathcal{H}_i$. Moreover, there is a commutative diagram\n$$\n\\xymatrix@R=1.2cm{\n\\ldots & \\mathcal{T}_i & \\mathcal{T}_{i+1} & \\ldots \\\\\n\\ldots \\ar[r] & \\mathcal{L}_{\\mathcal{H}_i} \\ar[u]^{\\iota_i} \\ar[d]_{\\mathrm{incl}} \\ar[r]^{\\mathrm{incl}} & \\mathcal{L}_{\\mathcal{H}_{i+1}} \\ar[u]_{\\iota_{i+1}} \\ar[d]^{\\mathrm{incl}} \\ar[r] & \\ldots \\\\\n\\ldots \\ar[r] & \\mathcal{L}_i \\ar[r]_{\\mathrm{incl}} & \\mathcal{L}_{i+1} \\ar[r] & \\ldots\n}\n$$\nwhere all the vertical arrows induce homotopy equivalences between the realizations of the corresponding nerves, by \\cite[Theorem B]{BCGLO1}. Thus, if we denote $B\\mathcal{G}_i = |\\mathcal{T}_i|^{\\wedge}_p$, then Lemma \\ref{approx0} implies that $B\\mathcal{G} \\simeq (\\mathrm{hocolim \\,} B\\mathcal{G}_i)^{\\wedge}_p$.\n\n\\end{rmk}\n\n\n\\subsection{Preliminary constructions}\\label{Sprelim}\n\nIn this subsection we establish the notation and basic facts necessary for the proof that every $p$-local compact group has an approximation by $p$-local finite groups, in the next subsection. \n\n\\begin{hyp}\\label{hyp1}\n\nFor the rest of this subsection, let $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $((-)^{\\star}_{\\mathcal{F}}, (-)^{\\star}_{\\mathcal{L}})$ be a finite retraction pair. Let also $\\widetilde{\\LL}$ be the associated telescopic transporter system, and let $\\Psi$ be a fine unstable Adams operation on $\\mathcal{L}$ (in the sense of \\ref{uAo}). By a slight abuse of notation, we denote by $\\Psi$ the corresponding extension of $\\Psi$ to $\\widetilde{\\LL}$ (see \\ref{extendL2}), which is again a fine unstable Adams operation. Set $\\Psi_0 = \\Psi$, and for all $i \\geq 0$, define\n\\begin{enumerate}[(a)]\n\n\\item $S_i = C_S(\\Psi_i) = \\{x \\in S \\, | \\, \\Psi_i(x) = x\\}$; and\n\n\\item $\\Psi_{i+1} = (\\Psi_i)^p$.\n\n\\end{enumerate}\n\n\\end{hyp}\n\n\\begin{lmm}\\label{SiS}\n\nThe following properties hold.\n\\begin{enumerate}[(i)]\n\n\\item $S = \\bigcup_{i \\geq 0} S_i$.\n\n\\item There is some $M_a \\in \\mathbb{N}$ such that $(S_i)^{\\star} = S$ for all $i \\geq M_a$.\n\n\\end{enumerate}\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $T\\leq S$ be the maximal torus of $S$, and set $T_i = T \\cap S_i$. Notice that by definition $T_i$ is the subgroup of $T$ of elements fixed by $\\Psi_i$. By hypothesis, $\\Psi = \\Psi_0$ has degree $\\zeta \\in 1 + p^m \\Z^{\\wedge}_p$ for some $m > 0$, and $\\Psi_{i+1} = (\\Psi_i)^p$. Thus, we have $T_i \\lneqq T_{i+1}$ for all $i \\geq 0$, and $T = \\bigcup_{i \\geq 0} T_i$. By \\cite[Lemma 2.6]{JLL}, there is a subgroup $H\\leq S_0$ such that $S = H \\cdot T$. Thus, $H\\leq S_i$ for all $i$, and we get\n$$\nS = H \\cdot T = \\bigcup_{i \\geq 0} H \\cdot T_i \\subseteq \\bigcup_{i \\geq 0} S_i.\n$$\nThis proves part (i). To prove part (ii), suppose otherwise that no such $M_a \\in \\mathbb{N}$ exists, that is, $(S_i)^{\\star} \\lneqq S$ for all $i \\geq 0$. Since $\\mathrm{Ob}(\\mathcal{F}^{\\star})$ contains only finitely many conjugacy classes of elements, this means that there exist some $R \\in \\mathrm{Ob}(\\mathcal{F}^{\\star})$ and some $M \\in \\mathbb{N}$ such that $R \\lneqq S$ and such that $(S_i)^{\\star} = R$ for all $i \\geq M$. Notice that this contradicts part (i): if $R \\lneqq S$, then there is some $x \\in S \\setminus R$. On the other hand, by part (i) we have $x \\in S_i$ for $i$ big enough, and thus $x \\in (S_i)^{\\star} = R$, hence a contradiction.\n\\end{proof}\n\nFor simplicity we may assume that $M_a = 0$. In particular, $S_i \\in \\mathrm{Ob}(\\widetilde{\\LL})$ for all $i \\geq 0$.\n\n\\begin{defi}\\label{Li}\n\nWith the conventions above, for each $i \\geq 0$ define $\\mathcal{L}_i$ as the category with object set $\\mathrm{Ob}(\\mathcal{L}_i) = \\{P\\leq S_i \\, | \\, P \\in \\mathrm{Ob}(\\widetilde{\\LL})\\}$, and with morphism sets\n$$\n\\mathrm{Mor}_{\\mathcal{L}_i}(P,Q) = \\{\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q) \\, | \\, \\Psi_i(\\varphi) = \\varphi\\}.\n$$\nDefine also $\\mathcal{F}_i$ as the fusion system over $S_i$ generated by the restriction of $\\4{\\rho} \\colon \\widetilde{\\LL} \\to \\mathcal{F}$ to $\\mathcal{L}_i$ (i.e. $\\mathcal{F}_i$ is $\\mathrm{Ob}(\\mathcal{L}_i)$-generated). Finally, define functors\n$$\n\\mathcal{T}_{\\mathrm{Ob}(\\mathcal{L}_i)}(S_i) \\Right3{\\varepsilon_i} \\mathcal{L}_i \\qquad \\mbox{and} \\qquad \\mathcal{L}_i \\Right3{\\rho_i} \\mathcal{F}_i\n$$\nas the obvious restrictions of the structural functors $\\4{\\varepsilon} \\colon \\mathcal{T}_{\\mathrm{Ob}(\\widetilde{\\LL})}(S) \\to \\widetilde{\\LL}$ and $\\4{\\rho} \\colon \\widetilde{\\LL} \\to \\mathcal{F}$.\n\n\\end{defi}\n\nDespite its simplicity, the following example illustrates why it is necessary to work with a telescopic linking system rather than a centric linking system.\n\n\\begin{expl}\\label{expl0}\n\nLet $T$ be a discrete $p$-torus, i.e. $T \\cong (\\Z\/p^{\\infty})^r$ for some $r \\geq 1$. Let also $\\mathcal{G} = (S, \\FF, \\LL)$ be the \\emph{trivial} $p$-local compact group associated to $T$. That is, $S = T$ and $\\mathcal{F} = \\mathcal{F}_T(T)$ is the fusion system over $T$ whose only morphisms are inclusions (since $T$ is abelian). This fusion system is obviously saturated, and has only one centric object, namely $T$ itself. Thus, $\\mathcal{L}$ has a single object, $T$, with $\\mathrm{Aut}_{\\mathcal{L}}(T) = T$. On the other hand, since $T = S$, we have $P^{\\bullet} = T$ for all $P\\leq T$, and the telescopic linking system $\\widetilde{\\LL}$ associated to $\\mathcal{L}$ in \\ref{expl1} is the actual transporter category of the group $T$. That is, $\\mathrm{Ob}(\\widetilde{\\LL}) = \\{P\\leq T\\}$, and $\\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q) = N_T(P,Q) = T$ for all $P, Q\\leq T$. Let now $\\Psi$ be an unstable Adams operation as fixed in \\ref{hyp1}. An easy computation reveals that $C_T(\\Psi)$ must be a finite subgroup of $T$, and thus is not an object in $\\mathcal{L}$. In particular, without replacing $\\mathcal{L}$ by $\\widetilde{\\LL}$, the subcategories $\\mathcal{L}_i$ defined above would be empty for all $i \\geq 0$.\n\n\\end{expl}\n\n\\begin{prop}\\label{fix1}\n\nThe following holds.\n\\begin{enumerate}[(i)]\n\n\\item For each $P \\in \\mathrm{Ob}(\\mathcal{F}^{\\star})$ there exists some $M_P \\in \\mathbb{N}$ such that $(P \\cap S_i)^{\\star} = P$ for all $i \\geq M_P$.\n\n\\item For each $\\varphi \\in \\mathrm{Mor}(\\widetilde{\\LL})$ there exists some $M_{\\varphi} \\in \\mathbb{N}$ such that $\\Psi_i(\\varphi) = \\varphi$ for all $i \\geq M_{\\varphi}$.\n\n\\item For each $i \\geq 0$ and each $x \\in S$, $x^{-1} \\cdot \\Psi_i(x) \\in T$.\n\n\\end{enumerate}\n\n\\end{prop}\n\n\\begin{proof}\n\nTo prove part (i), notice that $P^{\\star} = P = \\bigcup_{i \\geq 0} P \\cap S_i$ by Lemma \\ref{SiS} (i). Suppose that $(P \\cap S_i)^{\\star} \\lneqq P$ for all $i \\geq 0$. Since $\\mathcal{F}^{\\star}$ contains finitely many conjugacy classes of objects, this means that there is some $R \\in \\mathrm{Ob}(\\mathcal{F}^{\\star})$ such that $(P \\cap S_i)^{\\star} = R \\lneqq P$ for all $i$ big enough, contradicting the identity $P = \\bigcup_{i \\geq 0} P \\cap S_i$. Part (iii) follows immediately by definition of unstable Adams operation, since the  $\\Psi_i \\in \\mathrm{Aut}(S)$ induces the identity on $S\/T$.\n\nFinally, part (ii) follows by construction of the unstable Adams operations $\\Psi_i$. More specifically, as stablished in \\ref{hyp1}, the unstable Adams operation $\\Psi$ satisfies the following property (see Remark \\ref{uAo1}, or the proof of \\cite[Theorem 4.1]{JLL} for a more detailed explanation): there is a (finite) set $\\mathcal{M}$ of morphisms in $\\mathcal{L}^{\\star}$ such that the following holds\n\\begin{enumerate}[(1)]\n\n\\item $\\Psi(\\varphi) = \\varphi$ for all $\\varphi \\in \\mathcal{M}$; and\n\n\\item every morphism $\\psi$ in $\\mathcal{L}$ (and hence in $\\widetilde{\\LL}$ by definition) decomposes as $\\psi = \\varepsilon(g) \\circ \\varphi$, where $\\varphi$ is (the restriction of) a morphism in $\\mathcal{M}$, and $g$ is an element of $S$.\n\n\\end{enumerate}\nMoreover, these properties depend only on $\\mathcal{L}^{\\star}$ containing finitely many $S$-conjugacy classes, and on $\\mathcal{L}$ being a centric linking system, but not on the functor $(-)^{\\star}_{\\mathcal{L}}$. Fix $\\psi \\in \\mathrm{Mor}(\\widetilde{\\LL})$, and let $\\psi = \\varepsilon(g) \\circ \\varphi$ be the corresponding decomposition, as described above. By assumption, $\\Psi(\\varphi) = \\varphi$, and thus $\\Psi_i(\\varphi) = \\varphi$ for all $i \\geq 0$. Also, $S = \\bigcup_{i \\geq 0} S_i$ by Lemma \\ref{SiS} (i), and thus there exists some $M_{\\varphi} \\in \\mathbb{N}$ such that $g \\in S_i$ for all $i \\geq M_{\\varphi}$. It follows that $\\Psi_i(\\psi) = \\Psi_i(\\varepsilon(g)) \\circ \\varphi = \\varepsilon(g) \\circ \\varphi = \\psi$, and part (ii) follows.\n\\end{proof}\n\n\\begin{prop}\\label{fix2}\n\nThere exists some $M_b \\in \\mathbb{N}$ such that, for all $i \\geq M_b$, the triple $(\\mathcal{L}_i, \\varepsilon_i, \\rho_i)$ is a transporter system associated to the fusion system $\\mathcal{F}_i$.\n\n\\end{prop}\n\n\\begin{proof}\n\nWe have to check the axioms in Definition \\ref{defitransporter}. Notice that $\\mathcal{L}_i$ is a finite category, and thus we do not have to deal with axiom (III). Axioms (A1), (B) and (C) follow immediately by definition of $\\mathcal{L}_i$ as a subcategory of $\\widetilde{\\LL}$. We deal with the remaining axioms of transporter systems in separate steps for the reader's convenience.\n\nFor each $P \\in \\mathrm{Ob}(\\mathcal{L}_i)$, set\n$$\nE_i(P) = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{L}_i}(P) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(P)).\n$$\nNote that, via the inclusion $\\mathrm{Aut}_{\\mathcal{L}_i}(P) \\to \\mathrm{Aut}_{\\widetilde{\\LL}}(P^{\\star})$, and by property (a) in Definition \\ref{compsyst} of the functor $(-)^{\\star}_{\\mathcal{L}}$, the subgroup $E_i(P)$ is mapped to a subgroup of $E(P^{\\star}) = \\mathrm{Ker}(\\mathrm{Aut}_{\\widetilde{\\LL}}(P^{\\star}) \\to \\mathrm{Aut}_{\\mathcal{F}}(P^{\\star}))$. Indeed, if $\\varphi \\in E_i(P)$, then $f = \\rho_i(\\varphi) = \\mathrm{Id}$. Thus, $\\4{\\rho}((\\varphi)^{\\star}_{\\mathcal{L}}) = (\\mathrm{Id})^{\\star}_{\\mathcal{F}} = \\mathrm{Id}$.\n\n\\textbf{Step 1.} Axiom (A2) is satisfied: for each $P, Q \\in \\mathrm{Ob}(\\mathcal{L}_i)$, the group $E_i(P)$ acts freely on $\\mathrm{Mor}_{\\mathcal{L}_i}(P,Q)$ by right composition, and $\\rho_i \\colon \\mathrm{Mor}_{\\mathcal{L}_i}(P,Q) \\to \\mathrm{Hom}_{\\mathcal{F}_i}(P,Q)$ is the orbit map of this action. Also, $E_i(Q)$ acts freely on $\\mathrm{Mor}_{\\mathcal{L}_i}(P,Q)$ by left composition.\n\nThe freeness of both the left action by $E_i(P)$ and the right action of $E_i(Q)$ follows by Lemma \\ref{epimono}, which states that morphisms in $\\widetilde{\\LL}$ (and in particular in $\\mathcal{L}_i$ by definition) are monomorphisms and epimorphisms in the categorical sense. That $\\rho_i$ is the orbit map of the left conjugation action of $E_i(P)$ is now immediate.\n\n\\textbf{Step 2.} Axiom (I) is satisfied. In fact, since $\\mathcal{L}_i$ is a finite category, it is enough to show that there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, $\\mathcal{L}_i$ satisfies axiom (I') in Remark \\ref{rmktransp}: $\\varepsilon_i(S_i) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{L}_i}(S_i))$ or, equivalently, the group $\\mathrm{Out}_{\\mathcal{F}_i}(S_i)$ has trivial Sylow $p$-subgroup.\n\nFix a set $\\mathcal{N} \\subseteq \\mathrm{Aut}_{\\widetilde{\\LL}}(S)$ of representatives of the elements of $\\mathrm{Out}_{\\mathcal{F}}(S) = \\mathrm{Aut}_{\\widetilde{\\LL}}(S)\/S$. Then there exists some $M_b \\in \\mathbb{N}$ such that, for all $i \\geq M_b$, $(S_i)^{\\star} = S$ and $\\mathcal{N} \\subseteq \\mathrm{Aut}_{\\widetilde{\\LL}}(S_i)$ (by abuse of notation we consider $\\mathcal{N}$ as the restriction of its elements to $S_i$). Furthermore, by Proposition \\ref{fix1} (ii) we can assume that $\\mathcal{N} \\subseteq \\mathrm{Aut}_{\\mathcal{L}_i}(S_i)$ for all $i \\geq M_b$. Thus, there is a commutative diagram of group extensions\n$$\n\\xymatrix{\n\\4{\\varepsilon}_S(S)^{\\Psi_i} \\ar[r] \\ar[d]_{\\mathrm{res}} & \\mathrm{Aut}_{\\widetilde{\\LL}}(S)^{\\Psi_i} \\ar[r] \\ar[d]^{\\mathrm{res}} & \\mathrm{Out}_{\\mathcal{F}}(S) \\ar[d] \\\\\n\\varepsilon_i(S_i) \\ar[r] & \\mathrm{Aut}_{\\mathcal{L}_i}(S_i) \\ar[r] & \\mathrm{Out}_{\\mathcal{F}_i}(S_i)\n}\n$$\nwhere $G^{\\Psi_i} = \\{g \\in G \\, | \\, \\Psi_i(g) = g\\}$, for $G = \\varepsilon_S(S)$ or $G = \\mathrm{Aut}_{\\mathcal{L}}(S)$. Furthermore, note that the restrictions $\\mathrm{res} \\colon \\4{\\varepsilon}_S(S)^{\\Psi_i} \\to \\varepsilon_i(S_i)$ and $\\mathrm{res} \\colon \\mathrm{Aut}_{\\widetilde{\\LL}}(S)^{\\Psi_i} \\to \\mathrm{Aut}_{\\mathcal{L}_i}(S_i)$ are isomorphisms by definition. Thus, for all $i \\geq M_b$ we have $\\mathrm{Out}_{\\mathcal{F}_i}(S_i) \\cong \\mathrm{Out}_{\\mathcal{F}}(S)$, and axiom (I') follows since $\\{1\\} \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Out}_{\\mathcal{F}}(S))$.\n\n\\textbf{Step 3.} Axiom (II) is satisfied: let $\\varphi \\in \\mathrm{Iso}_{\\mathcal{L}_i}(P,Q)$, $P \\lhd \\4{P}\\leq S_i$, and $Q \\lhd \\4{Q}\\leq S_i$ be such that $\\varphi \\circ \\varepsilon_i(\\4{P}) \\circ \\varphi^{-1}\\leq \\varepsilon_i(\\4{Q})$; then there is some $\\4{\\varphi} \\in \\mathrm{Mor}_{\\mathcal{L}_i}(\\4{P}, \\4{Q})$ such that $\\4{\\varphi} \\circ \\varepsilon_i(1) = \\varepsilon_i(1) \\circ \\varphi$.\n\nFix some $i \\geq 0$, and let $\\varphi \\in \\mathrm{Iso}_{\\mathcal{L}_i}(P,Q)$, $P \\lhd \\4{P}\\leq S_i$, and $Q \\lhd \\4{Q}\\leq S_i$ be as above, and notice that in this case we have $\\varepsilon_i(1) = \\4{\\varepsilon} \\colon X \\to \\4{X}$, where $X = P,Q$. Since $\\widetilde{\\LL}$ is a transporter system, there is some $\\4{\\varphi} \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(\\4{P}, \\4{Q})$ such that $\\4{\\varphi} \\circ \\varepsilon_i(1) = \\varepsilon_i(1) \\circ \\varphi$. Applying $\\Psi_i$ to this equation we get\n$$\n\\4{\\varphi} \\circ \\varepsilon_i(1) = \\varepsilon_i(1) \\circ \\varphi = \\Psi_i(\\varepsilon_i(1) \\circ \\varphi) = \\Psi_i(\\4{\\varphi}) \\circ \\varepsilon_i(1),\n$$\nand thus Lemma \\ref{epimono} implies that $\\Psi_i(\\4{\\varphi}) = \\4{\\varphi}$.\n\\end{proof}\n\nAgain, we may assume that $M_b = 0$ for simplicity.\n\n\\begin{cor}\\label{fix3}\n\nFor all $i \\geq 0$, the fusion system $\\mathcal{F}_i$ is $\\mathrm{Ob}(\\mathcal{L}_i)$-generated and $\\mathrm{Ob}(\\mathcal{L}_i)$-saturated.\n\n\\end{cor}\n\n\\begin{proof}\n\nThe fusion system $\\mathcal{F}_i$ is $\\mathrm{Ob}(\\mathcal{L}_i)$-generated by definition, and the $\\mathrm{Ob}(\\mathcal{L}_i)$-saturation follows by \\cite[Proposition 3.6]{OV}.\n\\end{proof}\n\n\n\\subsection{Existence of finite approximations}\\label{Ssapprox}\n\nWe are ready to prove that every $p$-local compact group has an approximation by $p$-local finite groups. In fact, we prove something stronger: every fine unstable Adams operation (in the sense of \\ref{uAo}) defines an approximation by $p$-local finite groups.\n\n\\begin{hyp}\\label{hyp2}\n\nFor the rest of this section, let $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ be the finite retraction pair of \\ref{expl1}. Let also $\\widetilde{\\LL}$ be the associated telescopic linking system, and let $\\Psi$ be a fine unstable Adams operation on $\\widetilde{\\LL}$ (that is, $\\Psi$ is an unstable Adams operation whose degree $\\zeta \\neq 1$ is congruent to $1$ modulo $p$). Let also $\\{\\Psi_i\\}_{i \\geq 0}$ and $\\{S_i\\}_{i \\geq 0}$ be as defined in \\ref{hyp1}, and let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be the associated family of finite transporter systems defined in \\ref{Li}. Finally, for all $i \\geq 0$ let\n\\begin{enumerate}[(a)]\n\n\\item $\\Gamma_i =\\{P\\leq S_i \\, | \\, P^{\\bullet} \\notin \\mathrm{Ob}(\\mathcal{L}^{\\bullet}) \\mbox{ and } P^{\\bullet} \\cap S_i = P\\}$; and\n\n\\item $\\Omega_i = \\{R \\in \\mathrm{Ob}(\\mathcal{F}^{\\bullet})\\setminus \\mathrm{Ob}(\\mathcal{L}^{\\bullet}) \\, | \\, (R \\cap S_i)^{\\bullet} = R\\}$.\n\n\\end{enumerate}\nNote that $\\Psi_i$ is a fine unstable Adams operation for all $i$. Also, for each $i \\geq 0$ the sets $\\Gamma_i$ and $\\Omega_i$ are in one-to-one correspondence with each other for all $i \\geq 0$. The bijection is given in one direction by $P \\mapsto P^{\\bullet}$, and in the reverse direction by $R \\mapsto R \\cap S_i$. Also note that $\\Gamma_i \\cap \\mathrm{Ob}(\\mathcal{L}_i) = \\emptyset$ for all $i \\geq 0$.\n\n\\end{hyp}\n\n\\begin{prop}\\label{fix2-1}\n\nFor all $i \\geq 0$, $\\mathcal{L}_i$ is a linking system, and $P$ is $\\mathcal{F}_i$-centric for each $P \\in \\mathrm{Ob}(\\mathcal{L}_i)$.\n\n\\end{prop}\n\n\\begin{proof}\n\nBy definition of $\\mathcal{L}_i$, for all $P \\in \\mathrm{Ob}(\\mathcal{L}_i)$ we have $\\mathrm{Ker}(\\mathrm{Aut}_{\\widetilde{\\LL}}(P) \\to \\mathrm{Aut}_{\\mathcal{F}}(P)) = \\varepsilon_P(Z(P^{\\bullet}))$ by property (1) in \\ref{expl1}, and thus $\\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{L}_i}(P) \\to \\mathrm{Aut}_{\\mathcal{F}_i}(P)) = \\varepsilon_P(Z(P^{\\bullet}) \\cap S_i) = \\varepsilon_i(Z(P))$. Furthermore, we have $\\varepsilon_i(C_{S_i}(P))\\leq \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{L}_i}(P) \\to \\mathrm{Aut}_{\\mathcal{F}_i}(P))$ by axiom (C) of transporter systems on $\\mathcal{L}_i$, and the statement follows immediately.\n\\end{proof}\n\n\\begin{prop}\\label{fix4}\n\nLet $i \\geq 0$ and let $P\\leq S_i$ be such that $P \\lneqq P^{\\bullet} \\cap S_i$. Then\n$$\n\\mathrm{Out}_{S_i}(P) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(P)) \\neq 1.\n$$\nIn particular, $P$ is not $\\mathcal{F}_i$-centric $\\mathcal{F}_i$-radical.\n\n\\end{prop}\n\n\\begin{proof}\n\nSuppose $P$ is $\\mathcal{F}_i$-centric. Since $P \\notin \\Gamma_i$, we have $P \\lneqq P^{\\bullet} \\cap S_i \\stackrel{def} = Q$. Notice that the functor $(-)^{\\bullet}$ induces an inclusion $\\mathrm{Aut}_{\\mathcal{F}_i}(P)\\leq \\mathrm{Aut}_{\\mathcal{F}_i}(Q)$. Consider the subgroup $A = \\{c_x \\in \\mathrm{Aut}_{\\mathcal{F}_i}(P) \\, | \\, x \\in N_Q(P)\\}$. Via the above inclusion, we have $A = \\mathrm{Aut}_{\\mathcal{F}_i}(P) \\cap \\mathrm{Inn}(Q)$. Since $P \\lneqq Q$, it follows that $P \\lneqq N_Q(P)$, and hence $\\mathrm{Inn}(P) \\lneqq A$, since $P$ is $\\mathcal{F}_i$-centric by hypothesis. The group $\\mathrm{Aut}_{\\mathcal{F}_i}(P)$, seen as a subgroup of $\\mathrm{Aut}_{\\mathcal{F}_i}(Q)$, normalizes $\\mathrm{Inn}(Q)$, and thus $A \\lhd \\mathrm{Aut}_{\\mathcal{F}_i}(P)$, and\n$$\n\\{1\\} \\neq A\/\\mathrm{Inn}(P)\\leq \\mathrm{Out}_{S_i}(P) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(P)).\n$$\nThis finishes the proof.\n\\end{proof}\n\n\\begin{prop}\\label{fix5}\n\nAssume Hypothesis \\ref{hyp2}. Then, there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, the following holds: if $P \\in \\Gamma_i$, then either $P$ is not $\\mathcal{F}_i$-centric or $P$ is $\\mathcal{F}_i$-conjugate to some $Q\\leq S_i$ such that\n$$\n\\mathrm{Out}_{S_i}(Q) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(Q)) \\neq 1.\n$$\n\n\\end{prop}\n\n\\begin{proof}\n\nWe start with some general observations, after which we deduce a certain condition ($\\ddagger$) which will imply the statement. The rest of the proof consists of a series of steps to show that ($\\ddagger$) holds.\n\nBy Proposition \\ref{3.2BLO3} (i), the set $\\mathrm{Ob}(\\mathcal{F}^{\\bullet})$ contains finitely many $\\mathcal{F}$-conjugacy classes, and the same applies to $\\mathrm{Ob}(\\mathcal{F}^{\\bullet}) \\setminus \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$. Let $\\mathcal{P} = \\{X_1, \\ldots, X_n\\}$ be a set of representatives of the $\\mathcal{F}$-conjugacy classes in $\\mathrm{Ob}(\\mathcal{F}^{\\bullet}) \\setminus \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$. By Proposition \\ref{fix1} (i), there exists some $M' \\in \\mathbb{N}$ such that $\\mathcal{P} \\subseteq \\Omega_i$ for all $i \\geq M'$, and we can assume that $M' = 0$ without loss of generality.\n\nFor each $X \\in \\mathcal{P}$, fix also a set $\\mathcal{H}_X = \\{Y_1, \\ldots, Y_m\\}$ of representatives of the $T$-conjugacy classes in $X^{\\mathcal{F}}$. Again, there exists some $M'' \\in \\mathbb{N}$ such that $\\mathcal{H}_X \\subseteq \\Omega_i$ for all $i \\geq M''$, and once more we can assume $M'' = 0$ for simplicity.\n\nLet $i \\geq 0$, and let $Q\\leq S_i$. Then, $T_{Q^{\\bullet}} \\cap Q \\lhd Q$, and every automorphism in $\\mathcal{F}_i$ of $Q$ restricts to an automorphism of $T_{Q^{\\bullet}} \\cap Q$ (by the properties of $(-)^{\\bullet}$, see \\ref{3.2BLO3}). Set\n$$\nK_Q = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}_i}(Q) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(T_{Q^{\\bullet}} \\cap Q) \\times \\mathrm{Aut}(Q\/(T_{Q^{\\bullet}} \\cap Q))),\n$$\nwhich is a normal $p$-subgroup of $\\mathrm{Aut}_{\\mathcal{F}_i}(Q)$ by Lemma \\ref{Kpgp}. Since $\\Gamma_i$ and $\\Omega_i$ are in one-to-one correspondence (see \\ref{hyp2}), in order to prove the statement it is enough to prove the following, slightly stronger statement for each $X \\in \\mathcal{P}$ and each $Y \\in \\mathcal{H}_X$.\n\\begin{itemize}\n\n\\item[($\\ddagger$)] There exists some $M_Y \\in \\mathbb{N}$ such that the following holds for all $i \\geq M_Y$: if $R \\in Y^T \\cap \\Omega_i$, then either $R \\cap S_i$ is not $\\mathcal{F}_i$-centric, or $R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to some $Q\\leq S_i$ such that $K_Q \\cap \\mathrm{Aut}_S(Q)$ contains some element which is not in $\\mathrm{Inn}(Q)$.\n\n\\end{itemize}\n\nIndeed, suppose that ($\\ddagger$) holds for all $X \\in \\mathcal{P}$ and all $Y \\in \\mathcal{H}_X$. We claim that the statement follows with $M = \\max\\{M_Y \\, | \\, X \\in \\mathcal{P} \\mbox{ and } Y \\in \\mathcal{H}_X\\}$. To prove this, let $P \\in \\Gamma_i$, and let $R = P^{\\bullet} \\in \\Omega_i$. Then, there exist some $X \\in \\mathcal{P}$ and some $Y \\in \\mathcal{H}_X$ such that $R \\in Y^T \\cap \\Omega_i$. Thus, ($\\ddagger$) applies to $Y$, and it follows that either $P = R \\cap S_i$ is not $\\mathcal{F}_i$-centric, or $P = R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to some $Q\\leq S_i$ such that $K_Q \\cap \\mathrm{Aut}_S(Q)$ contains some element which is not in $\\mathrm{Inn}(Q)$, in which case we have\n$$\n\\mathrm{Out}_{S_i}(Q) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(Q)) \\neq 1.\n$$\n\nFor the rest of the proof, fix $X \\in \\mathcal{P}$ and $\\mathcal{H}_X$ as above. Since this proof is rather long, we have divided it into several steps, for the reader's convenience. We also include a brief summary of the steps in the proof. In Step 1, we give a general tool to deduce that ($\\ddagger$) holds in some cases. In Step 2 we show that we may reduce to prove that ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ with $T_Y$ fully $\\mathcal{F}$-normalized. In Step 3 we justify the reduction to assume that $A = T_X$ is normal in $\\mathcal{F}$. In Step 4 we show some properties regarding the quotient $\\mathcal{G}\/A = (S\/A, \\mathcal{F}\/A, \\mathcal{L}\/A)$. In Step 5 we introduce a certain subgroup $Z_Y$ for each $Y \\in \\mathcal{H}_X$, related to $C_{T\/A}(Y\/A)$, and prove some of its properties. In Step 6, we show that ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ such that $Z_Y \\not\\leq Y$. In Step 7 we prove that we may reduce to prove ($\\ddagger$) for all $Y \\in \\mathcal{H}_X$ such that $Z_Y$ is fully $\\mathcal{F}$-normalized. In Step 8 we show that ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ such that $C_S(Y) \\not\\leq Y$ (in particular, this applies to $Y \\in \\mathcal{H}_X$ such that $C_S(Y) \\not\\leq Y$ and such that $Z_Y$ is fully $\\mathcal{F}$-normalized). In step 9 we prove a technical property necessary for the last step of the proof. Finally, in Step 10 we show that ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ such that $Z_Y$ is fully $\\mathcal{F}$-normalized.\n\n\\textbf{Step 1.} Let $Y, Y' \\in \\mathcal{H}_X$, and suppose that ($\\ddagger$) holds for $Y$. Suppose in addition that there exists some $f \\in \\mathrm{Hom}_{\\mathcal{F}}(Y'T, YT)$ such that $f(Y') = Y$. Then ($\\ddagger$) holds for $Y'$.\n\nLet $f \\in \\mathrm{Hom}_{\\mathcal{F}}(Y'T, YT)$ as above. By Alperin's Fusion Theorem \\cite[Theorem 3.6]{BLO3}, there exist subgroups $A_0 = Y'T, A_1, \\ldots, A_n = YT\\leq S$, objects $B_1, \\ldots, B_n \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$, and automorphisms $\\phi_k \\in \\mathrm{Aut}_{\\widetilde{\\LL}}(B_k)$, for $k = 1, \\ldots, n$, such that\n$$\nA_{k-1}, A_k\\leq B_k \\qquad \\mbox{and} \\qquad \\rho(\\phi_k)(A_{k-1}) = A_k\n$$\nfor each $k = 1, \\ldots, n$, and such that $f = \\rho(\\phi_n) \\circ \\ldots \\rho(\\phi_1)$. By Proposition \\ref{fix1} (i) and (ii), there exists some $M_1 \\in \\mathbb{N}$ such that, for all $i \\geq M_1$ and all $k = 1, \\ldots, n$,\n$$\nB_k \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i) \\qquad \\mbox{and} \\qquad (\\phi_k)|_{B_k \\cap S_i} \\in \\mathrm{Aut}_{\\mathcal{L}_i}(B_k \\cap S_i).\n$$\nMoreover, since $T\\leq A_0$, it follows that $T\\leq A_k, B_k$ for all $k = 1, \\ldots, n$. Note that for each $k$, there exists some $t_k \\in T$ such that $t_k A_k t_k^{-1} \\in \\mathcal{H}_X$ (in particular, $t_0 = t_n = 1$). Since $T\\leq B_k$ for each $k$, we may replace $\\phi_k$ by $\\varepsilon(t_k) \\circ \\phi_k \\circ \\varepsilon(t_{k-1})$, and this way we may assume that $A_k \\in \\mathcal{H}_X$ for each $k$.\n\nSuppose now that ($\\ddagger$) holds for $Y$. That is, there exists some $M_Y \\in \\mathbb{N}$ (we may choose $M_Y \\geq M_1$) such that, for all $i \\geq M_Y$, the following holds: if $R \\in Y^T \\cap \\Omega_i$, then $R \\cap S_i$ satisfies the conclusion of ($\\ddagger$). Let $i \\geq M_Y$, and let $K \\in (Y')^T \\cap \\Omega_i$, and let $H = f(K)$. Since $f \\colon Y'T \\to YT$, it follows that $H \\in Y^T$. Moreover, $f(K \\cap S_i) = H \\cap S_i$ (since each $\\phi_k$ above is fixed by $\\Psi_i$). Thus, $K \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $H \\cap S_i$, and ($\\ddagger$) holds for $Y'$.\n\n\\textbf{Step 2.} We show that if ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$ such that $T_Y$ is fully $\\mathcal{F}$-normalized, then ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$.\n\nIndeed, let $Y' \\in \\mathcal{H}_X$, and suppose that $T_{Y'}$ is not fully $\\mathcal{F}$-normalized. Then, there exists some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S(T_{Y'}), S)$ such that $\\gamma(T_{Y'})$ is fully $\\mathcal{F}$-normalized. Since $Y'\\leq N_S(T_{Y'})$, we may define $Y = \\gamma(Y')$. Since $\\mathcal{H}_X$ contains representatives of all the $T$-conjugacy classes in $X^{\\mathcal{F}}$, there is some $t \\in T$ such that $tYt^{-1} \\in \\mathcal{H}_X$. Thus, upon replacing $\\gamma$ by $c_t \\circ \\gamma$, we may assume that $Y \\in \\mathcal{H}_X$. Note that $T_{Y} = \\gamma(T_{Y'})$. Notice also that $T\\leq N_S(T_{Y'})$, and thus $\\gamma$ restricts to some $f \\in \\mathrm{Hom}_{\\mathcal{F}}(Y'T, YT)$ such that $Y = \\gamma(Y')$. The claim follows by Step 1.\n\n\\textbf{Step 3.} Let $\\mathcal{M} = \\{A = T_Y\\leq T \\, | \\, Y \\in \\mathcal{H}_X \\mbox{ and $T_Y$ is fully $\\mathcal{F}$-normalized}\\}$. Since $\\mathcal{H}_X$ is finite, so is $\\mathcal{M}$. For each $A \\in \\mathcal{M}$, let $\\mathcal{H}_{X,A} = \\{U \\in \\mathcal{H}_X \\, | \\, T_U = A\\} \\subseteq \\mathcal{H}_X$. To prove the statement of \\ref{fix5}, it is enough to prove that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_{X,A}$, for a fixed $A \\in \\mathcal{M}$ at a time. Fix $A \\in \\mathcal{M}$, and let $\\mathcal{H}_{X,A}$ be as above. The main goal of this step is to justify the reduction to the case where $A$ is normal in $\\mathcal{F}$.\n\nSince $A$ is fully $\\mathcal{F}$-normalized, to prove that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_{X,A}$,  we can reduce to work on the normalizer $p$-local compact group $N_{\\mathcal{G}}(A) = (N_S(A), N_{\\mathcal{F}}(A), N_{\\mathcal{L}}(A))$ defined in \\ref{rmknorm}, instead of the whole $\\mathcal{G} = (S, \\FF, \\LL)$. Set for short $N = N_S(A)$, $\\mathcal{E} = N_{\\mathcal{F}}(A)$, and $\\mathcal{T} = N_{\\mathcal{L}}(A)$, and note that $T\\leq N$ since $A\\leq T$. Let $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ be the finite retraction pair for $\\mathcal{G}$ fixed in \\ref{hyp2}, and let $((-)^{\\star}_{\\mathcal{E}}, (-)^{\\star}_{\\mathcal{T}})$ be the finite retraction pair for $N_{\\mathcal{G}}(A)$ described in \\ref{expl3}. Recall that $P^{\\star} = P^{\\bullet}$ for all $P\\leq N$. Let also $\\widetilde{\\LL}$ and $\\4{\\mathcal{T}}$ be the corresponding associated telescopic transporter systems.\n\nWe start by stating and proving several general properties.\n\\begin{itemize}\n\n\\item[(3-a)] Let $U \\in \\mathcal{H}_{X,A}$ and let $R \\in U^T$. Then, $T_R = A$, $N_S(T_R) = N_S(A)$, and $N_S(R)\\leq N_S(A)$. Moreover, every automorphism of $R$ preserves $T_R$, and thus $\\mathrm{Aut}_{\\mathcal{F}}(R) = \\mathrm{Aut}_{\\mathcal{E}}(R)$.\n\n\\item[(3-b)] Since $A\\leq T$ is a subtorus, it follows that $\\Psi_i(A) = A$ for all $i \\geq 0$. Thus, each $\\Psi_i$ restricts to a fine unstable Adams operation (see \\ref{uAo}) on $N_{\\mathcal{G}}(A)$, which extends to $\\4{\\mathcal{T}}$ by \\ref{extendL2}. Let us denote by $\\Psi_i$ the resulting unstable Adams operation.\n\n\\end{itemize}\nConsider the family of transporter systems $\\{(N_i, \\mathcal{E}_i, \\mathcal{T}_i)\\}_{i \\geq 0}$ associated to $(N, \\mathcal{E}, \\4{\\mathcal{T}})$ in \\ref{Li}. As pointed out in \\ref{expl3}, $\\4{\\mathcal{T}}$ is not a subcategory of $\\widetilde{\\LL}$, and thus it is hard to compare the fusion systems $\\mathcal{E}_i$ and $\\mathcal{F}_i$. However, if we restrict to the full subcategory $\\4{\\mathcal{T}}_{\\geq A} \\subseteq \\4{\\mathcal{T}}$ of subgroups $P\\leq N$ such that $A\\leq P^{\\star} = P^{\\bullet}$, then $\\4{\\mathcal{T}}_{\\geq A}$ is a subcategory of $\\widetilde{\\LL}$, by \\ref{expl3} (a). In particular we have the following.\n\\begin{itemize}\n\n\\item[(3-c)] For all $i \\geq 0$, there is an inclusion $\\4{\\mathcal{T}}_{\\geq A} \\cap \\mathcal{T}_i \\subseteq \\mathcal{L}_i$.\n\n\\end{itemize}\nFinally, note that $A^{\\bullet} = A$, since $A$ is the maximal torus of $U$, for some $U \\in \\mathcal{H}_{X,A}$, and $U = U^{\\bullet}$ by hypothesis. The following holds.\n\\begin{itemize}\n\n\\item[(3-d)] For each $i \\geq 0$, let $A_i = A \\cap S_i$. By \\ref{fix1} (i), there exists some $M_2 \\in \\mathbb{N}$ such that $(A_i)^{\\bullet} = A$ for all $i \\geq M_2$. For simplicity we may assume that $M_2 = 0$.\n\n\\item[(3-e)] For each $U \\in \\mathcal{H}_{X,A}$ and each $R \\in U^T \\cap \\Omega_i$, we have $R \\cap N_i = R \\cap S_i$ and $\\mathrm{Aut}_{N_i}(R \\cap S_i) = \\mathrm{Aut}_{S_i}(R \\cap S_i)$. This follows since $N_S(R \\cap S_i)\\leq N_S(R)\\leq N_S(A) = N$ by (3-a).\n\n\\item[(3-f)] Let $H,K\\leq N_i$ be such that $A\\leq H^{\\bullet}, K^{\\bullet}$. Then, for all $f \\in \\mathrm{Hom}_{\\mathcal{E}}(H,K)$, we have $(f)^{\\star}_{\\mathcal{E}} = (f)^{\\bullet}_{\\mathcal{F}}$. Similarly, if $H, K \\in \\mathrm{Ob}(\\4{\\mathcal{T}}_{\\geq A})$ and $\\varphi \\in \\mathrm{Mor}_{\\4{\\mathcal{T}}}(H,K)$, then $H, K \\in \\mathrm{Ob}(\\widetilde{\\LL})$, and $(\\varphi)^{\\star}_{\\4{\\mathcal{T}}} = (\\varphi)^{\\bullet}_{\\widetilde{\\LL}}$. The first part follows by definition of $(-)^{\\star}_{\\mathcal{E}}$ in \\ref{expl3}, and the second part follows by \\ref{expl3} (c).\n\n\\end{itemize}\nIn fact, we deduce more. Let $W\\leq N_i$ be such that $A\\leq W^{\\bullet}$. Then, the following holds.\n\\begin{itemize}\n\n\\item[(3-g)] $\\mathrm{Hom}_{\\mathcal{E}_i}(W, N_i) \\subseteq \\mathrm{Hom}_{\\mathcal{F}_i}(W, N_i)$.\n\n\\end{itemize}\nLet $f \\in \\mathrm{Hom}_{\\mathcal{E}_i}(W, N_i)$. Since $\\mathcal{E}_i$ is $\\mathrm{Ob}(\\mathcal{T}_i)$-generated, there exist objects $H_k, H_k' \\in \\mathrm{Ob}(\\mathcal{T}_i)$ and morphisms $\\gamma_k \\in \\mathrm{Hom}_{\\mathcal{T}_i}(H_k, H'_k)$, for $k = 1, \\ldots, n$, and such that, upon taking the necessary restrictions,\n$$\nf = \\4{\\rho}_i(\\gamma_n) \\circ \\ldots \\circ \\4{\\rho}_i(\\gamma_1).\n$$\nNotice that $(A_i)^{\\bullet} = A\\leq W^{\\bullet}$. Thus, $A\\leq W^{\\bullet}\\leq (H_1)^{\\bullet}$, and $A = \\4{\\rho}(\\gamma_1)(A)\\leq (H_1')^{\\bullet}$ since $\\gamma_1 \\in \\mathrm{Mor}(\\4{\\mathcal{T}})$. Inductively, it follows that $A\\leq (H_k)^{\\bullet}, (H'_k)^{\\bullet}$ for all $k$, and thus $H_k, H_k' \\in \\mathrm{Ob}(\\4{\\mathcal{T}}_{\\geq A})$. By (3-c) above, for all $k = 1, \\ldots, n$ we have $H_k, H'_k \\in \\mathcal{L}_i$, and $\\gamma_k \\in \\mathrm{Mor}(\\mathcal{L}_i)$. Thus $f \\in \\mathrm{Mor}(\\mathcal{F}_i)$, and (3-g) follows. In particular, if $U \\in \\mathcal{H}_{X,A}$, and $P \\in U^{\\mathcal{E}} \\cap \\Omega_i$, then\n$$\n\\mathrm{Hom}_{\\mathcal{E}_i}(P \\cap S_i, N_i) \\subseteq \\mathrm{Hom}_{\\mathcal{F}_i}(P \\cap S_i, N_i).\n$$\n\nLet $U \\in \\mathcal{H}_{X,A}$. For each $P \\in U^{\\mathcal{E}} \\cap \\Omega_i$, note that $A \\cap S_i \\lhd P \\cap S_i$, and every automorphism of $P \\cap S_i$ in $\\mathcal{F}_i$ restricts to an automorphism of $A \\cap S_i$. Let $\\3{P} = P\/A$. Note that $P = (P \\cap S_i)A$ (since $P \\in \\Omega_i$), and thus we have $(P \\cap S_i)\/(A \\cap S_i) \\cong \\3{P}$. Set for short $Q = P \\cap S_i$, and let\n$$\nK_Q = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}_i}(Q) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(A \\cap S_i) \\times \\mathrm{Aut}(\\3{P}));\n$$\n$$\nK'_Q = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{E}_i}(Q) \\Right2{} \\mathrm{Aut}_{\\mathcal{E}_i}(A \\cap S_i) \\times \\mathrm{Aut}(\\3{P})).\n$$\nBy Lemma \\ref{Kpgp}, we have $K_Q\\leq O_p(\\mathrm{Aut}_{\\mathcal{F}_i}(Q))$ and $K'_Q\\leq O_p(\\mathrm{Aut}_{\\mathcal{E}_i}(Q))$. Moreover, by (3-g) there is an inclusion $\\mathrm{Aut}_{\\mathcal{E}_i}(Q)\\leq \\mathrm{Aut}_{\\mathcal{F}_i}(Q)$, and it follows that $K'_Q\\leq K_Q$. We claim that, in order to prove that ($\\ddagger$) holds for $U \\in \\mathcal{H}_{X,A}$, it is enough to prove that the following version of ($\\ddagger$) in terms of $\\mathcal{E}_i$ holds.\n\\begin{itemize}\n\n\\item[(3-h)] There exists some $M_U \\in \\mathbb{N}$ such that, for all $i \\geq M_U$ and all $R \\in U^T \\cap \\Omega_i$, either $R \\cap S_i$ is not $\\mathcal{E}_i$-centric, or $R \\cap S_i$ is $\\mathcal{E}_i$-conjugate to some $Q$ such that $K'_{Q} \\cap \\mathrm{Aut}_{N_i}(Q)$ contains some element that is not in $\\mathrm{Inn}(Q)$.\n\n\\end{itemize}\nIndeed, suppose that (3-h) holds for $U$, and let $i \\geq M_U$ and $R \\in U^T \\cap \\Omega_i$. Clearly, if $R \\cap S_i$ is not $\\mathcal{E}_i$-centric, then it is not $\\mathcal{F}_i$-centric by (3-g). Suppose that $R \\cap S_i$ is $\\mathcal{E}_i$-centric. Then, $R \\cap S_i$ is $\\mathcal{E}_i$-conjugate to some $Q$ such that $K'_Q \\cap \\mathrm{Aut}_{N_i}(Q)$ contains some element that is not in $\\mathrm{Inn}(Q)$. Then, $R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $Q$ by (3-g), and ($\\ddagger$) holds for $U$ since $K'_Q \\cap \\mathrm{Aut}_{N_i}(Q)\\leq K_Q \\cap \\mathrm{Aut}_{S_i}(Q)$.\n\nThe above just shows that, in order to prove ($\\ddagger$) for $U \\in \\mathcal{H}_{X, A}$ with respect to $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$, it is enough to prove that the conclusion ($\\ddagger$) holds for $U$ with respect to $\\{(N_i, \\mathcal{E}_i, \\mathcal{T}_i)\\}_{i \\geq 0}$. For the rest of the proof, we assume that $A = T_X$ is normal in $\\mathcal{F}$, so $\\mathcal{H}_{X, A} = \\mathcal{H}_X$ (and $T_R = A$ for all $R \\in X^{\\mathcal{F}}$). Moreover, notice that we are only concerned about subgroups $H\\leq S$ such that $A\\leq H^{\\bullet}$, and for these subgroups there is no difference between the finite retraction pairs $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$ and $((-)^{\\star}_{\\mathcal{E}}, (-)^{\\star}_{\\mathcal{T}})$, by (3-f). Thus, we may assume also that we are still working with $((-)^{\\bullet}_{\\mathcal{F}}, (-)^{\\bullet}_{\\mathcal{L}})$.\n\nTo finish this step, we prove the following.\n\\begin{itemize}\n\n\\item[(3-i)] If $A = T$ then ($\\ddagger$) holds for all $Y \\in \\mathcal{H}_X$.\n\n\\end{itemize}\nIndeed, in this case we have $Y^T = \\{Y\\}$ for each $Y \\in \\mathcal{H}_X$. Since $\\mathcal{H}_X$ is finite, it is not hard to check in this case that in fact there exists some $M_X \\in \\mathbb{N}$ such that, for all $i \\geq M_X$ and all $Y \\in \\mathcal{H}_X$, the subgroup $Y \\cap S_i$ is not $\\mathcal{F}_i$-centric.\n\n\\textbf{Step 4.} Since $A = T_X$ is normal in $\\mathcal{F}$, consider the quotient $(S\/A, \\mathcal{F}\/A, \\widetilde{\\LL}\/A)$. By Lemma \\ref{quotient22}, $\\mathcal{F}\/A$ is a saturated fusion system over $S\/A$, and $\\widetilde{\\LL}\/A$ is a transporter system associated to $\\mathcal{F}\/A$ which contains all the centric subgroups of $\\mathcal{F}\/A$. In this step we prove several properties relating $(S, \\mathcal{F}, \\widetilde{\\LL})$ to $(S\/A, \\mathcal{F}\/A, \\widetilde{\\LL}\/A)$, which we will need in later steps.\n\nSet for short $\\3{S} = S\/A$, $\\3{\\mathcal{F}} = \\mathcal{F}\/A$ and $\\3{\\mathcal{T}} = \\widetilde{\\LL}\/A$. Let also $\\widetilde{\\LL}_{\\geq A} \\subseteq \\widetilde{\\LL}$ be the full subcategory of subgroups that contain $A$, and let $\\tau \\colon \\widetilde{\\LL}_{\\geq A} \\to \\3{\\mathcal{T}}$ be the projection functor. By a slight abuse of notation, we also write $\\tau \\colon S \\to \\3{S}$ for the projection homomorphism. We adopt the notation $\\3{P}, \\3{Q}, \\ldots$ to denote subgroups of $\\3{S}$, $\\3{f}, \\3{f}', \\ldots$ to denote morphisms in $\\3{\\mathcal{F}}$, and $\\3{\\varphi}, \\3{\\psi}, \\ldots$ to denote morphisms in $\\3{\\mathcal{T}}$. In particular, $T\/A = \\3{T}\\leq \\3{S}$ denotes the maximal torus of $\\3{S}$. Also, for each $P\\leq S$ that contains $A$, we will write $\\3{P}$ instead of $\\tau(P) = P\/A$, unless there is risk of confusion.\n\nThe following is easily check to hold since $A\\leq T$ is normal in $\\mathcal{F}$.\n\\begin{itemize}\n\n\\item[(4-a)] Let $P, Q\\leq S$ be such that $A\\leq P,Q$. Then, $Q \\in P^{\\mathcal{F}}$ if and only if $\\3{Q} \\in \\3{P}^{\\3{\\mathcal{F}}}$. Similarly, $Q \\in P^T$ if and only if $\\3{Q} \\in \\3{P}^{\\3{T}}$, since $A\\leq T$.\n\n\\end{itemize}\n\nBy property (3-b), $\\Psi_i(A) = A$ for all $i \\geq 0$. By (3-i) we may assume that $A \\lneqq T$, and thus, for each $i \\geq 0$, the unstable Adams operation $\\Psi_i$ induces a fine unstable Adams (see \\ref{uAo}) operation $\\3{\\Psi}_i$ on $\\3{\\mathcal{T}}$. By definition, if $\\varphi \\in \\mathrm{Mor}(\\widetilde{\\LL})$ is such that $\\Psi_i(\\varphi) = \\varphi$ for some $i$, then $\\3{\\Psi}_i(\\3{\\varphi}) = \\3{\\varphi}$. The following is some sort of converse of this statement.\n\\begin{itemize}\n\n\\item[(4-b)] Let $P, Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$ be such that $A\\leq P, Q$, and let $\\3{\\varphi} \\in \\mathrm{Mor}_{\\3{\\mathcal{T}}}(\\3{P}, \\3{Q})$ be such that $\\3{\\Psi}_i(\\3{\\varphi}) = \\3{\\varphi}$. Then, there exists some $\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$ such that $\\tau(\\varphi) = \\3{\\varphi}$ and such that $\\Psi_i(\\varphi) = \\varphi$. Similarly, let $\\3{x} \\in \\3{S}$ be such that $\\3{\\Psi}_i(\\3{x}) = \\3{x}$. Then there exists some $x \\in S$ such that $\\Psi_i(x) = x$ (i.e. $x \\in S_i$) and $\\tau(x) = \\3{x}$.\n\n\\end{itemize}\nLet $\\3{\\varphi}$ be as above, and let $\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$ be such that $\\tau(\\varphi) = \\3{\\varphi}$. Since $\\tau(\\varphi) = \\3{\\varphi} = \\3{\\Psi}_i(\\3{\\varphi}) = \\tau(\\Psi_i(\\varphi))$, it follows that $\\Psi_i(\\varphi) = \\varphi \\circ \\varepsilon_P(a)$, for some $a \\in A$ (see Definition \\ref{quotient1}). Consider the map $A \\to A$ defined by $t \\mapsto t^{-1} \\Psi_i(t)$. Since $A$ is abelian, this turns out to be a group homomorphism, which is in fact surjective, since $\\mathrm{Ker}(A)$ is the subgroup of fixed points of $A$, and this is a finite subgroup of $A$ for all $i \\geq 0$. In particular, there exists some $t \\in A$ such that $t^{-1} \\Psi_i(t) = a^{-1}$, and we get\n$$\n\\Psi_i(\\varphi \\circ \\varepsilon_P(t)) = \\Psi_i(\\varphi) \\circ \\varepsilon_P(\\Psi_i(t)) = \\varphi \\circ \\varepsilon_P(a) \\circ \\varepsilon_P(\\Psi_i(t)) = \\varphi \\circ \\varepsilon_P(t).\n$$\nA similar argument shows that, for each $\\3{x} \\in \\3{S}$ such that $\\3{\\Psi}_i(\\3{x}) = \\3{x}$, there is some $x \\in S_i$ such that $\\tau(x) = \\3{x}$.\n\nFor each $i \\geq 0$, let $\\3{S}_i\\leq \\3{S}$ be the subgroup of fixed points by $\\3{\\Psi}_i$. By (4-b), we deduce that $S_iA\/A = \\3{S}_i$, and thus $S_i\/(S_i \\cap A) \\cong \\3{S}_i$. Let $U \\in \\mathcal{H}_X$. By (4-a), $R \\in U^T$ if and only if $\\3{R} \\in \\3{U}^{\\3{T}}$.\n\\begin{itemize}\n\n\\item[(4-c)] For all $i \\geq 0$ and all $R \\in U^{T}$, $R \\in \\Omega_i$ if and only if $\\3{R}\\leq \\3{S}_i$.\n\n\\end{itemize}\nSuppose first that $R \\in \\Omega_i$, and let $P = R \\cap S_i$. Then, $R = PA$ (since $A = T_R$), and $\\3{R} = PA\/A\\leq S_iA\/A = \\3{S}_i$. Conversely, let $R \\in U^{T}$ be such that $\\3{R}\\leq \\3{S}_i$. Then, it follows by (4-b) that $P = R \\cap S_i$ contains representatives of all the elements in $\\3{R}$. Since $A_i = A \\cap S_i\\leq R \\cap S_i = P$ and $(A_i)^{\\bullet} = A$, it follows that $P^{\\bullet} \\geq R$. The inclusion $P^{\\bullet}\\leq R$ follows from $P = R \\cap S_i\\leq R$ and the fact that $R^{\\bullet} = R$. Thus $R \\in \\Omega_i$.\n\n\\textbf{Step 5.} For each $R \\in X^{\\mathcal{F}}$, recall the notation $\\3{R} = R\/A$. Set also\n$$\nZ_{\\3{R}} \\stackrel{def} = C_{\\3{T}}(\\3{R}) \\qquad \\mbox{and} \\qquad Z_R \\stackrel{def} = \\{t \\in N_T(R) \\, | \\, \\tau(t) \\in Z_{\\3{R}}\\}.\n$$\nNote that $A\\leq Z_R$ and $Z_{\\3{R}} = Z_R\/A$. The main goal of this step is to show the following: if $Z_U$ is fully $\\mathcal{F}$-normalized for some $U \\in \\mathcal{H}_X$, then $Z_{\\3{U}}$ is fully $\\3{\\mathcal{F}}$-normalized.\n\nIndeed, let $U \\in \\mathcal{H}_X$ be such that $Z_U$ is fully $\\mathcal{F}$-normalized. Note that $T\\leq N_S(Z_U)$, and\n$$\n\\3{T}\\leq N_{\\3{S}}(Z_{\\3{U}}) = N_S(Z_U)\/A.\n$$\nSuppose that $Z_{\\3{U}}$ is not fully $\\3{\\mathcal{F}}$-normalized, and let $\\3{\\gamma} \\in \\mathrm{Hom}_{\\3{\\mathcal{F}}}(N_{\\3{S}}(Z_{\\3{U}}), \\3{S})$ be such that $\\3{H} = \\3{\\gamma}(Z_{\\3{U}})$ is fully $\\3{\\mathcal{F}}$-normalized, and $|N_{\\3{S}}(Z_{\\3{U}})| < |N_{\\3{S}}(\\3{\\gamma}(Z_{\\3{U}}))|$. Then $\\3{\\gamma}$ lifts to a map $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S(Z_U), S)$ such that $|N_S(Z_U)| = |\\gamma(N_S(Z_U))| < |N_S(\\gamma(Z_U))|$, since\n$$\n\\3{T}\\leq N_{\\3{S}}(\\3{\\gamma}(Z_{\\3{U}})) = N_S(\\gamma(Z_U))\/A,\n$$\nand this contradicts the maximality of $|N_S(Z_U)|$.\n\n\\textbf{Step 6.}  We show that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $Z_U \\not\\leq U$.\n\nLet $U \\in \\mathcal{H}_X$ be such that $Z_U \\not\\leq U$, and let $a \\in Z_U \\setminus U$. Then, by Lemma \\ref{SiS} there exists some $M_U \\in \\mathbb{N}$ such that $a \\in S_i$ for all $i \\geq M_U$. Fix some $i \\geq M_U$, and let $R \\in U^T \\cap \\Omega_i$. Since $R$ is $T$-conjugate to $U$, we have $Z_R = Z_U$. Let\n$$\nK_{R \\cap S_i} = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}_i}(R \\cap S_i) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(T_R \\cap S_i) \\times \\mathrm{Aut}(\\3{R})).\n$$\nBy Lemma \\ref{Kpgp} we know that $K_{R \\cap S_i}\\leq O_p(\\mathrm{Aut}_{\\mathcal{F}_i}(R \\cap S_i))$.\n\nThe element $a \\in Z_U \\cap S_i$ fixed above satisfies $a \\in Z_R \\cap S_i$, since $Z_R = Z_U$, and in particular $a \\in N_{S_i}(R \\cap S_i)$. Moreover, $c_a \\in K_{R \\cap S_i} \\cap \\mathrm{Aut}_{S_i}(R \\cap S_i)$, since $a \\in T \\cap S_i$. In particular, if $R \\cap S_i$ is $\\mathcal{F}_i$-centric then $c_a \\notin \\mathrm{Inn}(R \\cap S_i)$, and this shows that ($\\ddagger$) holds for $U$.\n\n\\textbf{Step 7.} Suppose ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized. Then we claim that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$.\n\nLet $U \\in \\mathcal{H}_X$ be such that $Z_U$ is not fully $\\mathcal{F}$-normalized. Then there exists some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S(Z_U), S)$ such that $\\gamma(Z_U)$ is fully $\\mathcal{F}$-normalized. Since $U\\leq N_S(Z_U)$, we may set $V = \\gamma(U)$. Moreover, since $\\mathcal{H}_X$ contains representatives of all the $T$-conjugacy classes in $X^{\\mathcal{F}}$, we may assume that $V \\in \\mathcal{H}_X$. Finally, note that $T\\leq N_S(Z_U)$, since $Z_U\\leq T$. Thus, $Z_V = \\gamma(Z_U)$.\n\nSuppose now that ($\\ddagger$) holds for all $V \\in \\mathcal{H}_X$ such that $Z_V$ is fully $\\mathcal{F}$-normalized, and let $U \\in \\mathcal{H}_X$. By the above discussion, there exists some $f \\in \\mathrm{Hom}_{\\mathcal{F}}(UT, S)$ such that $V = f(U) \\in \\mathcal{H}_X$ is such that $Z_V$ is fully $\\mathcal{F}$-normalized, and the claim follows by Step 1.\n\n\\textbf{Step 8.} We show that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $C_S(U) \\not\\leq U$. \n\nBy Step 6, if $Z_U \\not\\leq U$ then ($\\ddagger$) holds for $U$. Thus we may assume that $Z_U\\leq U$. Also, since $C_S(U) \\not\\leq U$, it follows that $C_{\\3{S}}(\\3{U}) \\not\\leq \\3{U}$. Notice that $C_{\\3{T}}(\\3{U}) = Z_{\\3{U}}\\leq \\3{U}$ and $\\3{U}$ is finite, which implies that $C_{\\3{S}}(\\3{U})$ is a finite group. Thus, there exists some $M_U \\in \\mathbb{N}$ such that $C_{\\3{S}}(\\3{U})\\leq \\3{S}_i$. Moreover, since $U \\in \\Omega_i$ by assumption, we have $\\3{U}\\leq \\3{S}_i$ by (4-c).\n\nLet $i \\geq M_U$ and let $R \\in U^T \\cap \\Omega_i$. Then $C_{\\3{T}}(\\3{R}) = Z_{\\3{U}}$. By (4-a) we have $\\3{R} \\in \\3{U}^{\\3{T}}$, and by (4-c) we know that $\\3{R}\\leq \\3{S}_i$. Thus, by Lemma \\ref{invar1} (i), together with Proposition \\ref{fix1} (iii), we have\n$$\n\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in C_{\\3{T}}(\\3{R}) = Z_{\\3{R}} = Z_{\\3{U}}\n$$\nfor some $\\3{x} \\in N_{\\3{T}}(\\3{R}, \\3{U})$. Fix such $\\3{x} \\in N_{\\3{T}}(\\3{R}, \\3{U})$, and note that $\\3{x}$ conjugates $C_{\\3{S}}(\\3{R})$ to $C_{\\3{S}}(\\3{U})$, and in particular $C_{\\3{S}}(\\3{R}) \\not\\leq \\3{R}$. Moreover, since $Z_{\\3{U}} = C_{\\3{T}}(\\3{R})\\leq \\3{R}$, it follows that\n$$\n\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in C_{\\3{T}}(\\3{R})\\leq \\3{R} \\cap \\3{T}\\leq C_{\\3{T}}(C_{\\3{S}}(\\3{R})).\n$$\nThus, by \\ref{invar1} (i) we deduce that $C_{\\3{S}}(\\3{R})\\leq \\3{S}_i$.\n\nSet $X_R = C_S(R)A$. Then, by (4-b) $X_R \\cap S_i$ contains representatives of all the elements in $C_S(R)A\/A\\leq C_{\\3{S}}(\\3{R})\\leq \\3{S}_i$. Note that $C_S(R)A \\not\\leq R$, and thus $X_R \\cap S_i$ contains elements which are not in $R \\cap S_i$. If $(X_R \\cap S_i)\\setminus (R \\cap S_i)$ contains some element of $C_S(R)$, then clearly $R \\cap S_i$ is not $\\mathcal{F}_i$-centric. Suppose then that $(X_R \\cap S_i) \\cap C_S(R)\\leq R \\cap S_i$, and let\n$$\nK_{R \\cap S_i} = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}_i}(R \\cap S_i) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}_i}(T_R \\cap S_i) \\times \\mathrm{Aut}(\\3{R})),\n$$\nwhich is a normal $p$-subgroup of $\\mathrm{Aut}_{\\mathcal{F}_i}(R \\cap S_i)$ by Lemma \\ref{Kpgp}. If $R \\cap S_i$ is $\\mathcal{F}_i$-centric, then, for every $x \\in (X_R \\cap S_i)\\setminus(R \\cap S_i)$, we have $c_x \\in K_{R \\cap S_i} \\cap \\mathrm{Aut}_{S_i}(R \\cap S_i)$ and $c_x \\notin \\mathrm{Inn}(R \\cap S_i)$, and ($\\ddagger$) holds for $U$.\n\n\\textbf{Step 9.} Recall the notation $\\3{\\mathcal{T}} = \\widetilde{\\LL}\/A$, introduced in Step 4, which is a transporter system associated to $\\3{\\mathcal{F}}$. Fix $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized. Then $Z_{\\3{U}}$ is fully $\\3{\\mathcal{F}}$-normalized by Step 5, and in particular it is fully $\\3{\\mathcal{F}}$-centralized. Thus we may consider the centralizer fusion system $\\3{\\mathcal{E}} = C_{\\3{\\mathcal{F}}}(Z_{\\3{U}})$ over $\\3{Z} = C_{\\3{S}}(Z_{\\3{U}})$, and note that $\\3{T}\\leq \\3{Z}$. Since $\\3{\\mathcal{F}}$ is saturated, it follows that $\\3{\\mathcal{E}}$ is also saturated. The main goal of this step is to prove the following property.\n\\begin{itemize}\n\n\\item[(9-a)] Suppose that $Z_U\\leq U$. Then, there exists $V \\in \\mathcal{H}_X$ such that $C_S(V) \\not\\leq V$ and such that $\\3{V} \\in \\3{U}^{\\3{\\mathcal{E}}}$.\n\n\\end{itemize}\n\nFirst, note that the following holds.\n\\begin{itemize}\n\n\\item[(9-b)] For each $\\3{R} \\in \\3{U}^{\\3{T}}$, we have $Z_{\\3{R}} = C_{\\3{T}}(\\3{R}) = Z_{\\3{U}}$.\n\n\\item[(9-c)] Since $Z_{\\3{U}}\\leq \\3{T}$ is abelian, every $\\3{\\mathcal{E}}$-centric subgroup of $\\3{Z}$ must contain $Z_{\\3{U}}$. Thus, by Lemma \\ref{centricNFKA}, every $\\3{\\mathcal{E}}$-centric subgroup of $\\3{Z}$ is also an $\\3{\\mathcal{F}}$-centric subgroup of $\\3{S}$, and hence also an object in $\\3{\\mathcal{T}}$, since $\\3{\\mathcal{T}}$ contains all the $\\3{\\mathcal{F}}$-centric subgroups of $\\3{S}$.\n\n\\end{itemize}\nWe are ready now to prove (9-a). Let $K = \\mathrm{Aut}_U(Z_U)$, and let $N_S^K(Z_U) = \\{x \\in S \\, | \\, c_x \\in K\\}$. Then,\n$$\n\\mathrm{Aut}_S^K(Z_U) = K \\cap \\mathrm{Aut}_S(Z_U) = \\mathrm{Aut}_U(Z_U) = K \\cap \\mathrm{Aut}_{\\mathcal{F}}(Z_U) = \\mathrm{Aut}_{\\mathcal{F}}^K(Z_U).\n$$\nIn particular, $Z_U$ is fully $\\mathcal{F}$-centralized (since it is fully $\\mathcal{F}$-normalized), and clearly $\\mathrm{Aut}_S^K(Z_U) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{F}}^K(Z_U))$. By \\cite[Lemma 2.2]{BLO6} it follows that $Z_U$ is fully $K$-normalized in $\\mathcal{F}$ (see Section \\ref{Squotient}). Thus, the fusion system $N_{\\mathcal{F}}^K(Z_U)$ over $N_S^K(Z_U)$ (as defined in \\ref{definorm}) is saturated.\n\nNote that $U, T\\leq N_S^K(Z_U) = U C_S(Z_U)$. Since $U$ is not $\\mathcal{F}$-centric and $Z_U\\leq U$, it follows that $U$ is not $N_{\\mathcal{F}}^K(Z_U)$-centric by Lemma \\ref{centricNFKA}. Let $V$ be $N_{\\mathcal{F}}^K(Z_U)$-conjugate to $U$ and such that $C_{N_S^K(Z_U)}(V) \\not\\leq V$. Since $T\\leq N_S^K(Z_U)$, we may choose $V \\in \\mathcal{H}_X$. Then,\n$$\nC_{N_S^K(Z_U)}(V)\\leq C_S(V) \\not\\leq V.\n$$\nAlso, if $f \\in \\mathrm{Hom}_{N_{\\mathcal{F}}^K(Z_U)}(U,V)$, then by definition $f|_{Z_U} \\in \\mathrm{Aut}_U(Z_U)$. Hence, the induced morphism $\\3{f} \\in \\mathrm{Hom}_{\\3{\\mathcal{F}}}(\\3{U}, \\3{V})$ satisfies $\\3{f}|_{Z_{\\3{U}}} = \\mathrm{Id}$, and thus $\\3{V} \\in \\3{U}^{\\3{\\mathcal{E}}}$. Note that in particular $Z_U\\leq Z_V$, although this may not be an equality.\n\n\\textbf{Step 10.} We show that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized.\n\nFix $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized. By Step 6, we may assume that $Z_U\\leq U$, and thus $Z_{\\3{U}}\\leq \\3{U}$. By Step 5, we know that $Z_{\\3{U}}$ is fully $\\3{\\mathcal{F}}$-normalized (and in particular fully $\\3{\\mathcal{F}}$-centralized). Let $\\3{Z} = C_{\\3{S}}(Z_{\\3{U}})$ for short, and let $\\3{\\mathcal{E}} = C_{\\3{\\mathcal{F}}}(Z_{\\3{U}})$ be the centralizer fusion system of $Z_{\\3{U}}$ in $\\3{\\mathcal{F}}$, which is saturated by Step 9. By (9-a), there exists $V \\in \\mathcal{H}_X$ such that $\\3{V} \\in \\3{U}^{\\3{\\mathcal{E}}}$ and $C_S(V) \\not\\leq V$.\n\nNotice that the set $\\{T_K \\, | \\, K \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})\\}$ is finite since $\\mathrm{Ob}(\\mathcal{L}^{\\bullet})$ contains finitely many $T$-conjugacy classes. Moreover, since $K^{\\bullet} = K$, it follows that $(T_K)^{\\bullet} = T_K$ for all $K \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$. By Proposition \\ref{fix1} (i) there exists some $M_3 \\in \\mathbb{N}$ such that $(T_K \\cap S_i)^{\\bullet} = T_K = (T_K)^{\\bullet}$ for all $i \\geq M_3$ and all $T_K$ in the set above. Without loss of generality we may assume that $M_3 = 0$.\n\nFix $\\3{f} \\in \\mathrm{Hom}_{\\3{\\mathcal{E}}}(\\3{U}, \\3{V})$. By Alperin's Fusion Theorem \\cite[Theorem 3.6]{BLO3}, there exist sequences of subgroups $\\3{W}_0 = \\3{U}, \\3{W}_1, \\ldots, \\3{W}_n = \\3{V}\\leq\\3{Z}$ and $\\3{K}_1, \\ldots, \\3{K}_n \\in \\mathrm{Ob}((\\3{\\mathcal{E}})^c) \\subseteq \\mathrm{Ob}(\\3{\\mathcal{F}})$, and morphisms $\\3{\\gamma}_j \\in \\mathrm{Aut}_{\\3{\\mathcal{E}}}(\\3{K}_j)$ for each $j = 1, \\ldots, n$, such that\n$$\n\\3{W}_{j-1}, \\3{W}_j\\leq \\3{K}_j \\qquad \\mbox{and} \\qquad \\3{\\gamma}_j(\\3{W}_{j-1}) = \\3{W}_j\n$$\nfor each $j = 1, \\ldots, n$. For each $j = 1, \\ldots, n$, let $\\3{\\varphi}_j \\in \\mathrm{Aut}_{\\3{\\mathcal{T}}}(\\3{K}_j)$ be such that $\\3{\\rho}(\\3{\\varphi}_j) = \\3{\\gamma}_j$. Let also $K_j \\in \\mathrm{Ob}(\\widetilde{\\LL})$ be the preimage of $\\3{K}_j$, and let $\\varphi_j \\in \\mathrm{Aut}_{\\widetilde{\\LL}}(K_j)$ be such that $\\tau(\\varphi_j) = \\3{\\varphi}_j$. Note that $U\\leq K_1$ and $V\\leq K_n$.\n\nWe claim that we may assume that $K_j \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$ for all $j$ without loss of generality. Indeed, we have $\\3{K}_j \\in \\mathrm{Ob}(\\3{\\mathcal{T}})$ by (9-c) (where $\\3{\\mathcal{T}} = \\widetilde{\\LL}\/A$), and thus $K_j \\in \\mathrm{Ob}(\\widetilde{\\LL})$. This implies that $(K_j)^{\\bullet} \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$ by definition of $\\widetilde{\\LL}$. Furthermore, we have $A\\leq (K_j)^{\\bullet}$, and it follows that $(K_j)^{\\bullet}\/A\\leq \\3{Z}$, since $(K_j)^{\\bullet}\\leq K_j T$ and $\\3{T}\\leq \\3{Z}$. By Proposition \\ref{fix1} (i) and (ii), there exists some $M_U \\in \\mathbb{N}$ such that\n$$\nK_j \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i) \\qquad \\mbox{and} \\qquad \\Psi_i(\\varphi_j) = \\varphi_j\n$$\nfor all $i \\geq M_U$ and all $j = 1, \\ldots, n$. In particular, this implies that $U \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $V \\cap S_i$. Note that this also implies that $\\3{\\Psi}_i(\\3{\\varphi}_j) = \\3{\\varphi}_j$ for all $i \\geq M_U$.\n\nLet $i \\geq M_U$, and let $R \\in U^T \\cap \\Omega_i$. By assumption, $\\3{U}\\leq \\3{S}_i$. By (4-a) we have $\\3{R} \\in \\3{U}^{\\3{T}}$, which implies that $C_{\\3{T}}(\\3{R}) = Z_{\\3{R}} = Z_{\\3{U}}$, and by (4-c) we know that $\\3{R}\\leq \\3{S}_i$. Thus, by Lemma \\ref{invar1} (i), together with Proposition \\ref{fix1} (iii), we have\n$$\n\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in C_{\\3{T}}(\\3{R}) = Z_{\\3{U}}\\leq \\3{R}\n$$\nfor some $\\3{x} \\in N_{\\3{T}}(\\3{R}, \\3{U})$. Fix such $\\3{x} \\in N_{\\3{T}}(\\3{R}, \\3{U})$, set $\\3{W}'_0 = \\3{R}$, and for all $j = 1, \\ldots, n$ set\n$$\n\\3{W}'_j = \\3{x}^{-1} \\cdot \\3{W}_j \\cdot \\3{x} \\qquad \\mbox{and} \\qquad \\3{K}'_j = \\3{x}^{-1} \\cdot \\3{K}_j \\cdot \\3{x}.\n$$\nSet also $\\3{\\varphi}_j' = \\3{\\varepsilon}(\\3{x})^{-1} \\circ \\3{\\varphi}_j \\circ \\3{\\varepsilon}(\\3{x}) \\in \\mathrm{Aut}_{\\3{\\mathcal{T}}}(\\3{K}'_j)$. Let also $W_j'$ and $K_j'$ be the corresponding preimages in $S$. The following holds.\n\\begin{itemize}\n\n\\item[(10-a)] Since $Z_{\\3{U}}$ is abelian and $\\3{K}_j, \\3{K}_j'$ are $\\3{\\mathcal{E}}$-centric, it follows that $Z_{\\3{U}}\\leq \\3{K}_j, \\3{K}_j'$ for each $j$.\n\n\\item[(10-b)] Since $\\3{x} \\in \\3{T}\\leq \\3{Z} = C_{\\3{S}}(Z_{\\3{U}})$ and $\\3{K}_j\\leq \\3{Z}$, it follows that $\\3{K}'_j\\leq \\3{Z}$. Moreover, since $\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in Z_{\\3{U}}$ and $Z_{\\3{U}}\\leq C_{\\3{T}}(\\3{K}_j), C_{\\3{T}}(\\3{K}'_j)$, it follows from Lemma \\ref{invar1} (i) that $\\3{x} \\in N_{\\3{T}}(\\3{K}'_j \\cap \\3{S}_i, \\3{K}_j \\cap \\3{S}_i)$.\n\n\\item[(10-c)] Since $Z_{\\3{U}}\\leq \\3{K}_j, \\3{K}'_j$, $\\3{\\varphi}_j \\in \\mathrm{Mor}(\\3{\\mathcal{E}})$, and $\\3{x}^{-1} \\cdot \\3{\\Psi}_i(\\3{x}) \\in Z_{\\3{U}}$, it follows from axiom (C) of transporter systems for $\\3{\\mathcal{T}}$ that $\\3{\\varepsilon}(\\3{x}^{-1} \\Psi_i(\\3{x})) \\circ \\3{\\varphi_j}' = \\3{\\varphi}'_j \\circ \\3{\\varepsilon}(\\3{x}^{-1}\\Psi_i(\\3{x}))$. Thus, $\\3{\\Psi}_i(\\3{\\varphi}'_j) = \\3{\\varphi}'_j$ by Lemma \\ref{invar1} (ii). By (4-b), this implies that there is some $\\varphi'_j \\in \\mathrm{Aut}_{\\widetilde{\\LL}}(K'_j)$ such that $\\Psi_i(\\varphi'_j) = \\varphi'_j$ and $\\tau(\\varphi'_j) = \\3{\\varphi}'_j$.\n\n\\end{itemize}\nLet $\\3{R'} = \\3{W}'_n \\in \\3{V}^{\\3{T}}$, and let $R'$ be its preimage in $S$, and note that $R' \\in V^T$ by (4-a). Note that $R\\leq K'_1$ and $R'\\leq K'_n$. Thus, if $K'_j \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ for each $j$, then $R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $R' \\cap S_i$.\n\nIt remains to show that $K'_j \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ for each $j$. Recall that, by definition, $K_j' \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ if $(K'_j \\cap S_i)^{\\bullet} \\in \\mathrm{Ob}(\\mathcal{L})$. For each $j = 1, \\ldots, n$, let $T_j$ be the maximal torus of $K_j$. Note that $K_j'$ is $T$-conjugate to $K_j$, so in particular $K_j' \\in \\mathrm{Ob}(\\mathcal{L}^{\\bullet})$. As noted above, we have $(T_j \\cap S_i)^{\\bullet} = T_j$ for all $i \\geq M_U$. Also, $T_j \\cap S_i\\leq K'_j \\cap S_i$ and\n$$\n(T_j \\cap S_i)^{\\bullet} = T_j\\leq (K'_j \\cap S_i)^{\\bullet}.\n$$\nIn particular, this implies that $K'_j \\cap S_i, T_j\\leq (K'_j \\cap S_i)^{\\bullet}$. Since $(K_j \\cap S_i)^{\\bullet} = K_j$, it follows that $K_j \\cap S_i$ contains representatives of all the elements in $\\3{K}_j \\cap \\3{S}_i$. Thus, by (10-b) we deduce that $K_j' \\cap S_i$ contains representatives of all the elements in $\\3{K}'_j \\cap \\3{S}_i$, and thus $K'_j = (K'_j \\cap S_i)T_j\\leq (K'_j \\cap S_i)^{\\bullet}$. It follows that $(K'_j \\cap S_i)^{\\bullet} = K_j'$, and thus $K'_j \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$.\n\nBy the above, $R \\cap S_i$ is $\\mathcal{F}_i$-conjugate to $R' \\cap S_i$. Moreover, $R' \\in V^T$, and $C_S(V) \\not\\leq V$ by assumption. By Step 8, ($\\ddagger$) holds for $V$, and thus the conclusion of ($\\ddagger$) holds for $R$ and $R'$. This shows that ($\\ddagger$) holds for $U$ as well. Repeating this step, we deduce that ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ such that $Z_U$ is fully $\\mathcal{F}$-normalized, and thus ($\\ddagger$) holds for all $U \\in \\mathcal{H}_X$ by Step 7. This finishes the proof.\n\\end{proof}\n\n\\begin{thm}\\label{fix6}\n\nEvery $p$-local compact group admits an approximation by $p$-local finite groups.\n\n\\end{thm}\n\n\\begin{proof}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\widetilde{\\LL}$ and $\\Psi$ be as fixed in \\ref{hyp2}. Let also $\\Psi_0 = \\Psi$, and let $\\{\\Psi_i\\}_{i \\geq 0}$ be such that $\\Psi_{i+1} = \\Psi_i^p$. Let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be the family of transporter systems defined in \\ref{Li}. For simplicity, we can assume that the degree of $\\Psi$ is high enough so that Lemma \\ref{SiS} and Propositions \\ref{fix2} and \\ref{fix5} hold (otherwise replace $\\Psi$ by an appropriate power of it), and we claim that $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ is an approximation of $\\mathcal{G}$ by $p$-local finite groups.\n\nCondition (i) in \\ref{defiapprox} is satisfied, by Lemma \\ref{SiS} (i). Also, $S_i$ is a finite $p$-group for all $i \\geq 0$, and $\\mathcal{L}_i$ is a linking system associated to $\\mathcal{F}_i$ by Proposition \\ref{fix2-1}, and there are inclusions $\\mathcal{L}_i \\subseteq \\mathcal{L}_{i+1}$ and $\\mathcal{L}_i \\subseteq \\widetilde{\\LL}$ for all $i \\geq 0$. Thus, to show that condition (ii) in \\ref{defiapprox} is satisfied, it remains to check that $\\mathcal{F}_i$ is saturated and $\\mathrm{Ob}(\\mathcal{F}_i^{cr}) \\subseteq \\mathrm{Ob}(\\mathcal{L}_i)$. By Propositions \\ref{fix4} and \\ref{fix5}, if $P\\leq S_i$ is $\\mathcal{F}_i$-centric but $P \\notin \\mathrm{Ob}(\\mathcal{L}_i)$, then $P$ is $\\mathcal{F}_i$-conjugate to some $Q$ such that\n$$\n\\mathrm{Out}_{S_i}(Q) \\cap O_p(\\mathrm{Out}_{\\mathcal{F}_i}(Q)) \\neq 1.\n$$\nMoreover, $\\mathcal{F}_i$ is $\\mathrm{Ob}(\\mathcal{L}_i)$-generated and $\\mathrm{Ob}(\\mathcal{L}_i)$-saturated by Corollary \\ref{fix3}. Thus, the conditions of Theorem \\ref{5A} are satisfied: $\\mathcal{F}_i$ is saturated, and $\\mathrm{Ob}(\\mathcal{L}_i)$ contains all the centric radical subgroups of $\\mathcal{F}_i$. Thus condition (ii) in \\ref{defiapprox} is satisfied.\n\nFinally, we have to check condition (iii): for each $P, Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$ and each $\\varphi \\in \\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$, there exists some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, there are objects $P_i, Q_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ and morphisms $\\varphi_i \\in \\mathrm{Mor}_{\\mathcal{L}_i}(P_i, Q_i)$, such that $P = \\bigcup_{i \\geq M} P_i$ and $Q = \\bigcup_{i \\geq M} Q_i$, and $\\4{\\varepsilon}_{Q_i, Q}(1) \\circ \\varphi_i = \\varphi \\circ \\4{\\varepsilon}_{P_i, P}(1)$. By Proposition \\ref{fix1} (i) and (ii), there is some $M \\in \\mathbb{N}$ such that $P \\cap S_i, Q \\cap S_i \\in \\mathrm{Ob}(\\mathcal{L}_i)$ for all $i \\geq M$, and the restriction $\\varphi_i = \\varphi|_{P \\cap S_i}$ is a morphism in $\\mathrm{Mor}_{\\mathcal{L}_i}(P \\cap S_i, Q \\cap S_i)$. Since $S = \\bigcup_{i \\geq 0} S_i$ by Lemma \\ref{SiS} (i), it follows that\n$$\nP = \\bigcup_{i \\geq 0} P_i \\qquad \\mbox{and} \\qquad Q = \\bigcup_{i \\geq 0} Q_i.\n$$\nThe condition $\\4{\\varepsilon}_{Q_i, Q}(1) \\circ \\varphi_i = \\varphi \\circ \\4{\\varepsilon}_{P_i, P}(1)$ is easily checked.\n\\end{proof}\n\n\\begin{rmk}\\label{fix7}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\mathcal{O}(\\mathcal{F})$ be its \\emph{orbit category}: the category $\\mathcal{O}(\\mathcal{F})$ with object set $\\mathrm{Ob}(\\mathcal{F})$, and with morphism sets\n\\begin{equation}\\label{orbitcat}\n\\mathrm{Mor}_{\\mathcal{O}(\\mathcal{F}_{\\mathcal{H}})}(P,Q) = \\mathrm{Inn}(Q)\\backslash \\mathrm{Hom}_{\\mathcal{F}}(P,Q).\n\\end{equation}\nNotice that the subcategory $\\mathcal{O}(\\mathcal{F}^{\\bullet c}) \\subseteq \\mathcal{O}(\\mathcal{F})$ has a finite (full) subcategory as a skeletal subcategory, which in addition contains a representative of each $\\mathcal{F}$-conjugacy class of centric radical subgroups. Let $\\widetilde{\\LL}$ be the associated telescopic linking system, and let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups. Denote by $\\mathcal{L}_i^{\\bullet} \\subseteq \\mathcal{L}^{\\bullet}$ the image of $\\mathcal{L}_i$ through the functor $(-)^{\\bullet} \\colon \\widetilde{\\LL} \\to \\mathcal{L}^{\\bullet}$ for each $i \\geq 0$. Then, by Definition \\ref{defiapprox} there is some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, the category $\\mathcal{L}_i^{\\bullet}$ contains representatives of all the morphisms in $\\mathcal{O}(\\mathcal{F}^{\\bullet c})$ up to $S$-conjugation. In particular, this implies that\n\\begin{enumerate}[(a)]\n\n\\item the fusion system $\\mathcal{F}$ is generated by $\\mathcal{F}_i$ and $\\mathrm{Inn}(S)$; and\n\n\\item the linking system $\\widetilde{\\LL}$ is generated by $\\mathcal{L}_i^{\\bullet}$ and $S$ (and thus so is $\\mathcal{L}$).\n\n\\end{enumerate}\nMore precisely, property (b) means that every object in $\\mathcal{L}^{\\bullet}$ is $S$-conjugate to an object in $\\mathcal{L}_i^{\\bullet}$, and every morphism in $\\mathcal{L}^{\\bullet}$ is the composition of a morphism in $\\mathcal{L}_i^{\\bullet}$ with (the restriction of) a morphism in $\\varepsilon(S)\\leq \\mathrm{Aut}_{\\mathcal{L}}(S)$. Since $\\mathcal{L}^{\\bullet}$ is a deformation retract of $\\widetilde{\\LL}$, we can say that $\\widetilde{\\LL}$ is generated by $\\mathcal{L}_i^{\\bullet}$ and $S$.\n\n\\end{rmk}\n\n\n\\subsection{An example}\\label{Ssexample}\n\nIn this subsection we analyze our constructions in detail on a specific example: the $2$-local compact group associated to $SO(3)$. In particular, this example reveals that there are approximations by $p$-local finite groups that do not appear as fixed points of any (family of) fine unstable Adams operation.\n\nLet us first fix some notation and facts. The reader is referred to \\cite[Example 3.7]{Gonza2} for further details. Let $\\mathcal{G} = (S, \\FF, \\LL)$ be the $2$-local compact group associated to $SO(3)$. Then,\n$$\nS = \\gen{\\{t_n\\}_{n \\geq 1}, \\, x \\, | \\, \\forall n, \\, t_n^{2^n} = x^2 = 1, \\, t_{n+1}^2 = t_n, \\, x \\cdot t_n \\cdot x^{-1} = t_n^{-1}} \\cong D_{2^{\\infty}}.\n$$\nFor each $n \\geq 1$, set $T_n = \\gen{t_n}$, so that the maximal torus of $S$ is $T = \\gen{\\{t_n\\}_{n \\geq 1}}\\leq S$. Set also $V = \\gen{t_1, x} \\cong \\Z\/2 \\times \\Z\/2$. Then $\\mathcal{F}$ is generated by $\\mathrm{Aut}_{\\mathcal{F}}(S) = \\mathrm{Inn}(S)$ and $\\mathrm{Aut}_{\\mathcal{F}}(V) = \\mathrm{Aut}(V) \\cong \\Sigma_3$. Regarding the linking system $\\mathcal{L}$, the only centric radical subgroups (up to $S$-conjugation) are $S$ and $V$, with\n$$\n\\mathrm{Aut}_{\\mathcal{L}}(S) = S \\qquad \\mbox{and} \\qquad \\mathrm{Aut}_{\\mathcal{L}}(V) \\cong \\Sigma_4.\n$$\nLet $\\widetilde{\\LL}$ be the associated telescopic linking system. An easy computation shows that the only new objects in $\\widetilde{\\LL}$ are the subgroups $T_n$, for $n \\geq 2$, with $\\mathrm{Aut}_{\\widetilde{\\LL}}(T_n) = S$ for all $n \\geq 2$.\n\nLet now $\\Psi \\in \\mathrm{Aut}_{\\mathrm{typ}}^{I}(\\mathcal{L})$ be an unstable Adams operation. As usual, set $\\Psi_0 = \\Psi$ and $\\Psi_{i+1} = (\\Psi_i)^2$. Then, for each $i \\geq 0$ we have $S_i = C_S(\\Psi_i) \\cong D_{2^{n_i}}$ for some $n_i \\in \\mathbb{N}$, and we may assume without loss of generality that $V\\leq S_i$ for all $i \\geq 0$. Notice that $S_i$ contains two different $S_i$-conjugacy classes of maximal elementary abelian subgroups, and $V$ is a representative of one of them. Fix a representative $W_i\\leq S_i$ of the other $S_i$-conjugacy class of maximal elementary abelian subgroups. Notice that, after embedding $S_i$ into $S_{i+1}$, the subgroup $W_i$ becomes $S_{i+1}$-conjugate to $V$.\n\nLet also $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be the approximation by $2$-local finite groups associated to $\\{\\Psi_i\\}_{i \\geq 0}$. A careful inspection reveals that, in order to describe $(S_i, \\mathcal{F}_i, \\mathcal{L}_i)$ it is enough to specify the groups $\\mathrm{Aut}_{\\mathcal{L}_i}(S_i)$, $\\mathrm{Aut}_{\\mathcal{L}_i}(V)$ and $\\mathrm{Aut}_{\\mathcal{L}_i}(W_i)$. By construction, we have\n$$\n\\mathrm{Aut}_{\\mathcal{L}_i}(S_i) = S_i \\qquad \\mbox{and} \\qquad \\mathrm{Aut}_{\\mathcal{L}_i}(V) \\cong \\Sigma_4,\n$$\nand we have to determine the group $\\mathrm{Aut}_{\\mathcal{L}_i}(W_i)$. Let $\\varphi \\in \\mathrm{Aut}_{\\mathcal{L}_i}(V)$ be an automorphism of order $3$ that conjugates $t_1$ to $x$. An easy computation in $S$ shows that there is some $t \\in N_T(W_i, V)$ such that $t^{-1} \\cdot \\Psi_i(t) = t_1$. By Lemma \\ref{invar1} (ii), it follows that $\\varepsilon(x)^{-1} \\circ \\varphi \\circ \\varepsilon(x)$ is not fixed by $\\Psi_i$, and thus\n$$\n\\mathrm{Aut}_{\\mathcal{L}_i}(W_i) = N_{S_i}(W_i) \\cong D_8.\n$$\nThese computations imply that, for all $i$, $(S_i, \\mathcal{F}_i, \\mathcal{L}_i)$ is the $2$-local finite group associated to $PGL_2(\\mathbb{F}_q)$, where $q$ is some power of some odd prime $p$.\n\nLet now $(S_i, \\mathcal{E}_i, \\mathcal{T}_i)$ be the $2$-local finite group associated to $PSL_2(\\mathbb{F}_q)$. This $2$-local finite group is determined by $\\mathrm{Aut}_{\\mathcal{T}_i}(S_i) = S_i$ and $\\mathrm{Aut}_{\\mathcal{T}_i}(V) \\cong \\Sigma_4 \\cong \\mathrm{Aut}_{\\mathcal{T}_i}(W_i)$. Clearly, $\\{(S_i, \\mathcal{E}_i, \\mathcal{T}_i)\\}_{i \\geq 0}$ is an approximation of $(S, \\FF, \\LL)$ by $2$-local finite groups, but our computations above show that it cannot be the product of our constructions in (\\ref{Li}).\n\n\n\\section{Stable Elements Theorem for \\texorpdfstring{$p$}{p}-local compact groups}\\label{Sstable}\n\nIn this section we use the approximations constructed in the previous section to prove a version of the Stable Elements Theorem for $p$-local compact groups, which computes the cohomology of the classifying space of a $p$-local compact group with coefficients in a trivial $\\Z_{(p)}$-module as the \\emph{stable} elements of the cohomology of its Sylow $p$-subgroup with the same module of coefficients. The finite version of this result was proved in \\cite[Theorem 5.8]{BLO2} for coefficients in the trivial module $\\mathbb{F}_p$, and then generalized to any trivial $\\Z_{(p)}$-module in \\cite[Lemma 6.12]{BCGLO2}.\n\nLet $S$ be a discrete $p$-toral group, and let $\\mathcal{F}$ be a saturated fusion system. In this section we consider certain contravariant functors $A \\colon \\mathcal{F} \\to \\mathcal{C}$, where $\\mathcal{C}$ is either $\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, the category of $\\Z_{(p)}$-modules, or $\\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, the category of graded $\\Z_{(p)}$-modules.  We start with a brief discussion of some general properties of such functors. For each $X\\leq S$, let $\\iota_X \\colon X \\to S$ denote the inclusion homomorphism. This way, given a contravariant functor $A \\colon \\mathcal{F} \\to \\mathcal{C}$ and a subcategory $\\mathcal{E} \\subseteq \\mathcal{F}$, we define\n\\begin{equation}\\label{Aee}\nA^{\\mathcal{E}} \\stackrel{def} = \\{z \\in A(S) \\, | \\, A(\\iota_P)(z) = A(\\iota_Q \\circ f)(z), \\, \\forall P, Q \\in \\mathrm{Ob}(\\mathcal{E}) \\mbox{ and } \\forall f \\in \\mathrm{Hom}_{\\mathcal{E}}(P,Q)\\}.\n\\end{equation}\nGiven $\\mathcal{E}_1 \\subseteq \\mathcal{E}_2$ two subcategories of $\\mathcal{F}$, there is an obvious inclusion $A^{\\mathcal{E}_2} \\subseteq A^{\\mathcal{E}_1}$.\n\n\\begin{lmm}\\label{aux34}\n\nLet $S$ be a discrete $p$-toral group, let $\\mathcal{F}$ be a fusion system over $S$, and let $\\{(S_i, \\mathcal{F}_i)\\}_{i \\geq 0}$ be a family of finite fusion subsystems of $\\mathcal{F}$ with $\\mathcal{F}_i \\subseteq \\mathcal{F}_{i+1}$ for all $i$ (in particular, $S_i\\leq S_{i+1}$ are finite subgroups of $S$ for all $i$), and satisfying the following properties:\n\\begin{enumerate}[(i)]\n\n\\item $S = \\bigcup_{i \\geq 0} S_i$; and\n\n\\item for all $P\\leq S$ and for all $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P, S)$ there exists some $M_f \\in \\mathbb{N}$ such that, for all $i \\geq M_f$, $f|_{P \\cap S_i} \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P \\cap S_i, S_i)$.\n\n\\end{enumerate}\nLet also $\\mathcal{C}$ be either $\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, the category of $\\Z_{(p)}$-modules, or $\\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, the category of graded-$\\Z_{(p)}$-modules, and let $A \\colon \\mathcal{F} \\to \\mathcal{C}$ be a contravariant functor satisfying the following property:\n\\begin{itemize}\n\n\\item[(\\textasteriskcentered)] For each $P\\leq S$, the natural map $A(P) \\to \\varprojlim_i A(P \\cap S_i)$ is an isomorphism.\n\n\\end{itemize}\nThen, upon setting $\\mathcal{F}^{\\circ} = \\bigcup_{i \\geq 0} \\mathcal{F}_i \\subseteq \\mathcal{F}$, there are equalities\n$$\nA^{\\mathcal{F}} = A^{\\mathcal{F}^{\\circ}} = \\varprojlim_i A^{\\mathcal{F}_i}\n$$\nas subsets of $A(S)$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nWe claim first that $\\mathcal{F}^{\\circ} \\subseteq \\mathcal{F}$ is the full subcategory of $\\mathcal{F}$ whose objects are the finite subgroups of $S$. Indeed, if $P\\leq S$ is a finite subgroup, then there exists some $M_P \\in \\mathbb{N}$ such that $P\\leq S_i$ for all $i \\geq M_P$, since $S = \\bigcup_{i \\geq 0} S_i$. Similarly, if $P, Q\\leq S$ are finite subgroups and $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$, then by condition (ii) there exists some $M_f \\in \\mathbb{N}$ such that $P, Q\\leq S_i$, and $f = f|_{P \\cap S_i} \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P,Q)$ for all $i \\geq M_f$.\n\nBy (\\textasteriskcentered), we have $A(S) = \\varprojlim_i A(S_i)$, which implies that $A^{\\mathcal{F}^{\\circ}} = \\varprojlim_i A^{\\mathcal{F}_i}$. The inclusion $\\mathcal{F}^{\\circ} \\subseteq \\mathcal{F}$ implies that $A^{\\mathcal{F}} \\subseteq A^{\\mathcal{F}^{\\circ}}$ by (\\ref{Aee}). To show the reverse inclusion, let $P, Q \\in \\mathrm{Ob}(\\mathcal{F})$ and $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$. For $X = P, Q, S$, set $X_i = X \\cap S_i$, and notice that\n$$\nX = \\bigcup_{i \\geq 0} X_i \\qquad \\mbox{and} \\qquad A(X) = \\varprojlim_i A(X_i),\n$$\nby (i) and (\\textasteriskcentered) respectively. By condition (ii), there exists some $M_f \\in \\mathbb{N}$ such that,  for all $i \\geq M_f$, $f|_{P_i} \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P_i, Q_i)$. Thus, if $z \\in A^{\\mathcal{F}^{\\circ}}$, then\n$$\nA(\\iota_{P_i})(z) = A(\\iota_{Q_i} \\circ f|_{P_i})(z)\n$$\nfor all $i \\geq M_f$, and thus $A(\\iota_P)(z) = A(\\iota_Q \\circ f)(z)$. Hence $A^{\\mathcal{F}^{\\circ}} \\subseteq A^{\\mathcal{F}}$.\n\\end{proof}\n\n\\begin{rmk}\n\nIf $\\mathcal{G} = (S, \\FF, \\LL)$ is a $p$-local compact group and $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ is an approximation of $\\mathcal{G}$ by $p$-local finite groups, then conditions (i) and (ii) in Lemma \\ref{aux34} follow from condition (i) in \\ref{defiapprox} and \\ref{finmorph}, respectively. Hence, in this case \\ref{aux34} applies to any functor $A \\colon \\mathcal{F} \\to \\mathcal{C}$ as above that satisfies condition (\\textasteriskcentered).\n\n\\end{rmk}\n\n\\begin{prop}\\label{stable1}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $M$ be a (finite) $\\Z_{(p)}$-module with trivial $S$-action. Let also $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups, and let $P\\leq S$. Then, there are isomorphisms\n$$\nH^{\\ast}(BP; M) \\cong \\varprojlim H^{\\ast}(B(P \\cap S_i); M) \\qquad \\mbox{and} \\qquad H^{\\ast}(B\\mathcal{G}; M) \\cong \\varprojlim H^{\\ast}(B\\mathcal{G}_i; M).\n$$\nIn particular, the functor $H^{\\ast}(-; M) \\colon \\mathcal{F} \\to \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$ satisfies condition (\\textasteriskcentered) in Lemma \\ref{aux34}.\n\n\\end{prop}\n\n\\begin{proof}\n\nFix some $P\\leq S$, and set $P_i = P \\cap S_i$ for all $i \\geq 0$. Let $X$ be either $B\\mathcal{G}$ or $BP$, and similarly let $X_i$ be either $B\\mathcal{G}_i$ or $BP_i$, depending on which case we want to prove. Note that the following holds.\n\\begin{enumerate}[(i)]\n\n\\item If $X = BP$, then $X = \\mathrm{hocolim \\,} X_i$, since $P = \\bigcup_{i \\geq 0} P_i$ by hypothesis.\n\n\\item If $X = B\\mathcal{G}$, then $X \\simeq (\\mathrm{hocolim \\,} X_i)^{\\wedge}_p$ by Lemma \\ref{approx0} and Remark \\ref{approx-1}. In particular, $H^{\\ast}(X; M) \\cong H^{\\ast}(\\mathrm{hocolim \\,} X_i; M)$.\n\n\\end{enumerate}\nConsider the homotopy colimit spectral sequence for cohomology \\cite[XII.5.7]{BK}:\n$$\nE^{r,s}_2 = \\varprojlim \\!\\! \\phantom{i}^rH^s(X_i;M) \\Longrightarrow H^{r+s}(X;M).\n$$\nWe will see that, for $r \\geq 1$, $E_2^{r,s} = \\{0\\}$, which, in particular, will imply the statement.\n\nFor each $s$, let $H^s_i = H^s(X_i;M)$, and let $F_i$ be the induced morphism in cohomology (in degree $s$) by the map $|\\Theta_i| \\colon |\\mathcal{L}_i| \\to |\\mathcal{L}_{i+1}|$ induced by the inclusion $\\mathcal{L}_i \\subseteq \\mathcal{L}_{i+1}$. The cohomology ring $H^{\\ast}(X_i;M)$ is noetherian by \\cite[Theorem 5.8]{BLO2}, and in particular $H^s_i$ is finite for all $s$ and all $i$. Thus, the inverse system $\\{H^s_i;F_i\\}$ satisfies the Mittag-Leffler condition \\cite[3.5.6]{Weibel}, and hence the higher limits $\\varprojlim^rH^s_i$ vanish for all $r \\geq 1$. This in turn implies that the differentials in the above spectral sequence are all trivial, and thus it collapses.\n\\end{proof}\n\nWe are ready to prove Theorem \\ref{thmB}, the Stable Elements Theorem for $p$-local compact groups, which we restate below.\n\n\\begin{thm}\\label{stable2}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $M$ be a finite $\\Z_{(p)}$-module $M$ with trivial $S$-action. Then, the natural map\n$$\nH^{\\ast}(B\\mathcal{G}; M) \\Right3{\\cong} H^{\\ast}(\\mathcal{F};M) \\stackrel{def} = \\varprojlim_{\\mathcal{F}} H^{\\ast}(-; M) \\subseteq H^{\\ast}(BS; M)\n$$\nis an isomorphism.\n\n\\end{thm}\n\n\\begin{proof}\n\nLet $\\{\\mathcal{G}_i = (S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups, with respect to some telescopic transporter system $\\widetilde{\\LL}$ satisfying the conditions in \\ref{defiapprox}. By Remark \\ref{approx-1}, the space $B\\mathcal{G}_i \\stackrel{def} = |\\mathcal{L}_i|^{\\wedge}_p$ is the classifying space of a $p$-local finite group, and we can apply the Stable Elements Theorem for $p$-local finite groups: there is a natural isomorphism\n$$\nH^{\\ast}(B\\mathcal{G}_i; M) \\stackrel{\\cong} \\longrightarrow H^{\\ast}(\\mathcal{F}_i;M).\n$$\nBy Proposition \\ref{stable1} there are natural isomorphisms\n$$\nH^{\\ast}(B\\mathcal{G};M) \\cong \\varprojlim_i H^{\\ast}(B\\mathcal{G}_i; M) \\cong \\varprojlim_i H^{\\ast}(\\mathcal{F}_i;M) \\subseteq \\varprojlim_i H^{\\ast}(BS_i; M) \\cong H^{\\ast}(BS;M),\n$$\nand we have to show that $\\varprojlim_i H^{\\ast}(\\mathcal{F}_i;M) \\cong H^{\\ast}(\\mathcal{F};M)$. If we set $\\mathcal{F}^{\\circ} = \\bigcup_{i \\geq 0} \\mathcal{F}_i$, then Lemma \\ref{aux34} and Proposition \\ref{stable1} combined imply that there are isomorphisms\n$$\n\\varprojlim_i H^{\\ast}(\\mathcal{F}_i;M) \\cong H^{\\ast}(\\mathcal{F}^{\\circ};M) \\cong H^{\\ast}(\\mathcal{F};M),\n$$\nand this finishes the proof.\n\\end{proof}\n\n\\begin{rmk}\\label{stable21}\n\nThe reader may have noticed the difference between the original statement of the Stable Elements Theorem for $p$-local finite groups, \\cite[Theorem 5.8]{BLO3}, in terms of the orbit category of $\\mathcal{F}$ defined in (\\ref{orbitcat}), and our statement, in terms of $\\mathcal{F}$. In fact, a formulation of the Stable Elements Theorem in terms of $\\mathcal{F}$, rather than $\\mathcal{O}(\\mathcal{F})$, is already found in \\cite[6.12]{BCGLO2}, as well as in other papers, and it is rather straightforward to justify the equivalence of statements. Consider the projection functor $\\tau \\colon \\mathcal{F} \\to \\mathcal{O}(\\mathcal{F})$, which is the identity on objects. Since conjugation by elements of $S$ induces the identity on cohomology, we have a commutative triangle\n$$\n\\xymatrix{\n\\mathcal{F} \\ar[rrr]^{H^{\\ast}(-)} \\ar[d]_{\\tau} & & & \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}\\\\\n\\mathcal{O}(\\mathcal{F}) \\ar[rrru]_{H^{\\ast}(-)} & & &\n}\n$$\nand an induced morphism between the corresponding inverse limits\n$$\n\\varprojlim_{\\mathcal{O}(\\mathcal{F})} H^{\\ast}(-;\\mathbb{F}_p) \\Right2{} \\varprojlim_{\\mathcal{F}} H^{\\ast}(-;\\mathbb{F}_p).\n$$\nThis morphism is easily checked to be an isomorphism upon considering both groups as subgroups of stable elements in $H^{\\ast}(S; \\mathbb{F}_p)$, since every element of $H^{\\ast}(S; \\mathbb{F}_p)$ is stable by any morphism in $\\mathcal{F}_S(S)$.\n\n\\end{rmk}\n\nWe finish this section proving Theorem \\ref{thmD}, restated as Theorem \\ref{thmd} below, which states the existence of a certain spectral sequence associated to a strongly closed subgroup of a given saturated fusion system. We first fix some notation.\n\nLet $\\mathcal{F}$ be a saturated fusion system over a discrete $p$-toral group $S$, let $R\\leq S$ be a strongly $\\mathcal{F}$-closed subgroup, and let $M$ be an $\\Z_{(p)}$-module with trivial $S$-action. For each $X\\leq S$, set $\\overline{X} = X\/(X \\cap R) \\cong XR\/R\\leq S\/R$. Then, each $f \\in \\mathrm{Hom}_{\\mathcal{F}}(P,Q)$ induces a morphism of extensions\n$$\n\\xymatrix{\nP \\cap R \\ar[r] \\ar[d]_{f_0} & P \\ar[d]^f \\ar[r] & \\overline{P} \\ar[d]^{\\overline{f}} \\\\\nQ \\cap R \\ar[r] & Q \\ar[r] & \\overline{Q},\n}\n$$\nand thus also homomorphisms\n$$\n\\gamma(f) \\colon H^n(\\overline{Q}; H^m(Q \\cap R;M)) \\Right2{f_0^{\\ast}} H^n(\\overline{Q}; H^m(P \\cap R; M))  \\Right2{\\overline{f}^{\\,\\ast}} H^n(\\overline{P}; H^m(P \\cap R;M))\n$$\nfor all $n, m \\geq 0$. This defines a contravariant functor\n\\begin{equation}\\label{frakx}\n\\mathfrak{X}^{n,m} \\colon \\mathcal{F} \\Right3{} \\Z_{(p)}\\mathrm{\\mathbf{-Mod}},\n\\end{equation}\nwhich satisfies condition (\\textasteriskcentered) in Lemma \\ref{aux34}, since, for each $P\\leq S$ and each $i \\geq 0$, the group $H^n(\\overline{P \\cap S_i}; H^m(P \\cap S_i \\cap R; M))$ is finite and the Mittag-Leffler Condition \\cite[3.5.6]{Weibel} applies. Set\n$$\nH^n(S\/R; H^m(R; M))^{\\mathcal{F}} = (\\mathfrak{X}^{n,m})^{\\mathcal{F}}\n$$\nin order to match the notation of \\cite{Diaz}\n\n\\begin{thm}\\label{thmd}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, let $R\\leq S$ be a strongly $\\mathcal{F}$-closed subgroup, and let $M$ be a finite $\\Z_{(p)}$-module with trivial $S$-action. Then there is a first quadrant cohomological spectral sequence with second page\n$$\nE^{n,m}_2 = H^n(S\/R; H^m(R;M))^{\\mathcal{F}}\n$$\nand converging to $H^{n+m}(B\\mathcal{G}; M)$.\n\n\\end{thm}\n\n\\begin{proof}\n\nThe spectral sequence of the statement is constructed as an inverse limit of spectral sequences. For the reader's convenience, the proof is divided into smaller steps.\n\n\\textbf{Step 1.} Construction of inverse systems of spectral sequences. Let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups. Notice that $R_i \\stackrel{def} = R \\cap S_i$ is strongly $\\mathcal{F}_i$-closed for all $i \\geq 0$, since $R$ is strongly $\\mathcal{F}$-closed.\n\nFor each $i \\geq 0$, consider the group extension\n$$\nR_i \\Right2{} S_i \\Right2{} S_i\/R_i,\n$$\nand its associated Lyndon-Hochschild-Serre spectral sequence $\\{E(i)_k^{\\ast, \\ast}, d_k^i\\}_{k \\geq 2}$, with second page $E(i)_2^{n,m} = H^n(S_i\/R_i; H^m(R_i;M))$, and converging to $H^{n+m}(S;M)$. For all $k \\geq 2$ and all $i, n, m \\geq 0$, the inclusion homomorphism $\\iota_{i,i+1} \\colon S_i \\to S_{i+1}$ induces homomorphisms of spectral sequences for each $i \\geq 0$,\n$$\n\\gamma(\\iota_{i, i+1}) \\colon \\{E(i+1)_k^{\\ast,\\ast}, d_k^{i+1}\\}_{k \\geq 2} \\Right3{} \\{E(i)_k^{\\ast,\\ast}, d_k^i\\}_{k \\geq 2}\n$$\nand thus an inverse system of spectral sequences $\\{\\{E(i)_k^{\\ast,\\ast}, d_k^i\\}_{k \\geq 2}, \\gamma(\\iota_{i, i+1})\\}_{i \\geq 0}$.\n\nNote that $R_i$ is strongly $\\mathcal{F}_i$-closed, and thus for each $n, m \\geq 0$ there is a functor\n$$\n\\mathfrak{X}_i^{n,m} \\colon \\mathcal{F}_i \\Right3{} \\Z_{(p)}\\mathrm{\\mathbf{-Mod}},\n$$\ndefined by similar arguments as those used in the definition the functor $\\mathfrak{X}^{n,m}$ in (\\ref{frakx}). Furthermore, we may consider the spectral sequence $\\{\\4{E}(i)_k^{\\ast, \\ast}, \\4{d}_k^i\\}_{k \\geq 2}$ of \\cite[Theorem 1.1]{Diaz}, whose second page is $\\4{E}(i)_2^{n,m} = H^n(S_i\/R_i; H^m(R_i;M))^{\\mathcal{F}_i} = (\\mathfrak{X}_i^{n,m})^{\\mathcal{F}_i}$, and which converges to $H^{n+m}(B\\mathcal{G}_i;M)$. We may see this spectral sequence as the restriction of the spectral sequence $\\{E(i)_k^{\\ast, \\ast}, d_k^i\\}_{k \\geq 2}$. We claim that $\\gamma(\\iota_{i,i+1})$ restricts to a morphism of spectral sequences\n$$\n\\gamma(\\alpha_{i, i+1}) \\colon \\{\\4{E}(i+1)_k^{\\ast,\\ast}, \\4{d}_k^{i+1}\\}_{k \\geq 2} \\Right3{} \\{\\4{E}(i)_k^{\\ast,\\ast}, \\4{d}_k^i\\}_{k \\geq 2}.\n$$\n\nFor each $i,n, m \\geq 0$ and for $X\\leq S_i$, let $A^{n,m}_i(X) = \\mathrm{Hom}_X(\\mathcal{B}^n_{\\overline{X}} \\otimes \\mathcal{B}^m_X, M)$ be the double complex defined in \\cite[Section 3]{Diaz} (although we do not give an explicit description here, notice that the complex itself does not actually depend on $i$). With this notation, for each $k \\geq 2$, the group $\\4{E}(i)_k^{n,m}$ can be seen as $(\\xi(i)^{n,m}_k)^{\\mathcal{F}_i}$, for a certain functor\n$$\n\\xi(i)_k^{n,m} \\colon \\mathcal{F}_i \\Right3{} \\Z_{(p)}\\mathrm{\\mathbf{-Mod}},\n$$\ndefined in terms of the double complexes $A^{n,m}_i(X)$ above. In particular, the functor $\\mathfrak{X}_i^{n,m}$ defined above corresponding to the case $k = 2$ (for the sake of brevity, the reader is referred to \\cite[Section 4]{Diaz} for details). Furthermore, we have a commutative triangle of functors\n$$\n\\xymatrix{\n\\mathcal{F}_i \\ar[rrr]^{\\xi(i)_k^{n,m}} \\ar[d]_{\\mathrm{incl}} & & & \\Z_{(p)}\\mathrm{\\mathbf{-Mod}} \\\\\n\\mathcal{F}_{i+1} \\ar[rrru]_{\\xi(i+1)_k^{n,m}} & & &\n}\n$$\n\nFix some $k \\geq 2$, and some $i, n, m \\geq 0$. For a subgroup $X\\leq S_i\\leq S_{i+1}$, let $\n\\iota_X \\colon X \\to S_i$ and $\\4{\\iota}_X \\colon X \\to S_{i+1}$ be the corresponding inclusion monomorphisms, and note that $\\4{\\iota}_X = \\iota_{i, i+1} \\circ \\iota_X$. Also, recall that $\\4{E}(i+1)_k^{n,m}$ is the subgroup of elements $z \\in E(i+1)_k^{n,m}$ such that\n$$\n\\xi(i+1)_k^{n,m}(\\4{\\iota}_P)(z) = \\xi(i+1)_k^{n,m}(\\4{\\iota}_Q \\circ f)(z)\n$$\nfor all $P,Q\\leq S_{i+1}$ and all $f \\in \\mathrm{Hom}_{\\mathcal{F}_{i+1}}(P,Q)$, and a similar, equation with $\\xi(i)_k^{n,m}$ replacing $\\xi(i+1)_k^{n,m}$, describes the elements of $\\4{E}(i)_k^{n,m}$.\n\nFix some $z \\in E(i+1)_k^{n,m}$, and set $w = \\gamma(\\iota_{i, i+1})(z) \\in E(i)_k^{n,m}$. Then, for all $P, Q\\leq S_i$ and all $f \\in \\mathrm{Hom}_{\\mathcal{F}_i}(P,Q) \\subseteq \\mathrm{Hom}_{\\mathcal{F}_{i+1}}(P,Q)$, the commutativity of the above triangle implies that\n$$\n\\begin{aligned}\n\\xi(i)_k^{n,m}(\\iota_P)(w) & = \\xi(i+1)_k^{n,m}(\\iota_{i, i+1} \\circ \\iota_P)(z) = \\xi(i+1)_k^{n,m}(\\4{\\iota}_P)(z) = \\\\\n & = \\xi(i+1)_k^{n,m}(\\4{\\iota}_Q \\circ f)(z) = \\xi(i+1)_k^{n,m}(\\iota_{i,i+1} \\circ \\iota_Q \\circ f)(z) = \\\\\n & = \\xi(i)_k^{n,m}(\\iota_Q \\circ f)(w).\n\\end{aligned}\n$$\nThus $w \\in \\4{E}(i)_k^{n,m}$ and the claim follows.\n\n\\textbf{Step 2.} The inverse limit spectral sequences and their convergence. Consider the inverse systems of spectral sequences defined in Step 1, $\\{\\{E(i)_k^{\\ast,\\ast}, d_k^i\\}_{k \\geq 2}, \\gamma(\\iota_{i, i+1})\\}_{i \\geq 0}$ and $\\{\\{\\4{E}(i)_k^{\\ast,\\ast}, \\4{d}_k^i\\}_{k \\geq 2}, \\gamma(\\alpha_{i, i+1})\\}_{i \\geq 0}$. For each $k \\geq 2$, and for each $n, m \\geq 0$, define\n$$\nE_k^{n,m} \\stackrel{def} = \\varprojlim_i E_k^{n,m}(i) \\qquad \\qquad \\4{E}_k^{n,m} \\stackrel{def} = \\varprojlim_i \\4{E}_k^{n,m}(i).\n$$\n\nFor each $k \\geq 2$ and each $i, n,m  \\geq 0$, the group $E(i)_k^{n,m}$ is finite by definition, and thus the higher limits of $\\{E(i)_k^{n,m}\\}_{i \\geq 0}$ all vanish, by the Mittag-Leffler Condition \\cite[3.5.6]{Weibel}. A similar conclusion applies to $\\{\\4{E}(i)_k^{n,m}\\}_{i \\geq 0}$, since it is a restriction of $\\{E(i)_k^{\\ast, \\ast}\\}_{i \\geq 0}$. Furthermore, the differentials $\\{d_k^i\\}_{i \\geq 0}$ and $\\{\\4{d}_k^i\\}_{i \\geq 0}$ induce differentials $d_k$ (on $E_k^{\\ast, \\ast}$) and $\\4{d}_k$ (on $\\4{E}_k^{\\ast, \\ast}$) respectively, and an easy computation shows that\n$$\n\\begin{array}{c}\n\\mathrm{Ker}(d_k) \\cong \\varprojlim_i \\mathrm{Ker}(d_k^i) \\qquad \\qquad \\Im(d_k) \\cong \\varprojlim_i \\Im(d_k^i)\\\\[4pt]\n\\mathrm{Ker}(\\4{d}_k) \\cong \\varprojlim_i \\mathrm{Ker}(\\4{d}_k^i) \\qquad \\qquad \\Im(\\4{d}_k) \\cong \\varprojlim_i \\Im(\\4{d}_k^i).\n\\end{array}\n$$\nHence, $\\{E_k^{\\ast, \\ast}, d_k\\}_{k \\geq 2}$ and $\\{\\4{E}_k^{\\ast, \\ast}, \\4{d}_k\\}_{k \\geq 2}$ are well defined spectral sequences. Moreover, their corresponding second and infinity pages are, respectively,\n$$\n\\begin{array}{c}\nE_2^{n,m} \\cong \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M)) \\qquad \\4{E}_2^{n,m} \\cong \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M))^{\\mathcal{F}_i}\\\\[6pt]\nE_{\\infty}^{n,m} \\cong \\varprojlim_i H^{k}(S_i;M) \\cong H^{k}(S; M) \\qquad \\4{E}_{\\infty}^{n,m} \\cong \\varprojlim_i H^{k}(B\\mathcal{G}_i; M) \\cong H^{k}(B\\mathcal{G};M)\n\\end{array}\n$$\nwhere $k = n+m$, and where the last isomorphism on each position of the bottom row holds by Proposition \\ref{stable1}.\n\nIt remains to show that the corresponding second pages of the limit spectral sequences satisfy, respectively,\n$$\n\\begin{array}{l}\nE_2^{n,m} = \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M)) \\cong H^n(S\/R; H^m(R;M)) \\\\[2pt]\n\\4{E}_2^{n,m} = \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M))^{\\mathcal{F}_i} \\cong H^n(S\/R; H^m(R;M))^{\\mathcal{F}}\n\\end{array}\n$$\nBy Proposition \\ref{stable1}, there are isomorphisms \n$$\nH^{\\ast}(R; M) \\cong \\varprojlim H^{\\ast}(R_i;M) \\quad \\mbox{and} \\quad H^{\\ast}(S\/R; H^{\\ast}(R;M))\\cong \\varprojlim H^{\\ast}(S_i\/R_i; H^{\\ast}(R;M)).\n$$\nFurthermore, \\cite[Proposition B.2.3]{Rubin} implies that there are isomorphisms\n$$\n\\begin{aligned}\nH^n(S\/R; H^m(R;M)) & \\cong \\varprojlim_i H^n(S_i\/R_i; H^m(R;M)) \\cong \\\\\n & \\cong \\varprojlim_i \\varprojlim_j H^n(S_i\/R_i; H^m(R_j;M)) \\cong \\varprojlim_i H^n(S_i\/R_i; H^m(R_i;M)),\n\\end{aligned}\n$$\nsince $H^n(S_i\/R_i; H^m(R_j;M))$ is finite for all $i, n, m \\geq 0$. To finish the proof, we have to show that the $H^n(S\/R; H^m(R;M))^{\\mathcal{F}}$ is isomorphic to $\\varprojlim_i H^n(S_iR_i; H^m(R_i;M))^{\\mathcal{F}_i}$, and this follows from Lemma \\ref{aux34}, since the functor $\\mathfrak{X}^{n,m}$ in (\\ref{frakx}) satisfies the required condition (\\textasteriskcentered).\n\\end{proof}\n\n\n\\section{Mapping spaces}\\label{Smap}\n\nIn this section we describe the mapping space $\\operatorname{Map}\\nolimits(BP, B\\mathcal{G})$, where $P$ is a discrete $p$-toral group and $\\mathcal{G}$ is a $p$-local compact group, in terms of centralizers in $\\mathcal{G}$ of subgroups of $S$. Such mapping spaces where described in \\cite{BLO3} when $P\\leq S$ is centric, and in full generality in \\cite{BLO2} when $P$ is a finite $p$-group and $\\mathcal{G}$ is a $p$-local finite group. Our proof follows the same lines as the proof in \\cite[Theorem 6.3]{BLO2}.\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $H^{\\ast}(\\mathcal{F}) \\subseteq H^{\\ast}(BS)$ be the subring of stable elements for $\\mathcal{F}$. Let also $E\\leq S$ be an elementary abelian subgroup which is fully $\\mathcal{F}$-centralized, and let $j_E \\colon H^{\\ast}(\\mathcal{F}) \\to H^{\\ast}(BE)$ be the map induced by inclusion. Let $T_E$ be Lannes' $T$-functor (see \\cite{Lannes}), and let $T_E(H^{\\ast}(\\mathcal{F}); j_E)$ be the component in $T_E(H^{\\ast}(\\mathcal{F}))$ of $j_E \\in T_E^0(H^{\\ast}(\\mathcal{F})) \\cong \\mathrm{Hom}_{\\mathcal{K}}(H^{\\ast}(\\mathcal{F}), H^{\\ast}(BE))$, where $\\mathcal{K}$ is the category of unstable algebras over the mod $p$ Steenrod algebra.\n\n\\begin{lmm}\\label{mapping1}\n\nThere is an isomorphism\n$$\nT_E(H^{\\ast}(\\mathcal{F}); j_E) \\Right3{\\cong} H^{\\ast}(C_{\\mathcal{F}}(E)) \\stackrel{def} = \\varprojlim_{C_{\\mathcal{F}}(E)} H^{\\ast}(-),\n$$\nwhich is the restriction of the homomorphism $T_E(H^{\\ast}(BS);j_E) \\to H^{\\ast}(C_S(E))$ induced by the natural homomorphism $C_S(E) \\times E \\to S$.\n\n\\end{lmm}\n\n\\begin{proof}\n\nLet $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups. For the reader's convenience, we have divided the proof into several shorter steps.\n\n\\textbf{Step 1.} Given a discrete $p$-toral group $X$ and a morphism $\\sigma \\colon E \\to X$, we claim that\n$$\nT_E(H^{\\ast}(BX);\\sigma^{\\ast}) \\cong H^{\\ast}(\\mathrm{Map}(BE, BX)_{B\\sigma}) \\cong H^{\\ast}(BC_X(\\sigma E)).\n$$\nIndeed, since $X$ is discrete $p$-toral, it follows that $BX$ is $p$-good and $H^{\\ast}(BX)$ is of finite type (i.e. finite in every dimension). Moreover, since both $E$ and $X$ are discrete groups, it follows that $\\mathrm{Map}(BE, BX)_{B\\sigma} \\simeq BC_X(\\sigma E)$. The claim follows immediately by \\cite[Proposition 3.4.4]{Lannes}.\n\n\\textbf{Step 2.} Next we claim that there is some $M \\in \\mathbb{N}$ such that $E$ is fully centralized in $\\mathcal{F}_i$, for all $i \\geq M$.\n\nTo prove this, fix representatives $V_0 = E, V_1, \\ldots, V_n\\leq S$ of the different $S$-conjugacy classes in $E^{\\mathcal{F}}$. Note that, since $E$ is abelian, we have $VC_S(V) = C_S(V)$ for each $V \\in E^{\\mathcal{F}}$. We will just write $C_S(V)$ instead of $VC_S(V)$ for simplicity. There is some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$, we have $V_0, \\ldots, V_n\\leq S_i$, and the following holds for each $j = 0, \\ldots, n$:\n\\begin{enumerate}[(a)]\n\n\\item $|C_{S_i}(V_j)|\\leq |C_{S_i}(E)|$; and\n\n\\item $|C_{S_i}(V_j)\/C_{T_i}(V_j)| = |C_S(V_j)\/C_T(V_j)|$.\n\n\\end{enumerate}\nIndeed, since $V_j$ is $\\mathcal{F}$-conjugate to $E$, there exists some $f_j \\in \\mathrm{Hom}_{\\mathcal{F}}(C_S(V_j), C_S(E))$ such that $f_j(V_j) = E$. Moreover, by Lemma \\ref{finmorph}, there is some $M \\in \\mathbb{N}$ such that, for all $i \\geq M$ and all $j = 0, \\ldots, n$, the restriction of $f_j$ to $C_S(V_j) \\cap S_i$ is a morphism in $\\mathcal{F}_i$. Property (a) follows immediately. Property (b) is easily checked, since $S\/T \\cong S_i\/T_i$ for all $i \\geq 0$ by assumption.\n\nFor each $i \\geq M$, set $T_i = T \\cap S_i$ for short. If $V\\leq S_i$ is $\\mathcal{F}_i$-conjugate to $E$, then $V$ is $S$-conjugate to some $V_j$, for some $j \\in \\{1, \\ldots, n\\}$. Fix $x \\in N_S(V, V_j)$. By Lemma \\ref{invar1} (i), together with Proposition \\ref{fix1} (iv), it follows that $x^{-1} \\Psi_i(x) \\in C_T(V)$. Since $T$ is abelian and $T_i\\leq T$, it follows that\n$$\nx^{-1} \\Psi_i(x) \\in C_T(C_{T_i}(V)) = T.\n$$\nThus, $C_{T_i}(V)$ is $S$-conjugate (by $x$) to a subgroup of $C_{T_i}(V_j)$, again by Lemma \\ref{invar1} (i), and a similar argument with $x^{-1}$ instead of $x$ shows that in fact the element $x$ conjugates $C_{T_i}(V)$ onto $C_{T_i}(V_j)$. Moreover, via the inclusion $C_{S_i}(V)\\leq C_S(V)$, the quotient $C_{S_i}(V)\/C_{T_i}(V)$ can be identified with a subgroup of $C_S(V)\/C_T(V)$, and\n$$\n|C_{S_i}(V)\/C_{T_i}(V)|\\leq |C_S(V)\/C_T(V)| = |C_S(V_j)\/C_T(V_j)| = |C_{S_i}(V_j)\/C_{T_i}(V_j)|.\n$$\nSince $C_{S_i}(V)$ and $C_{S_i}(V_j)$ are finite groups, it follows that\n$$\n\\begin{aligned}\n|C_{S_i}(V)| & = |C_{T_i}(V)| \\cdot |C_{S_i}(V)\/C_{T_i}(V)|\\leq \\\\\n &\\leq |C_{T_i}(V_j)| \\cdot  |C_{S_i}(V_j)\/C_{T_i}(V_j)| = |C_{S_i}(V_j)|\\leq |C_{S_i}(E)|,\n\\end{aligned}\n$$\nand $E$ is fully centralized in $\\mathcal{F}_i$.\n\n\\textbf{Step 3.} For each $P\\leq S$, set $\\mathcal{T}_P = \\mathrm{Rep}_{\\mathcal{F}}(E,P) = \\mathrm{Inn}(P)\\backslash \\mathrm{Hom}_{\\mathcal{F}}(E,P)$. Notice that $\\mathcal{T}_P$ is finite by \\cite[Lemma 2.5]{BLO3}. Consider the functor $\\4{T}_E \\colon \\mathcal{O}(\\mathcal{F}) \\Right2{} \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$, defined on objects by\n$$\n\\4{T}_E(P) = \\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}).\n$$\nWe claim that there is an isomorphism\n$$\nT_E(H^{\\ast}(\\mathcal{F});j_E) \\cong \\varprojlim_{P \\in \\mathcal{O}(\\mathcal{F})} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big).\n$$\n\nTo prove this, consider the orbit category $\\mathcal{O}(\\mathcal{F})$, defined in (\\ref{orbitcat}). By \\cite[Lemma 2.5]{BLO3}, all morphism sets in $\\mathcal{O}(\\mathcal{F})$ are finite. Recall also that the full subcategory $\\mathcal{F}^{\\bullet} \\subseteq \\mathcal{F}$ contains only finitely many $\\mathcal{F}$-conjugacy classes by \\cite[Lemma 3.2 (a)]{BLO3}, and thus the full subcategory $\\mathcal{O}(\\mathcal{F}^{\\bullet}) \\subseteq \\mathcal{O}(\\mathcal{F})$ contains a finite skeletal subcategory. Furthermore, $\\mathcal{F}^{\\bullet}$ contains all the $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups of $S$ by \\cite[Corollary 3.5]{BLO3}.\n\nFix a finite skeletal subcategory $\\mathcal{O}_{sk}$ of $\\mathcal{O}(\\mathcal{F}^{\\bullet})$. The functor $T_E$ is exact and commutes with direct limits. As a consequence, it also commutes with inverse limits over finite categories, and we have\n$$\n\\begin{aligned}\nT_E(H^{\\ast}(\\mathcal{F})) & = T_E(\\varprojlim_{\\mathcal{O}(\\mathcal{F})} H^{\\ast}(-)) = \\\\\n & =  T_E(\\varprojlim_{\\mathcal{O}_{sk}} H^{\\ast}(-)) \\cong \\varprojlim_{\\mathcal{O}_{sk}} T_E(H^{\\ast}(-)) = \\varprojlim_{\\mathcal{O}(\\mathcal{F})} T_E(H^{\\ast}(-)),\n\\end{aligned}\n$$\nRestricting the above to $T_E(H^{\\ast}(\\mathcal{F});j_E)$, we obtain\n$$\n\\begin{aligned}\nT_E(H^{\\ast}(\\mathcal{F});j_E) & \\cong \\varprojlim_{P \\in \\mathcal{O}_{sk}} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big) \\cong \\\\\n & \\cong \\varprojlim_{P \\in \\mathcal{O}(\\mathcal{F})} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big) = \\varprojlim_{\\mathcal{O}(\\mathcal{F})} \\4{T}_E(-).\n\\end{aligned}\n$$\n\n\\textbf{Step 4.} Consider the functor $\\4{T}_E \\colon \\mathcal{O}(\\mathcal{F}) \\Right2{} \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}}$ defined in Step 3. By precomposing with the projection functor $\\tau \\colon \\mathcal{F} \\to \\mathcal{O}(\\mathcal{F})$, we obtain a functor\n$$\n\\mathcal{F} \\Right2{} \\mathrm{\\mathbf{Gr}}-\\Z_{(p)}\\mathrm{\\mathbf{-Mod}},\n$$\nwhich by abuse of notation we also denote by $\\4{T}_E$. This will not lead to confusion since both functors take the same values on objects (as $\\tau$ is the identity on objects), as well as the same value on any two morphisms in $\\mathcal{F}$ representing a given morphism in $\\mathcal{O}(\\mathcal{F})$. Note also that, for each $P \\in \\mathrm{Ob}(\\mathcal{F}) = \\mathrm{Ob}(\\mathcal{O}(\\mathcal{F}))$, the set $\\mathcal{T}_P$ does not depend on the choice of category between $\\mathcal{F}$ and $\\mathcal{O}(\\mathcal{F})$, and an argument similar to that used in Remark \\ref{stable21} implies that\n$$\n\\begin{aligned}\n\\varprojlim_{\\mathcal{O}(\\mathcal{F})} \\4{T}_E(-) & = \\varprojlim_{P \\in \\mathcal{O}(\\mathcal{F})} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big) \\cong\\\\\n & \\cong \\varprojlim_{P \\in \\mathcal{F}} \\big(\\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) \\big) = \\varprojlim_{\\mathcal{F}} \\4{T}_E(-).\n\\end{aligned}\n$$\nFrom now on we consider $\\4{T}_E$ as a functor on $\\mathcal{F}$. In this step we prove that the functor $\\4{T}_E$ satisfies condition (\\textasteriskcentered) in Lemma \\ref{aux34}: for each $P\\leq S$, the natural map $\\4{T}_E(P) \\to \\varprojlim_i \\4{T}_E(P \\cap S_i)$ is an isomorphism.\n\nFix $P\\leq S$, and let $\\Omega \\subseteq \\mathrm{Hom}_{\\mathcal{F}}(E,P)$ be a set of representatives of the classes in $\\mathcal{T}_P$. Note that, by definition, there is an equality\n\\begin{equation}\\label{TEP}\n\\4{T}_E(P) \\stackrel{def} = \\bigoplus_{\\rho \\in \\mathcal{T}_P} T_E(H^{\\ast}(BP);\\rho^{\\ast}) = \\bigoplus_{f \\in \\Omega} T_E(H^{\\ast}(BP); f^{\\ast}).\n\\end{equation}\nFor each $i \\geq 0$, set $P_i = P \\cap S_i$. Since $\\Omega$ is a finite set, there exists some $M_P \\in \\mathbb{N}$ such that $f(E)\\leq P_i$ for all $i \\geq M_P$ and all $f \\in \\Omega$. For simplicity we may assume that $M_P = 0$.\n\nFor each $0\\leq i\\leq j$, consider the maps $\\mathcal{T}_{P_i} \\Right1{\\alpha_i} \\mathcal{T}_P$ and $\\mathcal{T}_{P_i} \\Right1{\\beta_{i,j}} \\mathcal{T}_{P_j}$, defined by $\\alpha_i(\\rho) = \\mathrm{incl}_{P_i}^P \\circ \\rho$ and $\\beta_{i,j}(\\rho) = \\mathrm{incl}_{P_i}^{P_j} \\circ \\rho$, respectively. Notice that $\\alpha_i = \\alpha_j \\circ \\beta_{i,j}$ for all $0\\leq i\\leq j$. Moreover, we claim that $\\mathcal{T}_P$ is the colimit of the system $\\{\\mathcal{T}_{P_i}, \\beta_{i,j}\\}$. Indeed, surjectivity of $\\operatornamewithlimits{colim} \\mathcal{T}_{P_i} \\to \\mathcal{T}_P$ follows from the discussion above. To prove injectivity, fix some $i \\geq 0$, and let $\\rho, \\rho' \\in \\mathcal{T}_{P_i}$ be such that $\\alpha_i(\\rho) = \\alpha_i(\\rho')$. Let also $f, f' \\in \\mathrm{Hom}_{\\mathcal{F}}(E, P_i)$ be representatives of $\\rho$ and $\\rho'$ respectively. Then, by definition of $\\mathcal{T}_P$ there exists some $x \\in P$ such that\n$$\n\\mathrm{incl}_{P_i}^P \\circ f' = c_x \\circ \\mathrm{incl}_{P_i}^P \\circ f.\n$$\nSince $P = \\bigcup_{i \\geq 0} P_i$, it follows that $x \\in P_j$ for some $j \\geq i$, and thus $\\beta_{i,j}(\\rho) = \\beta_{i,j}(\\rho') \\in \\mathcal{T}_{P_j}$.\n\nFor each $i \\geq 0$ and each $\\rho \\in \\mathcal{T}_P$, set $\\mathcal{T}_{P_i}^{\\rho} = \\alpha_i^{-1}(\\rho)$. We also fix the following.\n\\begin{enumerate}[(a)]\n\n\\item For each $f \\in \\Omega$, let $f_i \\in \\mathrm{Hom}_{\\mathcal{F}}(E,P_i)$ be the restriction of $f$.\n\n\\item For each $\\rho \\in \\mathcal{T}_P$, fix a set $\\Omega_i^{\\rho} \\subseteq \\mathrm{Hom}_{\\mathcal{F}}(E, P_i)$ of representatives of the classes in $\\mathcal{T}_{P_i}^{\\rho}$. In particular, if $f \\in \\Omega$ represents the class $\\rho \\in \\mathcal{T}_P$, then we choose $f_i$ as representative of its own class in $\\mathcal{T}_{P_i}^{\\rho}$. Let also $\\Omega_i = \\coprod_{\\rho \\in \\mathcal{T}_P} \\Omega_i^{\\rho}$. By definition there is an equality\n\\begin{equation}\\label{TEPi}\n\\4{T}_E(P_i) \\stackrel{def} = \\bigoplus_{\\gamma \\in \\mathcal{T}_{P_i}} T_E(H^{\\ast}(BP_i);\\gamma^{\\ast}) = \\bigoplus_{\\omega \\in \\Omega_i} T_E(H^{\\ast}(BP_i); \\omega^{\\ast}).\n\\end{equation}\n\n\\item For each $\\rho \\in \\mathcal{T}_P$, each $i \\geq 0$, and each $\\omega \\in \\Omega_i^{\\rho}$, fix an element $x_{\\omega} \\in P_{i+1}$ such that $c_{x_{\\omega}} \\circ \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega \\in \\Omega_{i+1}^{\\rho}$ (such an element must exist since $\\beta_{i,i+1}[\\omega] \\in \\mathcal{T}_{P_{i+1}}^{\\rho}$). In the particular case where $\\omega = f_i$ (see (a) above), we choose $x_{\\omega} = 1$. Although the element $x_{\\omega}$ clearly depends on $i$, we omit this dependence from the notation since it will be clear at all times which $i$ is involved.\n\n\\end{enumerate}\nNote that, since $x_{\\omega} \\in P_{i+1}$, we have\n\\begin{equation}\\label{samemap}\n\\mathrm{Map}(BE, BP_{i+1})_{\\4{\\omega}} = \\mathrm{Map}(BE, BP_{i+1})_{\\sigma},\n\\end{equation}\nwhere $\\4{\\omega} = \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega$ and $\\sigma = c_{x_{\\omega}} \\circ \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega$.\n\nFix $\\rho \\in \\mathcal{T}_P$. For each $i \\geq 0$ and each $\\omega \\in \\Omega_i^{\\rho}$, consider the homomorphism\n$$\n\\Gamma_{\\omega} \\colon C_{P_i}(\\omega E) \\times E \\Right2{} P_i,\n$$\ndefined by $\\Gamma_{\\omega}(a,y) = a \\cdot \\omega(y) = \\omega(y) \\cdot a$. Fix $i \\geq 0$ and $\\omega \\in \\Omega_i^{\\rho}$, and let $x_{\\omega}$ be as fixed in (c) above. Let also $\\4{\\omega} = \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega$ and $\\sigma = c_{x_{\\omega}} \\circ \\mathrm{incl}_{P_i}^{P_{i+1}} \\circ \\omega$. Then there is a commutative diagram\n$$\n\\xymatrix@C=2cm{\nC_{P_i}(\\omega E) \\times E \\ar[d]_{(\\mathrm{incl}, \\mathrm{Id})} \\ar[r]^{\\Gamma_{\\omega}} & P_i \\ar[d]^{\\mathrm{incl}} \\\\\nC_{P_{i+1}}(\\4{\\omega} E) \\times E \\ar[d]^{\\cong}_{(c_{x_{\\omega}}, \\mathrm{Id})} \\ar[r]^{\\Gamma_{\\4{\\omega}}} & P_{i+1} \\ar[d]_{\\cong}^{c_{x_{\\omega}}}\\\\\nC_{P_{i+1}}(\\sigma E) \\times E \\ar[r]_{\\Gamma_{\\sigma}} & P_{i+1}\n}\n$$\nwhich in turn induces the commutative diagram below by first passing to classifying spaces and then applying adjunction.\n\\begin{equation}\\label{diagrams}\n\\vcenter{\n\\xymatrix@C=2cm{\nBC_{P_i}(\\omega E) \\ar[r]^{\\simeq} \\ar[d]_{B\\mathrm{incl}} & \\mathrm{Map}(BE, BP_i)_{\\omega} \\ar[d]^{B\\mathrm{incl}_{\\ast}}\\\\\nBC_{P_{i+1}}(\\4{\\omega} E) \\ar[r]^{\\simeq} \\ar[d]_{Bc_{x_{\\omega}}}^{\\cong} & \\mathrm{Map}(BE, BP_{i+1})_{\\4{\\omega}} \\ar[d]_{\\cong}^{(Bc_{x_{\\omega}})_{\\ast}}\\\\\nBC_{P_{i+1}}(\\sigma E) \\ar[r]_{\\simeq} & \\mathrm{Map}(BE, BP_{i+1})_{\\sigma}\n}\n}\n\\end{equation}\nNote that the horizontal maps in the diagram above are homotopy equivalences by Step 1. Moreover, recall from (\\ref{samemap}) that $\\mathrm{Map}(BE, BP_{i+1})_{\\4{\\omega}} = \\mathrm{Map}(BE, BP_{i+1})_{\\sigma}$.\n\nLet $\\rho \\in \\mathcal{T}_P$, and let $f \\in \\Omega$ be its representative. We claim that there is an isomorphism\n\\begin{equation}\\label{isoTP}\nH^{\\ast}(BC_P(f E)) \\cong \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} H^{\\ast}(BC_{P_i}(\\omega E)) \\big),\n\\end{equation}\nwhere the limit is defined by the homomorphisms\n$$\nH_{\\omega} \\colon BC_{P_i}(\\omega E) \\Right2{B\\mathrm{incl}} BC_{P_{i+1}}(\\4{\\omega} E) \\Right2{Bc_{x_{\\omega}}} BC_{P_{i+1}}(\\sigma E).\n$$\nNote that the above limit does not depend on the choice of the elements $x_{\\omega}$ in (c), since a different choice would differ from $x_{\\omega}$ by an element in $C_{P_{i+1}}(\\sigma E)$.\n\nFor each $i \\geq 0$, let $f_i \\in \\mathrm{Hom}_{\\mathcal{F}}(E, P_i)$ be the restriction of $f$ as fixed in (a) above, which is an element of $\\Omega_i^{\\rho}$ by (b), and let $\\3{\\Omega}_i^{\\rho} = \\Omega_i^{\\rho} \\setminus \\{f_i\\}$. For each $\\omega \\in \\Omega_i^{\\rho}$, set for short $A_{\\omega} = H^{\\ast}(BC_{P_i}(\\omega E))$. Then, there are short exact sequences\n$$\n0 \\to \\bigoplus_{\\omega \\in \\3{\\Omega}_i^{\\rho}} A_{\\omega} \\Right2{\\iota_i} \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} A_{\\omega} \\Right2{\\pi_i} A_{f_i} \\to 0\n$$\nfor all $i \\geq 0$. Let also $\\kappa_{i+1} \\colon \\bigoplus_{\\sigma \\in \\Omega_{i+1}^{\\rho}} A_{\\sigma} \\to \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} A_{\\omega}$ be the morphism induced by the maps $H_{\\omega}$ described above, and let $\\3{\\kappa}_{i+1}$ be the restriction of $\\kappa_{i+1}$ to $\\bigoplus_{\\sigma \\in \\3{\\Omega}_{i+1}^{\\rho}} A_{\\sigma}$. In order to prove (\\ref{isoTP}) we use the above exact sequences to construct an exact sequence of inverse limits.\n\nRecall that $\\mathcal{T}_P$ is the colimit of the sets $\\mathcal{T}_{P_i}$. Thus, given $i \\geq 0$, there exists some $M \\in \\mathbb{N}$ such that $[\\mathrm{incl}_{P_i}^{P_{i+M}} \\circ \\omega] = [f_{i + M}] \\in \\mathcal{T}_{P_{i+M}}^{\\rho}$ for all $\\omega \\in \\Omega_i^{\\rho}$, where $f_{i+M}$ is the restriction of $f$ to $P_{i+M}$ as fixed in (a) above. For simplicity let us assume that $M = 1$. In terms of mapping spaces, this means that, composing with the inclusion map $BP_i \\to BP_{i+1}$, we have\n$$\n\\coprod_{\\omega \\in \\Omega_i^{\\rho}} \\mathrm{Map}(BE, BP_i)_{\\omega} \\Right2{} \\mathrm{Map}(BE, P_{i+1})_{f_{i+1}},\n$$\nand it follows that the morphism $\\3{\\kappa}_{i+1}$, defined in the previous paragraph, is simply the trivial homomorphism. The morphism $\\kappa_{i+1}$ also induces a morphism $A_{f_{i+1}} \\to A_{f_i}$, which is easily seen to coincide with the restriction homomorphism induced by the inclusion $C_{P_i}(f_iE)\\leq C_{P_{i+1}}(f_{i+1}E)$. Summarizing, we obtain commutative diagrams\n$$\n\\xymatrix@C=1.5cm{\n0 \\ar[r] & \\bigoplus_{\\sigma \\in \\3{\\Omega}_{i+1}^{\\rho}} A_{\\sigma} \\ar[d]_{\\3{\\kappa}_{i+1}= 0} \\ar[r]^{\\iota_{i+1}} & \\bigoplus_{\\sigma \\in \\Omega_{i+1}^{\\rho}} A_{\\sigma} \\ar[r]^{\\pi_{i+1}} \\ar[d]^{\\kappa_{i+1}} & A_{f_{i+1}} \\ar[r] \\ar[d]^{\\mathrm{res}_{i+1}} & 0 \\\\\n0 \\ar[r] & \\bigoplus_{\\omega \\in \\3{\\Omega}_i^{\\rho}} A_{\\omega} \\ar[r]_{\\iota_i} & \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} A_{\\omega} \\ar[r]_{\\pi_i} & A_{f_i} \\ar[r] & 0\n}\n$$\nThis produces a morphism of inverse systems, and hence, since the inverse limit functor is left exact, there is an exact sequence\n\\begin{equation}\\label{exactlim}\n0 \\to \\varprojlim_i \\big(\\bigoplus_{\\omega \\in \\3{\\Omega}_i^{\\rho}} A_{\\omega} \\big) \\Right1{} \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} A_{\\omega} \\big) \\Right2{\\pi} \\varprojlim_i A_{f_i},\n\\end{equation}\nwhere $\\varprojlim_i \\big(\\bigoplus_{\\omega \\in \\3{\\Omega}_i^{\\rho}} H^{\\ast}(BC_{P_i}(\\omega E))\\big) = 0$ since the morphisms in the corresponding inverse system, namely $\\3{\\kappa}_{i+1}$, are trivial. Thus, $\\pi$ is a monomorphism, and it remains to prove the it is also surjective. Let $(x_i)_{i \\in \\mathbb{N}}$ be an element in $\\varprojlim_i H^{\\ast}(BC_{P_i}(f_i E))$. That is, $\\mathrm{res}_{i+1}(x_{i+1}) = x_i$ for all $i \\geq 0$. Let also\n$$\nX_i = \\pi_i^{-1}(\\{x_i\\}) \\subseteq \\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} H^{\\ast}(BC_{P_i}(\\omega E)).\n$$\nRestricting the maps $\\kappa_{i+1}$ to the sets $X_{i+1}$ produces an inverse system of sets, and the resulting inverse limit $X = \\varprojlim_i X_i$ is nonempty by \\cite[Proposition 1.1.4]{RZ}. Moreover, by definition of the maps $\\kappa_i$, any element in $X$ is a preimage of $(x_i)_{i \\in \\mathbb{N}}$, and $\\pi$ is surjective. Thus, we have\n$$\nH^{\\ast}(BC_P(f E)) \\cong \\varprojlim_i H^{\\ast}(BC_{P_i}(f_i E)) \\cong \\varprojlim_i \\big(\\bigoplus_{\\omega \\in \\Omega_i^{\\rho}} H^{\\ast}(BC_{P_i}(\\omega E))\\big),\n$$\nwhere the leftmost isomorphism follows from Proposition \\ref{stable1}, and the rightmost isomorphism corresponds to $\\pi$ in (\\ref{exactlim}). This proves (\\ref{isoTP}).\n\n\nCombining Step 1 and the above discussion, we deduce the following.\n$$\n\\begin{aligned}\n\\varprojlim_i \\4{T}_E(P_i) & = \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i} T_E(H^{\\ast}(BP_i); \\omega^{\\ast}) \\big) \\cong \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i} H^{\\ast}(\\mathrm{Map}(BE, BP_i)_{\\omega}) \\big) \\cong \\\\\n & \\cong \\varprojlim_i \\big( \\bigoplus_{\\omega \\in \\Omega_i} H^{\\ast}(BC_{P_i}(\\omega E)) \\big) \\cong \\bigoplus_{f \\in \\Omega} H^{\\ast}(BC_P(f E)) \\cong \\\\\n & \\cong \\bigoplus_{f \\in \\Omega} H^{\\ast}(\\mathrm{Map}(BE, BP)_{f}) \\cong \\bigoplus_{f \\in \\Omega} T_E(H^{\\ast}(BP); f^{\\ast}) = \\4{T}_E(P).\n\\end{aligned}\n$$\nMore precisely, the first equality holds by (\\ref{TEPi}), while the last equality corresponds to (\\ref{TEP}), the first isomorphism in the first row follows from Step 1, the isomorphism between the last term in the first row and the first term in the second row follows from Step 1 together with the leftmost diagram in (\\ref{diagrams}), the middle isomorphism in the second row holds by (\\ref{isoTP}), and the rest follows again from Step 1.\n\n\n\\textbf{Step 5.} The isomorphism $T_E(H^{\\ast}(\\mathcal{F}); j_E) \\Right3{\\cong} H^{\\ast}(C_{\\mathcal{F}}(E))$. Consider the fusion system $C_{\\mathcal{F}}(E)$ over $C_S(E)$, and consider also the set of fusion subsystems $\\{C_{\\mathcal{F}_i}(E)\\}_{i \\geq 0}$. This setup satisfies the conditions of Lemma \\ref{aux34}: clearly, $C_S(E) = \\bigcup_{i \\geq 0} C_{S_i}(E)$, since $S = \\bigcup_{i \\geq 0} S_i$, and every morphism in $C_{\\mathcal{F}}(E)$ eventually restricts to a morphism in $C_{\\mathcal{F}_i}(E)$ for $i$ big enough, since the fusion systems $\\mathcal{F}_i$ are part of an approximation of $(S, \\FF, \\LL)$ by $p$-local finite groups. It follows that there is a sequence of isomorphisms\n$$\nT_E(H^{\\ast}(\\mathcal{F}); j_E) \\cong \\varprojlim_{\\mathcal{F}} \\4{T}_E(-) \\cong \\varprojlim_i \\big(\\varprojlim_{\\mathcal{F}_i} \\4{T}_E(-) \\big) \\cong \\varprojlim_i H^{\\ast}(C_{\\mathcal{F}_i}(E)) \\cong H^{\\ast}(C_{\\mathcal{F}}(E))\n$$\nwhere the first isomorphism follows from Steps 3 and 4 combined, the second isomorphism is a consequence of Lemma \\ref{aux34}, since we have checked in Step 4 that $\\4{T}_E$ satisfies condition (\\textasteriskcentered), the third isomorphism follows from \\cite[Lemma 5.7]{BLO2}, since we have shown in Step 2 that $E$ can be assumed to be fully $\\mathcal{F}_i$-centralized for all $i$, and the last isomorphism is a consequence of Proposition \\ref{stable1}.\n\\end{proof}\n\n\\begin{rmk}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ be an approximation of $\\mathcal{G}$ by $p$-local finite groups. Let also $E\\leq S$ be an elementary abelian subgroup which is fully centralized, and let $j_E$ be as above. Without loss of generality we may assume that $E\\leq S_i$ for all $i \\geq 0$, and that $E$ is fully $\\mathcal{F}_i$-centralized for all $i \\geq 0$. Let $j_{E,i} \\colon H^{\\ast}(\\mathcal{F}_i) \\to H^{\\ast}(BE)$ be the map induced by the inclusion $E\\leq S_i$, for all $i \\geq 0$. In this situation, we have just proved that\n$$\nT_E(H^{\\ast}(\\mathcal{F}); j_E) \\cong T_E(\\varprojlim_i H^{\\ast}(\\mathcal{F}_i); j_E) \\cong \\varprojlim_i T_E(H^{\\ast}(\\mathcal{F}_i);j_{E,i}).\n$$\nIn other words, the functor $T_E$ commutes with the inverse limit $\\varprojlim_i H^{\\ast}(\\mathcal{F}_i)$. We do not know of any general result about the functor $T_E$ commuting with infinite inverse limits.\n\n\\end{rmk}\n\nThe proof of Theorem \\ref{mapping} below requires the following result involving the quotient of a linking system by a normal discrete $p$-toral subgroup. The following is a generalization of \\cite[Lemma 5.6]{BLO2} to the compact case.\n\n\\begin{lmm}\\label{quotient2}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $\\widetilde{\\LL}$ be the telescopic linking system associated to $\\mathcal{L}$ in \\ref{expl1}. Let also $V\\leq S$ be a central subgroup in $\\mathcal{F}$ of order $p$, and let $(\\widetilde{\\LL}\/V, \\3{\\varepsilon}, \\3{\\rho})$ be the quotient transporter system, associated to the saturated fusion system $\\mathcal{F}\/V$. Finally, let $(\\widetilde{\\LL}\/V)^c \\subseteq \\widetilde{\\LL}\/V$ and $\\widetilde{\\LL}_0 \\subseteq \\widetilde{\\LL}$ be the full subcategories with object sets\n$$\n\\begin{array}{l}\n\\mathrm{Ob}((\\widetilde{\\LL}\/V)^c) = \\{P\/V \\in \\mathrm{Ob}(\\widetilde{\\LL}\/V) \\, | \\, P\/V \\mbox{ is $\\mathcal{F}\/V$-centric}\\} \\\\[2pt]\n\\mathrm{Ob}(\\widetilde{\\LL}_0) = \\{P \\in \\mathrm{Ob}(\\widetilde{\\LL}) \\, | \\, P\/V \\in \\mathrm{Ob}((\\widetilde{\\LL}\/V)^c)\\}\n\\end{array}\n$$\nrespectively. Then the following holds.\n\\begin{enumerate}[(i)]\n\n\\item $(\\widetilde{\\LL}\/V)^c$ is a linking system associated to $\\mathcal{F}\/V$.\n\n\\item $BV \\to |\\widetilde{\\LL}_0|^{\\wedge}_p \\to |(\\widetilde{\\LL}\/V)^c|^{\\wedge}_p$ is a fibration sequence.\n\n\\item The inclusion $\\widetilde{\\LL}_0 \\subseteq \\widetilde{\\LL}$ induces a homotopy equivalence $|\\widetilde{\\LL}_0|^{\\wedge}_p \\simeq B\\mathcal{G}$.\n\n\\end{enumerate}\n\n\\end{lmm}\n\n\\begin{proof}\n\nClearly, all the $\\mathcal{F}$-centric subgroups of $S$ must contain $V$, since $V$ is abelian and $\\mathcal{F}$-central. Thus, for all $P, Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$, it follows by axiom (C) of transporter systems that the left and right actions of $V$ on $\\mathrm{Mor}_{\\widetilde{\\LL}}(P,Q)$ (via composition with $\\varepsilon_Q(V)$ and $\\varepsilon_P(V)$ respectively) are the same. Since the proof is rather long, it is divided into steps for the reader's convenience.\n\n\\textbf{Step 1.} For each $P\\leq S$ which contains $V$, we claim that\n$$\nP \\mbox{ fully $\\mathcal{F}$-normalized } \\Longrightarrow P\/V \\mbox{ fully $\\mathcal{F}\/V$-centralized } \\Longrightarrow \\Gamma_P\\leq \\mathrm{Aut}_S(P).\n$$\n\nLet $P\\leq S$ be such that $V\\leq P$, and note that $\\mathrm{Aut}_{\\mathcal{F}}(V) = \\{\\mathrm{Id}\\}$, since $V$ is central in $\\mathcal{F}$. Thus, the subgroup\n$$\n\\Gamma_P \\stackrel{def} = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}}(P) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}\/V}(P\/V))\n$$\nis a discrete $p$-toral normal subgroup of $\\mathrm{Aut}_{\\mathcal{F}}(P)$ by Lemma \\ref{Kpgp}. Since $\\mathcal{F}$ is saturated, every subgroup of $S$ is $\\mathcal{F}$-conjugate to a fully $\\mathcal{F}$-normalized subgroup. Similarly, since $\\mathcal{F}\/V$ is saturated, every subgroup of $S\/V$ is $\\mathcal{F}\/V$-conjugate to a fully $\\mathcal{F}\/V$-centralized subgroup. Thus it is enough to show the following: if $P, Q\\leq S$ are $\\mathcal{F}$-conjugate subgroups such that $P\/V$ is fully $\\mathcal{F}\/V$-centralized and $Q$ is fully $\\mathcal{F}$-normalized, then $Q\/V$ is fully $\\mathcal{F}\/V$-centralized and $\\Gamma_P\\leq \\mathrm{Aut}_S(P)$.\n\nAs show above, the group $\\Gamma_Q$ is a normal discrete $p$-toral subgroup of $\\mathrm{Aut}_{\\mathcal{F}}(Q)$. Furthermore, since $Q$ is fully $\\mathcal{F}$-normalized we have $\\mathrm{Aut}_S(Q) \\in \\operatorname{Syl}\\nolimits_p(\\mathrm{Aut}_{\\mathcal{F}}(Q))$, and thus $\\Gamma_Q\\leq \\mathrm{Aut}_S(Q)$. By axiom (II) of saturated fusion systems, every isomorphism $f \\in \\mathrm{Iso}_{\\mathcal{F}}(P,Q)$ extends to some $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_f, N_S(Q))$, where\n$$\nN_f = \\{g \\in N_S(P) \\, | \\, f \\circ c_g \\circ f^{-1} \\in \\mathrm{Aut}_S(Q)\\}.\n$$\nSet $N_S^0(P) = \\{g \\in N_S(P) \\, | \\, c_g \\in \\Gamma_P\\}$, and notice that $N_S^0(P)\/V = C_{S\/V}(P\/V)$. We claim that $N_S^0(P)\\leq N_f$. To prove that, fix $g \\in N_S^0(P)$ and $a \\in Q$, and set $b = f^{-1}(a) \\in P$. By definition, $c_g(b) = bv$ for some $v \\in V$, and we have\n$$\n(f \\circ g \\circ f^{-1})(a) = (f \\circ c_g)(b) = f(bv) = f(b) v = a v,\n$$\nwhere $f(bv) = f(b) v$ since $V$ is central in $\\mathcal{F}$ and $V,\\leq P,Q$ (and thus the morphism $f$ restricts to the identity on $V$).\n\nThe above implies that $\\gamma$ restricts to $\\gamma \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S^0(P), N_S^0(Q))$, which in turn factors through a homomorphism\n$$\n\\3{\\gamma} \\in \\mathrm{Hom}_{\\mathcal{F}\/V}(C_{S\/V}(P\/V), C_{S\/V}(Q\/V)).\n$$\nSince $P\/V$ is fully $\\mathcal{F}\/V$-centralized, it follows that $\\3{\\gamma}$ is an isomorphism and $Q\/V$ is also fully $\\mathcal{F}\/V$-centralized. Furthermore, $\\gamma$ must be an isomorphism too, and thus $\\Gamma_P\\leq \\mathrm{Aut}_S(P)$.\n\n\\textbf{Step 2.} We show now that the category $(\\widetilde{\\LL}\/V)^c$ is a centric linking system associated to $\\mathcal{F}\/V$. First notice that $(\\widetilde{\\LL}\/V)^c$ is a transporter system associated to $\\mathcal{F}\/V$ by the above remarks, and thus we only have to check that, for each object $P\/V$ of $(\\widetilde{\\LL}\/V)^c$,\n$$\nE(P\/V) \\stackrel{def} = \\mathrm{Ker}(\\mathrm{Aut}_{(\\widetilde{\\LL}\/V)^c}(P\/V) \\to \\mathrm{Aut}_{\\mathcal{F}\/V}(P\/V)) = \\3{\\varepsilon}_{P\/V}(Z(P\/V)).\n$$\n\nFix $P\/V \\in \\mathrm{Ob}((\\widetilde{\\LL}\/V)^c)$, and consider the following commutative diagram\n$$\n\\xymatrix{\n\\mathrm{Aut}_{\\widetilde{\\LL}}(P) \\ar[rr]^{\\tau_P} \\ar[d]_{\\rho_P} & & \\mathrm{Aut}_{\\widetilde{\\LL}\/V}(P\/V) \\ar[d]^{\\3{\\rho}_{P\/V}} \\\\\n\\mathrm{Aut}_{\\mathcal{F}}(P) \\ar[rr]_{\\omega_P} & & \\mathrm{Aut}_{\\mathcal{F}\/V}(P\/V),\n}\n$$\nwhere $\\rho_{P}$ and $\\3{\\rho}_{P\/V}$ denote the corresponding structural functors in the transporter systems $\\widetilde{\\LL}$ and $\\widetilde{\\LL}\/V$, respectively. By definition, we have\n$$\n\\mathrm{Ker}(\\tau_P) = \\varepsilon_P(V) \\qquad \\qquad \\mathrm{Ker}(\\rho_P) = \\varepsilon_P(C_S(P)) \\qquad \\qquad \\mathrm{Ker}(\\omega_P) = \\Gamma_P,\n$$\nand an easy computation shows then that $E(P\/V) = \\tau_P(\\gen{\\varepsilon_P(C_S(P)), \\varepsilon_P(N_S^0(P))})$. In particular it follows that $E(P\/V)$ is a discrete $p$-toral group. To finish the proof, recall that $P\/V$ is $\\mathcal{F}\/V$-centric, and in particular it is fully $\\mathcal{F}\/V$-centralized. Thus $Z(P\/V) \\in \\operatorname{Syl}\\nolimits_p(E(P\/V))$, which implies that $E(P\/V) = Z(P\/V)$.\n\n\\textbf{Step 3.} $BV \\to |\\widetilde{\\LL}_0|^{\\wedge}_p \\to |(\\widetilde{\\LL}\/V)^c|^{\\wedge}_p$ is a fibration sequence. Using \\cite[Lemma 4.3 (a)]{BLO3} it is easy to check that each undercategory for the projection of $\\widetilde{\\LL}_0$ onto $(\\widetilde{\\LL}\/V)^c$ contains a category equivalent to $\\mathcal{B}(V)$ as a deformation retract. Thus, by Quillen's Theorem B, the map $|\\widetilde{\\LL}_0| \\to |(\\widetilde{\\LL}\/V)^c|$ has homotopy fiber $BV$. By \\cite[II.5.1]{BK}, the fibration sequence $BV \\to |\\widetilde{\\LL}_0| \\to |(\\widetilde{\\LL}\/V)^c|$ is still a fibration sequence after $p$-completion.\n\n\\textbf{Step 4.} The inclusion $\\widetilde{\\LL}_0 \\subseteq \\widetilde{\\LL}$ induces a homotopy equivalence $|\\widetilde{\\LL}_0|^{\\wedge}_p \\simeq B\\mathcal{G}$. Notice that the functor $(-)^{\\bullet}$ restricts to a functor $(-)^{\\bullet}$ on $\\widetilde{\\LL}_0$. Indeed, let $P \\in \\mathrm{Ob}(\\widetilde{\\LL}^{\\bullet})\\setminus \\mathrm{Ob}(\\widetilde{\\LL}_0)$, and let $Q \\in \\mathrm{Ob}(\\widetilde{\\LL})$ be such that $V\\leq Q$ and $Q^{\\bullet} = P$. Then $Q\/V\\leq P\/V$, and since $P\/V$ is not $\\mathcal{F}\/V$-centric by assumption, neither is $Q\/V$. Thus $Q \\notin \\mathrm{Ob}(\\widetilde{\\LL}_0)$. Conversely, if $Q \\in \\mathrm{Ob}(\\widetilde{\\LL}_0)$, then clearly $Q^{\\bullet}\\in \\mathrm{Ob}(\\widetilde{\\LL}_0)$.\n\nAlso, note that if $\\widetilde{\\LL}_0$ contains $\\mathcal{L}^{\\bullet}$ then the claim follows easily. Thus, fix some $P\\leq S$ in $\\mathcal{L}^{\\bullet}$ but not in $\\widetilde{\\LL}_0$. In particular, $P$ is $\\mathcal{F}$-centric, and $V\\leq P$, but $P\/V$ is not $\\mathcal{F}\/V$-centric. By replacing $P$ by a conjugate if necessary, we may assume that $C_{S\/V}(P\/V) \\neq Z(P\/V)$, and thus there is some $gV \\in S\/V$ such that $gV \\notin P\/V$ and $[gV, P\/V] = 1$. Equivalently, there is some $g \\in S$ such that $g \\notin P$ and $[g, P] = V$. Furthermore, $c_g \\in \\mathrm{Aut}_{\\mathcal{F}}(P)$ cannot be an inner automorphism, because if this was the case then $c_g = c_x$ for some $x \\in P$, and $gx^{-1} \\in C_S(P) \\setminus P = \\emptyset$, since $P$ is $\\mathcal{F}$-centric. Thus $c_g$ is a nontrivial element of $\\mathrm{Ker}(\\mathrm{Out}_{\\mathcal{F}}(P) \\to \\mathrm{Out}_{\\mathcal{F}\/V}(P\/V))$, and thus $P$ is not $\\mathcal{F}$-radical. In other words, we have shown that $\\widetilde{\\LL}_0$ contains all $\\mathcal{F}$-centric $\\mathcal{F}$-radical subgroups. By \\cite[Corollary A.10]{BLO6} the inclusion $\\widetilde{\\LL}_0 \\subseteq \\widetilde{\\LL}$ induces a homotopy equivalence $|\\widetilde{\\LL}_0|^{\\wedge}_p \\simeq B\\mathcal{G}$.\n\\end{proof}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, and let $P\\leq S$ be a fully $\\mathcal{F}$-centralized subgroup. Let also $C_{\\mathcal{G}}(P) = (C_S(P), C_{\\mathcal{F}}(P), C_{\\mathcal{L}}(P))$ be the centralizer $p$-local compact group of $P$ defined in \\ref{rmknorm}, with classifying space $BC_{\\mathcal{G}}(P)$, and let $\\mathcal{B}(P)$ be the category with a single object $\\circ_P$ and $P$ as automorphism group. By Lemma \\ref{centricNFKA}, if $Q\\leq C_S(P)$ is $C_{\\mathcal{F}}(P)$-centric then $QP$ is $\\mathcal{F}$-centric. Thus we can define a functor\n$$\n\\Gamma_{\\mathcal{L}, P} \\colon C_{\\mathcal{L}}(P) \\times \\mathcal{B}(P) \\Right3{} \\mathcal{L}\n$$\nby setting $\\Gamma_{\\mathcal{L},P}(Q, \\circ_P) = QP$ for each $C_{\\mathcal{F}}(P)$-centric subgroup of $C_S(P)$. Given a morphism $(\\varphi, g) \\in \\mathrm{Mor}_{C_{\\mathcal{L}}(P) \\times \\mathcal{B}(P)}((Q,\\circ), (R,\\circ))$, the functor $\\Gamma$ is defined as follows\n$$\n\\Gamma_{\\mathcal{L},P}(\\varphi, g) = \\varphi \\circ \\varepsilon_{QP}(g) = \\varepsilon_{RP}(g) \\circ \\varphi,\n$$\nwhere the last equality follows from condition (C) of transporter systems, since the underlying homomorphism of $\\varphi \\in \\mathrm{Mor}_{\\mathcal{L}}(QP, RP)$ restricts to the identity on $P$ by definition of $C_{\\mathcal{L}}(P)$. By first realizing nerves and them $p$-completing, we obtain a map\n$$\nBC_{\\mathcal{G}}(P) \\times (BP)^{\\wedge}_p \\Right3{} B\\mathcal{G}\n$$\n(notice that $(BP)^{\\wedge}_p$ is not necessarily equivalent to $BP$ since $P$ is a discrete $p$-toral group). By taking adjoint first and then precomposing with the natural map $BP \\to (BP)^{\\wedge}_p$, we obtain a map\n$$\n\\Gamma'_{\\mathcal{L},P} \\colon |C_{\\mathcal{L}}(P)|^{\\wedge}_p \\Right3{} \\mathrm{Map}((BP)^{\\wedge}_p, B\\mathcal{G})_{\\mathrm{incl}} \\Right3{} \\mathrm{Map}(BP, B\\mathcal{G})_{\\mathrm{incl}}.\n$$\n\n\\begin{thm}\\label{mapping}\n\nLet $\\mathcal{G} = (S, \\FF, \\LL)$ be a $p$-local compact group, let $P$ be a discrete $p$-toral group, and $\\gamma: P \\to S$ be a group homomorphism such that $\\gamma(P)$ is fully $\\mathcal{F}$-centralized in $\\mathcal{F}$. Then,\n$$\n\\Gamma'_{\\mathcal{L}, \\gamma(P)} \\colon BC_{\\mathcal{G}}(\\gamma(P)) \\Right2{\\simeq} \\mathrm{Map}(BP, B\\mathcal{G})_{B\\gamma}\n$$\nis a homotopy equivalence.\n\n\\end{thm}\n\n\\begin{proof}\n\nOur proof follows the same strategy as the proof of \\cite[Theorem 6.3]{BLO2}. The referee suggested an alternative to the cases 1-3 below, which we briefly discuss after the proof. By \\cite[Proposition 6.2]{BLO3}, for each $\\gamma \\in \\mathrm{Hom}(P, S)$,\n\\begin{equation}\\label{zero}\n\\mathrm{Map}(BP, B\\mathcal{G})_{B\\gamma} \\simeq \\mathrm{Map}(B\\gamma(P), B\\mathcal{G})_{\\mathrm{incl}}.\n\\end{equation}\nThus, it suffices to prove the statement when $P\\leq S$ is fully $\\mathcal{F}$-centralized and $\\gamma$ is the inclusion. The proof is divided into several cases for the reader's convenience.\n\n\\textbf{Case 1.} Suppose that $P$ is elementary abelian. In this case, by Theorem 0.5 \\cite{Lannes}, it is enough to prove that $\\Gamma_{\\mathcal{L},P}$ induces an isomorphism\n$$\nT_P(H^{\\ast}(B\\mathcal{G}); \\mathrm{incl}^{\\ast}) \\Right2{\\cong} H^{\\ast}(BC_{\\mathcal{G}}(P)).\n$$\nBy Theorem \\ref{stable2}, $H^{\\ast}(B\\mathcal{G}) \\cong H^{\\ast}(\\mathcal{F})$, and $H^{\\ast}(BC_{\\mathcal{G}}(P)) \\cong H^{\\ast}(C_{\\mathcal{F}}(P))$, and the above isomorphism follows from Lemma \\ref{mapping1}.\n\n\\textbf{Case 2.} Suppose that $P$ is a normal subgroup of $\\mathcal{F}$, that is, $N_{\\mathcal{F}}(P) = \\mathcal{F}$. Let $\\mathcal{L}_0 \\subseteq \\mathcal{L}$ be the full subcategory whose objects are the subgroups $Q\\leq S$ such that $C_Q(P)$ is $C_{\\mathcal{F}}(P)$-centric.\n\nIn order to prove the statement in this case, we first need to show that the inclusion of $\\mathcal{L}_0$ into $\\mathcal{L}$ induces an equivalence $|\\mathcal{L}_0|^{\\wedge}_p \\simeq |\\mathcal{L}|^{\\wedge}_p$. By \\cite[Corollary A.10]{BLO6}, it is enough to check that $\\mathrm{Ob}(\\mathcal{F}^{cr}) \\subseteq \\mathrm{Ob}(\\mathcal{L}_0)$.\n\nLet $Q \\in \\mathrm{Ob}(\\mathcal{L})$ be an $\\mathcal{F}$-centric subgroup of $S$ which is not an object in $\\mathcal{L}_0$. We claim that $Q$ is not $\\mathcal{F}$-radical. Set $Q_0 = C_Q(P)$, and note that every element of $\\mathrm{Aut}_{\\mathcal{F}}(Q)$ restricts to an automorphism of $Q_0$ since $P$ is normal in $\\mathcal{F}$. Furthermore $Q_0 \\lhd Q$, and we can define\n$$\nK = \\mathrm{Ker}(\\mathrm{Aut}_{\\mathcal{F}}(Q) \\Right2{} \\mathrm{Aut}_{\\mathcal{F}}(Q_0) \\times \\mathrm{Aut}(Q\/Q_0)) \\lhd \\mathrm{Aut}_{\\mathcal{F}}(Q).\n$$\nNote that $K$ is a discrete $p$-toral subgroup of $\\mathrm{Aut}_{\\mathcal{F}}(Q)$ by Lemma \\ref{Kpgp}. In order to prove that $Q$ is not $\\mathcal{F}$-radical, it is enough to check that $1 \\neq K \\not\\leq \\mathrm{Inn}(Q)$.\n\nBy assumption $Q_0 \\notin \\mathrm{Ob}(\\mathcal{L}_0)$, and thus $Q_0$ is not $C_{\\mathcal{F}}(P)$-centric. Thus, we may assume that $C_{C_S(P)}(Q_0) \\not\\leq Q_0$, since otherwise $Q$ can be replaced by an $\\mathcal{F}$-conjugate $R$ such that the corresponding subgroup $R_0 = C_R(P)$ is $C_{\\mathcal{F}}(P)$-conjugate to $Q_0$ and satisfies the desired condition. Set\n$$\nQ_1 \\stackrel{def} = C_S(Q_0P) = C_{C_S(P)}(Q_0).\n$$\nAs discussed above we have $Q_1 \\not\\leq Q_0 = C_Q(P)$, and thus, $Q_1 \\cap Q = C_Q(Q_0P)\\leq Q_0$, and $Q_1 \\not\\leq Q$. Also, $Q\\leq N_S(Q_1)$ by definition, and thus $Q_1Q\\leq S$ is a subgroup. Note that $Q \\lneqq Q_1Q$, and thus $Q \\lneqq N_{Q_1Q}(Q)$. Choose some $x \\in N_{Q_1Q}(Q)$ such that $x \\notin Q$. Then,\n$$\n[x, Q] = \\{xax^{-1}a^{-1} \\,\\, | \\,\\, a \\in Q\\}\\leq Q_1 \\cap Q\\leq Q_0,\n$$\nand hence $c_x \\in K$. Since $x \\notin Q$ and $Q$ is $\\mathcal{F}$-centric, it follows that $c_x \\notin \\mathrm{Inn}(Q)$, and thus $Q$ is not $\\mathcal{F}$-radical.\n\nLet $(\\mathcal{L}_0)_{P, Id}$ be the category with object set the pairs $(Q, \\alpha)$, for $Q$ in $\\mathcal{L}_0$ and $\\alpha \\in \\mathrm{Hom}_{\\mathcal{F}}(P, Q)$ and such that\n$$\n\\mathrm{Mor}_{(\\mathcal{L}_0)_{P, Id}}((Q,\\alpha), (R, \\alpha')) = \\{\\varphi \\in \\mathrm{Mor}_{\\mathcal{L}}(Q,R) \\mbox{ } | \\mbox{ } \\alpha' = \\rho(\\varphi) \\circ \\alpha\\}.\n$$\nThis is equivalent to the component of the object $(P,Id)$ in the category $\\mathcal{L}_0^P$ of \\cite[Proposition 6.2]{BLO3}, and hence there is a homotopy equivalence\n$$\n\\mathrm{Map} (BP, |\\mathcal{L}_0|^{\\wedge}_p)_{\\mathrm{incl}} \\simeq |(\\mathcal{L})_{P, Id}|^{\\wedge}_p.\n$$\n\nAt this point, one can define functors\n$$\n\\xymatrix{\n(\\mathcal{L}_0)_{P,Id} \\ar @<  2pt> [r]^{\\sigma} & C_{\\mathcal{L}}(P) \\ar @< 2pt> [l]^{\\tau}\\\\\n}\n$$\nin the same way as they are constructed in Step 2 of the proof \\cite[Theorem 6.3]{BLO2}, and which are inverse to each other up to natural transformation. This implies that the composite\n$$\nBC_{\\mathcal{G}}(P) \\Right2{\\tau} |(\\mathcal{L}_0)_{P, Id}|^{\\wedge}_p \\Right2{\\simeq} \\mathrm{Map}(BP, |\\mathcal{L}_0|^{\\wedge}_p)_{\\mathrm{incl}} \\Right2{\\simeq} \\mathrm{Map}(BP, B\\mathcal{G})_{\\mathrm{incl}}\n$$\nis a homotopy equivalence, and by construction it is equal to $\\Gamma'_{\\mathcal{L},P}$.\n\n\\textbf{Case 3.} Suppose that $P$ is a finite subgroup of $S$. As an induction hypothesis, we can assume that the statement holds for all maps with source $BP'$, with $|P'| < |P|$, and all $p$-local compact groups.\n\nFix a subgroup $V\\leq  P \\cap Z(N_S(P))$ of order $p$, and note that in particular $N_S(P)\\leq C_S(V)$. By \\cite[Lemma 2.2 (b)]{BLO6}, there exists some $\\omega \\in \\mathrm{Hom}_{\\mathcal{F}}(N_S(V), S)$ such that $\\omega(V)$ is fully $\\mathcal{F}$-centralized. Hence, there is an inequality $|N_S(\\omega(P))| \\geq |N_S(P)|$ which is in fact an equality since we are assuming $P$ to be fully $\\mathcal{F}$-normalized.\n\nWe may replace $P$ and $V$ by $\\omega(P)$ and $\\omega(V)$, and assume that $V$ is fully $\\mathcal{F}$-centralized and $P$ is fully $\\mathcal{F}$-normalized. Furthermore, $P$ is fully normalized in the saturated fusion system $C_{\\mathcal{F}}(V)$, since $N_S(P) = N_{C_S(V)}(P)$.\n\nBy Case 1, the map $BC_{\\mathcal{G}}(V) \\to B\\mathcal{G}$, induced by the inclusion $C_{\\mathcal{L}}(V) \\subseteq \\mathcal{L}$, induces a homotopy equivalence $\\mathrm{Map}(BV, BC_{\\mathcal{L}}(V))_{\\mathrm{incl}} \\simeq \\mathrm{Map}(BV, B\\mathcal{G})_{\\mathrm{incl}}$, and hence also a homotopy equivalence\n$$\n\\mathrm{Map}((EP)\/V, BC_{\\mathcal{G}}(V))_{\\mathrm{incl}} \\Right2{\\simeq} \\mathrm{Map}((EP)\/V, B\\mathcal{G})_{\\mathrm{incl}}\n$$\nwhich is $P\/V$-equivariant (where the action of $P\/V$ is the action induced by the original action of $P$ on $EP$). This is still a homotopy equivalence after considering homotopy fixed point sets by \\cite[Remark 10.2]{DW0}, and thus we obtain another homotopy equivalence\n$$\n\\big[\\mathrm{Map}((EP)\/V, BC_{\\mathcal{G}}(V))_{\\mathrm{incl}}\\big]^{h(P\/V)} \\Right2{\\simeq} \\big[\\mathrm{Map}((EP)\/V, B\\mathcal{G})_{\\mathrm{incl}}\\big]^{h(P\/V)}.\n$$\nNotice that $E(P\/V) \\times_{P\/V} (EP)\/V \\simeq BP$ with the given actions (here $P\/V$ is acting diagonally on $E(P\/V) \\times (EP)\/V$). Let $X = BC_{\\mathcal{G}}(V)$ or $B\\mathcal{G}$. By definition of homotopy fixed point sets, we have\n$$\n\\begin{aligned}\n\\big[\\mathrm{Map}((EP)\/V, X)\\big]^{h(P\/V)} & = \\mathrm{Map}_{P\/V}(E(P\/V), \\mathrm{Map}((EP)\/V,X)) \\simeq \\\\\n & \\simeq \\mathrm{Map}_{P\/V}(E(P\/V) \\times (EP)\/V, X),\n\\end{aligned}\n$$\nwhere the rightmost equivalence follows by adjunction. Furthermore, $P\/V$ acts trivially on $X$, and thus it follows that $\\mathrm{Map}_{P\/V}(E(P\/V) \\times (EP)\/V, X) \\simeq \\mathrm{Map}(E(P\/V) \\times_{P\/V} (EP)\/V, X) \\simeq \\mathrm{Map}(BP,X)$. Thus, the equivalence of homotopy fixed point sets above induces the following equivalence\n$$\n\\mathrm{Map}(BP, BC_{\\mathcal{G}}(V))_{\\mathrm{incl}} \\Right2{\\simeq} \\mathrm{Map}(BP, B\\mathcal{G})_{\\mathrm{incl}}.\n$$\nWe can suppose that $\\mathcal{L} = C_{\\mathcal{L}}(V)$, and hence that $V$ is central in $\\mathcal{L}$. Let $\\mathcal{G}\/V$ be the quotient of $\\mathcal{G}$ by $V$, as described in Definition \\ref{quotient1}. In particular, $\\mathcal{F}\/V$ is a saturated fusion system on $S\/V$, and $\\mathcal{L}\/V$ is a transporter system.\n\nConsider the full subcategories $\\mathcal{L}_0 \\subseteq \\mathcal{L}$ and $(\\mathcal{L}\/V)^c \\subseteq \\mathcal{L}\/V$ whose objects are the subgroups $Q\\leq S$, respectively $Q\/V\\leq S\/V$, such that $Q\/V$ is $\\mathcal{F}\/V$-centric. In particular, $(\\mathcal{L}\/V)^c$ determines a centric linking system associated to $\\mathcal{F}\/V$, and there are homotopy equivalences $|\\mathcal{L}_0|^{\\wedge}_p \\simeq B\\mathcal{G}$ and $|(\\mathcal{L}\/V)^c|^{\\wedge}_p \\simeq |\\mathcal{L}\/V|^{\\wedge}_p$.\n\nFinally, let also $\\mathcal{F}' = N_{\\mathcal{F}}(P)$, $\\mathcal{L}' = N_{\\mathcal{L}}(P)$, and define $\\mathcal{L}'\/V$, $\\mathcal{L}_0' \\subseteq \\mathcal{L}'$ in a similar way as done above. It follows by Lemma \\ref{quotient2} that there are fibration sequences\n$$\n\\xymatrix@R=2mm{\nBV \\ar[rr] & & |\\mathcal{L}_0|^{\\wedge}_p \\ar[rr]^{\\Phi} & & |(\\mathcal{L}\/V)^c|^{\\wedge}_p \\\\\nBV \\ar[rr] & & |\\mathcal{L}_0'|^{\\wedge}_p \\ar[rr]^{\\Phi'} & & |(\\mathcal{L}'\/V)^c|^{\\wedge}_p,\\\\\n}\n$$\nand hence also a homotopy pull-back square\n$$\n\\xymatrix{\n\\mathrm{Map}(BP, |\\mathcal{L}_0'|^{\\wedge}_p)_{\\iota} \\ar[rr]^{I_1} \\ar[d]_{\\Phi' \\circ -} & & \\mathrm{Map}(BP, |\\mathcal{L}_0|^{\\wedge}_p)_{\\mathrm{incl}} \\ar[d]^{\\Phi \\circ -} \\\\\n\\mathrm{Map}(BP, |(\\mathcal{L}'\/V)^c|^{\\wedge}_p)_{\\Phi \\circ \\mathrm{incl}} \\ar[rr]_{I_2} & & \\mathrm{Map}(BP, |(\\mathcal{L}\/V)^c|^{\\wedge}_p)_{\\Phi \\circ \\mathrm{incl}},\n}\n$$\nwhere $\\mathrm{Map}(BP, |\\mathcal{L}_0'|^{\\wedge}_p)_{\\iota}$ is the union of the connected components which map to the inclusion in $|\\mathcal{L}_0|^{\\wedge}_p$ and to $\\Phi \\circ \\mathrm{incl}$ in $|(\\mathcal{L}'\/V)^c|^{\\wedge}_p$, and $I_1$, $I_2$ are inclusions.\n\nBy (\\ref{zero}), together with the induction hypothesis, and since $P\/V$ has strictly smaller order than $P$ (recall that $P$ is finite by hypothesis), there are homotopy equivalences\n$$\n\\begin{aligned}\n|C_{(\\mathcal{L}\/V)^c}(P\/V)|^{\\wedge}_p & \\RIGHT6{\\Gamma'_{\\mathcal{L}\/V, P\/V}}{\\simeq} \\mathrm{Map}(B(P\/V), |(\\mathcal{L}\/V)^c|^{\\wedge}_p)_{\\mathrm{incl}} \\Right1{} \\\\\n & \\RIGHT6{- \\circ \\operatorname{proj}\\nolimits}{\\simeq} \\mathrm{Map}(BP, |(\\mathcal{L}\/V)^c|^{\\wedge}_p)_{f \\circ \\mathrm{incl}}\n\\end{aligned}\n$$\nand similarly for maps to $|(\\mathcal{L}'\/V)^c|^{\\wedge}_p$. Since $\\Gamma'_{\\mathcal{L}\/V, P\/V}$ is the composite of $\\Gamma'_{\\mathcal{L}'\/V, P\/V}$ with the inclusion by definition , this shows that the map $I_2$ in the diagram above is a homotopy equivalence, and hence so is $I_1$. In particular, $\\mathrm{Map}(BP, |\\mathcal{L}_0'|^{\\wedge}_p)_{\\iota}$ is connected and contains the component of the inclusion. Thus, Case 3 follows from Case 2 applied to the mapping space $\\mathrm{Map}(BP, |\\mathcal{L}_0'|^{\\wedge}_p)_{\\mathrm{incl}}$.\n\n\\textbf{Case 4.} Suppose that $P$ is an infinite discrete $p$-toral group. By Lemma \\ref{central2}, there is a sequence of subgroups $P_0\\leq P_1\\leq \\ldots$ such that $P = \\bigcup_{n \\geq 0} P_n$, and such that $P_n$ is fully $\\mathcal{F}$-centralized with $C_{\\mathcal{G}}(P_n) = C_{\\mathcal{G}}(P)$ for all $n \\geq 0$. We have a sequence of homotopy equivalences\n$$\n\\begin{aligned}\n\\mathrm{Map}(BP, B\\mathcal{G})_{\\mathrm{incl}} & = \\mathrm{Map}(\\mathrm{hocolim \\,} \\mbox{ } BP_n, B\\mathcal{G})_{\\mathrm{incl}} \\simeq \\\\\n& \\simeq \\operatornamewithlimits{holim} \\mbox{ } \\mathrm{Map}(BP_n, B\\mathcal{G})_{\\mathrm{incl}} \\simeq \\operatornamewithlimits{holim} \\mbox{ } BC_{\\mathcal{G}}(P_n) = BC_{\\mathcal{G}}(P),\n\\end{aligned}\n$$\nwhere the equivalence $\\mathrm{Map}(\\mathrm{hocolim \\,} \\mbox{ } BP_n, B\\mathcal{G})_{\\mathrm{incl}} \\simeq \\operatornamewithlimits{holim} \\mbox{ } \\mathrm{Map}(BP_n, B\\mathcal{G})_{\\mathrm{incl}}$ follows from \\cite[Proposition 2, page 187]{D-F}. This finishes the proof.\n\\end{proof}\n\nThe reader may think of replacing Cases 1 to 3 in the proof above by the following argument. Given a $p$-local compact group $\\mathcal{G} = (S, \\FF, \\LL)$ and a fully $\\mathcal{F}$-centralized finite subgroup $P\\leq S$, consider the centralizer $p$-local compact group $C_{\\mathcal{G}}(P) = (C_S(P), C_{\\mathcal{F}}(P), C_{\\mathcal{L}}(P))$ of $P$ in $\\mathcal{G}$. Set for short $Z = C_S(P)$, $\\mathcal{E} = C_{\\mathcal{F}}(P)$, and $\\mathcal{T} = C_{\\mathcal{L}}(P)$. Given a fine unstable Adams operation (see \\ref{uAo}), we may assume that $\\Psi(P) = P$, since $P$ is a finite subgroup of $S$, and thus $\\Psi$ restricts to a fine unstable Adams operation on $C_{\\mathcal{G}}(P)$.\n\nThis way, $\\Psi$ defines approximations of $\\mathcal{G}$ and $C_{\\mathcal{G}}(P)$ by $p$-local finite groups, namely $\\{(S_i, \\mathcal{F}_i, \\mathcal{L}_i)\\}_{i \\geq 0}$ and $\\{(Z_i, \\mathcal{E}_i, \\mathcal{T}_i)\\}_{i \\geq 0}$ respectively. Moreover, we may assume that $P\\leq S_i$ for all $i \\geq 0$. It is not hard to see that $Z_i = C_{S_i}(P)$ for all $i \\geq 0$, since $Z_i = Z \\cap S_i = C_S(P) \\cap S_i$. The main difficulty of this argument is that it is not clear whether $\\mathcal{E}_i$ corresponds to the centralizer fusion system of $P$ in $\\mathcal{F}_i$ for any $i$. Essentially, the main problem is that the finite retraction pairs for $\\mathcal{G}$ and $C_{\\mathcal{G}}(P)$ given in \\ref{expl1} do not agree with each other in general, and \\ref{expl3} does not apply to this situation, since in general $C_S(P)$ does not have finite index in $S$. One could drop this last condition, but at the price of dealing with a more complicated situation.\n\n\\begin{rmk}\n\nWith the above description of the homotopy type of the mapping spaces $\\operatorname{Map}\\nolimits(BP, B\\mathcal{G})$, one could now generalize the results of C. Broto, N. Castellana, J. Grodal, R. Levi and B. Oliver \\cite{BCGLO1, BCGLO2}. More precisely, \\cite[Theorem A]{BCGLO1} has already been proved for $p$-local compact groups as \\cite[Theorem 4.2]{BLO6}, and \\cite[Theorem B]{BCGLO1} would follow easily now from our result above. Regarding \\cite{BCGLO2}, some results have already been extended to $p$-local compact groups in \\cite[Appendix B]{Gonza2}, and the rest would follow by the same arguments (with some minor modifications). We omit this for the sake of brevity of this paper.\n\n\\end{rmk}\n\n\n\\bibliographystyle{gtart}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nAstronomical observations have been traditionally done with visible light. From~the times of Galileo times until now we have been able to expand the observation range to all the electromagnetic spectrum, from~radio to gamma rays.\nApart from photons, the~discovery of other particles has opened the possibility of using them as cosmic messengers to explore the Universe. For~instance, cosmic rays (CRs) were first discovered in the early 1910s. We know that CRs are ionized nuclei of extraterrestrial origin, and~that they are produced and accelerated in a broad energy range, reaching energies above 10$^{20}$ eV. The~spectrum of the CRs detected at Earth decreases with the energy, following approximately a power law E$^{-n}$ with n$\\sim$3. The~Sun is the main source of CRs below GeV energies. Up~to PeV energies, CR are believed to be dominated by Galactic sources, and~then, at~the highest energies, CRs are likely of extragalactic origin~\\cite{ParticleDataGroup:2020ssz}. As CRs are charged, their directionality is lost due to the Galactic magnetic field. This is probably one of the main reasons why the origin of the most energetic CRs is still unknown.\nWe have also detected neutrinos from extraterrestrial origin. First from the Sun~\\cite{Davis:1968cp}, and~then from a nearby supernova explosion in 1987~\\cite{Hirata:1988ad,IMB:1987klg,Baksan:1987} which can be considered as the birth of neutrino astronomy. By~that time, the first project to construct a neutrino telescope was already ongoing~\\cite{DUMAND:1989dxw}. However, only after decades of research and development, neutrino astronomy had its turning point in 2014 with the discovery of a high-energy cosmic neutrino flux by the IceCube collaboration~\\cite{IceCube:2014stg}.\nThe most recent cosmic messengers discovered are gravitational waves (GWs). The~path to the first GW detection was also not easy, and~it took a century from their theoretical prediction to the first confirmation in 2015 by the LIGO and VIRGO collaborations~\\cite{LIGOScientific:2016aoc}.\n\n\nMultimessenger astronomy originates as a consequence of astrophysical neutrino and GW detection techniques reaching maturity. Multimessenger astronomy is based on the observation of four cosmic messengers, namely, photons, CRs, neutrinos, and~GWs. The~detection in coincidence of all, or~some, of~these messengers allows the study of a source in a similar fashion as it has been done with multiwavelength electromagnetic observations. Moreover, the~fact of involving different types of particles adds extra information coming from interactions involving all fundamental forces of~nature.\n\nWe have divided this review into three sections. In~Section~\\ref{Milestones}, we will discuss the two events that marked the start of the multimessenger era, four years ago. In~Section~\\ref{Results}, we will review the most recent results and what we have learned from them. To~conclude, in~Section~\\ref{Future}, we will discuss the future prospects and challenges in the~field.\n \n\\section{Multimessenger Astronomy~Milestones}\n\\label{Milestones}\nExcluding the solar neutrino detection, the~observation by chance of neutrinos coming from the supernova explosion SN1987A is the first astronomical event producing a multimessenger (photon--neutrino) coincidence. However, there are two main events, both happening in 2017, that really marked the birth of a new field in astrophysics, multimessenger~astronomy.\n\nThe first event was the result of two neutron stars merging into a black hole. This produced a GW (GW170817) that was detected by the LIGO and Virgo~\\cite{LIGOScientific:2017vwq} Scientific Collaborations. Less than 2 seconds after the event, a~short gamma-ray burst (GRB) (GRB 170817A) was detected by the Fermi and INTEGRAL satellites. This coincidence triggered a campaign where several observatories followed-up the event in an unprecedented way~\\cite{LIGOScientific:2017ync}.\nThanks to this coincident detection it was possible to determine the location and the type of sources involved, bringing the first experimental evidence of a kilonova~\\cite{Metzger:2017}, a~type of transient event, theoretically predicted more than two decades ago~\\cite{Lixin:1998}, where nucleosynthesis of the heavy elements is produced.\n\nThe second event (IC-170922A) was triggered by a high-energy neutrino, of~about 300 TeV, detected by the IceCube observatory on 22 September 2017. Again, this event was extensively followed up by other observatories. \nIn this case, observations from the Fermi-LAT satellite were able to point out a blazar (TXS0506+056) in active state which was in spatial and temporal coincidence with the neutrino event. The~event was rejected to be produced by background fluctuations at 3$\\sigma$ level~\\cite{IceCube:2018dnn}.\nAfter this detection, archival analysis of IceCube data prior to the IceCube-170922A event unveiled a potential flare in neutrinos~\\cite{IceCube:2018cha}, between~September 2014 and March 2015, with~3.5$\\sigma$ statistical significance and independent of the 2017 neutrino alert. In~this case, no gamma-ray counterpart was observed.\nAn extensive multiwavelength monitoring of TXS0506+056 started after the coincident event in September 2017 showed a low state emission except for December 1st and 3rd, 2018 with a flare comparable to the one in 2017~\\cite{Satalecka:2021}. However, no neutrino excess was observed.\nIt is also important to mention that TXS0506+056 came out as the second most significant source (2.8$\\sigma$ pre-trial) in the point source search analysis done with the ANTARES neutrino telescope using a pre-selected list of sources~\\cite{Illuminati:2021b}, which makes the case of TXS0506+056~stronger.\n\n\\section{Recent~Results}\n\\label{Results}\nOnce it has been well established that a flux of high-energy neutrinos of cosmic origin exist~\\cite{IceCube:2014stg}, the~next step is to disentangle it and identify which are the sources. The~most recent all-sky searches performed by ANTARES~\\cite{Illuminati:2021b} and IceCube~\\cite{IceCube:2019cia} did not reveal any significant detection above the discovery threshold of 5$\\sigma$, with the excess near the galaxy NGC 1068 observed by IceCube being the most interesting spot with a post-trial significance of 2.9$\\sigma$.\nThis type of high-energy searches benefit from multimessenger astronomy thanks to including the sky coordinates and timing information from potential cosmic messenger counterparts. That was how the first evidence of a cosmic neutrino source, TXS0506+056, was~found. \n\nUnderstanding the multimessenger emission from TXS0506+056 has been challenging from the theoretical point of view, as it is difficult to get a good agreement between the observed neutrino signal and other wavelength observations. \nIf one tries to explain it with a single-zone model, i.e.,~both gammas and neutrinos coming from the same region, one finds out that a leptonic scenario with a radiatively subdominant hadronic component provides the only physically consistent single-zone picture~\\cite{Keivani:2018rnh}.\nA higher neutrino flux would be expected if the source hosts two physically distinct emitting regions (see for instance~\\cite{Xue:2019txw}). However, current observations cannot discriminate between single- or multi-zone emission models.\nRelated to this, a~compelling neutrino--radio correlation~\\cite{Plavin:2020mkf} has been recently discovered. The~authors proposed that neutrinos and gamma rays may be indeed produced in different regions~\\cite{Plavin:2020emb}. If~this is actually the case, X-ray and radio may be better wavelengths when looking for photon--neutrino correlations. \nThe correlation with radio blazars is also supported by ANTARES observations~\\cite{ANTARES:2020zng}.\nIn this regard, ANTARES has recently reported the results from an untriggered search from radio blazars with an interesting association coming from J0242+1101~\\cite{Illuminati:2021}.\n\nThanks to the IceCube alert system~\\cite{IceCube:2016cqr}, which is presently providing on the order of 10 (20) gold (bronze) alerts per year\\endnote{Gold (bronze) alerts are neutrino events with >50\\% (>30\\%) probability of being from astrophysical origin.}, more coincidences between neutrino and blazars have been found lately. For~instance, PKS 1502+106 blazar was coincident with a 300~TeV neutrino~\\cite{Rodrigues:2020fbu}. In~this case, the blazar was in a quiescent state at the time of the neutrino alert. However, no more neutrinos were detected. \nAnother example is 3HSP J095507.9, which was also coincident with a high-energy neutrino detected by IceCube~\\cite{Paliya:2020mqm,Giommi:2020viy}. However, for~this event a lot of sources lay around the best position provided by IceCube, preventing the identification of a potential source candidate. This also underscores that sub-degree angular resolution, achievable by future neutrino observatories, will be key when looking for spatial coincidences.\nFrom Fermi-LAT observations we know that blazars are the most abundant extragalactic gamma-ray sources, constituting roughly 80\\% of the entire extragalactic source population~\\cite{Fermi-LAT:2019pir}. However, current predictions based on stacking catalog searches performed with IceCube data estimate that neutrinos emitted by blazars, in~the range between around 10 TeV and 2 PeV, can only contribute up to 27\\% to the total neutrino diffuse flux~\\cite{IceCube:2016qvd}. \nMore multimessenger observations with next generation experiments are required to test current theoretical models, and~therefore provide valuable information to understand the particle production and acceleration in~blazars.   \n\nApart from blazars other neutrino candidates have been already identified thanks to multimessenger observations. This is the case of Tidal Disruption Events (TDEs), which are the result of a star being ripped apart when passing next to a supermassive black hole. TDEs were already hypothesized as possible neutrino sources, e.g., in~\\cite{Wang:2011}. However, it was not until 2019 when an IceCube neutrino, with~59\\% probability of being of astrophysical origin (IC-191001A), triggered an alert that was followed-up by the Zwicky Transient Facility~\\cite{Bellm:2019}. In~spatial coincidence with the neutrino alert a TDE (AT2019dsg) was observed. Given that TDEs are rare events, the~chance probability of finding this coincident event was estimated to be less than 0.5\\%~\\cite{Stein:2020xhk}.\nAfter the first neutrino-TDE coincidence, more recently another possible association has been detected (IC200530A with AT2019fdr) which has been considered by some scientists as evidence of an emerging trend. The~chance probability of finding this second event in coincidence was also small.\nBoth events have been followed up by ANTARES, however did not produce any significant neutrino {excess}~\\cite{ANTARES:2021jmp}.\nThe~non detection does not contradict the observation by IceCube, as~the sensitivity of ANTARES was above the neutrino flux prediction.\nOn the other hand, there are preliminary indications of an excess in GVD-Baikal data~\\cite{Allakhverdyan:2021}.\nCurrent estimations predict that TDEs contribute at least 2\\% but not more than 40\\% of the total neutrino flux~\\cite{Stein:2021}. Again, more observations are needed to confirm TDEs as sources of high-energy neutrinos and~determine their precise contribution to the diffuse neutrino~flux.\n\nGRBs are another type of sources that have long been predicted as good candidates to emit high-energy neutrinos, see, for instance, in~\\cite{Waxman:1997ti}. However, so far, all the searches have been unsuccessful~\\cite{Albert:2016eyr,IceCube:2016ipa}.\nRecently, some studies have tried to infer what would be the relative contribution of the different neutrino candidate sources, see, e.g., in~\\cite{Bartos:2021tok}. However, it seems that there is no clear indication of a dominant type of source producing cosmic neutrinos. Interestingly enough, the~same study suggest that there is room for unknown~sources.\n\n\\section{Future Prospects and~Challenges}\n\\label{Future}\n\\unskip\n\\subsection{Future Instruments and Instrument~Upgrades}\n\nRegarding neutrino telescopes there are three major projects that will be operating in the near future.\nKM3NeT~\\cite{KM3Net:2016zxf} is a research infrastructure being built in the Mediterranean sea. KM3NeT is composed of two detectors, first ARCA (Astroparticle Research with Cosmics in the Abyss) which is designed to be sensitive to high-energy neutrinos in the TeV-PeV range, and~therefore with astrophysics studies as the main goal. The~second detector is called ORCA (Oscillation Research with Cosmics in the Abyss) and it is sensitive to GeV neutrinos. The~ORCA detector will primarily be used for the study of neutrino properties.\nBoth instruments will use the same technology and detection principle, i.e.,~array of photomultipliers tubes (PMTs) in sea water, being the main difference the volume covered, and~therefore the PMT density.\nARCA is expected to be fully operational in 2027 and ORCA in~2025.\n\nMoreover, in water there is the GVD-Baikal project that is in construction in the Baikal Lake in Russia. GVD-Baikal is currently operational with 2304 optical modules arranged in eight~clusters of eight strings each. In~the present configuration, it has an effective volume of 0.4~km$^{3}$ for cascades with energy above 100 TeV~\\cite{Baikal:2021}. Current plans are to deploy six~additional clusters for the period from 2022 to 2024 which should provide an additional 0.3~km$^{3}$ effective volume.\n\nThe leading project in neutrino telescopes in the last decade has been IceCube~\\cite{IceCube:2021} which is a neutrino telescope installed in the South Pole.\nAfter 10 years of successful operation there are plans for two major upgrades. One, called IceCube-Gen2~\\cite{IceCube-Gen2:2020qha}, expected to be completed by the early 2030s, will significantly increase the IceCube effective volume and energy range sensitivity. The~other, called IceCube-Upgrade~\\cite{IceCube:2019xdf}, represents a fraction of a larger project called Precision IceCube Next Generation Upgrade (PINGU)~\\cite{IceCube:2016xxt}, which aims to increase the sensitivity to lower energies even more than with IceCube-DeepCore, mainly thanks to a higher string density, and~that will mostly study neutrino~properties.\n\nWe can also add to the list of future intended neutrino telescopes the Pacific Ocean Neutrino Experiment (P-ONE)~\\cite{PONE:2021}. The~P-ONE project is presently in research and development phase. The~goal of the collaboration is to install a multi-cubic-kilometer neutrino telescope in the Pacific Ocean, which is expected to be operational in the next~decade.\n\n\\textcolor{black}{Other neutrino experiments, foreseen to be operational by the end of this decade, like Hyper-Kamiokande~\\cite{Hyper-Kamiokande:2018ofw} and DUNE~\\cite{DUNE:2020lwj}, will be sensitive to lower neutrino energies (MeV-GeV). However, they can still contribute to get the whole picture of some astrophysical events. For~instance, the~explosion of a nearby supernova, where low-energy neutrinos are expected to be produced.}\n\nThere are also very exciting plans for next generation experiments aiming to detect other cosmic messengers. Just to mention a few of them, we have the Cherenkov Telescope Array (CTA)~\\cite{CTA:2021} with two planned sites (Northern and Southern Hemisphere), and~LHAASO~\\cite{LHAASO:2021}, fully operational since July 2021, detecting gamma rays. KAGRA~\\cite{KAGRA:2019}, detecting GWs, will join LIGO and Virgo for the next GW data taking run (O4), which is expected to start in late 2022. Finally, detecting ultra-high-energy CRs, there will be AugerPrime~\\cite{PierreAuger:2016qzd}, the~upgrade of the Pierre Auger Observatory.\nSuch a network of observatories, distributed in different locations around the globe (see Figure~\\ref{fig1}), will provide a full multimessenger coverage of the~sky.\n\n\\subsection{Alert Systems and~Strategies}\n\nConsidering the number of experiments currently being under construction and planned, it seems clear that an efficient communication between collaborations is crucial. Moreover, for~the case of pointing instruments, like CTA, this communication also needs to be fast. \n\\textcolor{black}{Neutrinos are actually very good messengers to trigger alerts because they are able to easily escape from sources. These neutrino alerts can give an early warning to other observatories of an incoming event, allowing for a prompt follow up of transient~phenomena.}\n\nThere are already ways to announce, in~real-time, interesting events to the astrophysics community, e.g., the~GammaRay Coordinates Network (GCN)~\\cite{GCN:2021}. In~addition to these announcements, there are also sites, like the Astronomers Telegram (ATEL)~\\cite{ATEL:2021}, where a brief report about recent observations made by the experiments is posted online.\nStill in beta testing phase there is a project called {Astro-COLIBRI}~\\cite{Reichherzer:2021pfe}\nwhose goal is to act as a central platform where a large set of information coming from different experiments is gathered. This can be accessed via web or smartphone interface. The~data will be immediately available and will contain relevant information such as the visibility of the event for a given observatory, the~false alarm rate, or~the probability of the event to be of astrophysical~origin.\n\n\\begin{figure}[H]\n\\includegraphics[width=0.95\\linewidth]{Figures\/MM_WorldMap.pdf}\n\\caption{Earth map indicating the location of a selection of multimessenger observatories that are either currently operating (circles) or planned (triangles). Satellites are depicted outside to the~map.}\n\\label{fig1}\n\\end{figure}   \n\nIn addition to these prompt alert systems across collaborations, there are alert follow-up programs like TaToO~\\cite{ANTARES:2015fce}, where the most promising ANTARES events trigger a prompt optical, radio, and X-ray follow-up on the sky region where the neutrino candidate comes from, looking for any potential transient counterpart within hours, days, and months since the event. This is done using a network of optical telescopes at different locations (at a rate of 25 alerts per year) and, for~the most energetic ones (around 6 alerts per year), the~XRT instrument aboard the Swift satellite and the Murchison Wide field Array radio telescope~\\cite{tatoo:2021}.\n\nAs another example of how data from different experiments is shared, we have the Astrophysical Multimessenger Observatory Network (AMON)~\\cite{AyalaSolares:2019iiy}. One of the main ideas of AMON is to use sub-threshold data from different experiments to exploit the fact that a combined detection is expected to increase the significance of the event. Therefore, an~event that by itself is not enough to claim a detection with a single experiment, and~could have been rejected, can actually become significant when detected in coincidence with other observatories. An~example of a recent analysis done through this network is~\\cite{Hugo:2021}, focused on gamma-ray and neutrino~coincidences.\n\n\\subsection{Future~Challenges}\nConcerning the multimessenger astronomy goals in the next few years, one thing that should be attainable very soon is a firm confirmation of a source of cosmic neutrinos above the discovery threshold of 5$\\sigma$. To~this end, the~selection of the most promising sources, to~reduce the amount of trials in the search, thanks to multimessenger observations will be crucial.\nThis is quite likely to be accomplished by more than one project which will provide an unbiased way of measuring the spectrum of the sources, bringing key information to understand the high-energy neutrino production and acceleration mechanisms in the source. \nAlso combined analyses are possible, as~has been already done in the past~\\cite{ANTARES:2020srt}.\n\nOne multimessenger observation that is most awaited, and~can probably be achieved thanks to the improved sensitivity of the future experiments, is the coincident detection of a GW event with high-energy neutrinos. \nThanks to recent GW observations, we have a better understanding of the link between neutron star mergers and short GRBs, and~the physics involved. We already discussed the particular case of the binary neutron star merger GW170817, which led to the GRB170817A coincidence. This type of event should produce a GW signature together with gamma rays and neutrinos. However, at~present, the~only confirmed coincident detection is the GW-gamma correlation, while no evidence of neutrino emission was found~\\cite{ANTARES:2017bia,Baikal:2019}.\nThe theoretical estimations of neutrino production~\\cite{Kimura:2018vvz} from GW170817 show that the expected flux should be already detectable, with~current neutrino telescopes, under~favorable circumstances, see Figure~\\ref{fig2}.\n\n\\begin{figure}[H]\n\\vspace{-9pt}\n\\includegraphics[width=\\linewidth]{Figures\/GW.pdf}\n\\caption{Fluence upper limits, per flavor, on~the high-energy neutrino emission from experimental data assuming a $\\pm$500s time window around the GW170817 {event}\n~\\cite{ANTARES:2017bia}. Baikal limits are from~\\cite{Baikal:2019}. KM3NeT preliminary sensitivity, computed for an optimal zenith angle, is from~\\cite{Palacios:2021}. Theoretical models for comparison are from~\\cite{Kimura:2018vvz}. Figure adapted from the work in ~\\cite{ANTARES:2017bia}.}\n\\label{fig2}\n\\end{figure}\n\nApart from the detection of high-energy neutrinos, the~detection of the prompt emission in gamma-rays by observatories like HAWC or LHAASO will be essential to understand how these energetic explosions~work.\n\nAnother open question to be addressed by multimessenger observations is the connection between ultra-high-energy CRs, gamma-rays, and~high-energy neutrinos. \nIntensity of gamma-rays, neutrinos, and UHECRs has been shown to be comparable (see \\mbox{Figure~\\ref{fig3}}), suggesting that they may be powered by the same sources. \nAs blazars are the most abundant extragalactic sources, they are by default the most promising candidates. However, blazars do not seem to fit the bill, as they are subdominant in the high-energy neutrino flux~\\cite{Oikonomou:2021}.\n\n\\begin{figure}[H]\n\\includegraphics[width=0.97\\linewidth]{Figures\/DF.pdf}\n\\caption{High-energy fluxes of gamma {rays}~\\cite{Fermi-LAT:2014ryh}, neutrinos~\\cite{IceCube:2020wum}, and~cosmic rays~\\cite{Auger:2017}. Figure adapted from~the work in \\cite{IceCube:2020wum}.}\n\\label{fig3}\n\\end{figure}\n\nApart from the usual suspects (e.g., GRBs and TDEs), the~case for star-forming galaxies as common sources has recently grown in popularity thanks to the work in~\\cite{Roth:2021lvk} where the authors claim that the diffuse gamma-ray flux detected by Fermi-LAT~\\cite{Fermi-LAT:2014ryh} is actually dominated by star-forming galaxies.\nAt the same time, the~data collected in the Pierre Auger Observatory showed an indication of anisotropy at 4.0$\\sigma$ level in the arrival direction of CRs with E > 39 EeV showing an excess from the direction of nearby starburst galaxies~\\cite{PierreAuger:2018qvk}. Claiming that this type of high-rate star-forming galaxies can be the cause of $\\sim$~10\\% of the ultra-high-energy CR flux.\nIt has been also shown that the contribution of the starburst galaxies to the neutrino diffuse flux is sub-dominant and constrained to be at the level of $\\sim$10\\%~\\cite{Lunardini:2019zcf}. Therefore, it remains unanswered for now whether there is a dominant type of source accelerating all cosmic~messengers.\n\n\n\nFinally, if~we restrict the search to our Galaxy, one of the questions that will be solved in the near future thanks to multimessenger observations is what are the Galactic CR sources and how those CR propagate through the Galaxy.\nRegarding this, it has been shown that a sub-PeV diffuse Galactic gamma-ray emission exists~\\cite{TibetASgamma:2021tpz}.\nBased on this result in~\\cite{Fang:2021ylv}, the authors showed that the Galactic neutrino contribution should constitute roughly 5--10$\\%$ of the IceCube diffuse flux and that, in~the 10--100 TeV range, the~expected Galactic neutrino flux should be comparable to the total neutrino diffuse flux. If~so, the~next-generation neutrino telescopes should be sensitive enough to detect it.\nIt is possible that part of the measured gamma-ray diffuse flux comes from individual sources, being the obvious candidates the very-high-energy sources detected by HAWC~\\cite{HAWC:2019tcx} and LHAASO~\\cite{Cao:2021}. \nCombined multimessenger observations should be able to confirm whether this is actually the case.\nIn fact, there have been claims for evidence for such a joint production, of~high-energy neutrinos and gamma rays, in~the Cygnus Coocon region, based on the correlation of a high-energy IceCube neutrino and a high-energy (>300 TeV) photon flare observed by the Carpet\u20132 experiment~\\cite{Carpet-3Group:2021ygp}. \n\n\\section{Outlook}\nMultimessenger astronomy is a new branch of astroparticle physics, for~which neutrinos are expected to play a key role. It took just a few events, detected in coincidence, to~demonstrate that combining different messengers has a great potential for~discoveries.\n\nQuestions like what is the origin of the ultra-high energy cosmic rays are likely to be answered thanks to multimessenger~observations.\n\nConsidering the numerous facilities planned for the near future, or~already taking data (KM3NeT, IceCube-Gen2, GVD-Baikal, P-ONE, CTA, LHAASO, KAGRA, AugerPrime, etc.), multimessenger astronomy has a bright future and is expected to revolutionize our understanding of the Universe in the next~decade. \n\n\n\\vspace{6pt} \n\\authorcontributions{{All authors contributed equally to this article. All authors have read and agreed to the published version of the manuscript.}\n\n\\funding{This work was funded by Generalitat Valenciana, Spain (CIDEGENT\/2018\/034 and CIDEGENT\/2020\/049 grants).}\n\n\\institutionalreview{{Not applicable.}\n\n\\informedconsent{{Not applicable.}\n\n\\dataavailability{{Not applicable.}}\n\n\n\\acknowledgments{The authors thank the Valencia Experimental Group on Astroparticle Physics (VEGA) and the Ministerio de Ciencia, Innovaci\u00f3n, Investigaci\u00f3n y Universidades (MCIU): Programa Estatal de Generaci\u00f3n de Conocimiento (ref. PGC2018-096663-B-C41) (MCIU\/FEDER). }\n\n\\conflictsofinterest{The authors declare no conflict of interest.}\n\n\n\n\\abbreviations{Abbreviations}{\nThe following abbreviations are used in this manuscript:\\\\\n\n\\noindent \n\\begin{tabular}{@{}ll}\nGW & Gravitational Wave\\\\\nCR & Cosmic Ray\\\\\nGRB & Gamma-Ray Burst\\\\\nTDE & Tidal Disruption Event\\\\\nPMT & Photomultipliers tubes\n\n\\end{tabular}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}\nIntroduction\n}\n\nThe antiferromagnetic (AF) Ising model on a triangular lattice is one of the most fundamental models for geometrically frustrated systems. \nWhen the interaction is restricted to the nearest-neighbor (NN) pairs, frustration in each triangle prevents the system from forming a long-range order (LRO) down to zero temperature, and the ground state has extensive degeneracy and associated residual entropy~\\cite{Wannier1950,Houtappel1950,Husimi1950}.\nThe degenerate ground state is extremely sensitive to perturbations.\nFor instance, an infinitesimal second-neighbor interaction lifts the degeneracy and induces a LRO in the ground state; a two-sublattice stripe order [Fig.~\\ref{fig:model}(a)] is selected as the ground state when the additional interaction is AF, while a three-sublattice ferrimagnetic (FR) order [Fig.~\\ref{fig:model}(b)] is selected for the ferromagnetic (FM) interaction.\n\nIn such a degenerate situation, thermal fluctuations also play an interesting role. \nIn general, there is a possibility that a high-entropic state is selected out of the ground state manifold by raising temperature ---this is called the order by disorder~\\cite{Villain1977}. \nFor the AF Ising model, a candidate for such an emergent state is a partially disordered (PD) state. \nThe PD state is peculiar coexistence of magnetically ordered moments and thermally-fluctuating paramagnetic moments.\nSuch possibility was first discussed by the mean-field study in the presence of second-neighbor FM interaction~\\cite{Mekata1977}; the mean-field study predicted that a three-sublattice PD phase with an AF ordering on the honeycomb subnetwork and paramagnetic moments at the remaining sites [Fig.~\\ref{fig:model}(c)] was induced at finite temperature from the degenerate manifold in the limit of vanishing second-neighbor interaction. \nAlthough such PD state was experimentally observed in several Co compounds~\\cite{Kohmoto1998,Niitaka2001} and theoretically shown to present in a stacked triangular lattice model~\\cite{Todoroki2004}, Monte Carlo (MC) simulations in two-dimensional triangular lattice models have indicated that PD is fragile and remains at most as a quasi-LRO; namely, in most cases, the PD state is taken over by another peculiar intermediate state, the Kosterlitz-Thouless (KT) state~\\cite{Wada1982,Fujiki1983,Landau1983,Takayama1983,Fujiki1984,Takagi1995}.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig0v1.eps}\n   \\caption{(Color online).\n   Schematic pictures of (a) stripe order, (b) ferrimagnetic (FR) order, and (c) partial disorder (PD) on a triangular lattice.\n   The arrows show magnetically ordered sites and the open circles are thermally fluctuating paramagnetic sites.\n   }\n   \\label{fig:model}\n\\end{figure}\n\nOn the other hand, recently, the authors have studied Ising-spin Kondo lattice models on a triangular lattice~\\cite{Ishizuka2012} and kagome lattice~\\cite{Ishizuka2012-3} by MC simulation, and showed the presence of PD state in the purely two-dimensional models.\nIn these models, the interplay between localized moments and itinerant electrons plays a crucial role in the following points. \nFirst, the kinetic motion of electrons induces effective interactions known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism~\\cite{Ruderman1954,Kasuya1956,Yosida1957}. \nThe long-ranged and oscillating nature of the interactions drives keen competition between different magnetic states.\nFurthermore, the change of magnetic states affects the electronic state in a self-consistent manner through the spin-charge coupling; the system can gain the energy by forming some particular electronic state associated with magnetic ordering.  \nIn the previous study,  the authors suggested that the PD state is stabilized by the non-perturbative role of itinerant electrons~\\cite{Ishizuka2012}.\n\nIn this contribution, we present our comprehensive numerical results on the magnetic and electronic properties of the Ising-spin Kondo lattice model on a triangular lattice. \nTo further clarify the stabilization mechanism of PD, we analyze the evolution of band structure under the PD type magnetic texture on the basis of a simple mean-field argument. \nThe analysis suggests that the spin-charge coupling can stabilize the PD state by the Slater mechanism.\nBearing this mean-field picture in mind, we present and discuss the results of MC simulation in details. \nWe distinguish the two intermediate-temperature states, PD and KT-like states, from the two-sublattice stripe and three-sublattice FR LRO states, and identify the range of the phases by varying the electron filling and the strength of spin-charge coupling. \nAnalyzing the phase diagram and electronic states in comparison with the mean-field picture, we conclude that the two-dimensional PD state is stabilized through the Slater mechanism.\n\nThe organization of this paper is as follows.\nIn Sec.~\\ref{sec:model_and_method}, we introduce the model and method.\nThe definitions of physical quantities we calculated are also given.\nIn Sec.~\\ref{sec:mft}, we present the mean-field analyses on the band structure in the PD state.\nMC results are presented for magnetic properties in Sec.~\\ref{sec:mc} and for electronic properties in Sec.~\\ref{sec:estruct}.\nSection~\\ref{sec:summary} is devoted to summary.\n\n\\section{\\label{sec:model_and_method}\nModel and Method\n}\n\nIn this section, we introduce the model and method.\nThe model is given in Sec.~\\ref{sec:model} and the MC method is described in Sec.~\\ref{sec:method}.\nIn Sec.~\\ref{sec:pmoment}, we give the definitions of physical quantities that we used to elaborate the phase diagram and thermodynamic properties.\n\n\\subsection{\nModel\n\\label{sec:model}\n}\n\nWe consider a single-band Kondo lattice model on a triangular lattice with localized Ising spin moments.\nThe Hamiltonian is given by\n\\begin{eqnarray}\nH = -t \\! \\sum_{\\langle i,j \\rangle, \\sigma} \\! ( c^\\dagger_{i\\sigma} c_{j\\sigma} + \\text{H.c.} ) + J \\sum_{i}\\sigma_i^z S_i.\n\\label{eq:H}\n\\end{eqnarray}\nThe first term represents hopping of itinerant electrons, where $c_{i\\sigma}$ ($c^\\dagger_{i\\sigma}$) is\nthe annihilation (creation) operator of an itinerant electron with spin $\\sigma= \\uparrow, \\downarrow$ at\n$i$th site, and $t$ is the transfer integral.\nThe sum $\\langle i,j \\rangle$ is taken over nearest-neighbor (NN) sites on the triangular lattice.\nThe second term is the onsite interaction between localized spins and itinerant electrons, where $\\sigma_i^z = c_{i\\uparrow}^\\dagger c_{i\\uparrow} -  c_{i\\downarrow}^\\dagger c_{i\\downarrow}$ represents the $z$-component of itinerant electron spin, and $S_i = \\pm 1$ denotes the localized Ising spin at $i$th site; $J$ is the coupling constant (the sign of $J$ does not matter in the present model). \nHereafter, we take $t=1$ as the unit of energy, the lattice constant $a = 1$, and the Boltzmann constant $k_{\\rm B} = 1$.\n\n\\subsection{\nMonte Carlo simulation\n\\label{sec:method}\n}\n\nTo investigate thermodynamic properties of the model (\\ref{eq:H}), we adopted a MC simulation which is widely used for similar models~\\cite{Yunoki1998}.\nThe model belongs to the class of models in which fermions are coupled to classical fields.\nFor this class of models, the partition function is given by\n\\begin{eqnarray}\nZ={\\rm Tr}_f{\\rm Tr}_c \\exp[\\beta(H-\\mu\\hat{N_e})],\n\\end{eqnarray}\nwhere $\\beta=1\/T$ is the inverse temperature, $\\mu$ is the chemical potential, and $\\hat{N_e}$ is the total number operator for fermions.\nHere, ${\\rm Tr}_f$ is the trace over classical degree of freedom (in the current case, Ising spin configurations), and ${\\rm Tr}_c$ is the trace over itinerant fermions. \nIn the MC simulation, ${\\rm Tr}_f$ is calculated by using the Markov-chain MC sampling. \nMC updates are done by the usual single-spin flip on the basis of the standard METROPOLIS algorithm. \nThe MC weight is calculated by taking the fermion trace ${\\rm Tr}_c$ for each configuration of classical variables in the following form, \n\\begin{eqnarray}\nP(\\{S_i \\}) = \\exp[ -S_{\\rm eff}(\\{S_i \\}) ],\n\\label{eq:P}\n\\end{eqnarray}\nwhere $S_{\\rm eff}$ is the effective action calculated as\n\\begin{eqnarray}\nS_{\\rm eff}(\\{S_i \\}) = - \\sum_\\nu \\log[1 + \\exp\\{-\\beta(E_\\nu(\\{S_i \\})-\\mu)\\}].\n\\label{eq:S_eff}\n\\end{eqnarray}\nHere, $E_\\nu(\\{S_i \\})$ are the energy eigenvalues for the configuration $\\{S_i \\}$, which are readily calculated by the exact diagonalization as it is a one-particle problem in a static potential.\n\nThe calculations were conducted for the system sizes $N=12 \\times 12$, $15 \\times 15$, $12 \\times 18$, and $18 \\times 18$ under the periodic boundary conditions.\nThermal averages of physical quantities were calculated for typically 4300-9800 MC steps after 1700-5000 steps for thermalization. \nThe results are shown in the temperature range where the acceptance ratio is roughly larger than 1\\%.\nWe divide the MC measurements into five bins and estimate the statistical errors by the standard deviations among the bins.\n\n\\subsection{\nPhysical quantities\n\\label{sec:pmoment}\n}\n\nAs we will see later, the model (\\ref{eq:H}) exhibits phase transitions to various magnetic states including different types of three-sublattice orders: ferrimagnetic (FR) state [Fig.~\\ref{fig:model}(b)] and partially disordered (PD) state [Fig.~\\ref{fig:model}(c)].\nThese magnetic states, in principle, are distinguishable by the spin structure factor for the Ising spins,\n\\begin{eqnarray}\nS({\\bf q}) = \\frac{1}{N} \\sum_{i,j} \\langle S_i S_j \\rangle \\exp({\\rm i} {\\bf q}\\cdot{\\bf r}_{ij}),\n\\label{eq:Sq}\n\\end{eqnarray}\nwhere the braket denotes the thermal average in the grand canonical ensemble, and ${\\bf r}_{ij}$ is the position vector from $i$ to $j$th site. \nThe PD order is signaled by peaks of $S({\\bf q})$ at ${\\bf q}=\\pm(2\\pi\/3,-2\\pi\/3)$, while the FR order develops a peak at ${\\bf q}=0$ in addition to ${\\bf q}=\\pm(2\\pi\/3,-2\\pi\/3)$.\nNo Bragg peaks develop in the KT state as it is a quasi-LRO.\nHowever, in finite-size calculations, it is difficult to distinguish these phases solely by the structure factor, as the correlation length in the KT state is divergent and easily exceeds the system size at low temperature.\n\nFor distinguishing the FR, PD, and KT instabilities, it is helpful to use the pseudospin defined for each three-site unit cell:\n\\begin{eqnarray}\n\\tilde{\\bf S}_m = \n\\left(\n\\begin{array}{ccc}\n\\frac2{\\sqrt6} & -\\frac1{\\sqrt6} & -\\frac1{\\sqrt6} \\\\\n0              &  \\frac1{\\sqrt2} & -\\frac1{\\sqrt2} \\\\\n\\frac1{\\sqrt3} &  \\frac1{\\sqrt3} &  \\frac1{\\sqrt3} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nS_i  \\\\\nS_j  \\\\\nS_k  \\\\\n\\end{array}\n\\right),\n\\end{eqnarray}\nand its summation \n\\begin{eqnarray}\n\\tilde{\\bf M} = \\frac{3}{N} \\sum_m  \\tilde{\\bf S}_m \n\\end{eqnarray}\nwhere $m$ is the index for the three-site unit cells, and $(i,j,k)$ denote the three sites in the $m$th unit cell belonging to the sublattices (A,B,C), respectively~\\cite{Takayama1983,Fujiki1984}.\nThen, the three-sublattice PD state [Fig.~\\ref{fig:model}(c)] is characterized by a finite $\\tilde{\\bf M} = (\\tilde{M}_x,\\tilde{M}_y,\\tilde{M}_z)$ parallel to $(\\sqrt{3\/2},1\/\\sqrt2,0)$, $(0,\\sqrt2,0)$, or their threefold symmetric directions around the $z$-axis.\nOn the other hand, the three-sublattice FR state [Fig.~\\ref{fig:model}(b)] is characterized by a finite $\\tilde{\\bf M}$ along $(\\sqrt{2\/3},\\sqrt2,1\/\\sqrt3)$, $(2\\sqrt{2\/3},0,-1\/\\sqrt3)$, or their threefold symmetric directions around the $z$-axis.\nHence, the two states are distinguished by the azimuth of $\\tilde{\\bf M}$ in the $xy$-plane as well as $M_z$.\nIn the MC calculations, we measure \n\\begin{eqnarray}\nM_{xy} &=& \\langle (\\tilde{M}_x^2 + \\tilde{M}_y^2)^{1\/2} \\rangle, \n\\label{eq:Mxy} \\\\\nM_z &=& \\langle |\\tilde{M}_z| \\rangle,\n\\label{eq:Mz}\n\\end{eqnarray}\nand the corresponding susceptibilities,\n\\begin{eqnarray}\n\\chi_{xy} &=& \\frac{N}{T} (\\langle \\tilde{M}_x^2 + \\tilde{M}_y^2 \\rangle - M_{xy}^2 ), \\\\\n\\chi_z &=& \\frac{N}{T} (\\langle \\tilde{M}_z^2 \\rangle - M_z^2 ).\n\\end{eqnarray}\nWe also introduce the azimuth parameter of $\\tilde{{\\bf M}}$ defined by\n\\begin{eqnarray}\n\\psi = {\\cal M}^3 \n\\cos{6 \\phi_M},\n\\label{eq:psi}\n\\end{eqnarray}\nwhere $\\phi_M$ is the azimuth of $\\tilde{\\bf M}$ in the $xy$ plane and ${\\cal M} = \\frac38 M_{xy}^2$.\nThe parameter $\\psi$ has a negative value and $\\psi \\to -\\frac{27}{64}$ for the perfect PD ordering, while it becomes positive and $\\psi \\to 1$ for the perfect FR ordering; $\\psi=0$ for both paramagnetic and KT phases in the thermodynamic limit $N \\to \\infty$.\n\nIn addition, we calculate the spin entropy to distinguish the three-sublattice orderings.\nThe spin entropy per site is defined by\n\\begin{eqnarray}\n{\\cal S}(T) = -\\frac{1}{N} \\sum_{\\{S_i\\}} P(\\{S_i\\})\\log P(\\{S_i\\}),\n\\label{eq:Sdef}\n\\end{eqnarray}\nwhere $P(\\{S_i\\})$ is the probability for spin configuration $\\{S_i\\}$ to be realized, given in Eq.~(\\ref{eq:P}).\nIn the actual MC calculation, instead of directly calculating Eq.~(\\ref{eq:Sdef}), ${\\cal S}$ is evaluated by calculating its temperature derivative\n\\begin{eqnarray}\n\\frac{\\partial{\\cal S}(T)}{\\partial T} = \\frac{1}{NT^2} \\left\\{ \\langle S_{\\rm eff} H \\rangle - \\langle S_{\\rm eff}\\rangle \\langle H \\rangle \\right\\},\n\\label{eq:delSdelT}\n\\end{eqnarray}\nand integrating it as \n\\begin{equation}\n{\\cal S}(T) = \\int_0^T \\frac{\\partial{\\cal S}(T)}{\\partial T} dT = \\log 2 - \\int_T^\\infty \\frac{\\partial{\\cal S}(T)}{\\partial T} dT. \n\\label{eq:Sint}\n\\end{equation}\nIn Eq.~(\\ref{eq:delSdelT}), $S_{\\rm eff}$ is the effective action in Eq.~(\\ref{eq:S_eff}). \nIn the following calculations, we set the cutoff $T=1$ for the upper limit of the last integral in Eq.~(\\ref{eq:Sint}).\n\nOn the other hand, in order to identify the two-sublattice stripe order [Fig.~\\ref{fig:model}(a)], we calculate the order parameter\n\\begin{eqnarray}\nM_{{\\rm str}} = \\left[ \\sum_{{\\bf q}^{*}_{\\rm str}} \\left\\{ \\frac{S({\\bf q}^{*}_{\\rm str})}{N} \\right\\}^2 \\right]^{1\/2},\n\\label{eq:Mstr}\n\\end{eqnarray}\nand its susceptibility $\\chi_{\\rm str}$. \nHere, the sum is taken for the characteristic wave vectors of the stripe orders running in three different directions, ${\\bf q}^{*}_{\\rm str}= (\\pi,0)$ and $(\\pm\\frac12\\pi,\\frac{\\sqrt3}2\\pi)$.\n\nWe also examine the thermodynamic behavior of electronic states for itinerant electrons.\nThere, we computed the charge modulation defined by\n\\begin{eqnarray}\nn_{\\rm CO} = \\left\\{\\frac{N({\\bf q}^*_{\\rm CO})}{N}\\right\\}^{1\/2}\n\\label{eq:n_CO}\n\\end{eqnarray}\nat ${\\bf q}^*_{\\rm CO}=(-2\\pi\/3,2\\pi\/\\sqrt3)$, which corresponds to the wave numbers for the three-sublattice orders.\nHere, $N({\\bf q})$ is the charge structure factor for itinerant electrons,\n\\begin{eqnarray}\nN({\\bf q}) = \\frac{1}{N} \\sum_{i,j} \\langle n_i n_j \\rangle \\exp({\\rm i} {\\bf q}\\cdot{\\bf r}_{ij}),\n\\end{eqnarray}\nwhere $n_i = \\frac12\\sum_\\sigma c_{i\\sigma}^\\dagger c_{i\\sigma}$.\n\n\\section{\nMean-field band structure\n\\label{sec:mft}\n}\n\nBefore going to the MC results, we here discuss how one particle band structure is modulated by PD ordering in a mean-field picture. \nWe consider a three-sublattice LRO state, in which the localized spins give a mean-field local magnetic field to itinerant electrons.\nNamely, we consider a mean-field Hamiltonian given by\n\\begin{eqnarray}\n{\\cal H}^{\\rm MF} = \\sum_{\\bf k}\n\\begin{pmatrix}\n\\Delta_{\\mathrm{A}}\\sigma^z_\\alpha & \\tau_{\\bf k} & \\tau_{\\bf k}^\\ast \\\\\n \\tau_{\\bf k}^\\ast & \\Delta_{\\mathrm{B}}\\sigma^z_\\alpha & \\tau_{\\bf k} \\\\\n \\tau_{\\bf k} & \\tau_{\\bf k}^\\ast & \\Delta_{\\mathrm{C}}\\sigma^z_\\alpha\n\\end{pmatrix}\\label{eq:mfh}\n.\n\\end{eqnarray}\nHere, three rows correspond to the different sublattices A, B, and C in the three-site unit cell; $\\Delta_\\alpha$ is a mean field given by $J\\langle S_\\alpha\\rangle$ ($\\alpha=\\mathrm{A}, \\mathrm{B}, \\mathrm{C}$).\nThe sum is taken in the first Brillouin zone for the magnetic unit cell for three-sublattice order.\n$\\tau_{\\bf k}$ is the hopping term for itinerant electrons given by\n\\begin{eqnarray}\n\\tau_{\\bf k} = -t[e^{{\\rm i}k_x} + e^{{\\rm i}\\left(-\\frac{k_x}2 + \\frac{\\sqrt3}2 k_y\\right)} + e^{{\\rm i}\\left(-\\frac{k_x}2 - \\frac{\\sqrt3}2 k_y\\right)}]\n\\end{eqnarray}\nand $\\sigma^z_\\alpha$ corresponds to the $z$ component of itinerant electron spin in each sublattice $\\alpha$.\n\nThe band structure for a FR order, $(\\Delta_{\\rm A},\\Delta_{\\rm B},\\Delta_{\\rm C})=( \\Delta, \\Delta, -\\Delta)$, was recently studied by the authors~\\cite{Ishizuka2012-2}.\nThere, it was reported that the electronic structure in the FR order is semimetallic with forming Dirac nodes at the electron filling $n= \\frac{1}{2N}\\sum_{i\\sigma} \\langle c_{i\\sigma}^\\dagger c_{i\\sigma} \\rangle=1\/3$ for $J>t$.\n\n\\begin{figure}\n   \\includegraphics[width=0.92\\linewidth]{fig11v1.eps}\n   \\caption{(Color online).\n   Mean-field band structure calculated by Eq.~(\\ref{eq:mfh}) for the local magnetic field of PD type, $(\\Delta_{{\\rm A}},\\Delta_{{\\rm B}},\\Delta_{{\\rm C}})=(2,0,-2)$.\n   Each of the three bands shown is doubly degenerate, and there are totally six bands.\n   The gray hexagon on the basal plane shows the first Brillouin zone for the magnetic supercell.\n   }\n   \\label{fig:band1}\n\\end{figure}\n\nHere, we discuss the band structure for the PD case, $(\\Delta_{\\rm A},\\Delta_{\\rm B},\\Delta_{\\rm C})=(\\Delta,0,-\\Delta)$.\nThe band structure for $\\Delta=2$ is shown in Fig.~\\ref{fig:band1}.\nIn this case, all three bands shown in the figure are doubly degenerate and there are six bands in total.\nThe first Brillouin zone is shown by the gray shade in the bottom surface. \nThe result shows the presence of an energy gap at the Fermi level corresponding to $n=1\/3$, that opens between the lowest energy band and the middle band [see also Fig.~\\ref{fig:band2}(c)].\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig12v3.eps}\n   \\caption{(Color online).\n   Mean-field band structure along the symmetric lines in the local magnetic field of PD type, $(\\Delta_{\\rm A},\\Delta_{\\rm B},\\Delta_{\\rm C})=(\\Delta,0,-\\Delta)$: (a) $\\Delta=1\/3$, (b) $\\Delta=2\/3$, and (c) $\\Delta=2$.\n   The dashed horizontal lines indicate the Fermi level for $n=1\/3$.\n   }\n   \\label{fig:band2}\n\\end{figure}\n\nWe next look into the conditions for the energy gap formation in the mean-field PD band.\nFigure~\\ref{fig:band2} shows the results of band structure while varying $\\Delta$.\nThe results are plotted along the symmetric line in the Brillouin zone shown in the bottom surface in Fig.~\\ref{fig:band1}.\nFor small $\\Delta$, the system is metallic at $n=1\/3$, as shown in the case of $\\Delta=1\/3$ in Fig.~\\ref{fig:band2}(a); both electron and hole pockets are present at the Fermi level.\nThe pockets shrink as increasing $\\Delta$, and disappear at the same time at $\\Delta=2\/3$, as shown in Fig.~\\ref{fig:band2}(b).\nFor larger $\\Delta$, an energy gap opens between the lowest and middle bands, corresponding to $n=1\/3$, as stated above [Fig.~\\ref{fig:band2}(c)]. \nHence, $\\Delta_c=2\/3$ is the critical point for the metal-insulator transition in this mean-field PD state.\n\n\\begin{figure}\n   \\includegraphics[width=0.9\\linewidth]{fig13v2.eps}\n   \\caption{(Color online).\n   $\\Delta$ dependences of the mean-field energy gap and associated charge modulation $n_{\\rm CO}$ at $n=1\/3$.\n   }\n   \\label{fig:gap}\n\\end{figure}\n\nFigure~\\ref{fig:gap} shows $\\Delta$ depedences of the energy gap and associated charge modulation $n_{\\rm CO}$ [Eq.~(\\ref{eq:n_CO})] at $n=1\/3$.\nThe charge gap develops for $\\Delta > 2\/3$ and monotonically increases, approaching asymptotically a $\\Delta$-linear form as $\\Delta \\gg t$.\nThe charge modulation is induced by the inhomogeneity of local potential; the local charge density at B sites (the site corresponds to paramagnetic sites) becomes dilute compared to those at A and C sites (the magnetically ordered sites).\nIn the limit of $\\Delta \\gg t$, $n_{\\rm CO}$ approaches $n_{\\rm CO}=1\/\\sqrt{12}\\sim 0.289$.\n\nThe results above suggest a stabilization mechanism of PD which is absent in the localized spin only model.\nIn the previous studies on the Ising spin models~\\cite{Takayama1983,Fujiki1983,Wada1982} and an equivalent classical particle model~\\cite{Landau1983} on a triangular lattice, PD was shown to be unstable against thermal fluctuations and taken over by a KT state.\nIn the case of our model, however, as the KT state lacks a long-range periodic magnetic structure, it is expected that the KT state does not open an energy gap in the electronic state of itinerant electrons.\nTherefore, in contrast to the case of localized spin only models, there is a chance for the current model to stabilize the PD state by the Slater mechanism, that is, by forming an energy gap at the Fermi level with folding the Brillouin zone under a periodic magnetic order.\n\nIn addition, the formation of an energy gap for $\\Delta > 2\/3$ implies that, if the PD state is stabilized by the Slater mechanism, it should appear from a finite $J$, and not remain stable down to $J\\to0$.\nThis is in sharp contrast to magnetic ordering by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction~\\cite{Ruderman1954,Kasuya1956,Yosida1957}; as the RKKY interaction is given by the second-order perturbation in terms of $J\/t$, if the PD state is stabilized by the RKKY interaction, it should appear for an infinitesimal $J$.\nHence, the phase diagram in the small $J$ region gives an idea on how the PD state is stabilized.\nWe will discuss this point by showing the MC results while changing $J$ in the next section.\n\n\n\\section{\nMonte Carlo simulation\n\\label{sec:mc}\n}\nIn this section, we present the results of MC simulation introduced in Sec.~\\ref{sec:method}.\nWe first show the finite-temperature phase diagrams in Sec.~\\ref{sec:pdiag}, which include four magnetic phases: stripe, PD, FR, and KT-like states.\nThe details of numerical data for the PD state are elaborated in Sec.~\\ref{sec:pd}.\nThe results for stripe, KT-like, and FR states are discussed  in Sec.~\\ref{sec:stripe_ferri}.\n\n\\subsection{\nPhase diagrams\n\\label{sec:pdiag}\n}\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig2v3.eps}\n   \\caption{(Color online).\n   Phase diagrams of the model~(\\ref{eq:H}) while varying $n$ at (a) $J=1$ and (b) $J=2$.\n   The symbols show phase boundaries for the four phases: stripe, partially disordered (PD), KT-like (``KT\"), and ferrimagnetic (FR) phases.\n   PS represents a phase separation. The lines are guides for the eyes.\n   The strips at $T=0$ show the ground states obtained by comparing the energy of stripe and FR states.\n   }\n   \\label{fig:ndiag}\n\\end{figure}\n\nFigure~\\ref{fig:ndiag}(a) shows the phase diagram around the electron filling $n= 1\/3$ at $J=1$ obtained by MC calculations. \nThere are four dominant ordered phases ---stripe, FR, PD, and KT-like phases, in addition to an electronic phase separation (PS).\nThe strip at the bottom of the figure shows the ground state obtained by variational calculation comparing the ground state energy of the stripe and FR states (the details of variational calculation is given in Appendix~\\ref{sec:pseparation}).\nFor the relatively low filling of $n \\lesssim 0.29$, the stripe order with period two [Fig.~\\ref{fig:model}(a)] develops in the low temperature region.\nOn the other hand, for the higher filling of $n \\gtrsim 0.32$, the system exhibits the three-sublattice FR order at low temperature [Fig.~\\ref{fig:model}(b)]. \nMC data for the stripe and FR orders will be discussed in Sec.~\\ref{sec:stripe_ferri}.\nIn addition to these two states, the numerical results show two intermediate-temperature states depending on the electron filling $n$.\nFor $0.29 \\lesssim n \\lesssim 0.34$, we identify the intermediate phase as the three-sublattice PD state [Fig.~\\ref{fig:model}(c)]. \nThe details will be discussed in Sec.~\\ref{sec:pd}.\nMeanwhile, for $n \\gtrsim 0.34$, we find KT-like behavior similar to the one discussed in the Ising models~\\cite{Takayama1983,Fujiki1984,Wada1982,Fujiki1983,Landau1983}, as presented in Sec.~\\ref{sec:stripe_ferri}.\nIn these intermediate-temperature phases, the numerical data indicate a LRO for PD but a quasi-LRO in the KT-like region.\n\nA similar phase diagram is obtained at $J=2$, as shown in Fig.~\\ref{fig:ndiag}(b).\nIn this case also, the PD phase emerges in the intermediate-temperature region.\nHowever, in contrast to the case with $J=1$ where PD is found widely above the FR state as well as PS, the PD phase dominantly appears above the PS region between the stripe and FR states.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig3v4.eps}\n   \\caption{(Color online).\n   Phase diagram of the model~(\\ref{eq:H}) at $n=1\/3$ while varying $J$.\n   The notations are common to those in Fig.~\\ref{fig:ndiag}.\n   The boundary between PD and PS is difficult to determine by MC calculations, and supposed to be located at lower temperature than indicated by the gray arrows.\n   }\n   \\label{fig:jdiag}\n\\end{figure}\n\nWe also investigated the phase diagram of the model in Eq.~(\\ref{eq:H}) while varying $J$.\nFigure~\\ref{fig:jdiag} shows the numerically obtained phase diagram at $n=1\/3$.\nThe result shows that the PD state is stable in a wide range of $0.8 \\lesssim J \\lesssim 5.6$.\nThe transition temperature first rapidly increases as increasing $J$, while it turns to a gradual decrease after showing a peak at $J\\sim 2$.\n\nAn important observation in this constant-$n$ phase diagram is that the PD state does not survive down to $J \\to 0$, and it is taken over by the KT-like and FR phases in the small $J$ region.\nThe absence of PD state in the $J\\rightarrow 0$ limit implies that the RKKY interaction in the second-order perturbation theory is insufficient in stabilizing the PD state.\nMoreover, the emergence of PD for $J > J_c \\ne 0$ is consistently understood within the Slater mechanism discussed in Sec.~\\ref{sec:mft}; the MC result of $J_c\\sim 0.8$ is in good accordance with the mean-field argument of the critical value $\\Delta_c = 2\/3$.\nThe result clearly indicates that a non-perturbative effect of itinerant electrons plays a crucial role in stabilizing the PD state.\n\nIn the PD region in Fig.~\\ref{fig:jdiag}, our MC data do not show clear sign of further transition while decreasing temperature before the MC calculations become unstable.\nIn the low temperature region, however, it becomes difficult to determine the chemical potential $\\mu$ for $n=1\/3$.\nThe lowest temperature of MC calculations are shown in the phase diagram by the gray downward arrows. \nOn the other hand, the analysis of the ground state indicates that the ground state for $J \\lesssim 1.68$ is the FR state, while the region for $J \\gtrsim 1.68$ is PS between the stripe and FR states.\nIn addition, we observe the PS instability by carefully investigating the change of $n$ as a function of $\\mu$ at $J=5.4$ (see also Appendix~\\ref{sec:pseparation}). \nFrom these facts, we conclude that the PD for $J \\gtrsim 1.68$ is taken over by PS between the stripe and FR states.\nSince it is tedious to determine the PS boundary from $\\mu$-$n$ plot for all the values of $J$, we merely plot the lowest temperature we reached in our constant-$n$ calculations as the upper limit of temperature for the PS instability.\n\n\\subsection{\nPartial disorder\n\\label{sec:pd}\n}\n\n\\begin{figure*}\n   \\includegraphics[width=\\linewidth]{fig42v6.eps}\n   \\caption{(Color online).\n   MC results for (a1)-(c1) $M_{xy}$, $M_z$, and $\\psi$, (a2)-(c2) $\\chi_{xy}$ and $\\chi_z$, and (a3)-(c3) $\\cal{S}$ and its temperature derivative $\\partial {\\cal S}\/\\partial T$ at $n=1\/3$;\n   (a1)-(a3) $J=1$, (b1)-(b3) $J=2$, and (c1)-(c3) $J=4$.\n   The calculations were done for the system sizes $N=12\\times 12$, $12\\times 18$, and $18\\times18$.\n   $\\cal S$ is calculated from numerical integration of $\\partial {\\cal S}\/\\partial T$ by assuming ${\\cal S}(T=1)=\\log2$.\n   }\n   \\label{fig:mcpd}\n\\end{figure*}\n\nHere, we present the details of MC data for identifying the PD state.\nFigure~\\ref{fig:mcpd} shows $T$ dependences of MC results for different $J$ at $n=1\/3$.\nTo fix $n$, we tuned $\\mu$ for each temperature; the errors for $n$ at each temperature are controlled within 0.001.\nFigure~\\ref{fig:mcpd}(a1) is the result for the pseudomoments $M_{xy}$ and $M_z$ at $J=1$ [see the definitions in Eqs.~(\\ref{eq:Mxy}) and (\\ref{eq:Mz}), respectively].\n$M_{xy}$ shows two anomalies while decreasing temperature at $T_c^{\\rm (PD)} = 0.086(4)$ and $T_c^{\\rm (FR)} = 0.019(2)$.\nThe critical temperatures are determined by the peaks of the susceptibilities, $\\chi_{xy}$, and $\\chi_z$, as mentioned below.\nAt $T_c^{\\rm (PD)}$, $M_{xy}$ rapidly increases and approaches $\\sqrt{2}$ at lower temperature. \nIn addition, it shows a kink at $T_c^{\\rm (FR)}$ and further increase to $8\/3$ at lower temperature.\nMeanwhile, $M_z$ shows no anomaly at $T_c^{\\rm (PD)}$, while it shows a rapid increase to $1\/\\sqrt3$ at $T_c^{\\rm (FR)}$.\nCorrespondingly, $\\chi_{xy}$ and $\\chi_z$ in Fig.~\\ref{fig:mcpd}(a2) also show divergent peaks increasing with the system size;\npeaks of $\\chi_{xy}$ appear at both $T_c^{\\rm (PD)}$ and $T_c^{\\rm (FR)}$, while $\\chi_z$ shows a peak only at $T_c^{\\rm (FR)}$.\nThese results signal the presence of two successive phase transitions at $T_c^{\\rm (PD)} = 0.086(4)$ and $T_c^{\\rm (FR)} = 0.019(2)$.\nThe error bars are estimated by the range of temperature where the standard deviation of the MC data exceeds the difference of expectation value from the peak value.\nThe transition temperatures and error bars shown in Figs.~\\ref{fig:ndiag} and \\ref{fig:jdiag} are given by this criterion.\nMeanwhile, most of the calculations in Fig.~\\ref{fig:ndiag} were done by fixing $\\mu$ instead of $n$.\nHence, we also give the error bars in terms of $n$, as $n$ changes with $T$ in a fixed $\\mu$ calculation.\n\nTo determine the nature of low temperature phases at $n=1\/3$, we also computed the azimuth parameter $\\psi$ [Eq.~(\\ref{eq:psi})] shown in Fig.~\\ref{fig:mcpd}(a1).\nWhile increasing the system sizes, $\\psi$ apparently deviates from zero to a negative value below $T_c^{\\rm (PD)}$, indicating that the intermediate phase for $T_c^{\\rm (FR)} < T < T_c^{\\rm (PD)}$ has a PD type order.\nOn the other hand, $\\psi$ shows a sign change at $T_c^{\\rm (FR)}$, and rapidly increases to $\\psi=1$ at lower temperature.\nThis is a signature of the FR transition, which will be discussed in detail in Sec.~\\ref{sec:stripe_ferri}.\n\nThe emergence of PD is also seen in the results for the spin entropy $\\cal{S}$ and its temperature derivative [Eqs.~(\\ref{eq:Sint}) and (\\ref{eq:delSdelT}), respectively], as shown in Fig.~\\ref{fig:mcpd}(a3).\nIn the intermediate-temperature region for $T_c^{\\rm (FR)} < T < T_c^{\\rm (PD)}$, $\\cal{S}$ appears to approach $\\frac13\\log 2$ as decreasing temperature, which is the value expected for the ideal PD state where one out of three spins in the magnetic unit cell remains paramagnetic.\nThe remaining entropy is released rapidly at $T_c^{\\rm (FR)}$ and ${\\cal S} \\to 0$ at lower temperature due to the ordering of paramagnetic spins in the FR state.\n\nSimilar phase transitions to the PD state are observed in the wide range of $J$, as shown in Figs.~\\ref{fig:mcpd}(b) and \\ref{fig:mcpd}(c) at $J=2$ and $J=4$, respectively.\nIn these results, however, we could not confirm the presence of another phase transition at a lower temperature in the range of temperature we calculated, in contrast to the FR transition found in the case of $J=1$.\nAs the PD state retains a finite $\\cal{S}$, it is unlikely that this phase survives to $T\\rightarrow0$.\nHence, it is presumably taken over by other ordered phases or PS at a lower temperature. \nAs shown in Fig.~\\ref{fig:jdiag}, the ground state is deduced to be PS for the values of $J$ in Figs.~\\ref{fig:mcpd}(b)  and \\ref{fig:mcpd}(c).\nWe, therefore, expect that the PD state is taken over by PS below $T=0.02$ for $J \\gtrsim 2$. \nThe situation is indicated by the gray arrows in the phase diagram in Fig.~\\ref{fig:jdiag}, as discussed in Sec.~\\ref{sec:pdiag}.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig6v1.eps}\n   \\caption{(Color online).\n   MC results for $S({\\bf q})$ along the ${\\bf q}=(q_x,0)$ line at $T=0.02$.\n   The calculations were done for the system size $N=18\\times18$.\n   }\n   \\label{fig:sq}\n\\end{figure}\n\nAnother point to be noted is the systematic change in $\\cal{S}$ in the PD state by changing $J$.\nWhile the result at $J=1$ appears to show plateau like behavior at ${\\cal S}\\sim \\frac13 \\log2$, the plateau value of ${\\cal S}$ in the PD state decreases while increasing $J$, as shown in Figs.~\\ref{fig:mcpd}(a3), \\ref{fig:mcpd}(b3), and \\ref{fig:mcpd}(c3).\nThe decrease in ${\\cal S}$ is presumably attributed to the development of spatial correlations between paramagnetic sites in the PD state; \nthe ideal value ${\\cal S} = \\frac13 \\log2$ is for completely uncorrelated paramagnetic spins, and correlations between them reduces the entropy.\nSuch development of correlatins are observed in the spin structure factor $S({\\bf q})$ defined in Eq.~(\\ref{eq:Sq}).\nFigure~\\ref{fig:sq} shows a profile of $S({\\bf q})$ calculated by MC simulation at $T=0.02$.\nThe peaks at ${\\bf q}=(4\\pi\/3,0)$ and $(8\\pi\/3,0)$ indicates that the system is in a three-sublattice ordered phase, while the absence of a sharp peak at ${\\bf q}=(0,0)$ indicates that there is no net magnetic moment; the result is consistent with PD order.\nWhen comparing the results at $J=2$ and $J=4$, the peak corresponding to the three-sublattice order gets sharper for $J=4$, while the height of the peak of $S({\\bf q})$ is almost the same. \nThis indicates that the PD order at $J=2$ shows more spin fluctuations than that at $J=4$, consistent with the trend of the plateau value of ${\\cal S}$.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig_p1v4.eps}\n   \\caption{(Color online).\n   MC results for $\\psi$ while varying $n$ at $T=0.08$ and $J=2$.\n   The calculations were done for the system sizes $N=12\\times 12$, $12\\times 18$, and $18\\times18$.\n   }\n   \\label{fig:conT}\n\\end{figure}\n\nThus far, we showed the results at $n=1\/3$.\nNext, we show how the PD evolves while changing $n$.\nFigure~\\ref{fig:conT} shows the MC result of $\\psi$ as a function of $n$ at $T=0.08$ and $J=2$.\n$\\psi$ becomes negative around $n=1\/3$ and takes the lowest value at $n\\simeq1\/3$. \nThe data indicate that $\\psi$ is almost system size independent or rather slightly decreases as the system size increases in the finite range of $n$ around $n=1\/3$. \nHence, the PD state is stabilized not only at $n=1\/3$ but for a finite range of $0.31 \\lesssim n \\lesssim 0.34$ in the thermodynamic limit. \nThe range well agrees with that for the PD phase estimated from the peak of susceptibilities shown in Fig.~\\ref{fig:ndiag}(b).\n\nWith regard to the order of the PD transition, the PD transition in our MC results appears to be continuous, as shown in Fig.~\\ref{fig:mcpd}. \nHowever, it needs careful consideration, as we will discuss here.\nIt is known that the Ising model on a triangular lattice with AF NN interactions is effectively described by a six-state model, in which the low-energy states with three up-up-down and three up-down-down configurations in the three-site unit cell are described by six-state variables. \nThe PD state in our model also retains six low-energy states with different up-down-paramagnetic configurations, and hence, the transition to PD is expected to be classified in the framework of six-state models.\nHowever, from the argument of duality properties, it is prohibited that the six-state models exhibit a single second-order transition for changing temperature~\\cite{Cardy1980}.\nFor instance, a two-dimensional six-state clock model shows two KT transitions at finite temperature, without exhibiting true LRO for $T\\ne0$~\\cite{Jose1977,Challa1986}.\nOn the other hand, a six-state Potts model shows a weak first order transition to LRO, in which the correlation length reaches the order of 1000 sites at the critical point~\\cite{Buffenoir1993}.\nIn our PD case, the apparently second-order transition at $T_c^{\\rm (PD)}$ is not expected to be a single one, but is always followed by another transition to FR or PS at a lower temperature. \nThis appears not to violate the general argument for the six-state models, although it is not clear to what extent the argument applies, as the electronic PS never takes place in the localized spin models.\nHence, the PD transition can be of second order, as indicated in our numerical results. \nOf course, we cannot exclude the possibility of a weak first order transition, similar to that of the Potts model.\nIn this case, due to a long correlation length at the critical temperature, the system sizes used in our calculations are likely to be insufficient to distinguish the first order transition from second order one.\n\n\\subsection{\nOther magnetic orders\n\\label{sec:stripe_ferri}\n}\n\n\\begin{figure}\n   \\includegraphics[width=0.9\\linewidth]{fig41v5.eps}\n   \\caption{(Color online).\n   MC results for (a) $M_{\\rm str}$ and (b) its susceptibility $\\chi_{\\rm str}$ at $J=2$ and $n=0.27$.\n   The inset in (b) shows $T_c^{\\rm (str)}$ for different sizes and the solid line is the extrapolation which gives $T_c^{\\rm (str)}=0.051(13)$.\n   The calculations were done for the system sizes $N=12\\times 12$, $14\\times 14$, $12\\times 18$, $16\\times 16$, and $18\\times18$.\n   }\n   \\label{fig:mcstripe}\n\\end{figure}\n\nFigure~\\ref{fig:mcstripe} presents the results for the relatively low filling where the stripe order is stabilized at low temperature.\nFigure~\\ref{fig:mcstripe}(a) shows the order parameter for the stripe order, $M_{\\rm str}$ [Eq.~(\\ref{eq:Mstr})], and Fig.~\\ref{fig:mcstripe}(b) shows the corresponding susceptibility $\\chi_{\\rm str}$ at $J=2$ and $n=0.27$.\nA phase transition to the stripe phase is signaled by a rapid increase of $M_{{\\rm str}}$ and corresponding peak of $\\chi_{\\rm str}$;\nwe determine the transition temperature $T_c^{\\rm (str)}$ by the peak temperature of $\\chi_{\\rm str}$ for each system size, and plot them in the phase diagram in Fig.~\\ref{fig:ndiag}(a). \nThe error bars are estimated in a similar manner to the case of $T^{\\rm (PD)}_c$ and $T^{\\rm (FR)}_c$.\nWe also show the system-size extrapolation of $T_c^{\\rm (str)}$ in the inset of Fig.~\\ref{fig:mcstripe}(b).\nAlthough the data are rather scattered, we fit them by $f(N) = a + b\/N^c$ with fitting parameters $a$, $b$, and $c$. \nThe extrapolation clearly shows that the phase transition takes place at a finite temperature, as expected for the two-dimensional Ising order.\n\nThe stripe ordered phase is a peculiar magnetic state, in which the sixfold rotational symmetry of the lattice is spontaneously broken and reduced to twofold.\nDue to the symmetry breaking, the transport property is expected to show strong spatial anisotropy; e.g., the longitudinal conductivity will be large in the direction along the stripes, while suppressed in the perpendicular direction.\nThis is an interesting topic on the control of transport by magnetism and vice versa.\n\n\\begin{figure}\n   \\includegraphics[width=0.76\\linewidth]{fig43v2.eps}\n   \\caption{(Color online).\n   MC results for (a) $M_{xy}$, $M_z$, and $\\psi$, (b) $\\chi_{xy}$ and $\\chi_z$, and (c) $\\cal{S}$ and its temperature derivative $\\partial {\\cal S}\/\\partial T$ at $n=0.38$ and $J=2$.\n   The calculations were done for the system sizes $N=12\\times 12$, $12\\times 18$, and $18\\times18$.\n   }\n   \\label{fig:mcferri}\n\\end{figure}\n\nFigure~\\ref{fig:mcferri} shows the results for the relatively high filling where the low temperature phase is FR, at $n=0.38$ and $J=2$.\nThe data indicate two successive transitions signaled by the peaks in $\\chi_{xy}$ and $\\chi_{z}$ at different temperature.\nThe peak of $\\chi_z$ corresponding to the increase of $M_z$ signals the phase transition to the FR phase at $T_c^{\\rm (FR)} = 0.098(4)$.\nAt the same time, $\\psi$ becomes finite below $T_c^{\\rm (FR)}$, and approaches 1, as expected for the FR ordering.\nSimilar behavior was observed at $T_c^{\\rm (FR)} = 0.019(2)$ in Figs.~\\ref{fig:mcpd}(a1) and \\ref{fig:mcpd}(a2).\nOn the other hand, at a higher $T_{\\rm KT} = 0.146(4)$, only $M_{xy}$ changes rapidly, and correspondingly, $\\chi_{xy}$ shows a peak.\n$M_{xy}$, however, shows a noticeable system-size dependence even below $T_{\\rm KT}$, in contrast with the results below $T_c^{\\rm (PD)}$. \nSimilar behavior was observed in the KT transition in Ising spin systems~\\cite{Takayama1983,Fujiki1984}.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig45v2.eps}\n   \\caption{(Color online).\n   Extrapolation of $\\psi$ to $N\\to\\infty$ at different temperatures.\n   The solid lines for $T\\le 0.104$ is the linear fitting of data.\n   }\n   \\label{fig:psifit}\n\\end{figure}\n\nOn the other hand, $\\psi$ does not show an anomaly at $T_{\\rm KT}$, while it shows a sharp rise around $T_{c}^{{\\rm (FR)}}$, as shown in Fig.~\\ref{fig:mcferri}(a).\nThe value of $\\psi$ extrapolated to large $N$ converges to zero in the intermediate-temperature range.\nFigure~\\ref{fig:psifit} shows the extrapolation of $\\psi$ for $N\\to \\infty$.\nThe results indicate that, $\\psi$ remains to be zero at $N\\to\\infty$ for $T\\gtrsim0.104$, which is far below $T_{\\rm KT}=0.146(4)$.\nOn the other hand, the extrapolated value becomes finite for $T\\lesssim 0.104$, reflecting the FR order; the transition temperature is estimated as $\\tilde{T}_c^{\\rm (FR)} = 0.102(2)$, which is in accordance with $T_c^{\\rm (FR)} = 0.098(4)$.\n\nThe results above indicate that there is no sixfold symmetry breaking in $M_{xy}$ at $T_{\\rm KT}$, as seen in the KT phase in the Ising spin models~\\cite{Takayama1983}.\nHence, we consider that the higher-temperature transition at $T_{\\rm KT}$ is of KT type.\nNamely, the system exhibits two successive transitions from the paramagnetic phase to the KT-like phase at $T_{\\rm KT}$, and the KT-like phase to the low-temperature FR phase at $T_{c}^{{\\rm (FR)}}$. \nHere, we call the intermediate-temperature phase the KT-like phase, as it is difficult to confirm either the KT universality class by critical behavior or the quasi-LRO behavior within the system sizes we reached, as seen below.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig44v2.eps}\n   \\caption{(Color online).\n   MC results for the real-space spin correlation function $C(r)$ at $J=2$ and $n=0.38$.\n   The results are shown only for the sites with $C(r) > 0$.\n   The calculations were done for the system size $N=18\\times18$.\n   }\n   \\label{fig:mcsk}\n\\end{figure}\n\nThe signature of two successive transitions is also observed in the real-space spin correlation function $C(r)$.\nHere $C(r)$ is the averaged correlations between the Ising spins in distance $r$, defined by \n\\begin{eqnarray}\nC(r) = \\sum_{i,j} \\frac{1}{N_{\\rm p}(r)} \\langle S_i S_j \\rangle  \\delta(|{\\bf r}_{ij}| - r),\n\\end{eqnarray}\nwhere $N_{\\rm p}(r) =\\sum_{i,j} \\delta(|{\\bf r}_{ij}| - r)$ is the number of spin pairs with distance $r$, and $\\delta(x)$ is the delta function.\nThe MC data while varying temperature are shown in Fig.~\\ref{fig:mcsk}. \nAlthough the results are not conclusive due to the limitation on accessible system sizes, they appear to be consistent with the two transitions discussed above.\nFor $T \\lesssim T_{c}^{{\\rm (FR)}} = 0.098(4)$, the spin correlation appears to approach constant for large distance, well corresponding to the FR LRO developed in this low temperature region.\nOn the other hand, for $T \\gtrsim T_{\\rm KT} = 0.146(4)$, it becomes concave downward with a steep decrease with respect to the distance, which reflects an exponential decay in the high temperature paramagnetic state.\nIn the intermediate region for $T_c^{\\rm (FR)} \\lesssim T \\lesssim T_{\\rm KT}$, the spin correlation also decays with increasing distance.\nThe decay, however, is much slower and appears to obey an asymptotic power law, which is characteristic to the quasi-LRO in the KT state.\nIn principle, the critical exponents can be estimated from the asymptotic power-law behavior, but it is difficult to be conclusive in the current system sizes.\n\n\\section{\nElectronic structure of partially disordered state\n\\label{sec:estruct}\n}\n\nIn the previous section, we discussed the thermodynamic behavior of the localized spin degree of freedom, with emphasis on the emergence of peculiar PD state.\nIn this section, we focus on the behavior in the charge degree of freedom of itinerant electrons in the PD phase. \n\n\\begin{figure}\n   \\includegraphics[width=0.80\\linewidth]{fig5chargev4.eps}\n   \\caption{(Color online).\n   MC results for $n_{\\rm CO}$ at ${\\bf q}=(2\\pi\/3,-2\\pi\/3)$ at $n=1\/3$ and (a) $J=1$, (b) $J=2$, and (c) $J=4$.\n   The calculations were done for the system sizes $N=12\\times 12$, $12\\times 18$, and $18\\times18$.\n   }\n   \\label{fig:mcnq}\n\\end{figure}\n\nFigure~\\ref{fig:mcnq} shows temperature dependence of the charge modulation $n_{\\rm CO}$ [Eq.~(\\ref{eq:n_CO})] at $n=1\/3$ for different $J$. \nFigure~\\ref{fig:mcnq}(a) is the result at $J=1$ for different system sizes.\nThe result shows an increase of $n_{\\rm CO}$ below $T \\simeq T_c^{\\rm (PD)} =0.086(4)$, indicating that the PD state is accompanied by charge modulation with period three.\nSimilar onsets of charge modulation at $T_c^{\\rm (PD)}$ are observed for larger $J$, as shown in Figs.~\\ref{fig:mcnq}(b) and \\ref{fig:mcnq}(c);\nthe amplitude of the modulation in the PD phase increases monotonically as $J$ increases.\nThe magnitude of the charge modulation is in the same order compared to the mean-field result in Fig.~\\ref{fig:gap}), while the growth is considerably suppressed by a factor of two to four.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig5dosv2.eps}\n   \\caption{(Color online).\n   MC results for DOS of itinerant electrons at $n=1\/3$ and $J=2$ for $N=18\\times 18$.\n   The Fermi level is set at $\\varepsilon = 0$.\n   The statistical errors are comparable to the width of the lines.\n   }\n   \\label{fig:mcdos}\n\\end{figure}\n\nWe next look into the electronic density of states (DOS) at different temperature. \nFigure~\\ref{fig:mcdos} shows the results for DOS while varying temperature at $J=2$ and $n=1\/3$.\nThe Fermi level is set at $\\varepsilon=0$.\nHere, DOS was calculated by counting the number of energy eigenvalues as the histogram with the energy interval of 0.0375.\nIn the paramagnetic region for $T \\gtrsim T_c^{\\rm (PD)} = 0.130(4)$, DOS is featureless near the Fermi level.\nOn the other hand, below $T_c^{\\rm (PD)}$, an energy gap develops at the Fermi level for $n=1\/3$.\nThe result shows that the PD state is an insulator, which supports the scenario that PD is stabilized by the Slater mechanism described in Sec.~\\ref{sec:mft}.\nSimilarly to the charge modulation, the energy gap in the MC results is largely suppressed compared to that obtained by the mean-field analysis in Fig.~\\ref{fig:gap}.\nThis appears to show the importance of appropriately taking into account of thermal fluctuations.\n\n\\section{\\label{sec:summary}\nSummary\n}\n\nTo summarize, by a combined analysis of the mean-field type calculation and Monte Carlo simulation, we have investigated the origin of the partial disorder in the Ising-spin Kondo lattice model in a two-dimensional triangular lattice. \nIn the mean-field type calculation, we have clarified that a local magnetic field of the partial disorder type induces a metal-insulator transition at 1\/3 filling at a critical value of the field. \nThe result suggests that the three-sublattice partial disorder can give rise to an energy gap, and therefore, it has a chance to be stabilized through the Slater mechanism. \nOn the other hand, in the Monte Carlo simulation, we have provided convincing numerical results on the emergence of partial disorder at finite temperatures where the stripe phase and the ferrimagnetic order compete with each other.\nThe Monte Carlo result shows that the partially disordered state appears above a nonzero value of the spin-charge coupling, and that it is insulating and accompanied by charge disproportionation.\nThe nonzero critical value of the spin-charge coupling and the opening of the charge gap are both qualitatively consistent with the mean-field analysis.\nThe results indicate that the partial disorder is stabilized by the Slater mechanism which is characteristic to itinerant magnets.\nOur results not only clarify the new mechanism of partial disorder in two dimensions but also pave the way for understanding of the interesting physics related to the peculiar coexistence of magnetic order and paramagnetic moments in itinerant electron systems.\n\n\\begin{figure}\n   \\includegraphics[width=0.8\\linewidth]{fig8v1.eps}\n   \\caption{\n   (Color online).\n   (a) The grand potetial $\\Omega$ and (b) electron filling $n$ with respect to the chemical potential $\\mu$, numerically calculated by exactly diagonalizing the one-body Hamiltonian for itinerant electrons.\n   The results are obtained at $J=2$ with $N_s=24\\times24$ site superlattice of $N=12\\times 12$ site unit cells.\n   The strip at the left side of (b) shows the ground state at the corresponding filling.\n   }\n   \\label{fig:nvc}\n\\end{figure}\n\nAn interesting extension of the current work would be to consider the effect of quantum fluctuation of localized spins.\nIn our result, the partial disorder remains stable down to very low temperature, implying that the paramagnetic spins are largely fluctuating and sensitive to perturbations at low temperatures.\nHence, an interesting possibility is that, by including quantum fluctuations, the partial disorder is further stabilized and remains stable even in the ground state.\nIndeed, a similar partial disorder was found in the ground state of the Kondo lattice model with quantum spins at half filling~\\cite{Motome2010}. \nTherefore, it is intriguing to examine the effect of quantum fluctuations on the present model with Ising spins. \nHowever, it is not straightforwardly calculated by the present Monte Carlo method.\nThe interesting problem is left for future study.\n\n\\begin{acknowledgements}\nThe authors are grateful to G.-W. Chern, H. Kawamura, M. Matsuda, S. Miyashita, and H. Yoshida for fruitful discussions.\nThe authors also thank S. Hayami and T. Misawa for helpful comments.\nPart of the calculations were performed on the Supercomputer Center, Insitute for Solid State Physics,\nUniversity of Tokyo. H.I. is supported by Grant-in-Aid for JSPS Fellows.\nThis research was supported by KAKENHI (No.19052008, 21340090, 22540372, and 24340076), Global COE Program ``the Physical Sciences Frontier\", the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}