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+{"text":"\\section{INTRODUCTION}\nAs the adoption of digitized histopathologic slide images became widespread, the use of Artificial Intelligence (AI) methods in the digital pathology field increased. In particular, computer vision and deep learning methods are used to automate and improve various tasks such as image analysis, diagnostic decision-making, and disease monitoring \\citep{czyzewski2021machine, daniel2022deep, larey2022fron}. \n\nHowever, data limitations pose a major challenge in digital pathology and include issues related to data scarcity, variability, privacy, annotation, bias, quality, and labeling. Data scarcity and variability can make it difficult to train and evaluate computer algorithms for digital pathology, as there may not be enough data available for certain decision thresholds.  Data bias is another concern, as digital pathology datasets may be biased toward certain populations, which can limit the generalizability of algorithms trained on them. Data labeling can also be subjective and dependent on the expertise of the labeler leading to inaccuracies. An emerging solution to these challenges is generating synthetic images.\n\nThe field of generating synthetic images became more popular during the last years after the Generative Adversarial Network (GAN) \\citep{goodfellow2020generative} was introduced. In this approach, a 'discriminator' model is designed to discriminate between real data to fake data. A different model coined 'generator' is trained to produce synthetic data that will be plugged into the 'discriminator' during training. On the one hand, the 'generator' is trained in a way where the discriminator doesn't distinguish between the real data and the generated synthetic data. On the other hand, the discriminator model is trained to discern between the two correctly. It means the generated data is challenging the discriminator to get the best results.\n\nIn the classic approach of image generation (coined Vanilla GAN), the input for the generator is sampled from a given distribution, and then is processed into a synthetic image. More advanced techniques called Conditional GAN (C-GAN) \\citep{mirza2014conditional} supply information about the required type of generated data and plug it into the different GAN models, to control the type of generated data. Some approaches such as pix2pix \\citep{isola2017image} took this technique further and supply the generator with more detailed information at the pixel level. In this approach, the generator receives a semantic label mask as an input, and each pixel is generated to belong to its corresponding label from the given semantic mask. This image translation approach has the advantage of yielding pairs of images and semantic labels, that could be used in different tasks that require these pairs (e.g. segmentation), unlike the classic approach where the synthetic images lack semantic information. Yet, in some cases, the scarcity of semantic masks will be caused due to their creation complexity.\n\nA special case is the generation of synthetic histology images, where the semantic masks consist of various tissue types and complicated patterns resulting from the complex nature of the tissue. A na\u00efve solution uses tissue masks extracted from real histology images in the image translation pipeline, but the dependency on limited real images during the generation process causes a limited number of semantic masks. Thus, in this case, image translation models will not be scalable since the semantic masks have an integral part in their pipeline, while Vanilla GANs are scalable due to their only dependency on the scalable sampling process.\n\nIn this study, we show how our dual-phase pipeline overcomes the tradeoff (Fig.~\\ref{f:f1}), and generates pairs of histology synthetic semantic masks and images in a scalable design. We introduce DEPAS, a generative model that captures tissue structure and generates high-resolution semantic masks with state-of-the-art quality for three different organs: skin, lung, and prostate. Moreover, we also show that these masks can be processed by pix2pixHD \\citep{wang2018high}, a generative image translation model that supports high-resolution images, to produce photorealistic RGB tissue images (Fig.~\\ref{f:f1}). We demonstrate it for two types of staining: H\\&E and immunohistochemistry. This pipeline, on the one hand, generates pairs of semantic masks and histology images, and on the other hand, is scalable since it does not require real masks during inference.\n\n\\begin{figure}[thpb]\n\\centering\n\t\\includegraphics[scale=0.6]{Figure1.png}\n\t\\caption{(A) Illustration of the tradeoff between Image Translation GANs to the Vanilla GANs. The former generates synthetic images based on their semantic labels. In this case, the scalability is bounded when the quantity of semantic labels is limited. On the Other hand, Vanilla GANs lack semantic information but can produce an unlimited number of synthetic images. (B) Our platform resolves this challenge in the histology domain with a dual-phase generative system. The first step includes generating semantic masks of tissue labels using a novel architecture of a Vanilla GAN coined DEPAS. Then, the generated masks are processed by a paired image translation GAN (such as pix2pixHD) to produce the synthetic histology RGB image.}\n\t\\label{f:f1}\n\\end{figure}\n\n\\section{RELATED WORK}\n\\subsection{Medical Synthetic Images}\nThe use of generative models to produce synthetic images was explored in numerous works in the medical field. Image translation frameworks are widely used, such as models that generate endoscopy images given binary semantic masks \\citep{adjei2020gan}, transform between radiological images \\citep{armanious2020medgan}, and convert between histology staining types \\citep{lysik2022he}. Another image translation work is DeepLIIF \\citep{ghahremani2022deepliif}, which provides for a given IHC image several outputs including stain deconvolution, segmentation masks, and different marker images. Other types of generative frameworks are common as well. DCGAN framework generates synthetic images from a sampled noise input and processes it through convolution-layers architecture. It was used in several applications that generate medical images such as MR images \\citep{divya2022medical}, eye diseases images \\citep{smaida2021dcgan}, X-ray images \\citep{puttagunta2022novel}, and breast cancer histological images \\citep{blanco2021medical}. PathologyGAN \\citep{quiros2019pathologygan} introduced a novel framework that generates high-quality pathology images in the size of 224X224 pixels. \\citep{guibas2017synthetic} introduced a two-step pipeline that is similar to ours. In the first step, they generate binary vessel segmentation masks using DCGAN, next they generate the RGB retinal image. Their pipeline provides synthetic images in the size of 512X512 pixels size. However, our pipeline provides higher-resolution (x2) synthetic images in the challenging field of histology. We focus on the first phase of learning the complex geometry structure that is reflected in the semantic label mask. We show that DCGAN is not sufficient for this task and introduce DEPAS as an improved architecture to overcome the challenges in the high-resolution histology domain.\n\n\\subsection{Discrete Predictions}\nThe first step of our pipeline requires predicting discrete semantic masks. In this work, we focus on the binary scenario where there are two labels in the semantic mask \u2013 tissue and air. However, the binary output of the generator should be obtained by a step-function, but this non-differentiable operation can break the backpropagation of the optimization objective's gradients through the discriminator to the generator. A reasonable solution is by replacing the discrete output operations with continuous relaxations such as Sigmoid during training, and applying the discrete operation only during the test \\citep{neff2017generative}.  \\citep{bengio2013estimating} proposed to use binary neurons in machine learning models via straight-threw-estimators, where the binary operator is applied during training in the forward pass, but is ignored and treated as an identity function in the backward pass. \\citep{dong2018training} explored the generative use of a deterministic binary neuron and stochastic binary neuron and introduced the BinaryGAN. We investigated the different approaches and found that in the high-resolution histology domain, the best performance was achieved by the Annealing-Sigmoid. In this approach, the last layer of DEPAS's generator is a Sigmoid whose slope is increased gradually during training toward the step-function \\citep{chung2016hierarchical}. \n\n\\begin{figure}[t]\n\\centering\n\n        \\includegraphics[ width=\\textwidth]{Figure2.png}\n\t\\caption{Architecture of DEPAS. (A) The generator decodes semantic masks from latent noise. It consists of five transpose convolution layers where each one of them followed by Batch Normalization and ReLU activation. Another element of stochasticity is added to the hidden layers in the spatial dimension after being scaled. Finally, the last feature-maps are processed by the Discrete Adaptive block that outputs a semantic mask in the two-label case, or multiple masks in the multi-label case. (B) For training, we use three discriminators that support different resolutions of images. Each one of them encodes the corresponding image into a scalar which represents the probability that the image is real. The encoding is processed by convolution layers where each one of them is followed by Batch-Normalization and LeakyReLU activation.}\n\t\\label{f:f2}\n\\end{figure}\n\n\\section{METHODS}\nOur pipeline includes two main phases. The main focus of this study is on the first step where we learn the internal geometry of the digitalized histology tissue. For this task, we designed a generative architecture coined DEPAS that captures the tissue's morphology and expresses it by a semantic mask. The second phase is an image translation task to transfer the discrete semantic mask to an RGB photorealistic image of the tissue.\n\n\\subsection{DEPAS Architecture}\nTo enhance scalability, the generative process of producing synthetic tissue masks is initialized by sampling noise from a given distribution and applying it to the model (Vanilla GAN). The mechanism is based on the DCGAN architecture that was used by \\citep{guibas2017synthetic} and consists of multiple convolution blocks in its generator and discriminator. In DEPAS we adjusted the DCGAN layers to the high-resolution size of 512X1024 pixels output and included three main extensions.\n\n(1) Discrete Adaptive Block. In our case, where the discriminator should obtain a binary mask during training, we require a binary output from the generator where every pixel indicates one of the two classes \u2013 Tissue or Air. Thus, we replaced the DCGAN's last block with this module (Fig.~\\ref{f:f2}A). Instead of the non-differential step function, we use a Sigmoid activation with a high slope to mimic the former and yield a pseudo-binary differential output. For optimal convergence, we initiate the Sigmoid with its base slope of $1$ and increase it gradually during training (Annealing-Sigmoid, AKA AS). That is, in every iteration, the generator produces a Bernoulli probability that becomes more deterministic during training in differentiating the two classes. Formally, at iteration $t$, the AS is:\n\\begin{equation}\n{AS}_{t} = \\frac{1}{1+e^{-\\delta_{t}*x}} \\label{eq_sig}\n\\end{equation}\n\nWhere $x$ is the input for the element-wise operation, and $\\delta_{t}$ determines the Sigmoid's slope at iteration $t$. To increase the slope, we require that $\\delta_{t+1}>\\delta_{t}$, and for initialization with the basic Sigmoid, we define $\\delta_{t=0}=1$. \nFurthermore, we extend this approach to cases where there are more than two labels in the desired semantic mask. For example, in the case where the tissue itself has several types of morphology (e.g. tumor tissue, non-tumor tissue, and air) we will use the multi-label approach. In this scenario, we generalize the binary distribution to the multinomial distribution by designing the 'Discrete adaptive Block' to produce a multi-channel feature map, where each channel represents a different class. The feature maps are then applied to an Annealing-Softmax-Temperature (AST) activation. Instead of the non-differential Argmax function, we use a channel-wise Softmax layer with a low temperature to mimic a deterministic decision of the generated class for each pixel. Similarly to the binary situation, we initiate the Softmax temperature with its base value of $1$, and decrease it gradually during training. I.e. every iteration, the generator produces for each pixel its classes probabilities that become more deterministic during training.  Formally, at iteration $t$, the probability for class $c$ provided by the AST is:\n\\begin{equation}\n{AST}_{t,c} = \\frac{e^{\\frac{x_{c}}{T_{t}}}}{\\sum_{j}^{} e^{\\frac{x_{j}}{T_{t}}}} \\label{eq_temp}\n\\end{equation}\n\nWhere $x_i$ is the input for the element-wise operation of the channel that corresponds to class $i$, and $T_{t}$ determines the Softmax's temperature at iteration $t$. To increase determinism, we require that $T_{t+1}<T_{t}$, and for initialization with the standard Softmax, we define $T_{t=0}=1$. \nBoth binary and multi-label scenarios consist of the 'step-function' and 'argmax' operations respectively for inference. However, for training, where gradients should backpropagate through these layers, the non-differential operations are replaced by differential operations that adapt the former's attributes gradually. \n\n(2) Spatial Noise. In the standard DCGAN's implementation, latent vectors $z$ are drawn from a Gaussian distribution as the input for the generator. The sampling is performed channel-wise. That is, a sampled input noise is a one-dimensional latent vector where each element represents an initial channel with a spatial size of 1X1 pixels without any noise diversity in the spatial domain. When the feature maps are spatially increasing in the forward pass (via the transpose convolution layers), they are prone to become repetitive in the spatial aspect. In our case, where DEPAS provides high-resolution semantic masks, this phenomenon is significant. We address it by adding noise in the spatial domain as well. We draw a 2D array from a Gaussian distribution and inject it spatially into the hidden layers of the generator after resolution and scale adjustments (Fig.~\\ref{f:f2}A).\n\n(3) Multi-Scale-Discriminators. Our pipeline generates high-resolution synthetic images in the size of 512X1024 pixels. For comparison,  BinaryGAN \\citep{dong2018training}, PathologyGAN \\citep{quiros2019pathologygan}, and the retinal vessel dual-phase pipeline \\citep{guibas2017synthetic} generate 28X28 pixels, 224X224 pixels, and 512X512 pixels size of images respectively. Inspired by pix2pixHD \\citep{wang2018high}, we implemented three discriminators each one of them receiving a different scale of the input mask \u2013 100\\%, 50\\%, and 25\\% (Fig.~\\ref{f:f2}B). This technique helps the discriminator to distinguish between real and fake masks from different levels of perspective. Low-resolution masks provide high-level information such as the general structure of the tissue. In contrast, low-level information, such as intercellular spaces, is obtained by high-resolution masks. The combined objective is provided by the following equation:\n  \n\\begin{equation}\n\\mathcal{L}_{DEPAS}=\\sum_{r=1}^{R} \\alpha_{r} \\cdot \\mathcal{L}_{GAN,r} \n\\label{eq_depas_loss}\n\\end{equation}\n\nWhere $\\alpha_{r}$ is the weight of $D_r$, the discriminator that corresponds to the image resolution $r \\in \\{25\\%, 50\\%, 100\\%\\}$, and $R$ is the number of discriminators ($R=3$). $\\mathcal{L}_{GAN,r}$ is the Vanilla GAN's loss obtained by $D_{r}$ that receives the real input mask $x$ and the generator's synthetic mask $G(z)$, and is defined as:\n\\begin{equation}\n\\mathcal{L}_{GAN,r} = log(D_{r}(x)) + log(1-D_{r}(G(z)))  \\label{eq_gan_loss}\n\\end{equation}\n\n\\subsection{Paired Image Translation}\nPaired image translation is a set of tasks that translate one domain of images to another domain given input-output image training pairs \\citep{isola2017image}. One of these kinds of tasks is to insert a semantic map and translate it to an image, based on the additional information, such as class labels, passed together with the image to the network during the training phase. In the second step of our pipeline, we used pix2pixHD, an image translation generative network \\citep{wang2018high}, to produce synthetic pathological images from the given semantic masks. Particularly, pix2pixHD consists of a generator which is a composition of convolutional residual layers that receives a 512X1024 pixels semantic mask as an input and generates a 512X1024 pixels RGB image. In addition, we used two multiscale discriminators with the same CNN architecture, but work on two different image scales. \n\n\\subsection{Datasets}\nIn this study, we perform our methodology on four histology realizations subjected to two different types of staining.\nThe first type is hematoxylin and eosin (H\\&E) where histology images were collected from three different types of cancer: Prostate Adenocarcinoma (PRAD), Skin Cutaneous Melanoma (SKCM), and Lung Squamous Cell Carcinoma (LUSC). The three datasets are part of the Cancer Genome Atlas (TCGA) research network, where for all datatypes we used only their imaging information \\citep{tomczak2015review}. We performed our methodology on 50 WSIs from each H\\&E realization. \n\nThe second staining type is immunohistochemistry (IHC), where histology images were collected from non-small cell lung carcinoma (NSCLC) patients. This data was originally part of a study that involved an Immune Checkpoint inhibitor therapy, where the detection of programmed death-ligand 1 (PD-L1) in the tissue biopsies was required for determining the course of therapy \\citep{wang2016pd}. $27$ WSIs were obtained from patients diagnosed with NSCLC who underwent biopsy at Rambam Health Care Campus. All procedures performed in this study and involving human participants were in accordance with the ethical standards of the Rambam Medical center institutional research committee,  approval 0522-10-RMB, and with the 1964 Helsinki declaration and its later amendments or comparable ethical standard.\n\nIn all realizations, the slides were split into patches with a size of 512X1024 pixels. Patches containing more than 85\\% background were filtered. In total, 6000 images were used from each one of the H\\&E datasets and 2012 images from the IHC dataset. For each realization, 85\\% of the data was used for training and the rest for evaluation. In all realizations, we create ground truth for binary tissue semantic masks by converting the patches to grayscale and applying a high threshold to extract air pixels. We found that 204 and 235 were the optimal thresholds (in the range of 0-255) to distinguish between tissue and air pixels, in H\\&E and IHC respectively.\n\n\\subsection{Training Procedure}\nFor each realization, we trained two models individually: pix2pixHD and DEPAS.\nPixPixHD was trained with pairs of images and their corresponding tissue masks, where the tissue masks served as the input for the generator and the images as the real ground truth for the discriminators. The training was conducted with a batch size of 1 for 400 epochs in the H\\&E realizations, and for 700 epochs in the IHC realization.  We linearly decay the learning rate to zero over the last 100 epochs for H\\&E and over 200 epochs for IHC.\n\nDEPAS was trained only with the tissue masks from the same training set used to train pix2pixHD. In this case, the generator generates synthetic tissue masks from a sample drawn from the standard normal distribution. The model was trained with a batch size of 8, for 100 epochs where every 10 epochs, the Annealing Sigmoid's $\\delta$ parameter in (\\ref{eq_sig}) increased by one. Each scale objective in the loss term (\\ref{eq_depas_loss}) was equally weighted where $\\alpha_{100\\%} = \\alpha_{50\\%} = \\alpha_{25\\%} = 1$. \nBoth DEPAS and Pix2pixHD models' weights were optimized by Adam optimizer \\citep{kingma2014adam} with a learning rate of 2e-4, and beta's coefficients range of (0.5, 0.999). They were developed in the PyTorch framework \\citep{paszke2019pytorch} and were trained on a single NVIDIA RTX A6000 GPU with 48GB GPU memory.\n\n\\section{Results}\n\\subsection{Synthetic Semantic Tissue Masks}\nFor each realization, we trained in addition to DEPAS, a standard DCGAN as a baseline since it was used before to generate semantic masks in the medical field \\citep{guibas2017synthetic}. We generated from both trained models the same amount of synthetic tissue masks as in the real test set (1000 for H\\&E realizations and 328 for the IHC). Examples of the different tissue masks for all four histology realizations are shown in Fig.~\\ref{f:f3} with their latent space 2D projection extracted from a pre-trained ResNet model. \n\nFor quantitative evaluation, we calculated the distance between DEPAS synthetic tissue masks to the real test tissue masks (that were excluded from training). The first two distance metrics, Kolmogorov\u2013Smirnov (KS) test and Kullback\u2013Leibler (KL) divergence, were applied to the TSNE projection of the masks' representations. These representations were extracted from the last hidden-layer of a pre-trained ResNet \\citep{he2016deep} model (size of 2048). For the third metric, we captured representation vectors from the last layer of a pre-trained Inception v3 model \\citep{szegedy2016rethinking} (size of 2048), to calculate 'Fr\u00e9chet inception distance' (FID) which is the gold-standard metric for evaluating the quality of synthetic images \\citep{heusel2017gans}.\n\nThe results for the different representation metrics are presented in Table \\ref{tissue_mask_tab}. They show that for all four realizations, DEPAS synthetic tissue masks have the least distance to the real ones compared to DCGAN, in all metrics. Particularly, FID scores of DEPAS were lower (closer to the real images) than the DCGAN baseline by factors of 20.0, 6.9, 30.2,  and 17.0, when referring to the PRAD, SKCM, LUSC, and NSCLC tissue masks respectively.\n\n\\begin{figure}[p]\n\\centering\n\t\\includegraphics[scale=0.4]{Figure3.png}\n        \n\t\\caption{Examples of tissue masks from four different types of cancer realizations. These realizations include three organs: skin, prostate and lung, and two types of staining: H\\&E and PD-L1 IHC. SKCM - Skin Cutaneous Melanoma. PRAD - Prostate Adenocarcinoma. LUSC - Lung Squamous Cell Carcinoma. NSCLC - Non-small cell lung carcinoma. In each realization, we show tissue masks taken from real biopsy slides (with their original RGB image). We compared the tissue masks to the ones produced by DEPAS and by a baseline DCGAN. The different types of tissue mask representations are projected into 2D via TSNE (right). We show that DEPAS provides tissue masks from a distribution that is closer to the Real images rather than DCGAN's outputs (as quantified in Table I).}\n\t\\label{f:f3}\n\\end{figure}\n\n\n\\begin{table*}[p]\n        \\centering\n\t\\begin{center}\n\t\t\\small\n    \t\t\\begin{tabular}{l c c c c c}\n\t\t\t\\hline\n\t\t  & & \\multicolumn{3}{c}{Image quality metrics}\\\\\n            \\cmidrule{3-5}\n             Method & Dataset & KS $\\downarrow$  & KL  $\\downarrow$  & FID $\\downarrow$  \\\\\n\t\t\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\tDEPAS & \\multirow{2}{*}{PRAD$^{a}$} & 1 (0.3) & 1 (0.9) &  1 (420.3) \\\\\n   \t\tDCGAN & & x2.6 & x42.8 & x20.0 \\\\\n\t\t\t\\hline\n            \\hline\n\t\t\tDEPAS & \\multirow{2}{*}{SKCM$^{a}$} & 1 (0.3) & 1 (0.7) & 1 (1006.6) \\\\\n   \t\tDCGAN & & x5.0 & x31.3 & x6.9  \\\\\n        \t\\hline\n            \\hline\n\t\t\tDEPAS & \\multirow{2}{*}{LUSC$^{a}$} & 1 (0.2) & 1 (0.6) & 1 (151.3) \\\\\n   \t\tDCGAN & & x11.5 & x67.7 & x30.2 \\\\\n        \t\\hline\n            \\hline\n\t\t\tDEPAS & \\multirow{2}{*}{NSCLC$^{b}$} & 1 (0.2) & 1 (0.3) & 1 (480.0) \\\\\n   \t\tDCGAN & & x6.6 & x128.0 & x17.0 \\\\\n            \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Similarity metrics between DEPAS synthetics masks, real masks, and current SOTA. $^a$ and $^b$ denote H\\&E and IHC stained tissues respectively. In addition, KS is Kolmogorov\u2013Smirnov test, KL is Kullback\u2013Leibler divergence, and FID is Fr\u00e9chet inception distance. Values are normalized by DEPAS results, where the parenthesis phrase presents its actual raw data. Downarrow symbol indicates lower is better.}\n\t\\label{tissue_mask_tab}\n\\end{table*}\n\n\\begin{figure}[p]\n\\centering\n\n         \\includegraphics[ width=\\textwidth]{Figure4.png}\n\t\\caption{Examples of tissue masks and their corresponding histology RGB images for two types of cancers. H\\&E stating of skin cutaneous melanoma (left) and IHC staining of non-small cell lung carcinoma (right). For each realization, we show pairs of masks-images taken from real biopsies slides. We compare them to the pairs produced by DEPAS and by a standard DCGAN. The different types of RGB image representations are projected into 2D via TSNE (bottom). We show that DEPAS provides images from a distribution that is closer to the Real images' distribution rather than DCGAN's outputs, or the negative histology-control outputs taken from real histology RGB images from a different realization.}\n\t\\label{f:f4}\n\\end{figure}\n\n\\begin{table*}[t]\n    \\centering\n\t\\begin{center}\n\t\t\\small\n    \t\t\\begin{tabular}{l c c c c c}\n\t\t\n\t\t\t\\hline\n\t\t  & & \\multicolumn{3}{c}{Image quality metrics}\\\\\n            \\cmidrule{3-5}\n             Method & Dataset & KS $\\downarrow$  & KL $\\downarrow$  & FID $\\downarrow$  \\\\\n\t\t\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\tDEPAS & \\multirow{4}{*}{SKCM$^{a}$} & 1 (0.3) & 1 (0.4) & 1 (592.5) \\\\\n   \t\t  DCGAN & & x2.3 & x16.5 & x6.4  \\\\\n                Pathology Control & & x1.8 & x8.8 & x9.5  \\\\\n                Realistic Control & & x1.1 & x9.2 & x10.0  \\\\\n        \t\\hline\n                \\hline\n\t\t\tDEPAS & \\multirow{4}{*}{NSCLC$^{b}$} & 1 (0.2) & 1 (0.4) & 1 (219.9) \\\\\n   \t\t  DCGAN & & x3.0 & x8.2 & x1.4 \\\\\n                Pathology Control & & x3.7 & x44.8 & x25.7  \\\\\n                Realistic Control & & x2.8 & x111.8 & x40.2  \\\\\n               \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Similarity metrics between synthetic images based on DEPAS synthetics masks, real histological images, and various baselines. $^a$ and $^b$ denote H\\&E and IHC stained tissues respectively. In addition, KS is Kolmogorov\u2013Smirnov test, KL is Kullback\u2013Leibler divergence, and FID is Fr\u00e9chet inception distance. Values are normalized by DEPAS results, where the parenthesis phrase presents its actual raw data. The down arrow symbol indicates lower is better.}\n\t\\label{rgb_tab}\n\\end{table*}\n\n\\subsection{Synthetic Photorealistic RGB Images}\nTo further evaluate the full pipeline in the photorealistic histology perspective, we applied the synthetic tissue masks to the image translation model and compared their outputs to the real histology images. We performed this over two datasets, the first is SKCM which represents a realization subjected to H\\&E staining, and the second is lung cancer (NSCLC) subjected to IHC staining. Examples of the different tissue masks and images for both H\\&E and IHC realizations are shown in Fig.~\\ref{f:f4} with their latent space 2D projection extracted from a pre-trained ResNet model. \nFurthermore, we performed the same quantitative evaluation methodology over the images at the RGB level, by calculating the distance between the synthetic images stem from DEPAS to the real RGB histology images (Table \\ref{rgb_tab}). For comparison, we performed the same for three other datasets: (1) synthetic images where their prior tissue masks are generated by DCGAN. (2) Real histology images from a different type of pathology as a histology control. I.e. in the case where the evaluation is performed on the H\\&E realization, we also calculated the distance between the real IHC images to the real H\\&E images as a histology baseline, and vice-versa. (3) We also used SegTrack Dataset \\citep{li2013video} as real-life scenario images for realistic control taken from the non-pathology field. \n\nWe performed the same evaluation for all control batches in RGB levels. The results in Table \\ref{rgb_tab} show that for both H\\&E and IHC images, DEPAS had the best performance over all metrics. Where DEPAS's FID score for H\\&E images was better than DCGAN by a factor of 6.36, and by a factor of  1.42 for IHC images.\n\n\\subsection{Multi-Label DEPAS}\nWe further show the ability of DEPAS to generate synthetic multi-label discrete semantic masks on the IHC realization. This task is performed on the NSCLC dataset as before, but where the tissue mask is divided into more-detailed labels, based also on its PD-L1 attributes. PD-L1 is a molecule expressed by tumor cells and enables them to evade the immune system's attack. Hence, a common immunotherapy treatment uses blocking antibodies that target PD-L1 to increase the immune system's effectiveness against the tumor cells. Evaluating the PD-L1 rate in the patients' biopsies is essential for determining the treatment type and its level. To achieve PD-L1-related labels, all IHC patches were annotated by expert pathologists. For every patch, each tissue pixel was assigned to one of the four PD-L1 feature classes: Inflammation, PD-L1- and PD-L1+, or non of them. Two more classes were assigned using computer vision techniques. Air class was assigned as before where grayscale pixels with values higher than 235 were considered Air. The cells class was assigned to the darker values of brown. Empirically, they were captured in the RGB image representation where the green and blue channels were smaller than 200 and the red channel was higher than the other channels. \n\nAfter creating the multi-label Ground truth, we trained the image translation model and DEPAS using the same methodology and hyperparameters, with only one exception. In the multi-label case, we train DEPAS by adjusting the discrete adaptive block to produce a multi-channel feature map. In this case, we set the initial value of the temperature in (\\ref{eq_temp}) to 1, and divide it every 10 epochs by 1.25. Examples of synthetic semantic masks generated from DEPAS, and their corresponding synthetic histology images, are presented in Fig.~\\ref{f:f5}A and visualized near real examples. We projected the images' inception representation via TSNE into a 2D space and show that the real images get mixed up with the synthetic ones. We also present representing pairs of real and synthetic images that had the smallest Euclidian distance in the 2D space (Fig.~\\ref{f:f5}B). Furthermore, synthetic RGB images generated by the multi-label approach are closer to the real images than the ones generated in the binary approach by 14\\% when referring to FID scores.\n\n\\begin{figure}[thpb]\n\\centering\n\t\\includegraphics[scale=0.47]{Figure5.png}\n        \n\t\\caption{(A) Examples of a multi-label task that consists of PD-L1 tissue attributes as semantic labels. DEPAS's synthetic labels and images are shown alongside real PD-L1 examples as a reference. (B) presents a TSNE projection of inception's representations taken from both real and synthetic images' (after being cropped into 224X224 pixels). Autumn and Winter colormaps represent synthetic and real images projection respectively. As the colormaps are brighter, the histology images contain more air. Several pairs of real images and synthetic images that had the smallest Euclidian distance between them are shown as well. (C) displays the same TSNE projection but with a single coloring of the data types to emphasize the mixture between the synthetic (red) and real (blue).\n}\n\t\\label{f:f5}\n\\end{figure}\n\n\\section{CONCLUSION}\nOne of the main challenges of generating synthetic images of tissues is controlling the distribution of features within them. Paired GANs provide a good way to improve the synthetics image quality by introducing semantic masks that account for the spatial structure of the tissue. However,  Unlike other domains, such as autonomous vehicles or face recognition, simulating or generating synthetic masks that represent the actual biological complexity of the image with high fidelity are lacking.\nA reasonable compromise, is augmenting semantic information taken from real tissues to generate photorealistic histology images. Yet, this approach suffers from bounded scalability, when a limited amount of real data are required for inference.\n\nHere we introduce an architecture for generating high-resolution binary masks of tissue structure that can be used as semantic prior knowledge for image translation models. Our work copes with the main challenge of generating binary synthetic images by adding noise along the different stages of decoding blocks and adding annealing temperature blocks to overcome the undifferentiability that is associated with binary masks while processing by a differential pipeline through the generator and discriminator. Our approach allows us not only to generate synthetic binary masks but also to produce multilabel masks that are critical for many applications that require labeling different cellular regions (such as cancer cells). \n\nWe show that our synthetic mask indeed captures the real cell distribution and spatial orientation within various pathological realizations histology slides.  Moreover, we show the synthetic images that result from our masks resemble real histological images better than the baseline in two types of cancers subjected to H\\&E and IHC staining types. Furthermore, we show that performing our pipeline with more detailed tissue information reflected in the multi-label semantic masks improves the quality of the synthetic images.\n\nOverall, our work provides a state-of-the-art solution for the challenging task of synthetic histological image generation with their semantic information in a form that is both scalable and controllable.\n\\\\\n\\\\\n\n\\section*{ACKNOWLEDGMENT}\nThe authors would like to thank Tanya Wasserman for her technical support and valuable discussions. The results shown here are in part based upon data generated by the TCGA Research Network: https:\/\/www.cancer.gov\/tcga.\n\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the central needs of any wide-coverage parser is a large lexicon that\ncontains the syntactic information for various lexical items.  The creation of\nsuch a lexicon has traditionally been a very large and daunting task and most\nuniversities have shied away from it, leaving the creation of wide-coverage\nparsers to commercial institutions that could afford the time and personnel to\ndevote to the creation of such a lexicon.  The release of several\nmachine-readable dictionaries (MRDs) into the public domain has opened new\npossibilities to grammar developers at research institutions, but the task\ndid not become trivial.  The problem of creating large scale lexicons changed\nfrom the tiresome, painstaking task of trying to develop individual word lists\nfor various syntactic phenomena to the task of `simply' extracting the\ninformation from the on-line dictionaries.  This, however, has not turned out\nto be as simple or straight-forward as researchers may have hoped.  Machine\nreadable dictionaries present numerous problems in terms of errors and in-\\\\\n\n\\vspace*{1.7in}\n\\noindent\nconsistencies in the various components of the lexical entries, making\nextraction quite difficult.  Many researchers abandon the extraction process\naltogether because it consumes too many scarce resources.\n\nAlthough a number of researchers have extracted information out of the\nvarious dictionaries available, the resulting lexicons have not, in general, been made\nfreely available to the NLP research community.  In at least some cases\n(\\cite{carroll89},\n\\cite{guthrie93}) this is due to licensing restrictions on the source \ndictionaries.  In response to the related problems of duplication of effort and\nnon-availability of needed lexicons, there are currently several on-going\nprojects to create syntactic lexicons and make them generally available.\n\n\\begin{itemize}\n\n\\item  The Proteus Project at New York University is developing the Comlex \nSyntactic Dictionary from scratch for release as one of the lexical resources \nin COMLEX (available through the Linguistic Data Consortium) \\cite{macleod94}.\n\n\\item The IITLEX project at Illinois Institute of Technology has an\non-going project to extract and release the information in the Collins English\nDictionary, along with information from various other word lists that will\ninclude both syntactic and semantic information.  That system is still under\ndevelopment, however, and currently uses an expensive relational database\npackage, a drawback which they plan to correct. \\cite{conlon94}\n\n\\end{itemize}\n\nThe syntactic lexicon described here contains approximately 37,000 entries\nextracted from the {\\it Oxford Advanced Learner's Dictionary of Current\nEnglish} \\cite{oald74} and the {\\it Oxford Dictionary for Current Idiomatic\nEnglish} \\cite{cie75}.  It is available via FTP in both an ASCII and a database\nformat.  The database format uses a UNIX hash table facility \\cite{sy91} that\nis freely distributed, and comes with an X-windows based interface for\nmodifying the database and doing searches.  C and Lisp hooks to allow other\nprograms to use the database are also included.\n\n\\section{Syntactic Lexicon}\n\nThe syntactic lexicon has entries for 8 part-of-speech categories: Adjective,\nAdverb, Complementizer, Conjunction, Determiner, Noun, Preposition, and Verb.\nEach entry consists of the following required and optional fields:\n\n\\begin{itemize}\n\n\\item {\\sc index} field (required) -- the uninflected form under which\nthe lexical item is compiled in the database; \n\n\\item {\\sc entry} field (required) -- contains all of the lexical items \nassociated with the {\\sc index}\\footnote{For example, a verb particle\nconstruction would be {\\sc index}ed under the verb, but would contain both the\nverb and the verb particle in the {\\sc entry} field.};\n\n\\item {\\sc pos} field (required) -- gives the part-of-speech for the lexical \nitem(s) in the {\\sc entry} field;\n\n\\item {\\sc frame} field (required) -- contains the syntactic information about \nthat entry;\n\n\\item {\\sc fs} field (optional) -- the Feature Structure field may provide\nadditional information about the {\\sc frame} field.  \n\n\\item {\\sc ex} field (optional) --  may be used for any number of example \nsentences.\n\n\\end{itemize}\n\nNote that lexical items may have more than one entry in the database (e.g. {\\it\nhave}) and that they may select the same {\\sc frame} field more than once,\nusing the {\\sc fs} to capture lexical idiosyncrasies (e.g. {\\it map}).\nTable~\\ref{syn-entries} shows selected entries from the database.\n\n\\begin{table}[ht]\n\\begin{tabular}{ll}\n\nINDEX:& have \\\\\nENTRY:& have \\\\ \nPOS:& Verb \\\\ \nFRAME:& Auxiliary\\_Verb \\\\\nFS:& Goes\\_on\\_Infinitive  \\\\ \nEX:& John has to go to the store. \\\\\n\\\\\nINDEX:&have\\\\\nENTRY:& have \\\\\nPOS:& V \\\\\nFRAME: &Transitive\\_Verb \\\\\nFS:& Non-Ergative \\\\\nEX:& John has a problem. \\\\\n\\\\\nINDEX:&map \\\\\nENTRY:&map out \\\\\nPOS:&Verb Verb\\_Particle \\\\\nFRAME: & Transitive\\_Verb\\_Particle \\\\\n\\\\\nINDEX:& map \\\\\nENTRY:& map \\\\\nPOS:& Noun \\\\\nFRAME:& Base\\_Noun \\\\\n&Noun\\_Determiner\\_required\\\\\n&Noun\\_Modifier\\\\\nFS: &wh$-$, reflexive$-$\\\\\n\\\\\nINDEX:&map \\\\\nENTRY:&map \\\\\nPOS:&Noun \\\\\nFRAME:&Noun\\_Determiner\\_not\\_required\\\\\nFS:&wh$-$, reflexive$-$, plural\n\n\\end{tabular}\t\n\\caption{Selected Syntactic Database Entries}\n\\label{syn-entries}\n\\end{table}\n\n\nBecause the syntactic database is part of the XTAG project \\cite{xtag-notes94},\na on-going project to develop a wide-coverage parser for English (see Section\n\\ref{related-work}), some entries in the syntactic lexicon reflect specific \nXTAG analyses.  In fact, the graphical interface for the syntactic lexicon\n(described in Section~\\ref{interface}) can run in two modes - {\\bf xtag} and\n{\\bf verbose}.  Tables \\ref{syn-entries}, \\ref{verb-entries}, and\n\\ref{aux-verbs} were all generated in {\\bf verbose} mode.\n\nThe vast majority of lexical items in the database fall into just 3 categories\n- Adjectives, Nouns, and Verbs.  These three categories plus Adverbs are\npresented in more detail in the following subsections.\n\n\\subsection{Adjectives}\n\nThere are 3,303 lexical adjectives in the database, of which 80 are `Proper\nName' adjectives, such as {\\it Chinese} and {\\it American}.  Adjectives have 5\nframes that they can select, which are listed below.  Possible values for the\n{\\sc fs} field are {\\bf wh$-$} and {\\bf wh$+$}.\n\n\\begin{itemize}\n\n\\item {\\bf Base adjective}:  All adjectives.\n\n\\item {\\bf Modifying adjective}: Adjectives that can occur in direct\nmodification contexts.  Ex. {\\it the {\\bf Chinese} man.}\n\n\\item {\\bf Predicative adjective}: Adjectives that can occur as the complement\nof a predicative verb. Ex. {\\it John was {\\bf happy}.}\n\n\\item {\\bf Predicative adjective w\/ sentential complement}: Adjectives that can\noccur as the complement of a predicative verb and that take a sentential\ncomplement.  Ex. {\\it John was {\\bf happy} that Mary left Bill.}\n\n\\item {\\bf Predicative adjective w\/ sentential subject}:  Adjectives that can\noccur as the complement of a predicative verb and that take a sentential\nsubject.  Ex. {\\it That John loves Mary is {\\bf great}!}\n\n\\end{itemize}\n\n\\subsection{Nouns}\n\nNouns are by far the largest category in the syntactic database, accounting for\nwell over 50\\% of the entries.  Proper nouns and pronouns both have the\npart-of-speech Noun.  Proper names, such as {\\it Danielle} and {\\it Nicholas}\nare not well-represented in the database, but geographic names, particularly\nplaces in England, generally are\\footnote{This reflects the origin of the\ndictionary from which the lexicon was originally extracted.}.  The frames for\nnouns are similar in many ways to the frames for adjectives, since nouns can\nmodify other nouns and occur in predicative sentences.  Other frames provide\ninformation about the use of the noun with determiners when forming noun\nphrases.  The frames for noun are presented below:\n\n\\begin{itemize}\n\n\\item {\\bf Base noun}:  All nouns.\n\n\\item {\\bf Noun Phrase with Determiner}: Nouns that can take a determiner when\nforming a noun phrase.  Ex. {\\it a {\\bf man}}; *{\\it a {\\bf jealousy}}\n\n\\item {\\bf Noun Phrase without Determiner}: Nouns that can appear without a\ndeterminer when forming a noun phrase.  Ex. {\\it {\\bf envy}}; *{\\it {\\bf\nplant}}\n\n\\item {\\bf Modifying noun}: Nouns that can modify other nouns.  Note that not\nall nouns can modify other nouns.  Proper nouns in general cannot modify other\nnouns, and specific lexical items may be restricted as well. Ex. {\\it\n{\\bf basketball} game}; *{\\it {\\bf John} car}\n\n\\item {\\bf Noun with sentential complement}: Nouns that take sentential\ncomplements.  Ex. {\\it the {\\bf fact} that Mary loves John...}\n\n\\item {\\bf Predicative noun}: Nouns that can occur as the complement\nof a predicative verb. Ex. {\\it John was a {\\bf man}.}\n\n\\item {\\bf Predicative noun w\/ sentential subject}:  Nouns that can\noccur as the complement of a predicative verb and that take a sentential\nsubject.  Ex. {\\it That John loves Mary is a {\\bf crime}.}\n\n\\end{itemize}\n\nBecause this lexicon is used in the XTAG system, the lexicon often indicates\nprecise syntactic behavior, rather than simply placing a general label on a\nlexical item.  For the class of nouns, this is seen in the specification of\nnouns with respect to their co-occurrence with determiners.  Instead of\nassigning a general label as as `common noun' or `mass noun', the noun frames\nexplicitly indicate whether certain forms of the noun can appear with or\nwithout a determiner.  However, since the syntactic database is indexed on root\nforms only, the morphology of the lexical item is not available.  Instead, the\n{\\sc FS} field is used to indicate any restrictions on a particular use of a\nlexical item.  For example, in Table~\\ref{syn-entries}, the noun {\\it map}\noccurs twice.  The first time that it appears, it selects the {\\bf\nNoun\\_Determiner\\_required} frame.  The feature structures associated with it\nindicates only that the noun is not a wh-word, and that it is not reflexive.\nNo restrictions are made with respect to its morphology.  In contrast, the\nsecond entry, which selects the {\\bf Noun\\_Determiner\\_not\\_required} has {\\bf\nplural} as part of its {\\sc FS}.  This indicates that the noun for this frame\nis restricted to its plural form.  Hence {\\it map} can only occur with a\ndeterminer, but {\\it maps} is free to occur both with or without one.  Nouns\nthat belong to the class of so-called `mass nouns' would not have the {\\bf\nplural} restriction on the entry that selects the {\\bf\nNoun\\_Determiner\\_not\\_required} frame, thereby indicating that the singular\nform is also allowed to occur without a determiner.\n\n\\subsection{Verbs}\n\nVerbs, with their varied subcategorization frames, are perhaps the most\ninteresting lexical items in a syntactic lexicon.  There are over 8100\nverbs (not including auxiliary verbs) that make up almost 9000 entries\nin the database.  There are 19 different frames that the verbs can select,\nincluding transitive, intransitive, sentential complement, sentential subject,\nverb particle constructions (transitive and intransitive), double objects with\nshifting, double objects without shifting, and light verb constructions.\n\nAs with the nouns, the {\\sc FS} field is used to provide a more concise format\nfor specifying the frames for each lexical item.  For the verbs, the {\\sc FS}\nfield is used to specify the difference between ergative and non-ergative\ntransitive verbs, as can be seen in the {\\it have} entry in\nTable~\\ref{syn-entries}, and is also used heavily for further differentiating\nthe frames for verbs that take sentential complements.  There are two frames\nfor sentential complements - {\\bf Sentential\\_Complement} and {\\bf\nNP\\_and\\_Sentential\\_Complement}.  Either of these can occur with the feature\nstructures {\\bf Infinitive\\_Complement}, {\\bf Indicative\\_Complement}, or {\\bf\nPredicative\\_Comp\\-le\\-ment}.  This reduces the number of values for {\\sc\nFRAME} that are necessary to cover all of the possible lexical environments,\nand also allows for easier searches across categories.  To find all the verbs\nthat take infinitive complements, one can simply search on the {\\bf\nInfinitive\\_Complement} feature structure, rather than having to specify each\nframe that could fill this role.  Table~\\ref{verb-entries} shows some values\nfor various verbs that take sentential complements.\n\n\\begin{table}[ht]\n\\begin{tabular}{ll}\n\nINDEX: &want \\\\\nENTRY: &want \\\\ \nPOS: &Verb \\\\ \nFRAME: &Sentential\\_Complement \\\\\nFS: &Infinitive\\_Complement   \\\\ \nEX: &Dan wants to finish this paper. \\\\\n\\\\\nINDEX: & want \\\\\nENTRY:& want \\\\ \nPOS:& Verb \\\\ \nFRAME:& NP\\_and\\_Sentential Complement \\\\\nFS:& Infinitive\\_Complement\\\\ \nEX:& Dan wants Al to finish this paper. \\\\\n\\\\\nINDEX: &think \\\\\nENTRY: &think \\\\ \nPOS:& Verb\\\\ \nFRAME:& Sentential\\_Complement \\\\\nFS:& Indicative\\_Complement\\\\ \nEX:& Dan thought that the paper was done. \\\\\n\\\\\nINDEX:& think \\\\\nENTRY:& think \\\\ \nPOS:& Verb\\\\ \nFRAME:& Sentential\\_Complement \\\\\nFS:& Infinitive\\_Complement\\\\ \nEX:& Doug thought to clean the kitchen.\\\\\n\\\\\nINDEX:& think \\\\\nENTRY:& think \\\\ \nPOS:& Verb\\\\ \nFRAME:& Sentential\\_Complement \\\\\nFS:& Predicative\\_Complement\\\\ \nEX:& Dan thought Carl a jerk.\n\n\\end{tabular}\t\n\\caption{Verbs with Sentential Complements}\n\\label{verb-entries}\n\\end{table}\n\n\\subsubsection{Auxiliary verbs}\n\nThe lexical entries for auxiliary verbs are very closely tied to the XTAG\nanalysis, which orders the auxiliary verbs based on their morphological\nforms.  Each entry in the lexicon is restricted via the {\\sc FS} field to only\na certain form of the auxiliary verb ({\\bf present}, {\\bf past}, {\\bf\nppart}, etc), which also indicates what other forms that it can go\non\\footnote{For a more detailed description of this and other XTAG analyses,\nplease see the XTAG Technical Report \\cite{xtag-tech}.}.  Table\n\\ref{aux-verbs} shows the entries for the auxiliary verbs for the sentence {\\it\nJohn should have been waiting.}\n\n\\begin{table}[ht]\n\\begin{tabular}{ll}\n\nINDEX: &should \\\\\nENTRY: &should \\\\ \nPOS:& Verb \\\\ \nFRAME:& Auxiliary\\_Verb\\\\\nFS:& Indicative, Present, Goes\\_on\\_Base  \\\\ \n\\\\\nINDEX:& have \\\\\nENTRY:& have \\\\ \nPOS:& Verb \\\\ \nFRAME:& Auxiliary\\_Verb\\\\\nFS:& Base, Goes\\_on\\_Past\\_Participle  \\\\ \n\\\\\nINDEX:& be \\\\\nENTRY:& be\\\\ \nPOS:& Verb \\\\ \nFRAME:& Auxiliary\\_Verb\\\\\nFS:& Past\\_Participle, Goes\\_on\\_Gerund\n\n\\end{tabular}\t\n\\caption{Example Auxiliary Verb Entries}\n\\label{aux-verbs}\n\\end{table}\n\n\n\\subsection{Adverbs}\n\nA syntactic lexicon for adverbs is particularly useful because adverbs are so\nidiosyncratic as to where they can occur in a sentence.  Although there are\nonly 169 adverbs in the syntactic lexicon, but there are 15 different {\\sc\nframe} values that they can select.  These include basic adverb, pre and post\nverb phrases, pre and post sentences, pre and post adjective, pre-adverb,\npre-preposition, pre-noun, etc.  Table \\ref{adv-entries} shows some selected\nadverb entries.\n\n\\begin{table}[ht]\n\\begin{tabular}{ll}\n\nINDEX:& ahead \\\\\nENTRY:& ahead \\\\\nPOS:&\tAdverb \\\\\nFRAME:& Base\\_Adverb \\\\\n& Post-VP \\\\\n& Pre-PP \\\\\n\\\\\nINDEX:& essentially \\\\\nENTRY:& essentially \\\\\nPOS:& Adverb \\\\\nTREES:& Base\\_Adverb \\\\\n& Pre-VP \\\\\n& Pre-S \\\\\n& Post-S \\\\\n\\\\\nINDEX:& even \\\\\nENTRY:& even \\\\\nPOS:& Adverb \\\\\nFRAME:& Base\\_Adverb \\\\\n& Pre-VP \\\\\n& Pre-Adj \\\\\n& Pre-Noun \\\\\n& Pre-PP \\\\\n\\\\\nINDEX:& very \\\\\nENTRY:& very \\\\\nPOS:& Adverb \\\\\nFRAME:& Base\\_Adverb \\\\\n& Pre-Adj \\\\\n& Pre-Adv\n\n\\end{tabular}\t\n\\caption{Some Adverb Lexical Entries}\n\\label{adv-entries}\n\\end{table}\n\n\\section{File Formats}\n\nThe information in the syntactic database is available both in an ASCII 'flat'\nfile, and a hashed database format.  The ASCII file contains one entry per\nline, and each field is clearly marked.  This format is easily usable by\nvarious UNIX$^{tm}$ utilities such as {\\it grep} and {\\it awk}, and it can be\neasily parsed by custom programs.  \n\nThe hashed database format is very useful for programs that need quick access\nto the information in the database.  Each entry is indexed under the {\\sc\nindex} key, and a single call to the database for a particular index returns\nall of the entries that share that index.  This makes it particular useful for\nparsers.  The database uses an encoding scheme for the {\\sc pos}, {\\sc frame},\nand {\\sc FS} fields, which condenses the space required for the database and\nshortens the search time for non-index fields.  All of the entries for a given\nlexical index can be retrieved in 1.6 msecs, on average.\n\n\\section{Interface}\n\\label{interface}\n\nAlthough the format of the flat file is excellent for various file utilities\nprograms, and the database format works well for retrieving entries quickly,\nneither is particularly well-suited for human readability.  The X-windows\ninterface{\\footnote{The interface uses the MIT Athena Toolkit, which is\ndistributed with the standard MIT X release.} for the syntactic database allows\nusers to easily look at the database.  Searching is available not only on the\n{\\sc index} under which the lexical item is stored, but also on all other\nfields, with the exception of the {\\sc ex} field.  Searches may also be done on\ncombinations\\footnote{We hope to add expand this in the future to include full\nregular expression searches.} of fields.  For instance,\none could search on {\\sc POS}~=~{\\bf Noun} and {\\sc FS}~=~{\\bf wh$+$} to find\nthe set of all wh+ nouns ({\\it what}, {\\it who}, {\\it whom}, {\\it which}, {\\it\nwhen}).  Figure \\ref{interface-fig} shows the interface after a search has been\ndone on the index {\\it need}.  All of the entries with that index are listed in\na scroll window, which can be browsed through using the {\\bf Next} and {\\bf\nPrevious} buttons, or specific entries can be clicked on, and the entire record\nwill show in the upper window.  The results of searches can be saved to a file\nto create smaller `custom' lexicons.  In addition to searching the database,\nusers can also easily add, delete and modify individual entries, tailoring the\nsyntactic database to fit their needs.  Users may also delete all entries found\nin a given search, and we hope to add the capacity to modify a entire set of\nentries in the future.\n\n\\begin{figure}[htb]\n\\centering\n{\\psfig{figure=xsyn-image.eps,height=4.2in}}\n\\caption{Result of a search on the index {\\it need}}\n\\label{interface-fig}\n\\end{figure}\n\n\n\\section{Statistics}\n\nStatistics were gathered on the coverage of the syntactic lexicon on the IBM,\nATIS, WSJ, and Brown corpora.  These corpora were chosen because they have been\ntagged and hand corrected by the TreeBank project \\cite{santorini90}.  The data\nin Table \\ref{corpora-stats} show the coverage of the lexicon on various\ncorpora.  A lexical item\/part-of-speech pair is counted as a {\\bf hit} if the\nlexical item is in the syntactic lexicon with the indicated tag.  No attempt\nwas made to determine if the lexicon had the correct frame needed to parse the\nsentence.\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{|l|r|r|c|} \\hline\n& Number & Total \\# & Percent\\\\\nCorpus & of Hits & of Words & Hit\\\\ \\hline\n\nWSJ &  1974528 &  2462557 & 80.18\\% \\\\\\hline\n\nBrown & 799904 & 991008 & 80.72\\% \\\\\\hline\n\nIBM  & 60944 & 68800 & 88.58\\%\\\\ \\hline\n\nATIS &  10156 &  13791 & 73.64\\%\\\\ \\hline\n\\end{tabular}\n\\caption{Percentage of Hits for various corpora}\n\\label{corpora-stats}\n\\end{center}\n\\end{table}\n\n\\begin{table*}[ht]\n\\begin{center}\n\\begin{tabular}{|l|r|c|c|c|c|c|} \\hline\n& Number of & Percent & Percent & Percent & Percent & Percent\\\\\nCorpus &  Non-hits & Proper N & Nouns & Adj & Adv & Verbs\\\\ \\hline\n\nWSJ & 488029  & 43.8\\%  & 30.7\\% & 13.8\\% & 5.7\\% & 1.3\\% \\\\ \\hline\n\nBrown & 191104 & 26.2\\% & 40.6\\% & 14.8\\% & 7.4\\% & 1.8\\% \\\\ \\hline\n\nIBM  & 7856 & 17.1\\% & 56.9\\% & 11.3\\% & 2.8\\% & 2.5\\% \\\\ \\hline\n\nATIS & 3635  & 67.4\\%  & 14.0\\% & 1.6\\% & 0.6\\% & 2.4\\% \\\\ \\hline\n\\end{tabular}\n\\caption{Percentage of missing words for various Parts of Speech}\n\\label{missing-stats}\n\\end{center}\n\\end{table*}\n\nBecause the syntactic lexicon contains only the root form of lexical entries,\nthe inflected form was first looked up in the morphology database \\cite{karp92}\nto retrieve the root form, and then that was used for the syntactic lexicon.\nItems that were not found in the morphological database were counted against\nthe syntactic lexicon, as the morphology database is a superset of the\nsyntactic database\\footnote{Because these databases are being used in an actual\nparser, an attempt was made some time ago to make ensure that all words in the\nsyntactic lexicon appear in the morphological database.  Although the databases\nmay have diverged slightly since then, it should not be statistically\nsignificant.}.  The statistics in Table \\ref{corpora-stats} are over all word\noccurrences in the corpora\\footnote{Numbers and the genitive marker ('s) were\ntaken out before the statistics were compiled.}, so words that occur frequently\nare given more weight.\n\nNot surprisingly, nouns and proper nouns\\footnote{Although we do not\ndistinguish nouns and proper nouns in the syntactic lexicon, the TreeBank tags\ndo make this distinction, and it seemed useful to continue this distinction for\nthis part of the analysis.} comprise the largest category of words missed,\nfollowed by adjective, adverbs, and verbs.  Table \\ref{missing-stats} shows the\npercentage of each of these categories in the list of items not found.  Again,\nthis is a percentage of word occurrences in the corpora.\n\nAs Table \\ref{missing-stats} indicates, the majority of the missing items are\neither nouns or proper nouns (66.8\\% - 81.4\\%).  This is not surprising, nor\nparticularly distressing, as nouns tend to be the easiest items to `guess'\ninformation about.  Verbs, which tend to be the hardest, are reasonably\nwell-covered in this lexicon.  The number of adjectives not covered, however,\nseems fairly high, and we plan to add a number of those missing to the\nsyntactic lexicon.\n\n\\section{Future Work}\n\nThe lexicon in its present form does not provide a mechanism to specify\npreferences of lexical items for certain syntactic structures.  As part of\nfuture enhancements to the lexicon we hope to associate probabilities with each\nentry.  The probabilities will reflect the affinity of the lexical item for the\nsyntactic structure associated with that entry.  These probabilities will be\ncomputed from parsed corpora.\n\nIt has been observed quite conclusively in recent work in lexicography that\ncertain combinations of words co-occur more often than would be expected if\nthey corresponded to arbitrary usages of the individual words.  Collocational\ninformation has been shown to be of immense use in pruning the search space for\na parser. We hope to eventually extract collocational\ninformation from the corpora and make it a part of the syntactic lexicon.\n\n\\section{Related Work}\n\\label{related-work}\n\nThe syntactic lexicon was developed as part of the XTAG project\n\\cite{xtag-notes94} at the University of Pennsylvania under the direction of \nDr.  Aravind Joshi.  The XTAG system is a wide-coverage parser and grammar for\nEnglish based on the Tree Adjoining Grammar (TAG) formalism \\cite{joshi75}.\nThe English grammar consists of 3 sections - a morphology database, a syntactic\ndatabase, and a tree grammar.  Together with a parser and an X-windows\ninterface, they comprise the XTAG system.  Both the morphology \\cite{karp92}\nand syntactic databases are available separately.  The entire XTAG system is\nalso freely available to the NLP research community.  Information about the\nentire XTAG system and FTP instructions may be obtained by writing\nxtag-request@linc.cis.upenn.edu.\n\n\\section{Computer Platform}\n\nThe syntactic lexicon and accompanying interface were developed on the Sun\nSPARC station series, as were the other tools mentioned in\nSection~\\ref{related-work}.  All of the XTAG tools, including the syntactic\nlexicon and interface, are freely available without limitation through\nanonymous FTP to {\\bf ftp.cis.upenn.edu}.  The syntactic lexicon and\naccompanying programs together require about 9MB of space (for both the ASCII\nand DB versions of the lexicon).  Please send mail to\nlex-request@linc.cis.upenn.edu for current FTP instructions or for more\ninformation.\n\n\\newpage\n\\bibliographystyle{aaai-named}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and statement of result}\nConsider the initial value problem (IVP) of the \\emph{Korteweg-de Vries} (KdV) equation:\n\\begin{equation}\\label{KdV}\n\\left\\{\\begin{array}{l}\nu_t + u_{xxx} - 6 u u_x  =0\\\\\nu(0,x) = u_0(x)\n\\end{array}\\right.,\n\\end{equation}\nfor $x\\in \\mathbb{R}$ and rough initial data $u_0$ in the Sobolev space $H^s$.\n\nIt is well known that the KdV equation exhibits special travelling\nwave solutions, known as solitons -- indeed such solutions provided\nmuch of the historical impetus to study the equation.  Explicitly, up\nto a spatial translation, these solutions may be written in the form\n\\begin{equation}\nu:=R_c(x-ct),\n\\end{equation}  \nwhere $c>0$ and\n\\begin{equation}\nR_c:=-\\frac{c}{2}\\sech^2\\left(\\frac{\\sqrt{c}x}{2}\\right).\n\\end{equation}\n\nLet us summarise the well-posedness theory and stability results. \nThe initial value for KdV is known to be globally\nwell-posed\\footnote{See \\cite{MR2233925} for a discussion on the\n  subtleties in the definition of well-posedness.} for $s\\geq\n-\\frac{3}{4}$, (see \\cite{MR2531556}, \\cite{MR1969209}, \n\\cite{MR2233925} and \\cite{MR2501679}).  The problem is known to be ill-posed for $s<\n-\\frac{3}{4}$ in the sense that the flow map cannot be uniformly\ncontinuous  \\cite{MR1813239}.  One may hope for Hadamard\nwell-posedness for $s\\ge -1$, (cf. \\cite{MR2353092}, \\cite{2011arXiv1112.5177L} and\n\\cite{MR2267286}). Using the inverse scattering transform, Kappeler and Topalov proved that the flow map extends \ncontinuously to $H^{-1} $ in the periodic case, \nwhich provides motivation to address  the supposedly simpler question of well-posedness in $H^{-1}$ \non the real line.  On the other hand Molinet \\cite{MOLINET} has shown that \nno well-posedness can possibly hold below $s=-1$:\nthe solution map $u_0\n\\to u(t)$ does not extend to a continuous map from $H^s$, for $s < -1$, to\ndistributions. \n\n\nOrbital stability of the soliton in the energy space $H^1$ follows\nfrom Weinstein's convexity argument \\cite{MR820338}, this argument even holds for other\nsub-critical gKdV equations. Weinstein's argument is at the basis of a\nconsiderable amount of work since then, with one direction culminating \nin the seminal work of Martel and Merle to some version of asymptotic\nstability, again in the energy space \\cite{MR1826966}.  Merle and Vega\nproved orbital stability and asymptotic stability of the soliton\nmanifold in $L^2$ using the Miura map \\cite{MR1949297} in a similar\nfashion to our approach.  Their approach to the stability of the kink\nhowever is closer to the arguments of Martel and Merle for generalised\nKdV.\n\nWe now present our principal results.\n\n\\begin{thm}[New $H^{-1}$ a priori estimate for KdV]\\label{aprioriTHM}\nSuppose $s \\geq -\\frac34$ and $u\\in C([0,\\infty);H^s(\\mathbb{R}))$ is a solution to \\eqref{KdV}, then\n\\begin{equation}\\label{apriori}\n\\HN{-1}{u(t,\\cdot)} \\lesssim \\footnotemark  \\HN{-1}{u_0}+ \\HN{-1}{u_0}^3   ~~\\text{ for }~~ t\\in[0,\\infty).\n\\end{equation}\\footnotetext{Throughout this article we will adopt the notation $a\\lesssim b$ to signify $a\\leq Cb$, where $C$ is an insignificant constant.}\n\\end{thm}\n\\begin{rem}\nApplying scaling, the dependence on the $H^{-1}$ norm of the initial data in \\eqref{apriori} can be made more explicit, i.e. if $\\lambda$ is such that\n\\[ 0 < \\lambda  \\le    \\HN {-1}{ u_0}^{-2},   \\]\nthen we have \n\\begin{equation*}\n  \\HN{-1}{u(t,\\lambda \\cdot)} \\lesssim  \\HN{-1}{u_0(\\lambda \\cdot ) }  ~~\\text{ for }~~ t\\in[0,\\infty).\n\\end{equation*}\n\\end{rem}\n\n\\begin{thm}[Orbital stability of KdV solitons]\\label{KdVstab} \nThere exists an $\\varepsilon >0$ such that \nif $u\\in C([0,\\infty);H^s(\\mathbb{R})\\cap H^{-3\/4}(\\mathbb{R}))$ is a solution to (\\ref{KdV}), for some integer $s\\geq-1$, satisfying $\\HN{-1}{R_c-u_0}<\\varepsilon c^{1\/4}$ for some $c>0$, then there is a continuous function $y:[0,\\infty)\\to\\mathbb{R}$ such that \n\\[ \\HN{s}{ u - R_{ c}(x-y(t))} \\leq \\gamma_{s}(c,\\HN{s} {R_c-u_0})\\] \nfor any $t\\geq 0$, where $\\gamma_{s}:(0,\\infty)\\times [0,\\infty)$ is a continuous function, \npolynomial in the second variable, which  satisfies $\\gamma(\\cdot,0)=0$.\n\\end{thm} \n\n\\begin{rem}\nIf we rescale $c$ to $4$, we obtain a more precise result. The smallness assumption becomes \n\\[ \\HN {-1}{ 4c^{-1} u_0(2c^{-1\/2}x) - R_4 } \\le \\varepsilon, \\]\nwhich is weaker and more natural than the assumption of the theorem. \n\\end{rem}\n\n\n\\begin{thm}[Asymptotic stability of KdV solitons]\\label{KdVAsymStab}\n  Given real $\\gamma>0$ and integer $s\\geq -1$, there exists an\n  $\\varepsilon_{\\gamma} >0$ such that if $u\\in\n  C([0,\\infty);H^s(\\mathbb{R})\\cap H^{-3\/4}(\\mathbb{R}))$, is a solution to\n  (\\ref{KdV}), satisfying \n\\[ \\HN{-1}{R_c-u_0}<\\varepsilon_{\\gamma}  c^{1\/4} \\]\nfor $c>0$, then there is a continuous function\n  $y:[0,\\infty)\\to\\mathbb{R}$ and $\\tilde c>0$ such that\n\\begin{equation*}  \n \\lim_{t \\to \\infty} \\norm{ u - R_{\\tilde c}(x-y(t)) }_{H^{s}( (\\gamma t , \\infty) ) } =  0 \n\\end{equation*} \nfor any $t\\geq 0$.  Moreover we have the bound $\\abs{c-\\tilde  c}\\lesssim  c^{\\frac34} \\HN{-1}{R_c-u_0}   $.\n\\end{thm} \nThe decay follows from an explicit quantitative estimate in\nProposition \\ref{stableftright2} for $H^{-1}$, and similar estimates for higher \nnorms in   Corollary \\ref{mKdVAsymStab} .  The estimates we obtain are\nsufficiently strong to obtain existence of weak solutions by a\nstandard approximation and compactness argument.\n\n\\begin{thm}[Existence of global $H^{-1}$ weak solutions to KdV IVP]\\label{mainTheorem}\nFor any $u_0\\in H^{-1}$, there exists a weak solution $u$ to\n(\\ref{KdV}) satisfying\n\\begin{align}\nu\\in C_{\\omega}([0,\\infty); H^{-1}(\\mathbb{R})), \\footnotemark\\\\\nu\\in L^2([0,T]\\times[-R,R]) ~~\\text{for any}~~ R,T<\\infty,\\\\\nu(t,\\cdot) \\to u_0 \\qquad \\text{ in } H^{-1} ~~\\text{as}~~t\\downarrow 0.\n\\end{align}\\footnotetext{Here $C_{\\omega}([0,\\infty); H^{-1}(\\mathbb{R}))$ denotes the space of weakly continuous functions from $\\mathbb{R}$ to $H^{-1}$.}\nFurthermore $u$ satisfies the bounds given in Theorem \\ref{aprioriTHM}.\n\\end{thm}\n\n\nA closely related problem to the initial value problem of the\nKorteweg-de Vries equation is that of the \\emph{modified} Korteweg--de\nVries (mKdV) equation:\n\\begin{equation}\\label{mKdV}\n\\left\\{\\begin{array}{l}\nu_t + u_{xxx} - 6 u^2 u_x=0\\\\\nu(0,x) = u_0(x)\n\\end{array}\\right.,\n\\end{equation}\nfor $x\\in \\mathbb{R}$ and initial data $u_0$.  \n\nAn explicit family of solutions of the mKdV equation, called \\emph{kink}\nsolutions, can be written up to translations as\n\\[Q_\\lambda(t,x):=\\lambda\\tanh\\left(\\lambda x+2\\lambda^3 t\\right),\\]\nfor any $\\lambda>0$.\n\nThe mKdV problem and the KdV may be connected via a differential transformation known as the \\emph{Miura map}:\n\\begin{equation}\nu\\mapsto u_x+u^2;\n\\end{equation}\nwhich sends solutions of (\\ref{mKdV}) to solutions of (\\ref{KdV}).  To see this property formally, set\n\\begin{align*}\n\\KdV(u)&=u_t + u_{xxx} - 6 u u_x,&\\\\\n\\mKdV(u)&=u_t + u_{xxx} - 6 u^2 u_x,&~\\text{and}\\\\\n\\miura(u)&=u_x+u^2.&\n\\end{align*}\nOne can then easily check that\n\\begin{equation}\\label{miuraid}\n\\KdV(\\miura(u))=(\\mKdV(u))_x+2u\\cdot \\mKdV(u),\n\\end{equation}\nfrom which it follows that $\\KdV(\\miura(u))=0$ whenever $\\mKdV(u)=0$.  Additionally, note the mKdV equation satisfies the reflection symmetry: if $u$ is a solution to \\eqref{mKdV}, then $-u$ is also a solution. Hence if we define\n\\begin{equation*}\n\\miura^*(u):=\\miura(-u)=-u_x+u^2,\n\\end{equation*}\nthen $\\miura^*(u)$ maps solutions to the mKdV equation to solutions to the KdV equation.\n\nThe Korteweg--de Vries equation is invariant under the \\emph{Galilean\n  transformation}:\n\\begin{equation}\nu(t,x)\\mapsto u(t,x-ht) - \\frac{h}{6},\n\\end{equation}\nfor $h\\in \\mathbb{R}$, i.e. if $u$ is a solution to the KdV equation, then\nits image under the above transformation is also a solution, which is\neasily verified.\n\n\n\n\n\n\nThe Korteweg--de Vries equation also satisfies the following scaling symmetry:\n\\begin{equation}\\label{scaling}\nu(t,x)\\mapsto\\frac{1}{\\lambda^2}u(\\frac{t}{\\lambda^3},\\frac{x}{\\lambda}), \n\\end{equation}\nfor $\\lambda>0$ and $\\dot H^{-\\frac32}$ is the critical space. \n\nIn Section \\ref{invmiurasec} we will show how to use the Miura map\ncombined with the Galilean symmetry to relate mKdV solutions near a\nkink solutions to either KdV solutions near $0$, or to KdV solutions\nnear a soliton.  This will afford us the freedom to choose the most\nconvenient setting in order to prove the stated results.  The $H^{-1}$\na priori estimate of Theorem \\ref{aprioriTHM} will then follow as a\nconsequence of the $L^2$ stability of the kink (Theorem \\ref{stab}).\nThe orbital (Theorem \\ref{KdVstab}) and asymptotic stability (Theorem\n\\ref{KdVAsymStab}) of the soliton in the $H^{-1}$ norm will follow\nfrom the corresponding statement for the mKdV kink in $L^2$ (Theorem\n\\ref{stab} and Theorem \\ref{mKdVAsymStabL2}).  Higher conserved\nenergies imply stability of the trivial solution in $H^s$ for\nnonnegative integers $s$, and Kato's local smoothing argument along a\nmoving frame implies asymptotic stability of the trivial solution to\nthe right. We use the Miura map to derive \n orbital and asymptotic stability \nof the soliton for KdV, as well as orbital and asymptotic  stability of the kink for mKdV in higher norms, requiring  smallness of the deviation only in $H^{-1}$, \nsee Corollary \\ref{highernorm} and the proof of Corollary\n\\ref{mKdVAsymStab}.\n\nThe Miura map has been used in a simpler setting by Kappeler et al.\\\n\\cite{MR2189502}. Their results are limited by the fact that the Miura\nmap is not invertible. Our additional ingredient is the shift of the\ninitial data using the Galilean invariance.  To the best of our knowledge\nthe corresponding results on the Miura map are new, and we believe\nthem to be appealing and of independent interest.\n\nOf course this is intimately related to the integrable structure of\nKdV and mKdV, and also the Lax-Pair is clearly in the\nbackground. Nevertheless we do not explicitly use the integrable\nmachinery, and the use of elementary key elements of the theory of\nintegrable systems in combination with a PDE oriented approach seems\nto be new and promising.\n\nIt is worthwhile to point out that unlike the corresponding asymptotic\nstability results for generalised KdV, the scale $\\tilde c$ is\nindependent of time. This holds since the scale of the kink is related\nto its size at infinity, and this does not change by adding $L^2$\nperturbations.\n\n\n\\section{Inverting the Miura map}\\label{invmiurasec}\n\nKappeler et al.\\ showed in the paper \\cite{MR2189502} that if the\ninitial data $u_0\\in H^{-1}$ is contained in the image of $L^2$ under\nthe Miura map restricted to $L^2$, then there exists a global weak\nsolution to the IVP \\eqref{KdV}. The proof consists of constructing a\nweak solution to mKdV corresponding to initial data in the preimage\nunder the Miura map of the original initial data, and then\ntransforming the solution back, under the Miura map, to a solution to\nKdV.  The following proposition is one of the key tools used by\nKappeler et al.\\ to characterise the range of the Miura map.\n\n\\begin{prop}\\cite{MR2189502}\\label{spectrum}\nLet $ u_0 \\in H^{-1}_{\\text{loc}}$. The following three statements are equivalent. \n\\begin{enumerate}\n\\item The Schr\u00f6dinger operator $H_{u_0}:=-\\partial_{xx}+ u_0$ is positive\n  semi-definite.\n\\item There exists a strictly positive function $\\phi $ with $-\\phi_{xx} + u_0 \\phi=0$.\n\\item  $u_0\\in H^{-1}_{\\text{loc}}$ is in the range of the Miura map on $L^2_{\\text{loc}}$.  \n\\end{enumerate}\n\\end{prop} \n\nIn order to remove the restrictions on the initial data imposed in \\cite{MR2189502}, we will employ the use of\nthe Galilean transformation in order to transform KdV into the range of the Miura map.    This allows us to\nlink rough $H^{-1}$ KdV initial data to mKdV initial data.  The\ncorresponding mKdV initial data will be in the form of a sum of an\n$L^2$ function and a $\\tanh$ kink.  The authors would like to note\nthat the original idea to use such an argument  was somewhat motivated by\nthe papers \\cite{MR1949297} and \\cite{TZV} -- related to the $L^2$\nstability of soliton solutions to the KdV equation and KP-II equation\nrespectively.\n\nAppendix \\ref{appenSchr} contains results for  Schr\u00f6dinger operators with $H^{-1}$ potentials, which we will use below.  \n\nOur aim now is to\nconstruct an ``inverse'' of the Miura map for Galilean transformed\ninitial data.  The Galilean transformation essentially adds a\nconstant to the potential of the Schr\u00f6dinger operator corresponding to\nthe initial data. We easily achieve positive definiteness by adding a\nlarge enough constant; the caveat being that the initial data will no\nlonger remain in $H^{-1}$.\n\nGiven initial data $u_0\\in H^{-1}$, applying the Galilean symmetry to\n$u_0$, with $h$ set to $-6\\lambda^2$, for some $\\lambda>0$ and $t=0$, yields the function $u_0+\\lambda^2$.  Now\nconsider the problem of finding a function $r\\in L^2$ that is in the\npreimage of $u_0+\\lambda^2$ under the Miura transformation.  Observe that\n$\\miura( \\lambda\\tanh (\\lambda \\cdot))=\\lambda^2$; it then seems natural to consider\nthe problem\n\\begin{equation}\\label{prob1}\n(r+\\lambda \\tanh \\lambda x)_x+(r +\\lambda\\tanh \\lambda x)^2=u_0+\\lambda^2.\n\\end{equation}\n\nFor the problem of stability of solitons, we assume we are given some initial data $u_0\\in H^{s}$, where $s\\geq -\\frac{3}{4}$.  Applying the Galilean\ntransform with $h$ as above, and   noting that $\\miura^*(\\lambda\n\\tanh(\\lambda\\cdot))=\\lambda^2-2\\lambda^2\\sech^2(\\lambda\\cdot)$,\nwe are led to consider the problem\n\\begin{equation}\\label{prob2}\n-(r+\\lambda \\tanh (\\lambda \\cdot))_x+(r +\\lambda\\tanh (\\lambda \\cdot))^2= u_0+\\lambda^2.\n\\end{equation}\n\nWe now state sufficient and necessary conditions for the problems \\eqref{prob1}) and \\eqref{prob2} to have a solution.\n\n\\begin{prop}\\label{groundstates} Let $\\lambda>0$. The  \n ground state energy  of $H_{u_0}$, $u_0 \\in H^{-1}$ \nis $-\\lambda^2$ if and only if there exists $r \\in L^2-\\lambda\\tanh(\\lambda \\cdot)$ such that \n\\[   \\miura(r) = u_0+\\lambda^2. \\]\nThe spectrum of $H_{u_0}$ is contained in $(-\\lambda^2,\\infty)$ if and only \nif there  exists $r \\in L^2 + \\lambda\\tanh(\\lambda \\cdot)$ with \n\\[ \\miura(r) =u_0+\\lambda^2. \\]\n\\end{prop} \n\n\\begin{proof} \nLet $\\phi$ be the ground state. Observe then $r= \\frac{d}{dx} \\ln \\phi$ satisfies the Ricatti equation (see Appendix \\ref{appenSchr}) \n\\[ r_x+r^2 = u_0 +\\lambda^2. \\]\nThen as a consequence of Lemma \\ref{asymp} we have either\n\\[ r-\\lambda  \\in L^2(0,\\infty) \\quad\\text{or}\\quad r+\\lambda \\in L^2(0,\\infty). \\]\nNote however the property that\n\\[  e^{\\int_0^{x} r } \\in L^2 \\]\nenforces $r+\\lambda  \\in L^2(0,\\infty)$. Similarly, we obtain $r-\\lambda \\in L^2(-\\infty,0)$ and thus \n\\[ r + \\lambda \\tanh(\\lambda x) \\in L^2. \\]\nHence $u_0+\\lambda^2$ is in the range of the Miura map on $ - \\lambda \\tanh(\\lambda x)+ L^2$ if the ground state energy is $-\\lambda^2$. \n\nNow assume that \n\\[  \\lambda^2+  u_0 = r_x + r^2 \\quad \\text{and}\\quad   r+\\lambda \\tanh(\\lambda x) \\in L^2.   \\]\nThen $ \\phi = e^{\\int_0^{x} r}$ is a strictly positive function in $H^1$ satisfying\n\\[(H_{u_0}+\\lambda^2)\\phi=0,\\]\ni.e. $\\phi$ is the ground state with ground state energy $-\\lambda^2$. \n\n  \nNow we turn to the case when the spectrum is contained in $(-\\lambda^2,\\infty)$. Since $H_{u_0+\\lambda^2}$ is positive semi-definite, by Proposition \\ref{spectrum}, there exists strictly positive $\\phi\\in L^2_{\\text{loc}}$ satisfying\n\\begin{equation}\\label{abc}\n(H_{u_0} + \\lambda^2)\\phi=0.\n\\end{equation} \nNote that $\\phi\\notin L^2$, otherwise $\\phi$ would be the ground state.  Since $\\frac{d}{dx} \\ln \\phi$ solves the Ricatti equation, it follows by Lemma \\ref{asymp} that either $\\phi$ grows exponentially as $x\\to \\infty$ or as $x\\to -\\infty$. \n\nWe aim to construct a solution $\\tilde \\phi$ to \\eqref{abc} satisfying\n\\begin{equation}\\label{expbds}\n\\tilde \\phi(x) \\to  \\infty  \\text{ as } x \\to \\pm \\infty. \n\\end{equation} \n\nSuppose $\\phi$ is not such a solution; then without loss of generality we can assume \n\\[ \\phi(x) \\to \\infty \\qquad \\text{ as } x \\to \\infty, \\]\nand\n\\[ \\phi(x) \\to 0 \\qquad \\text{ as } x \\to -\\infty. \\]\n\n We obtain using Lemma \\ref{asymp}  that \n$\\frac{d}{dx} \\ln \\phi - \\lambda \\in L^2$.   It it is then not difficult to show that\n\\begin{equation}\n\\tilde \\phi(x)=C\\phi(x)+ \\phi(x)\\int_0^x \\phi^{-2}(s) d\n\\end{equation}\nfor large $C>0$, is a solution to \\eqref{abc}, satisfying the growth conditions \\eqref{expbds}. \n\nWe now define\n   $r= \\frac{d}{dx} \\ln  \\tilde\\phi$.  It  satisfies \nthe Ricatti equation; moreover, by Lemma \\ref{asymp},  \n\\[ r-\\lambda \\tanh(\\lambda x) \\in L^2. \\] \nThus $u_0+\\lambda^2$ is in the range of the Miura map restricted to $L^2+\\lambda \\tanh(\\lambda x)$ if the\nspectrum of $u_0$ is contained in $(-\\lambda^2, \\infty)$.\n\nNow suppose that $u_0+\\lambda^2 = r_x + r^2$ for $r - \\lambda \\tanh (\\lambda x) \\in L^2$; hence $\\phi = e^{\\int_0^x r }$\nsatisfies the equation\n\\[ -\\phi'' + u_0 \\phi = -\\lambda^2 \\phi, \\]\nand \n\\[ \\phi(x) \\to \\infty \\text{ as } x \\to \\pm \\infty. \\]\nObserve that Proposition \\ref{spectrum} implies that the Schr\u00f6dinger\noperator $H_{u_0}$ has spectrum contained in $[-\\lambda^2, \\infty)$. We\n  want to show that $-\\lambda^2$ is not an eigenvalue. If it were,\n  then there would be a non-negative, strictly positive $L^2$ ground\n  state $\\psi$. This is not possible since $\\phi\/\\psi$ cannot attain a\n  minimum (see Lemma \\ref{maximum}). Therefore the spectrum is\n  contained in $(-\\lambda^2, \\infty)$.\n\\end{proof} \n\nWe now turn to the problem of \nrelating the two sides of \\eqref{prob1} and \\eqref{prob2} \nby analytic diffeomorphisms.  We begin with a technical statement. \n\n\n\\begin{lem}\\label{techlem} \nThe multiplication map $(u, v)\\to uv$ can be extended from the\nbilinear map $C_0^\\infty\\times C_0^\\infty\\to C_0^\\infty$ to continuous\nbilinear maps\n\\begin{align*}\n&L^2\\times L^2 \\to L^1\\subset H^{s},&\\text{ for any }\n  s<-\\frac12\\\\ &L^2\\times H^{s^{\\prime}} \\to H^{s},& ~-\\frac12 <s\\le\n  0,~s^{\\prime}>\\frac12+s\\\\ &H^{s_1}\\times H^{s_2} \\to H^{s_1},&\\text{\n    for any } s_2>\\frac12,~ 0\\le s_1, \\le s_2.\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nThe first two statements may be proved using Sobolev embedding\ninequalities and their corresponding dual inequalities.  The last case\nis a particular case of Theorem 1, of Section 4.6.1 of\n\\cite{MR1419319}, alternatively it may be proved by interpolating the\nsecond statement with the well known algebraic property of Sobolev\nspaces $H^s$ for $s>\\frac{1}{2}$.\n\\end{proof}\n\nFor $s\\geq -1$, let $F_{\\lambda}: H^{s+1} \\to H^{s}\\times \\mathbb{R} $\nand $F^*: H^{s+1} \\times (0,\\infty) \\to H^{s}$ to be the maps:\n\\begin{align*}\n  F_{\\lambda}(r)&= \\left( r^2+2r\\lambda\\tanh(\\lambda x)+r_x , \\int r \\sech^2(\\lambda x) ~dx \\right)  ,\n\\\\  F^*(r,\\lambda)&=r^2+2r\\lambda\\tanh(\\lambda x)-r_x-2\\lambda^2\\sech^2(\\lambda\\cdot) .\n\\end{align*}\nIt then follows from  Lemma \\ref{techlem} , that the above maps define\nquadratic (neglecting $\\lambda$ for $F^*$ here), and hence analytic maps from $H^{s+1} \\to H^{s}\\times \\mathbb{R}$ and $H^{s+1} \\times (0,\\infty) \\to H^{s}$, respectively.\nAnalyticity in $\\lambda$ (and joined analyticity) follows from the obvious \nholomorphic extension of  $\\lambda$ into the complex plane. \n\n\n\nThe equations\n\\begin{equation*}\nr^2+2r\\tanh x+r_x=u_0,\n\\end{equation*}\nand\n\\begin{equation*}\nr^2+2r\\lambda\\tanh (\\lambda\\cdot)-r_x-2\\lambda^2\\sech^2(\\lambda\\cdot)= u_0;\n\\end{equation*}\nrelating functions in range and image come from the expansion of the \nleft hand sides of  \\eqref{prob1} and \\eqref{prob2} respectively.\n\n\nNow let $L_{\\lambda,r}$ denote the first component of the Fr\u00e9chet\nderivative at $r$, and similarly let $L_{\\lambda,r}^*$ denote the\nFr\u00e9chet derivative of $F^*$ with respect to the first component at\n$(r,\\lambda)$, i.e.\n\\begin{equation*}\nL_{\\lambda,r}v := 2\\left(\\lambda\\tanh (\\lambda \\cdot)+r\\right)v+v_x,\n\\end{equation*}\nand $L_{\\lambda,r}^*$ is its formal adjoint: \n\\[ L_{\\lambda,r}^*  v :=  2\\left(\\lambda \\tanh(\\lambda \\cdot) +r\\right)v -  v_x . \\]\n\n\\begin{lem} \\label{opL}\nFor any $s \\in \\mathbb{R}$ and $r\\in H^{s}\\cap L^2$, the abstract operator\n$L_{\\lambda,r}$ and its formal adjoint operator $L_{\\lambda,r}^*$\ndefine bounded operators from $H^{s+1}(\\mathbb{R})$ to\n$H^s(\\mathbb{R})$, which we denote by $L_{\\lambda,r}$ and\n$L^*_{\\lambda,r}$, respectively, suppressing $s$ from the notation.\n\nBoth $L_{\\lambda,r}$ and $L_{\\lambda,r}^*$ are Fredholm operators of index $1$ and $-1$ respectively. \nMoreover, setting $\\phi_r=\\sech^2( \\lambda \\cdot)e^{-2\\int_0^{\\cdot}r~dy}$, the operator $L_{\\lambda,r}$ is surjective, with null space spanned by $\\phi_r$; and the formal adjoint $L_{\\lambda,r}^*$ is injective with closed range and cokernel spanned by $\\phi_r$. \n\n\\end{lem} \n\\begin{proof}\n\n  It follows from Lemma \\ref{techlem} that the operators\n  $L_{\\lambda,r}$ and $L_{\\lambda,r}^*$ define continuous linear\n  operators from $H^{s+1}$ to $H^s$. A simple calculation shows that\n  \\[ L_{\\lambda,r} \\sech^2(\\lambda x)\\exp\\left(-2\\int_0^{x}r~dy\\right) = 0. \\] Since $L_{\\lambda,r}\\phi=0$\n  is a scalar ordinary differential equation, every solution is a\n  multiple of $\\sech^2(\\lambda x)\\exp\\left(-2\\int_0^{x}r~dy\\right)$ and the null space is one\n  dimensional.  Similarly, one can easily check that $L_{\\lambda,r}^*$\n  is injective (since solutions to the homogeneous equation are\n  multiples of $\\cosh^2(\\lambda x)\\exp\\left(2\\int_0^{x}r~dy\\right)$) and $\\sech^2(\\lambda x)\\exp\\left(-2\\int_0^{x}r~dy\\right)$ spans\n  the cokernel of $L_{\\lambda,r}^*$.\n\nTo complete the proof we need to show $L_{\\lambda,r}$ is surjective, and $L_{\\lambda,r}^*$ is injective with closed range.\n  By scaling, it suffices to show the case when $\\lambda=1$.  For reasons of brevity we will use the shorthand $L_{r}:=L_{1,r}$, and $L:=L_{1,0}$.  \n  \n   We start by defining an integral operator from $L^2$ to $H^{1}$\n  which is a right inverse of $L_r$.  We begin with  the simpler case when $r=0$.\n\nLet $\\eta\\inC^{\\infty}_{0}([-2,2])$ be a non-negative function such that $\\eta\\equiv 1$ on $[-1,1]$ and consider the operator  $T$ defined by\n\\begin{align*}\nT(g)=& e^{-2\\int_{0}^x \\tanh s~ds} \\int_{-\\infty}^x e^{2\\int_{0}^y \\tanh s^{\\prime}~ds^{\\prime}}\\eta(y)g(y) ~dy+\\\\& e^{-2\\int_{0}^x \\tanh s ~ds} \\int_0^x e^{2\\int_{0}^y \\tanh s^{\\prime}~ds^{\\prime}} \\left(1-\\eta(y)\\right)g(y) ~dy.\n\\end{align*}\n\nNote that $T$ is a well defined operator for functions in $C^{\\infty}_{0}$.  \nIt can then be easily checked that $Tg$  satisfies $LTg=g$.  Now let $K(x,y)$ be the kernel of $T$:\n\\begin{equation}\n    K(x,y) \\equiv\n    \\left\\{ \\begin{array}{ll}\n        \\eta(y)\\cosh^2 y\\sech^2 x & y < x \\leq 0\\\\\n        \\eta(y)\\cosh^2 y\\sech^2 x & y < 0 < x\\\\\n        -(1-\\eta(y))\\cosh^2 y\\sech^2 x & x \\leq y \\leq 0\\\\\n        \\cosh^2 y\\sech^2 x & 0 \\leq y < x\\\\\n        0 & \\textit{otherwise}\n    \\end{array}\\right. \n\\end{equation}\n\nNow consider the case for general $r$: formally we have \n\\begin{equation*}\nT_r g := \\exp\\left(-2\\int_0^{\\cdot}r\\right)T\\exp\\left(2\\int_0^{\\cdot}r\\right)g,\n\\end{equation*}\nsatisfies $L_rT_r g=g$; furthermore the kernel of $T_r$ is given by\n\\begin{equation*}\nK_r(x,y)=K(x,y)\\exp\\left(-2\\int_0^x r + 2\\int_0^y r\\right).\n\\end{equation*}\n\nWe now claim that \n\\[ \\HN{1}{ T_rg } \\lesssim \\LPN 2 g . \\]\n\nObserve that \n\\begin{equation}\nK_r(x,y)\\lesssim e^{-2\\abs{y-x}+\\sqrt{\\abs{y-x}}\\LPN{2}{r}}\\lesssim e^{-\\abs{y-x}+\\LPN{2}{r}^2},\n\\end{equation} and hence\n\\begin{equation}\\label{L2OpBd}\n\\LPN{2}{T_r g}\\lesssim \\LPN{2}{e^{-\\abs{\\cdot}}*g}\\lesssim \\LPN{2}{g}.\n\\end{equation}\nThe equality  \n\\begin{equation}\\label{RegIter}\n\\partial_x T_rg +2\\left( \\tanh(x) +r\\right)T_r g = g, \n\\end{equation}  \nimplies \n\\begin{align*}\n\\LPN{2}{ \\partial_x T_rg } &\\le 2 \\LPN{2}{ T_rg }+2\\LPN{2}{rT_rg}+ \\LPN{2}{ g }\\\\\n &\\lesssim \\LPN{2}{g}+\\LPN{2}{r}\\LPN{\\infty}{T_rg}\\\\\n&\\lesssim \\LPN{2}{g}+\\LPN{2}{r}\\LPN{2}{\\partial_x T_rg}^{1\/2}\\LPN{2}{T_rg}^{1\/2},\n\\end{align*} \nwhere we used the $L^2$ estimate \\eqref{L2OpBd}, H\u00f6lder's inequality and Gagliardo-Nirenberg's inequality.  Finally applying Young's inequality and \\eqref{L2OpBd} again, we obtain \n\\begin{equation}\\label{thirds}\n\\LPN{2}{ \\partial_x T_rg} \\lesssim \\left(1+\\LPN{2}{r}^2\\right)\\LPN{2}{g}.\n\\end{equation} \n\nHence if $r\\in L^2$, then $T_r$ extends to a bounded operator from $L^2$ to $H^1$, satisfying $ L_r T_r g= g$, thus $L_r:H^1\\rightarrow L^2$ is surjective.\n\nBy duality, for every $r\\in L^2$, the adjoint operator $L^*_r: L^2 \\rightarrow H^1$ is injective with closed range, or equivalently\n\\begin{equation}\\label{zuiop}\n\\LPN 2 f \\lesssim \\HN {-1}{L^*_r f}   \n\\end{equation}\nfor all $f\\in L^2$.\n\nWe will now show that given any $f\\in H^{s+1}$ and $h\\in H^s\\cap L^2$, if \n\\[g:=2\\left(\\tanh(\\cdot) +h\\right)f \\pm  f_x,\\]\nthen we have the following inequality\n\\begin{equation}\\label{regiterstate}\n\\HN {s+1} f \\leq C\\left(\\LPN 2 f+ \\HN s g\\right), \n\\end{equation}\nfor some constant $C$ depending on $\\HN s h + \\LPN 2 h$.\n\nFirst note the trivial estimate\n\\begin{align}\n\\label{seconds}\\begin{split}\n\\HN{s+1}{f}&\\lesssim \\LPN 2 f+\\HN{s}{\\partial_x f}\\\\\n&\\lesssim \\LPN 2 f+\\HN{s}{g}+\\HN{s}{f}+\\HN{s}{h f}.\n\\end{split}\n\\end{align}\n\nConsider the case for $-1\\le s<-\\frac12$: it follows from Lemma \\ref{techlem} and \\eqref{seconds} that\n\\begin{align}\n\\HN{s+1}{f}&\\lesssim \\LPN 2 f+\\HN{s}{g}+\\left(1+\\LPN{2}{h}\\right)\\LPN{2}{f}.\\label{firsts}\n\\end{align}\n\nNow consider the case when $s\\ge-\\frac12$: again from Lemma \\ref{techlem} and \\eqref{seconds} we have \n\\[\\HN{s+1}{f}\\lesssim \\HN{s}{f}+\\HN{s}{g}+\\left(1+\\HN s h+ \\LPN 2 h\\right)\\HN{s+3\/4}{f}.\\]\nUsing \\eqref{firsts} and applying the above estimate recursively we obtain \\eqref{regiterstate}.\n\nCombining \\eqref{zuiop} with \\eqref{regiterstate} ($h=r$), it follows\nthat for all $s\\geq -1$, $f\\in H^{s+1}$ and $r\\in L^2\\cap H^s$\n\\[\n\\HN {s+1}{f} \\lesssim \\HN{s}{L^*_r f},\n\\]\ni.e. $L^*_r:H^{s+1}\\rightarrow H^{s}$ is injective with closed range.\n\nBy duality it follows that if $r\\in L^2$ then $L_r:L^2\\rightarrow\nH^{-1}$ is surjective; moreover, as a consequence of\n\\eqref{regiterstate} ($h=r$) we obtain $L_r:H^{s+1}\\to H^{s}$ is\nsurjective for all $s \\in \\mathbb{R}$ and $r\\in L^2\\cap H^s$, and\nagain the full statement for $L^*_r$ follows.\n\\end{proof}\n\n\n\nLet $U^s_{>\\kappa}\\subset H^s$ denote the set of all functions in $H^s$ whose\nassociated Schr\u00f6dinger operator has spectrum contained in\n$(\\kappa,\\infty)$.  Similarly, define $U^s_{<\\kappa}$ to be the subset\nof $H^s$ of all functions $f$ whose associated Schr\u00f6dinger operator\nhas spectrum $\\omega(T_f)$ such that $\\omega \\setminus\n(\\kappa,\\infty)\\neq\\emptyset$.\n\n\n\\begin{thm}\\label{inverse} \n  For any $s\\geq -1$ the map $F_{\\lambda}:H^{s+1}\\rightarrow H^s\\times\\mathbb{R}$ is an analytic diffeomorphism onto its range. Moreover, for any $f\\in U^s_{>-\\lambda^2}$, there exists $\\rho\\in \\mathbb{R}$ such that $(f,\\rho)$ is contained in the range of $F_{\\lambda}:H^{s+1}\\rightarrow H^s\\times\\mathbb{R}$.\n  \n  For any $s\\geq -1$ the map $F^*:H^{s+1} \\times (0,\\infty)\\rightarrow H^s$ is an analytic diffeomorphisms onto  $U^s_{<0}$.\n\\end{thm} \n\\begin{proof}\nFirst we show the two maps are local analytic diffeomorphisms.  \n  \nNote the second component of $DF_{\\lambda}|_r$ is simply the map $f\\mapsto \\ip{f,\\sech^2(\\lambda \\cdot)}$; hence from Lemma \\ref{opL} we obtain $DF_{\\lambda}|_r$ is\n  invertible -- here we use the fact \\[\\ip{\\phi_r,\\sech^2(\\lambda \\cdot)}>0,\\] where $\\phi_r$ is defined as in Lemma \\ref{opL}.  Hence by the inverse function theorem, $F_{\\lambda}$ is a local analytic  diffeomorphism.\n  \n  Let $G:H^{-1}\\rightarrow \\mathbb{R}$ be the map from potentials to the ground state energy of their corresponding Schr\u00f6dinger operator.  By Proposition \\ref{groundstates} we have that $G(F^*(f,\\lambda))=-\\lambda^2$, from which it follows that the derivative of $F^*$ with respect to the\n  second component is not contained in the range of the derivative with respect to the first component.  Then from Lemma \\ref{opL} and the implicit function theorem, it follows that $F^*$ is a local analytic  diffeomorphism. \n\nWe now prove the injectivity of the two maps.\nSuppose $r_i$, $i\\in\\{1,2\\}$ satisfy $F_{\\lambda}(r_1) = F_{\\lambda}(r_2)$. Then, with $w= r_2-r_1$ we have\n\\[ w_x + (2\\lambda \\tanh(\\lambda x) + r_2+r_1) w = 0 , \\qquad \\int \\sech^2(\\lambda x) w(x) dx =0. \\]\nThe same argument as for invertibility implies that $w=0$, hence $r_1=r_2$ \nand $F_{\\lambda}$ is injective. \n\nWe now show $F^*$ is injective. First of all, if $F^*(r_1,\\lambda_1)\n=F^*(r_2,\\lambda_2)$, then by Lemma \\ref{groundstates} we necessarily have $\\lambda_1=\\lambda_2$. Next,  with $w=\nr_2-r_1$\n\\[ w_x - (2\\lambda \\tanh(\\lambda x) + r_1+r_2) w = 0. \\]\n The only solution in $L^2$ to this equation is $w=0$. This implies injectivity.\n\nNow we make the following observation: if $F_{\\lambda}(r) = (g,\\rho)$\nfor $r \\in L^2$ and $g \\in H^s$, $s >-1$ then we also have $r\\in\nH^{s+1}$; similarly, if $F^*(r,\\lambda) = g$ for $r \\in L^2$ and $g\n\\in H^s$, $s >-1$ then we also have $r\\in H^{s+1}$. This follows by\niteratively applying \\eqref{regiterstate}, with $h=r\/2$.\n\nThus as a consequence of the Proposition \\ref{groundstates}, together with the above observation, we obtain that the range of the projection  of $F_{\\lambda}(r):H^{s+1}\\rightarrow H^s\\times \\mathbb{R}$ onto its first component is precisely $U^s_{>-\\lambda^2}$, and the range of $F^*:H^{s+1} \\times (0,\\infty)\\rightarrow H^s$ is $U^s_{<0}$.\n\\end{proof}\n\n\\begin{rem}\\label{wvest}\nNote that working out the details of the $H^{s+1}$ estimates in the proof above, one may show that if \n\\[v(t,x):= w(t,x-y_0)^2+2w(t,x-y_0)\\tanh(x-y_1)+w_x(t,x-y_0),\\]\nfor some $y_0,y_1\\in \\mathbb{R}$ then for every integer $s\\ge -1$ there exists a $N>0$ such that\n\\begin{equation*}\n\\HN{s+1}{ w } \\lesssim (1+\\LPN 2 w + \\HN s v )^N \\left(\\HN s v+\\LPN 2 w\\right).\n\\end{equation*} \nThis estimate will be used later in Section \\ref{seckink}.\n\\end{rem}\n\n\n\\section{The modified Korteweg-de Vries equation close to a kink}\\label{seckink}\n\nIn Section \\ref{invmiurasec}, we mapped $H^{s}$ KdV initial data to\nmKdV initial data -- the mKdV initial data being in the form of a sum\nof a $H^{s+1}$ function, and a kink of the form\n$\\lambda\\tanh(\\lambda \\cdot)$.  In this section we will study the\ncorresponding mKdV problem.\n\nFirst note by scaling, we may restrict to the case $\\lambda=1$.  Recall that $Q_1(t,x)\\equiv Q(t,x)\\equiv \\tanh(x+2t)$\nis an  explicit kink solution to \\eqref{mKdV}.  We\nnow consider solutions to mKdV equation:\n\\begin{equation}\\label{uKINKmKdV}\n\\left\\{\\begin{array}{l}\nu_t + u_{xxx} - 6u^2 u_x=0\\\\\nu(0,x) = v_0(x)+\\tanh(x)\n\\end{array}\\right.,\n\\end{equation}\nfor initial data $v_0\\in H^s$, $s\\geq 0$, such that $u-Q\\in H^s$.  Equivalently, writing $u=v+Q$, we have:\n\\begin{equation}\\label{KINKmKdV}\n\\left\\{\\begin{array}{l}\nv_t + v_{xxx} - 2 \\partial_x (3Q^2 v+ 3Q v^2+v^3 )=0\\\\\nv(0,x) = v_0(x)\n\\end{array}\\right..\n\\end{equation}\n\nIn order to construct global solutions to \\eqref{uKINKmKdV}, we will\nneed to prove a number of energy estimates.  In our discussions below,\nwe will use a number of formal calculations, which are not difficult\nto justify rigorously.\n\n\\begin{lem}\nLet $p$ be any $C^{\\infty}(\\mathbb{R}^2)$ function with uniformly bounded derivatives and assume $v\\in C([0,\\infty); H^{1}(\\mathbb{R}))$ to be a solution to \\eqref{KINKmKdV}.  Then\n\\begin{align}\\label{nifty}\n\\begin{split}\n\\frac{d}{dt}  \\left[ \\int p v^2~dx\\right]=    &\\int \\Big[ p_t v^2 -3 p_x v_x^2+ p_{xxx}v^2 \\\\\n                                           &-6p_x Q^2 v^2-8p_x Q v^3-3p_xv^4 \\\\\n                                           &+12p Q Q_{x} v^2+4pQ_x v^3 \\Big]~dx.\n\\end{split}\n\\end{align} \n\\end{lem}\n\\begin{proof}\nNote that by \\eqref{KINKmKdV}, $v$ also has some time regularity. Thus, equation \\eqref{nifty} follows by employing \\eqref{KINKmKdV} and applying a series of integrations by parts.\n\\end{proof}\n\nWe are now in the position to prove global bounds on the $L^2$ norm of smooth solutions to \\eqref{KINKmKdV}, as well as a ``Kato smoothing'' type estimate.\n\n\\begin{lem}\\label{localbd}\nSuppose $v\\in C([0,\\infty); H^{1}(\\mathbb{R}))$ is a solution to (\\ref{KINKmKdV}); then for any $t\\in [0,\\infty)$, we have \n\\begin{equation}\\label{badL2bd}\n\\LPN{2}{v(t,\\cdot)} \\lesssim \\LPN{2}{v_0}+t^{1\/2}.\n\\end{equation}\nMoreover for any $T>0$\n\\begin{equation}\\label{KATOSMOOTH}\n\\int_{0}^T\\int_{-\\infty}^{\\infty} Q_x(t,x)v_x(t,x)^2~dx~dt\\leq C(T,\\LPN{2}{v(0,\\cdot)}).\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nFrom (\\ref{nifty}), with $p\\equiv 1$, we have\n\\begin{equation}\\label{mass1}\n\\frac {d}{dt} \\left[ \\int  v^2~dx\\right]=    \\int 12 Q Q_{x} v^2+4Q_x v^3 ~dx.\n\\end{equation} \n\nUsing (\\ref{nifty}) again, but with $p\\equiv Q$, yields\n\\begin{align}\\label{lastest}\n\\begin{split}\n\\frac{d}{dt} \\left[ \\int Q v^2~dx\\right]=&    \\int \\Big[ Q_t v^2 -3 Q_x v_x^2+ Q_{xxx}v^2 \\\\\n                                       &    -6Q^2 Q_x v^2-8Q Q_x v^3-3Q_xv^4 \\\\\n                                        &   +12Q^2 Q_{x} v^2+4QQ_x v^3 \\Big]~dx.\n\\end{split}\n\\end{align} \n\nObserve that the terms $Q^2Q_x v^2$, $QQ_x v^2$, $Q_t v^2$ and $Q_{xxx} v^2$ are all bounded above by a multiple of $Q_x^{1\/2}v^2$. Furthermore, we have\n\\begin{equation}\\label{YOUNG1}\n\\begin{split} \\int Q_x v^2~dx\\leq & \\LPN{2}{Q_x^{1\/2}}\\LPN{2}{Q_x^{1\/2}v^2}\\\\ \\leq& \\frac{1}{\\tau}\\int Q_x v^4~dx \n + \\tau \\LPN{2}{Q_x^{1\/2}}^2,\n\\end{split}\n\\end{equation} \nfor $\\tau >0$ by  Young's inequality.\nNote also $\\abs{QQ_x v^3}\\leq \\abs{Q_x v^3}$ and\n\\begin{align*}\n\\int \\abs{Q_x v^3}~dx &\\leq \\LPN{2}{Q_x^{1\/2}v}\\LPN{2}{Q_x^{1\/2}v^2}\\\\\n&\\leq \\frac{1}{\\kappa}\\int Q_x v^4~dx + \\kappa \\int Q_x v^2~dx,\n\\end{align*}\nfor any $\\kappa>0$.  Applying (\\ref{YOUNG1}), we find that for any $\\kappa>0$, there exists a constant $C_{\\kappa}>0$, such that\n\\begin{equation}\\label{YOUNG2}\n\\int \\abs{Q_x v^3}~dx \\leq \\frac{1}{\\kappa}\\int Q_x v^4~dx + C_{\\kappa}.\n\\end{equation}\nCombining equations (\\ref{mass1}-\\ref{YOUNG2}) we obtain\n\\begin{equation}\\label{equality}   \n\\frac {d}{dt} \\left[ \\int \\left[v^2+\\frac{1}{10} Q v^2\\right]~dx\\right]\\lesssim 1.\n\\end{equation}\n\nSince $v^2\\lesssim v^2+ \\frac{1}{10} Q v^2$ , we can conclude that for any $t\\ge 0$\n\\begin{equation*}\n\\LPN{2}{v(t,\\cdot)} \\lesssim \\LPN{2}{v(0,\\cdot)}+t^{1\/2}.\n\\end{equation*}\n\nThe proof of \\eqref{KATOSMOOTH} follows from \\eqref{badL2bd} and the estimate \\eqref{lastest}.  \n\\end{proof}\n\nAs a consequence of the above estimates, we obtain the following well-posedness result for initial mKdV data near a kink. \n\n\\begin{thm}\\label{strongsol}\nLet $s\\in \\mathbb{N}$ satisfy $s\\geq 1$.  Then there exists a unique, global, strong solution to (\\ref{KINKmKdV}), for any initial data $v_0\\in H^{s}$.  Moreover, for any $T>0$, the solution map from $H^s$ to $C_t ([0,T];H^s(\\mathbb{R}))$ is continuous.\n\\end{thm}\n\nThe proof of this theorem follows essentially from the same arguments\nas those given in \\cite{MR1211741} and \\cite{MR1949297} -- since the\n$L^2$ estimate \\eqref{badL2bd} is available. \n\nWe now establish global a priori bounds on the deviation of a solution\n$u$ to \\eqref{uKINKmKdV} from a translated kink, i.e. we aim to\nestablish bounds on $w:=u-\\tanh(x-y(t))$ for some continuous function\n$y:\\mathbb{R}^+\\rightarrow \\mathbb{R}$ yet to be determined. We start by providing some motivation for the full argument given later.\n\n From \\eqref{uKINKmKdV} we obtain\n\\begin{equation}\\label{KINKmKdV2}\n \\begin{array}{rl}  \nw_t + w_{xxx} - 2 \\partial_x (3\\tanh^2(x-y(t))^2 w+ 3 \\tanh(x-y(t))  w^2+w^3 )\\hspace{-4cm} & \n\\\\\n& \n= \n(\\dot y + 2) \\sech^2(x-y(t)) \n\\end{array}.\n\\end{equation}\n\nTo define the position $y(t)$, we impose an orthogonality condition \n\\begin{equation} \\label{pos2}   \\langle w , \\psi(x-y) \\rangle = 0, \\end{equation} \nwhere for the moment we choose $\\psi(x)= e^x \\sech^2(x)$  for reason which will become clear \nbelow -- later we will actually choose $\\psi(x)=\\eta(x)\\sech^2(x)$, where $\\eta$ is defined by \\eqref{etadef}. If $w$ is sufficiently close to the kink then $y$ exists and is unique \nby an application of the implicit function theorem -- see Lemma \\ref{position} \nbelow\\footnote{In Lemma \\ref{position} the weight $\\psi(x)=\\eta(x)\\sech^2(x)$ is actually used; however, the proof may be easily adapted to the case $\\psi(x)= e^x \\sech^2(x)$.}.  It is not hard to work out an equation for $\\dot y +2$ by formally \ndifferentiating the condition \\eqref{pos2} \n with respect to $t$.\n\n\n\nIt is instructive to first consider the linearised problem at the $Q(x,t)=\\tanh(x+2t)$ kink,\n in a frame moving with the kink:\n\\begin{equation}\\label{lineareq}\n\\tilde w_t(t,x) -4\\tilde w_x(t,x)+ \\tilde w_{xxx}+6 \\partial_x(\\sech^2(x)\\tilde w(t,x))= \\alpha(t) \\sech^2(x),\n\\end{equation} \nwhere $\\alpha(t)$ is chosen as indicated above so that orthogonality condition \n\\begin{equation}\\label{motivortho}\n \\langle \\tilde w , e^{x} \\sech^2(x) \\rangle = 0 \n\\end{equation}\n is preserved over time. To obtain a formula for $\\alpha$, we differentiate the above orthogonality condition with respect to $t$.  Thus we obtain\n\\[  \\langle  \\tilde w, (-4 \\partial_x +\\partial_{xxx} +6 \\sech^2 \\partial_x )e^x \\sech^2(x)  \\rangle \n + \\alpha(t) \\langle \\sech^2(x) , e^x \\sech^2(x)  \\rangle = 0, \\]\nwhich expresses $\\alpha$ as a linear function of $\\tilde w$.\n\n From \\eqref{lineareq} we obtain\n\\begin{equation}\\label{motiveq}  \\frac{d}{dt} \\int e^x \\tilde w^2 dx = -3B(e^{x\/2} \\tilde w)+2\\alpha(t)\\int \\tilde w(t,x) e^{x} \\sech^2(x)~dx,\n\\end{equation} \nwhere here $B$ is the quadratic form defined by\n\\begin{equation}\\label{newquadform}B(f):=\\int f_x^2 + \\bigg(\\frac54 -2\\sech^2(x)  - 4 \\sech^2(x)\\tanh(x)\\bigg) f(x)^2~dx.\\end{equation}\nNote that the second term on the right hand side \nof \\eqref{motiveq} vanishes due to \\eqref{motivortho}.\n\nAccording to the bound \\eqref{assumption} of Proposition\n\\ref{quadform}, taking into account the choice of $\\psi$, we have\n\\begin{equation*} B(e^{x\/2} w) \\ge \\frac{1}{10} \\HN 1 { e^{x\/2} w }^2.\n \\end{equation*}\nAs a consequence $\\int e^x \\tilde w^2 dx $ decays monotonically, and the time \nderivative controls $B(e^{x\/2} \\tilde w)$.\n\n\nWe will pursue a non-linear variant of this simple strategy.  The\nweight $e^{x}$ will be replaced by the following bounded and monotone\nweight function\n \\begin{equation}\\label{etadef}\n  \\eta_{R,\\delta} (x)=\\eta (x) =\n \\tanh\\left(\\frac{x-R}{2}\\right)+1+\\delta.\n\\end{equation}  \nWe will also define $y$ in terms of an orthogonality condition, similar to \\eqref{motivortho}.\n\n\\begin{lem} \\label{position} \nThere exists an $\\epsilon >0$ and a unique \nanalytic  function $y$ on $B^{L^2}(\\tanh(\\cdot),\\epsilon)$ such that \n\\[  \\langle   f(\\cdot)-  \\tanh(\\cdot-y(f)) ,  \\eta( \\cdot- y(f)) \\sech^2(\\cdot-y(f)) \\rangle = 0, \n\\]\nand $y(\\tanh(\\cdot)) = 0 $. \n\\end{lem} \n\\begin{proof} \nConsider the mapping \n\\[ F:B^{L^2}(\\tanh(\\cdot),\\epsilon)\\times \\mathbb{R}\\rightarrow\\mathbb{R} \\]\n-- here $B^{L^2}(\\tanh(\\cdot),\\epsilon)$ denotes by an abuse of notation \nthe set of sums of $L^2$ functions of norm $<\\varepsilon$ and $\\tanh$ --\n defined by\n\\[ F(f,y)=\\ip{ f - \\tanh(\\cdot-y) , \\eta(\\cdot-y)\\sech^2(\\cdot-y)}. \n\\]\nClearly $F(\\tanh(\\cdot),0)=0$. Differentiating with respect to $y$ at $f:=\\tanh(\\cdot)$ we obtain \n\n\\begin{equation*} \\left. \\frac{d}{dy} F(\\tanh(\\cdot),y) \\right|_{y:=0} \n=     \n\\langle \\sech^2(\\cdot-y) ,\\eta(\\cdot-y) \\sech^2(\\cdot-y)\\rangle >0 . \n\\end{equation*}\nThe implicit function theorem then yields the assertion. \n\\end{proof} \n\n\\begin{thm}\\label{stab} \nThere exists a $\\delta >0$\nsuch that if $u$ the  solution to \\eqref{uKINKmKdV} of Theorem \\ref{strongsol} with initial data  satisfying $u(0,\\cdot) - \\tanh(\\cdot) \\in H^1(\\mathbb{R})$  and $\\LPN 2 {u(0,\\cdot)-\\tanh(\\cdot)}<\\delta$, then there is a continuous function $y:[0,\\infty)\\to\\mathbb{R}$ such that \n\\begin{equation}\n\\LPN{2}{ u(t,.)  - \\tanh(x-y(t)) } \\lesssim \\LPN 2 {u(0,\\cdot)-\\tanh(\\cdot) }.\\label{apriorimkdv}\n\\end{equation}\nMoreover, writing $w:=u-\\tanh(\\cdot-y(t))$ we have the estimates\n\\begin{equation}\\label{virial}\n\\int_0^\\infty \\norm{\\eta_x(\\cdot-y(t))^{1\/2} w}_{H^1}^2~dt\\lesssim \\LPN{2}{u(0,\\cdot)-\\tanh(\\cdot)}^2,\n\\end{equation}\nand\n\\begin{align}\\label{dotyest3}\n\\begin{split}\n\\abs{\\dot{y}+2}\\lesssim &\\LPN 2 {\\eta_x(\\cdot-y(t))^{1\/2}w}+\\\\&\\qquad\n\\LPN 2 {\\eta_x(\\cdot-y(t))^{1\/2}w} \\norm{\\eta_x(\\cdot-y(t))^{1\/2} w}_{L^{\\infty}}^2.\n\\end{split}\n\\end{align}\n\\end{thm} \n\n\n\\begin{proof}\nFirst define\n\\begin{equation} \\label{psi}  \\psi(x) = \\eta(x) \\sech^2(x). \\end{equation}\n\nOur aim is to find a function $y$ satisfying the orthogonality condition:\n\\begin{equation}\\label{ortho}\n\\ip{  u(t,\\cdot)-\\tanh(\\cdot-y(t)) , \\psi (x-y(t)) }=0\n\\end{equation}\nfor all $t\\ge 0$ such that $y(0)=y_0$, where $y_0$ is given by Lemma\n\\ref{position}.  The existence of such a function, at least initially,\nfor $t\\in [0,T]$, for some $T>0$, is a consequence of Lemma\n\\ref{position} and the fact $u-\\tanh(x+2t)\\in C(\\mathbb{R};H^1(\\mathbb{R}))$.\n\nNow define $w(t,x)$ by\n\\[ w (x,t) =  u(t,x)- \\tanh(x-y(t)), \\]\nfrom which we obtain \n\\begin{multline}\\label{KINKmKdV3} \nw_t + w_{xxx}-2 \\partial_x( 3 \\tanh^2(x-y) w + 3 \\tanh(x-y) w^2+ w^3)=\\\\ \n(2+\\dot y)\\sech^2(x-y). \n\\end{multline} \n\nAgain by perhaps taking a smaller $T$ if necessary, we can assume for $t\\in [0,T]$\n\\[\\LPN 2 {w}\\leq 2\\epsilon.\\]\n\nBy (\\ref{KINKmKdV3}) and (\\ref{ortho}) we obtain:\n\\begin{multline}\\label{change}\n\\frac {d}{dt} \\int \\eta(x-y) w^2~dx=\\\\\n\\int \\bigg[ -3\\eta_x(x-y) w_x^2+ \\bigg(-\\dot y\\eta_x(x-y) + \\eta_{xxx}(x-y) \\\\-6\\tanh^2(x-y)\\eta_x(x-y) + 12\\eta(x-y)\\tanh(x-y)\\sech^2(x-y)\\bigg)w^2\\\\\n+(4\\eta(x-y) \\sech^2(x-y)-8\\eta_x(x-y)\\tanh(x-y))w^3-3\\eta_x(x-y)w^4\\bigg]~dx.\n\\end{multline}\nRewriting the quadratic part of the above equation by using the identity $\\sech^2(x)+\\tanh^2(x)=1$ numerous times, together with the observation\n\\begin{equation}\\label{uiop}\n\\eta_{xxx}=-3\\eta_x^2+\\eta_x,\n\\end{equation}\nalong with the trivial identity $\\dot y=-2+(\\dot y+2)$ we obtain\n\\begin{multline}\\label{change2}\n\\frac {d}{dt} \\int \\eta(x-y) w^2~dx=\\\\\n\\int \\bigg[ -3\\eta_x(x-y) w_x^2+ \\bigg(-3 +6\\sech^2(x-y)+\\\\24\\eta(x-y)\\tanh(x-y)\\sech^2(x-y)(\\eta_x(x-y))^{-1} \\bigg)\\eta_x(x-y)w^2\\\\\n -(2 +\\dot y)\\int \\eta_x(x-y) w^2~dx-\\int 3(\\eta_x(x-y))^2 w^2~dx\\\\   \n+(4\\eta(x-y) \\sech^2(x-y)-8\\eta_x(x-y)\\tanh(x-y))w^3-3\\eta_x(x-y)w^4\\bigg]~dx.\n\\end{multline}\n\nNow observe\n\\begin{align*}\n\\int \\left(\\eta_x^{1\/2}w\\right)^2_x~dx&=\\int\\left[\\eta_x w_x^2+\\frac{\\eta_{xx}^2}{4\\eta_x}w^2+\\eta_{xx}ww_x\\right]~dx\\\\\n&=\\int\\left[\\eta_x w_x^2+\\left(\\frac{\\eta_{xx}^2}{4\\eta_x}-\\frac12\\eta_{xxx}\\right)w^2\\right]~dx\\\\\n&=\\int\\left[\\eta_x w_x^2+\\left({\\eta_{x}^2} -\\frac14\\eta_x\\right)w^2\\right]~dx,\n\\end{align*}\nwhere in the last line we used \\eqref{uiop} in addition with the identity\n\\[\n\\frac{\\eta_{xx}^2}{\\eta_x}=\\eta_x-2\\eta_x^2.\n\\]\nWe define the quadratic form:\n\\begin{multline*}\nB_{\\varepsilon, R}(f):=\\int f_x^2 + \\bigg(\\frac54 -2\\sech^2(x)  \\\\- 8 \\sech^2(x)\\tanh(x)\\cosh^{2}\\left(\\frac{x-R}{2}\\right) \\left(1+\\varepsilon+\\tanh\\left(\\frac{x-R}{2}\\right)\\right)  f(x)^2~dx\n\\end{multline*}\nand rewrite the equation (\\ref{change2}) as\n\\[\n\\begin{split}\n\\frac {d}{dt} \\int  \\eta(x-y) & w^2~dx =    -3B_{\\varepsilon, R}(\\eta_x(x-y)^{1\/2}w)\n -(2 +\\dot y)\\int \\eta_x(x-y) w^2~dx\n \\\\ & +\\int (4\\eta(x-y) \\sech^2(x-y)-8\\eta_x(x-y)\\tanh(x-y))w^3~dx \n\\\\ &  -\\int 3\\eta_x(x-y)w^4 ~dx.\n\\end{split} \n\\]\nWe observe that $\\eta_x$ is positive and hence the last line is non-positive. We will now estimate the cubic term:\n\\begin{align*} \n\\Big|\\int (4\\eta(x-y) \\sech^2(x-y)-&8\\eta_x(x-y)\\tanh(x-y))w^3 dx\\Big| \\\\\n&\\lesssim C_R   \\int \\eta_x |w|^3 dx \n \\\\& \\lesssim C_R    \\LPN 2 w  \\LPN{4}{ \\eta_x^{1\/2} w }^2 \n\\\\ &\\lesssim C_R   \\LPN 2 w \\HN{1}{ \\eta_x^{1\/2} w}^2 .\n\\end{align*}\n\nWe turn to   bounding $\\abs{\\dot y +2}$. Note we have from \\eqref{KINKmKdV3} and \\eqref{psi}:\n\\begin{multline}\\label{dotyeq}\n0= \\frac{d}{dt} \\langle w, \\psi(x-y) \\rangle = \\int \\bigg[w\\psi_{xxx}(x-y)\\\\\n-2 ( 3 \\tanh^2(x-y) w + 3\\tanh(x-y) w^2+ w^3)\\psi_x(x-y)\\\\\n+ (2+\\dot y) \\sech^2(x-y)\\psi(x-y)   -\\dot y    w(x)\\psi_x(x-y)\\bigg]~dx.\n\\end{multline}\n\nThus we obtain\n\\begin{align}\\label{dotyest}\n\\begin{split}\n\\abs{\\dot{y}+2}&\\lesssim C_R (1+\\abs{\\dot y+2})\\LPN 2 {w}+\\LPN 2 {\\eta_x^{1\/2}w}^2+\\LPN 2 {\\eta_x^{1\/2}w} \\norm{\\eta_x^{1\/2} w}_{L^{\\infty}}^2 \\\\\n&\\lesssim C_R \\LPN 2 {w}+\\LPN 2 {w} \\norm{\\eta_x^{1\/2} w}_{L^{\\infty}}^2,\n\\end{split}\n\\end{align}\nwhere in the last line we use the fact that $\\LPN 2 w \\ll 1$.\n\nCollecting the above estimates together, we obtain:\n\\begin{equation}\\label{pos}\n\\frac {d}{dt} \\int \\eta(x-y) w^2 dx \\leq  -3B_{\\varepsilon, R}(\\eta_x(x-y)^{1\/2}w)+\\Lambda \\LPN 2 w \\norm {\\eta_x^{1\/2}w}_{H^1}^2,\n\\end{equation}\nfor some constant $\\Lambda$ depending on $R$, which will be a positive number. \n\nWe now compare the quadratic form $B_{\\varepsilon, R}$ with the quadratic form $B$ from Appendix B: \n\\[B(f)=\\int f_x^2 + \\bigg(\\frac54 -2\\sech^2(x)  - 4 \\sech^2(x)\\tanh(x)\\bigg) f(x)^2~dx.\\]\nThe difference $V(x)$ of the potentials in the quadratic forms is \n\\[ 4 \\sech^2(x)\\tanh(x)\\left(2\\cosh^{2}\\left(\\frac{x-R}{2}\\right) \\left(1+\\varepsilon+\\tanh\\left(\\frac{x-R}{2}\\right)\\right)- 1\\right).    \\]\nObserve that \n\\begin{equation*}\n\\cosh^2\\left(\\frac{x-R}{2}\\right)\\left(1+\\tanh\\left(\\frac{x-R}{2}\\right)\\right)= \\frac{1}{2}(e^{x-R}+1)\n\\end{equation*}\nhence the difference $V$ can be bounded by \n\\[ \\abs{V} \\le  8 \\varepsilon \\sech^2(x) \\cosh^2\\left(\\frac{x-R}2\\right) + 4\\sech^2(x) e^{x-R}   \n\\le  16 \\varepsilon e^{R} + 8 e^{-R}.\n \\]\nThus we obtain\n\\begin{equation} \\label{diffquad} \n|B(f) - B_{\\varepsilon,R}(f)| \\le \\left(16 \\varepsilon e^R   +4 e^{-R}\\right) \\LPN 2 f ^2. \n\\end{equation}\nNow define the modified quadratic form\n\\begin{equation*}\n\\hat{B}_{\\varepsilon, R}(f):=  B_{\\varepsilon, R}(f)+2 e^{R}\\left\\langle \\eta_x^{-1\/2}\\eta\\sech^2, f\\right\\rangle^2.\n\\end{equation*}\nObserve that\n\\[  \\eta_{R,\\varepsilon} - \\eta_{R,0} = \\varepsilon,\\]\nand\n\\begin{equation*}\n\\cosh\\left(\\frac{x-R}{2}\\right)\\left(1+\\tanh\\left(\\frac{x-R}{2}\\right)\\right)= e^{\\frac{x-R}{2}},\n\\end{equation*}\nfrom which it follows that\n\\[  (e^{R\/2} \\eta_x^{-1\/2} \\eta(x)  -e^{x\/2}) \\sech^2(x) =   \n\\sqrt{2} \\varepsilon  e^{R\/2} \\cosh\\left(\\frac{x-R}2\\right) \\sech^2(x) \\le 2 \\varepsilon  \ne^R,     \\]\nwhich yields the estimate\n\\[ \\abs{e^{R\/2} \\ip{ \\eta_x^{-1\/2}\\eta\\sech^2, f}- \\ip{  e^{x\/2}\\sech^2(x) ,f}} \\le 2 \\varepsilon e^R\\LPN 2 f . \\]\n\nBy estimate \\eqref{diffquad} \n together with the estimate \\eqref{assumption} we obtain: \n\\begin{lem} With $R=10$ we have  for all $f\\in H^1$\n\\[ \\hat{B}_{e^{-2R},R} (f)  \\geq \\frac{1}{20}  \\HN{1}{ f }^2. \\]\n\\end{lem} \nWe now fix $R=10$ and set $\\varepsilon:=e^{-2R}=e^{-20}$ -- noting that we\nonly require the existence of $R$ such that the conclusion of the Lemma holds,\nwith its size being neither optimal nor important.\n\n   If we for the moment assume that\n\\[ \\sup_{0\\le t \\le T} \\LPN 2 {w(t,\\cdot)}  \\le \\frac1{80 \\Lambda}, \\]\nthen it follows from the above lemma, and the orthogonality condition (\\ref{ortho}) that we can control the second term in the equation (\\ref{pos}) with the first term, which implies\n\\[\\int \\eta(x-y(t)) w(t,x)^2  dx \\le \\int \\eta(x-y(0)) w(0,x)^2 dx.\\] \n\nHence we obtain \n\\begin{align*}\n\\varepsilon  \\LPN{2} {w(t,\\cdot)}^2\\le&    \\int \\eta (x-y)w(t,x)^2  dx \\\\\\le&\n \\int \\eta(x-y(0)) w(0,x)^2 dx  \\\\\\le&  (1+ \\varepsilon)   \\LPN{2}{ w(0,\\cdot)}^2 \n\\end{align*}\nand thus \n\\[ \\LPN{2}{ w(t,\\cdot)} \\le  2 e^{R} \\LPN{2}{ w(0,\\cdot)}.  \n\\]\n\nA continuity argument gives the desired global bound provided (recall $\\varepsilon = e^{-2R}$)\n\\[ \\LPN{2}{ w(0,\\cdot)} \\le  \\frac1{120 \\Lambda} e^{-R}. \\]\n\\end{proof}\n\n\n\n\\begin{cor}\\label{highernorm}\nSuppose $u$ satisfies the conditions in the above Theorem, furthermore assume $u(0,\\cdot)- \\tanh(\\cdot)\\in H^s$, where $s$ is a positive integer, then there exist $C>0$ and $N >0$  depending only on $s$  such that\n\\begin{equation}  \\label{highernormineq}\n\\begin{split} \n\\HN{s}{u(t,\\cdot)-\\tanh(x-y(t))}\\le &  C \\HN{s}{u(0,\\cdot)-\\tanh(\\cdot)}\n\\\\ & \\times \\left(1+\\HN{s}{u(0,\\cdot)-\\tanh(\\cdot)}\\right)^N.\n\\end{split} \n\\end{equation} \n\\end{cor}\n\\begin{proof}\nLet $w$ be as in Theorem \\ref{stab} and define:\n\\begin{equation}\\label{uwrel}\nv(t,x):= w(t,x-6t)^2+2w(t,x-6t)\\tanh(x-y(t)-6t)+w_x(t,x-6t),\n\\end{equation}\ni.e. $v$ is a solution to \\eqref{KdV}.  \n\nNote as a consequence of infinite conservation laws associated with\nthe KdV equation (see \nAppendix \\ref{conserved}), we have\n\\begin{align}\\label{zut}\n\\begin{split}\n\\HN {s} {v(t,\\cdot)} &\\lesssim \\HN {s-1} {v(0,\\cdot)}\\left(1+\\HN {s-1} {v(0,\\cdot)}\\right)^{N^{\\prime}}\\\\\n&\\lesssim \\HN {s} {w(0,\\cdot)}\\left(1+\\HN {s} {w(0,\\cdot)}\\right)^{N^{\\prime\\prime}}\n\\end{split}\n\\end{align}\nfor some positive integers $N^{\\prime}$ and $N^{\\prime\\prime}$.\n\nThen from Remark \\ref{wvest} at the end of Section \\ref{invmiurasec}, Theorem \\ref{stab} and \\eqref{zut} we obtain \\eqref{highernormineq}.\n\\end{proof}\n\nWe now consider the problem of asymptotic stability of the mKdV equation near a kink.  We will require an additional weight function:\n\\[\\phi_{x_0,A}(t,x)=\\phi(t,x)=1+\\tanh\\left(\\frac{x-x_0+\\gamma t}{A}\\right).\\]  \n\n\\begin{prop}  \\label{stableftright2} \nLet $\\gamma < 6$, then there exists $\\delta, A>0$  such that if  $u$ is the  solution to \\eqref{uKINKmKdV} with initial data   satisfying $u(0,\\cdot) - \\tanh(\\cdot) \\in H^1(\\mathbb{R})$  and $\\LPN 2 {u(0,\\cdot)-\\tanh(\\cdot)}<\\delta$, and $x_0\\in \\mathbb{R}$ we have the bounds\n\\begin{equation}\\label{rightofsol2}\n\\int \\eta(x-y(t))\\phi_{A,x_0}(t,x) w(t,x)^2~dx  \\lesssim \\int \\eta(x-y(0)) \\phi_{A,x_0}(0,x)w(0,x)^2~dx ,\n\\end{equation}\nwhere $t>0$, $w:=u-\\tanh(\\cdot-y)$ and $y$ references to the continuous function constructed in Theorem \\ref{stab}.  Moreover, we have the following smoothing estimate:\n\\begin{equation}\\label{othersmooth}\n\\int_0^{\\infty} \\HN{1}{(\\eta(x-y)\\phi_{A,x_0})_x^{1\/2}w}^2~dt\\lesssim \\int\n\\eta(x-y(0)) \\phi_{A,x_0}(0,x)w(0,x)^2~dx.\n\\end{equation} \n\\end{prop}\n\\begin{proof}\nUsing the shorthand $\\eta=\\eta(x-y(t))$, $\\eta_x=\\eta_x(x-y(t))$, \\dots, $\\sech^2=\\sech^2(x-y(t))$, \\dots we obtain:\n\\[\n\\begin{split}\n\\frac {d}{dt} \\int & \\phi\\eta w^2~dx = -B_{\\varepsilon,\n  R}(\\left(\\phi\\eta_x\\right)^{1\/2}w)\\\\ \n& + \\int \\bigg[-(2 +\\dot\n  y)\\phi\\eta_x w^2 + \\phi\\left(4\\eta\n  \\sech^2-8\\eta_x\\tanh\\right)w^3-3\\phi\\eta_xw^4 \n\\\\ & + \\eta\n  \\bigg(-3\\phi_x w_x^2+ \\left(\\gamma\\phi_x + \\phi_{xxx}\n  -6\\tanh^2\\phi_x\\right)w^2-8\\phi_x\\tanh w^3-3\\phi_x w^4\\bigg) \n\\\\ &  +\n  4\\left(\\phi_{xx}\\eta_{x}+\\phi_{x}\\eta_{xx}\\right)w^2 + (2+\\dot\n  y)\\phi\\eta\\sech^2w\\bigg]~dx\n\\end{split}\n\\]\nWe first note that\n\\[\n\\begin{split} \n\\left(\\gamma  -6\\tanh^2(x-y)\\right)\\phi_x(t,x)\n=&\\! \\left((\\gamma-6)+6\\sech^2(x-y)\\right)\\phi_x(t,x)\\!\\\\\n \\leq &  (\\gamma-6)\\phi_x(t,x)  \n\\\\ &+ C A^{-1}\\phi (t,x)\\eta_x(x-y),\n\\end{split}\n\\]\nwhere we used the fact that $\\phi_x\\lesssim A^{-1}\\phi$.\nWe also have the estimate\n\\begin{equation*}\n\\int \\eta(x-y)\\phi_x(t,x)\\tanh(x-y)) w^3 \\lesssim \\LPN{2}{w}\\HN{1}{(\\eta(x-y)\\phi_x(t,\\cdot))^{1\/2}w}^2.\n\\end{equation*}\nThus if we assume $\\LPN{2}{w}$ to be suitably small and $A$ to be large,  then by the above estimates and the arguments in Theorem \\ref{stab} we obtain:\n\\begin{align}\\label{rightright}\n\\begin{split}\n\\frac {d}{dt} \\int \\phi \\eta w^2~dx \\leq &  -\\kappa \\HN{1}{(\\phi\\eta)_x^{1\/2}w}^2\\\\\n& +  c \\HN 1 {\\eta_x w}^2  \\HN{1}{(\\phi\\eta)_x^{1\/2}w}^2\n\\\\\n& +\\int \\bigg[4\\left(\\phi_{xx}\\eta_{x}+\\phi_{x}\\eta_{xx}\\right)w^2\\\\\n &+(2+\\dot y)\\phi\\eta\\sech^2(x-y)w \\bigg]~dx\n\\\\ &+C\\ip{\\phi^{1\/2} w,\\eta(x-y)\\sech^2(x-y)}^2.\n\\end{split}\n\\end{align} \n\nThe plan is to integrate \\eqref{rightright} to obtain our claim but first we will need estimate the last two terms.\n\nFirst note for large $A$ we have the following simple estimates\n\\begin{align*}\n\\abs{\\phi(t,x)-\\phi(t,y(t))}\\sech(x-y) &\\lesssim A^{-1}e^{-2\\abs{(y-x_0+\\gamma t)\/A}},\\\\\n\\phi(t,x)^{-1\/2}&\\lesssim e^{\\abs{(x-x_0+\\gamma t)\/A}},\\quad\\text{and}\\\\\n\\sech(x-y)&\\lesssim \\eta_x(x-y).\n\\end{align*}\nApplying the above estimates we obtain\n\\begin{align}\\label{orthest1}\n\\begin{split}\n\\abs{\\int \\phi\\eta\\sech^2(x-y)w~dx} &= \\abs{\\int \\left(\\phi(t,x)-\\phi(t,y)\\right)\\eta\\sech^2(x-y)w~dx}\n\\\\\n&\\lesssim A^{-1}e^{-2\\abs{(y-x_0+\\gamma t)\/A}}  \\LPN{2}{(\\phi\\eta_x)^{1\/2}w}\\times\\\\&\\qquad\\LPN{2}{\\eta_x^{1\/2}\\phi^{-1\/2}}\\\\\n&\\lesssim A^{-1}e^{-\\abs{(y-x_0+\\gamma t)\/A}} \\LPN{2}{(\\phi\\eta_x)^{1\/2} w}.\n\\end{split}\n\\end{align}\n\nBy (\\ref{dotyeq}) we have\n\\begin{align}\\label{orthest3}\n\\begin{split}\n\\abs{\\dot y +2}&\\lesssim \\int \\sech^2(x-y)\\left(\\abs{w}+\\abs{w}^3\\right)\\\\\n& \\lesssim \\LPN{2}{(\\phi\\eta_x)^{1\/2}w}\\LPN{2}{\\eta_x^{1\/2}\\phi^{-1\/2}}\\left(1+\\LPN{\\infty}{\\eta_x^{1\/2}w}^2\\right)\\\\&\\lesssim e^{\\abs{(y-x_0+\\gamma t)\/A}}\\LPN{2}{(\\phi\\eta_x)^{1\/2}w}\\left(1+\\HN{1}{\\eta_x^{1\/2}w}^2\\right).\n\\end{split}\n\\end{align}\nCombining (\\ref{orthest1}),  (\\ref{orthest3}) we get\n\\begin{multline*}\n\\abs{\\int(2+\\dot y)\\phi\\eta\\sech^2(x-y)w~dx}\\lesssim A^{-1}\\LPN{2}{(\\phi\\eta_x)^{1\/2}w}^2\\bigg(1+\\HN{1}{\\eta_x^{1\/2}w}^2\\bigg),\n\\end{multline*}\nand similarly for the last term we get\n\\[ \n\\begin{split}\n\\ip{\\phi^{1\/2} w,\\eta(x-y)\\sech^2(x-y)}^2 \\lesssim & \\LPN{\\infty}{\\left(\\phi(t,\\cdot)^{1\/2}-\\phi(t,y(t))^{1\/2}\\right)\\sech^{1\/2}(\\cdot-y)\\eta}^2\\\\&\\times \n\\LPN{2}{\\phi^{1\/2}\\sech^{1\/2}(\\cdot-y)w}^2\\LPN{2}{\\sech(\\cdot-y)\\phi^{-1\/2}}^2 \n\\\\\\lesssim  & A^{-1}\\HN{1}{(\\eta_x\\phi)^{1\/2} w}^2.\n\\end{split}\n\\]\nThen from the above estimates, if we assume $A$ to be suitably large we obtain\n\\begin{equation*}\n\\frac {d}{dt} \\int \\phi\\eta w^2~dx \\lesssim \\LPN{2}{(\\phi\\eta)_x^{1\/2}w}^2\\HN{1}{\\eta_x^{1\/2}w}^2.\n\\end{equation*}\n\nThus from Gronwall's inequality we have\n\\begin{equation*}\n\\LPN{2}{\\phi(t,x)^{1\/2}w(t,x)}^2\\lesssim \\LPN{2}{\\phi(0,x)^{1\/2}w(0,x)}\\exp\\left(\\int_0^t\\HN{1}{\\eta_x^{1\/2}w}^2\\right).\n\\end{equation*}\n\nThe claim then follows as a consequence of (\\ref{virial}).\n\\end{proof}\nAs a consequence of the  Proposition \\ref{stableftright2} and Theorem \\ref{stab} we obtain the following theorem.\n\n\\begin{thm}\\label{mKdVAsymStabL2}\nLet $ \\gamma < 6$. Then there exists  $\\delta_{\\gamma}>0$   such that if $u$ is a  solution to \\eqref{mKdV} \nwith initial data  $u_0$, satisfying $u_0 - \\tanh(x) \\in H^1(\\mathbb{R})$  and $\\LPN 2 {u(0,\\cdot)-\\tanh(x)}<\\delta_{\\gamma}$,  \n\\begin{equation}\\label{asymstab}  \n \\lim_{t \\to \\infty} \\norm{ u(t,.) - \\tanh(x-y(t)) }_{L^2( (-\\gamma t , \\infty) ) } =  0 \n\\end{equation} \nwhere $y:[0,\\infty)\\to\\mathbb{R}$ refers to the continuous function constructed in Theorem \\ref{stab}.\n\\end{thm}\n\nMaking use of the Miura transformation to relate mKdV near the kink \nwith KdV near zero,  we will  replace $L^2$ in the statement of the above theorem with $H^s$ for any non-negative integer $s$.  Specifically we have, denoting again $w= u-\\tanh(.-y(t))$:\n\n\\begin{cor}\\label{mKdVAsymStab}\nLet $ \\gamma < 6$ and $s$ any positive integer. Then there exists  $\\delta_{\\gamma}>0$   such that if $u$ is a  solution to \\eqref{uKINKmKdV} \nwith initial data  $u_0$, satisfying $u_0 - \\tanh(\\cdot) \\in H^s(\\mathbb{R})$  and $ \\LPN 2 { u(0,\\cdot)-\\tanh(x)} <\\delta_{\\gamma}$,  \n\\begin{equation}\\label{asymstab2}  \n \\lim_{t \\to \\infty} \\norm{ u(t,\\cdot) - \\tanh(x-y(t)) }_{H^{s}( (-\\gamma t , \\infty) ) } =  0 \n\\end{equation} \nwhere $y:[0,\\infty)\\to\\mathbb{R}$ refers to the continuous function constructed in Theorem \\ref{stab}.  Moreover we have the smoothing estimate\n\\begin{equation}\\label{othersmooth2}  \n\\int_0^\\infty \\HN{s+1}{\\rho_x(t,\\cdot+6t)^{1\/2}w(t,\\cdot)}^2~dt\\leq C\\HN s{\\rho(0,\\cdot)^{1\/2}w(0,\\cdot)}^2.\n\\end{equation}\nwhere C depends on $\\gamma$ and $\\HN s{u(0,\\cdot)-\\tanh(\\cdot)}$, and $\\rho$ is defined as\n\\[\\rho(x,t)=1+\\tanh\\left(\\frac{x-x_0+(\\gamma-6) t}{A}\\right),\\]\nfor some large constant $A>0$.\n \\end{cor}\n\\begin{proof}\nNote that the absolute values of the derivatives of $\\rho$ are bounded above by a constant multiple of $\\rho$.  The same property  is also true for the function $\\rho_x$.  This property of $\\rho$ and $\\rho_x$ will be used extensively below without further comment.  \n\n Define $v$ as in \\eqref{uwrel}, hence  $v$ is a solution to \\eqref{KdV}.  Fixing $t\\ge0$, observe from \\eqref{uwrel},\nLemma \\ref{techlem} we have for $f:=\\rho^{1\/2}$ or $f:=\\rho_x^{1\/2}$ the following estimate\n\\begin{align}\\label{wtov}\n\\begin{split}\n\\HN{s-1}{f(t,\\cdot)v(t,\\cdot)}\\lesssim&\\HN{s-1}{f(t,\\cdot+6t)w(t,\\cdot)^2}+\\\\&\\HN{s-1}{f(t,\\cdot+6t)w(t,\\cdot)\\tanh(\\cdot-y)}+\\\\&\\HN{s-1}{f(t,\\cdot+6t)w_x(t,\\cdot)}\\\\\n\\lesssim& \\left(1+ \\HN {s-1} {w(t,\\cdot)}\\right) \\HN{s}{f(t,\\cdot+6t) w(t,\\cdot)}\\\\\n\\leq &C \\HN{s}{f(t,\\cdot+6t) w(t,\\cdot)},\n\\end{split}\n\\end{align} \nfor all integers $s\\ge 1$, where $C$ depends on $\\HN {s-1} {w(t,\\cdot)}$.\n\nSimilarly we also have the estimate\n\\begin{align}\\label{vtow}\n\\begin{split}\n\\HN{s}{f(t,\\cdot+6t)w(t,\\cdot)}\\lesssim& \\LPN 2{f(t,\\cdot+6t) w(t,\\cdot)}+\\HN{s-1}{f(t,\\cdot+6t) w_x(t,\\cdot)}\\\\\n\\lesssim &\\LPN 2{f(t,\\cdot+6t) w(t,\\cdot)}+\\HN{s-1}{f(t,\\cdot) v(t,\\cdot)}+\\\\\n&\\HN{s-1}{f(t,\\cdot+6t) w(t,\\cdot)^2}+\\\\\n&\\HN{s-1}{f(t,\\cdot+6t) w(t,\\cdot)\\tanh(\\cdot-y)}\\\\\n\\lesssim & \\left(1+\\HN {s-1}{w(t,\\cdot)}+\\HN {1}{w(t,\\cdot)}\\right)\\times\\\\&\\HN{s-1}{f(t,\\cdot+6t) w(t,\\cdot)}+\\HN{s-1}{f(t,\\cdot) v(t,\\cdot)}\\\\\n\\leq & C \\left( \\HN{s-1}{f(t,\\cdot+6t) w(t,\\cdot)}+\\HN{s-1}{f(t,\\cdot) v(t,\\cdot)} \\right),\n\\end{split}\n\\end{align}\nfor all integers $s\\ge 1$, where $C$ depends on $\\HN {s-1} {w(t,\\cdot)}+\\HN {1} {w(t,\\cdot)}$\n\nThe inequalities \\eqref{wtov} and \\eqref{vtow} will essentially allow to shift our focus from a study of mKdV near a kink to that of KdV in a neighbourhood of zero.  In particular, note that by Theorem \\ref{stab} and Corollary \\ref{highernorm}, the constants in \\eqref{wtov} and \\eqref{vtow} depend only on the initial data $\\HN {s-1} {u(0,\\cdot)-\\tanh(\\cdot)}$ and $\\HN {s-1} {u(0,\\cdot)-\\tanh(\\cdot)}+\\HN {1} {u(0,\\cdot)-\\tanh(\\cdot)}$ respectively. \n\nNow consider the case $s=1$.  Below $C$ will denote a positive constant depending on $\\HN {1} {u(0,\\cdot)-\\tanh(\\cdot)}$ and $\\gamma$, which may change from line to line. \n\nA simple energy estimate yields \n\\begin{equation*}\n\\frac {d}{dt} \\int \\rho v^2 dx =\\int \\rho_t v^2+\\rho_{xxx}v^2-3\\rho_x v_x^2-4\\rho_x v^3~dx.\n\\end{equation*}\nNote that replacing $\\rho$ with $1$ we recover the $L^2$ conservation law for KdV. \n\n\nAlso, we have the simple estimate\n\\begin{align*}\n\\int \\rho_x v^3~dx &\\leq \\LPN 2 v \\LPN 2 {\\rho_x^{1\/2} v} \\LPN {\\infty} {\\rho_x^{1\/2} v}\\\\\n&\\lesssim \\LPN 2 {v(0,\\cdot)} \\LPN 2 {\\rho_x^{1\/2} v} \\HN 1 {\\rho_x^{1\/2} v}\\\\\n&\\leq C \\left(\\varepsilon^{-1} \\LPN 2 {\\rho_x^{1\/2} v}^2 +\\varepsilon \\HN 1 {\\rho_x^{1\/2} v}^2\\right),\n\\end{align*}\nfor any $\\varepsilon>0$.\n\nHence from the above estimates\n\\begin{align*}\n\\frac {d}{dt} \\int \\rho(t,x) v^2(t,x)~dx\\leq&- 2 \\LPN{2}{\\rho_x(t,\\cdot)^{1\/2}v_x(t,\\cdot)}^2\n \\\\&\\qquad+  C \\HN{1}{\\rho_x(t,\\cdot+6t)^{1\/2} w(t,\\cdot)}^2.\n\\end{align*}\n\nTherefore from \\eqref{othersmooth} we obtain \n\\begin{align*}\n&\\int \\rho(t,x) v^2(t,x)~dx+2\\int_0^{\\infty} \\LPN{2}{\\rho_x(t^{\\prime},\\cdot)^{1\/2}v_x(t^{\\prime},\\cdot)}^2~dt^{\\prime}\\lesssim \\\\&\\qquad \\int \\rho(0,x) v(0,x)^2~dx+ C\\int \\rho(0,x)w(0,x)^2~dx.\n\\end{align*}\n\nThen from the above inequality, \\eqref{vtow} (with $f=\\rho^{1\/2}$), and \\eqref{asymstab}, we obtain \\eqref{asymstab2} for $s=1$.  Similarly from the above inequality, \\eqref{vtow} (with $f=\\rho_x^{1\/2}$), and \\eqref{othersmooth}, we obtain \\eqref{asymstab2} for $s=1$.  \n\nWe now will provide a sketch of the proof for $s>1$.  We proceed by induction, assuming as our inductive hypothesis that \\eqref{asymstab2} and \\eqref{othersmooth2} holds for a given positive integer $s$. Below $C$ will denote a positive constant depending on $\\HN {s+1} {u(0,\\cdot)-\\tanh(\\cdot)}$ and $\\gamma$, which may change from line to line. \n\nFrom \\eqref{conslaw} we have\n\\begin{equation}\\label{iterhiest}\n\\frac{d}{dt} \\int \\rho  T^{(s)}dx =\\int \\rho_t T^{(s)}+\\rho_x X^{(s)}~dx.\n\\end{equation} \nObserve that from the two monomials $2\\partial_x^su\\partial_x^{s+2}u$ and $-(\\partial_x^{s}u)^2$ in $X^{(s)}$ we recover (after a couple of integration by parts) the terms \n\\begin{equation}\\label{goodterm}\n-3\\rho_x(\\partial_x^{s+1}u)^2+\\rho_{xxx}\\left(\\partial_x^s u\\right)^2,\n\\end{equation}\nin the integrand on the right hand side of \\eqref{iterhiest}.\n\nWe now proceed in a similar manner to the case of $s=1$, using extensively the properties of $X^{(s)}$ and $T^{(s)}$ as stated in Appendix \\ref{conserved}.  In this way, one can show that \n\\begin{align*}\n\\int \\rho_t T^{(s)}+\\rho_x X^{(s)}+3\\rho_x(\\partial_x^{s+1}u)^2~dx &\\lesssim \\left(1+\\HN{s}{v}^{s+1}\\right)\\HN{s}{\\rho_x^{1\/2} v}^2\\\\&\\leq C \\HN{s}{\\rho_x^{1\/2} v}^2.\n\\end{align*}\nIntegrating \\eqref{iterhiest} with respect to $t$, and using our induction hypothesis together with \\eqref{wtov} and \\eqref{vtow} leads to\n\\begin{multline*}\n\\int \\rho(t,x)  T^{(s)}(t,x)~dx+3\\int_0^\\infty \\LPN {2}{\\rho_x(t^{\\prime},\\cdot)^{1\/2}\\left(\\partial_x^{s+1} v\\right)(t^{\\prime},\\cdot)}^2~dt^{\\prime}\\leq \\\\C \\HN{s}{\\rho(0,\\cdot)^{1\/2} w(0,\\cdot)}^2+ \\int \\rho(0,x)  T^{(s)}(0,x)~dx.\n\\end{multline*}\n\nWe observe that the terms on the left hand side of the above equation\nresulting from lower order terms in $T^{(s)}$ can be bounded by a constant multiple of\n\\[\\left(1+\\HN{s}{v}^{s}\\right)\\HN{s-1}{\\rho^{1\/2} v}^2\\leq C \\HN{s-1}{\\rho^{1\/2} v}^2.\\]\nSimilarly, the terms on the right hand side \nresulting from lower order terms in $T^{(s)}$ can be bounded by $C \\HN{s-1}{\\rho(0,\\cdot)^{1\/2} v(0,\\cdot)}^2$.\nThus we obtain\n\\begin{multline*}\n\\int \\rho(t,x)  \\left(\\partial_x^s v\\right)^2(t,x)~dx+3\\int_0^\\infty \\LPN {2}{\\rho_x(t^{\\prime},\\cdot)^{1\/2}\\left(\\partial_x^{s+1} v\\right)(t^{\\prime},\\cdot)}^2~dt^{\\prime}\\leq \\\\ \\int \\rho(0,x)  \\left(\\partial_x^s v\\right)^2(0,x)~dx+\\\\C \\left(\\HN{s}{\\rho(0,\\cdot)^{1\/2} w(0,\\cdot)}^2+\\HN{s-1}{\\rho (t,\\cdot)^{1\/2}v(t,\\cdot)}^2+\\HN{s-1}{\\rho(0,\\cdot)^{1\/2} v(0,\\cdot)}^2\\right).\n\\end{multline*}\n Then applying \\eqref{wtov} and \\eqref{vtow}, together with our induction hypothesis, we obtain \\eqref{asymstab2} and \\eqref{othersmooth2} for $s+1$.\n\\end{proof}\n\n\n\n\n\n\\section{Existence of weak solutions to the Korteweg--de Vries equation with initial data in $H^{-1}$}\n\nWith the help of Theorem \\ref{strongsol}, Lemma \\ref{localbd} and Lemma \\ref{KATOSMOOTH}, we will now prove the existence of weak $L^2$ solutions to the IVP (\\ref{KINKmKdV}).  \n\n\\begin{prop}\\label{weakKINKmKdV}\nFor any $v_0\\in L^2$, there exists a weak solution $u=v+Q$ to (\\ref{KINKmKdV}) satisfying\n\\begin{align}\nv\\in C_{\\omega}([0,\\infty); L^2),\\label{prop1}\\\\\nv_x\\in L^2([0,T]\\times[-R,R]) ~~\\text{for any}~~ R,T<\\infty,\\label{prop2}\\\\\n\\LPN{2}{v(t,\\cdot)} \\lesssim \\LPN{2}{v(0,\\cdot)}+t^{1\/2} ~~\\text{for any}~~ t\\in[0,\\infty),\\label{prop4}\\\\\n v(t,\\cdot) \\to v_0 \\qquad \\text{ in } L^2 ~~\\text{as}~~t\\downarrow 0.\\label{prop3}\n\\end{align}\nFurthermore there exists a $\\delta>0$ such that if $\\LPN 2 {v_0}<\\delta$ then there exists a continuous function $y:\\mathbb{R}\\to\\mathbb{R}$ such that if we write $u=w+\\tanh(\\cdot+y(t))$, we have\n\\begin{equation}\\label{lastpro}\n\\LPN {2}{w}\\lesssim \\LPN 2 {v_0}.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nLet $v_0^{(j)}\\in H^{1}$ be a sequence such that $v_0^{(j)}\\rightarrow\nv_0$ in $L^2$, and $\\LPN{2}{v_0^{(j)}}=\\LPN{2}{v_0}$.  Define $v^{(j)}\\in\nC([0,\\infty); H^{1})$ to be the solution to (\\ref{KINKmKdV}) with\n  $v^{(j)}(0,\\cdot)=r_{0,j}$, corresponding to Theorem \\ref{strongsol}.  \n  \n   If in addition we have $\\LPN 2 {v_0}<\\delta$, and we write \n  \\[u^{(j)}(t,x)=v^{(j)}(t,x)+Q(t,x)=w^{(j)}(t,x)+\\tanh(x+y^{(j)}(t)),\\]\n  where $y^{(j)}$ is defined as in Theorem \\ref{stab}, then using \\eqref{dotyest3}, \\eqref{apriorimkdv}, and \\eqref{virial} we obtain a uniform bound of $y^{(j)}$ in $H^1([0,T])$, and thus by Morrey's inequality we have a uniform bound of $y^{(j)}$ in $C^{0,1\/2}([0,T])$, for any fixed $T>0$.  By the Azel\u00e0-Ascoli theorem, and a suitable diagonal argument we can construct a subsequence $(v^{(N_j)})$ such that for all $T>0$, $y^{(N_j)}$ converges uniformly to some continuous function $y:\\mathbb{R}^+\\rightarrow \\mathbb{R}$.  Moreover from \\eqref{apriorimkdv}, we have for any $t\\geq 0$, there exists a $k$ such that if\n$j>k$\n\\begin{equation}\\label{unibd}\n\\LPN {2}{u^{({N_j})}(t,\\cdot)-\\tanh(\\cdot-y(t))}\\lesssim \\LPN 2 {v_0}.\n\\end{equation}\n\n  \n  Now applying an almost identical argument to the one given in\n  \\cite{MR759907} to construct weak $L^2$ KdV solutions -- here the\n  smoothing estimate is replaced by \\eqref{KATOSMOOTH}, and $L^2$\n  conservation replaced by \\eqref{badL2bd} -- we obtain a subsequence\n  $(v^{(N^{\\prime}_j)})$ such that for any $R,T>0$ the sequence converges\n  weakly in $L^2([0,T]);H^1([-R,R]))$, strongly in\n  $L^2([0,T]\\times[-R,R]))$ and weak-* in $L^\\infty([0,\\infty);L^2)$ to\n    a limit $v$ satisfying (\\ref{prop1}-\\ref{prop4}), and solves\n    \\eqref{KINKmKdV} in the distributional sense. \n\nIn order to prove \\eqref{prop3} we set $\\tilde v= v\\sqrt{1+\\frac1{10}Q}$, and observe that $\\tilde v$ is continuous at  $t=0$ if and only \\eqref{prop3} is satisfied.  Note that weak continuity of $\\tilde v$ in $t$ follows from weak continuity of $v$.  Estimating we obtain\n\\begin{align*}\n\\LPN 2{\\tilde v(t,\\cdot)-\\tilde v(0,\\cdot)}^2&=\\LPN 2{\\tilde v(t,\\cdot)}^2+\\LPN 2{\\tilde v(0,\\cdot)}^2-2\\ip{\\tilde v(t,\\cdot),\\tilde v(0,\\cdot)}\\\\\n&\\le \\left(\\LPN 2{\\tilde v(t,\\cdot)}^2-\\LPN 2{\\tilde v(0,\\cdot)}^2\\right)+2\\LPN 2{\\tilde v(0,\\cdot)}^2-2\\ip{\\tilde v(t,\\cdot),v_0}\\\\\n&=\\left(\\LPN 2{\\tilde v(t,\\cdot)}^2-\\LPN 2{\\tilde v(0,\\cdot)}^2\\right)+2\\ip{\\tilde v(0,\\cdot)-\\tilde v(t,\\cdot),\\tilde v(0,\\cdot)}.\n\\end{align*}\nThen from \\eqref{equality} and the weak continuity of $\\tilde v$ we obtain \\eqref{prop3}.\n\nFinally, note \\eqref{lastpro} is a simple consequence of \\eqref{unibd}.\n\\end{proof}\n\nWe will now construct weak $H^{-1}$ solutions to the Korteweg--de\nVries equation.  Using the scaling symmetry, we may restrict to small\ninitial data in $H^{-1}$.\n\n\n\\begin{prop}\\label{lastprop}\nFor any $u_0\\in H^{-1}$ satisfying $\\HN{-1}{u_0}\\leq \\epsilon$, for $\\epsilon>0$ chosen suitably small, there exists a weak solution $u$ to (\\ref{KINKmKdV}), and a continuous function $y:[0,\\infty)\\to\\mathbb{R}$  satisfying\n\\begin{align}\nu\\in C_{\\omega}([0,\\infty); H^{-1}),\\label{propp1}\\\\\nu\\in L^2([0,T]\\times[-R,R]) ~~\\text{for any}~~ R,T<\\infty,\\label{propp2}\\\\\n\\HN{-1}{u}\\lesssim \\HN{-1}{f},\\label{propp3}\\\\\nu(t,\\cdot) \\to u_0 \\quad \\text{ in } H^{-1} ~~\\text{as}~~t\\downarrow 0.\\label{propp4}\n\\end{align}\n\\end{prop}\n\\begin{proof}\nDefine $v_0$ such that $F(v_0)=(u_0,0)$, where $F$ is defined as in Theorem \\ref{inverse}.  Then by Proposition \\ref{weakKINKmKdV}, there exists a weak solution $\\tilde u$ to the mKdV equation corresponding to initial data $v_0+\\tanh(\\cdot)$.  Let $u$ be map obtained by applying the Galilean transformation ($h=6$) to $\\miura(\\tilde u)$. It is then easy to check that $u$ satisfies (\\ref{propp1}-\\ref{propp4}).  What remains to be shown is that $u$ satisfies (\\ref{KdV}) in a distributional sense, which is equivalent to $\\miura(\\tilde u)$ satisfying (\\ref{KdV}) in a distributional sense -- this is the subject of Lemma \\ref{distkdvlem} below.\n\\end{proof}\n\n\n\\begin{lem}\\label{distkdvlem}\nLet $v_0\\in L^2$, and suppose $\\tilde u$ is a weak solution to (\\ref{mKdV}), satisfying the properties (\\ref{prop1}-\\ref{prop3}), then $u:=\\partial_x(\\tilde u)+(\\tilde u)^2$ satisfies (\\ref{KdV}), in a distributional sense, i.e.\n\\begin{equation*}\n\\int_{\\mathbb{R}^2}\\left[-u\\varphi_t - u\\varphi_{xxx} + 3 u^2\\varphi_x\\right]~dt~dx  =0.\n\\end{equation*}\nfor all $\\varphi\\in C^{\\infty}_0$.\n\\end{lem}\n\nFor a proof of the above lemma we refer the reader to the papers \\cite{MR990865} and \\cite{MR2189502}.  \n\nBy utilising the scaling symmetry of the KdV equation and Proposition\n\\ref{lastprop}, one easily obtains existence of weak solutions of\nTheorem \\ref{mainTheorem}.\n\n\n\\section{A priori bounds and soliton stability}\n\n\n\\begin{repthm}{aprioriTHM}\nSuppose $u\\in C([0,\\infty);H^s(\\mathbb{R}))$ is a solution to (\\ref{KdV}), for some $s\\geq -\\frac{3}{4}$, then\n\\begin{equation}\n\\HN{-1}{u(t,\\cdot)} \\lesssim  \\HN{-1}{u_0}+ \\HN{-1}{u_0}^3   ~~\\text{ for }~~ t\\in[0,\\infty).\n\\end{equation}\n\\end{repthm}\n\\begin{proof}\nFirst consider the case when $s=0$. By scaling, the problem reduces to showing that for all solutions $u\\in C([0,\\infty);L^2(\\mathbb{R}))$ to (\\ref{KdV}) satisfying $\\HN{-1}{u}\\leq \\epsilon$ for some suitably chosen $\\epsilon>0$, we have\n\\begin{equation}\\label{apriori3}\n\\HN{-1}{u(t,\\cdot)} \\lesssim 1  ~~\\text{for any}~~ t\\in[0,\\infty).\n\\end{equation}\nFrom Theorem \\ref{inverse}, Theorem \\ref{strongsol} and the well-posedness theory of the KdV equation, it follows that there exists a solution $\\tilde{u}\\in C([0,\\infty);H^{1}(\\mathbb{R}))$ to (\\ref{mKdV}), such that the Galilean transformation ($h=6$) of $\\miura(\\tilde{u})$ is   $u$. Assuming we chose $\\epsilon$ sufficiently  small, then as a consequence of Lemma (\\ref{localbd}) and Theorem \\ref{stab}, we obtain  (\\ref{apriori3}).\n\n The general case when $s\\geq -\\frac{3}{4}$ can be proven via approximation.\n\\end{proof}\n\n\\begin{repthm}{KdVstab} \nThere exists an $\\varepsilon >0$ such that \nif $u\\in C([0,\\infty);H^s(\\mathbb{R})\\cap H^{-3\/4}(\\mathbb{R}))$ is a solution to (\\ref{KdV}), for some integer $s\\geq-1$, satisfying $\\HN{-1}{R_c-u_0}<\\varepsilon c^{1\/4}$ for some $c>0$, then there is a continuous function $y:[0,\\infty)\\to\\mathbb{R}$ such that \n\\[ \\HN{s}{ u - R_{ c}(x-y(t))} \\leq \\gamma_{s}(c,\\HN{s} {R_c-u_0})\\] \nfor any $t\\geq 0$, where $\\gamma_{s}:(0,\\infty)\\times [0,\\infty)$ is a continuous function, \npolynomial in the second variable, which  satisfies $\\gamma(\\cdot,0)=0$.\n\\end{repthm} \n\\begin{proof}\n  The proof follows in a similar manner to that of Theorem\n  \\ref{aprioriTHM}.  Again, without loss of generality we may assume\n  $u_0\\in H^1$.  By scaling we may also assume that $c=4$.  Then assuming\n  $\\HN{-1}{R_c-u(0,\\cdot)}$ to be suitably small, and making use of\n  the arguments in Section \\ref{invmiurasec}, we may link the KdV IVP with initial\n  data $u(0,\\cdot)$ to the mKdV IVP with initial data $\\tilde\n  u_0:=\\lambda\\tanh(\\lambda\\cdot)+v_0$, for some $\\lambda\\approx 1$,\n  such that $v_0\\in H^{s+1}$ and $\\LPN{2}{v_0}\\lesssim\n  \\HN{-1}{R_4-u_0}$.  By scaling on the mKdV side, we can\n  assume $\\lambda=1$.  The conclusion then follows from Theorem\n  \\ref{strongsol}, the well posedness theory of the KdV equation,\n  Theorem \\ref{stab}, and Corollary \\ref{highernorm}.  \n\\end{proof}\n\nMaking use of Theorem \\ref{mKdVAsymStabL2}, Corollary \\ref{mKdVAsymStab}, and following a similar argument to that given above we obtain:\n\n\\begin{repthm}{KdVAsymStab} \nGiven real $\\gamma>0$ and integer $s\\geq -1$, there exists an\n  $\\varepsilon_{\\gamma} >0$ such that if $u\\in\n  C([0,\\infty);H^s(\\mathbb{R})\\cap H^{-3\/4}(\\mathbb{R}))$, is a solution to\n  (\\ref{KdV}), satisfying \n\\[ \\HN{-1}{R_c-u_0}<\\varepsilon_{\\gamma}  c^{1\/4} \\]\nfor $c>0$, then there is a continuous function\n  $y:[0,\\infty)\\to\\mathbb{R}$ and $\\tilde c>0$ such that\n\\begin{equation*}  \n \\lim_{t \\to \\infty} \\norm{ u - R_{\\tilde c}(x-y(t)) }_{H^{s}( (\\gamma t , \\infty) ) } =  0 \n\\end{equation*} \nfor any $t\\geq 0$.  Moreover we have the bound $\\abs{c-\\tilde  c}\\lesssim  c^{\\frac34} \\HN{-1}{R_c-u_0}   $.\n\\end{repthm}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\label{introduction}\n\nRecent work in higher gauge theory has revealed the importance of\ncategorifying the theory of bundles and considering `2-bundles', where\nthe fiber is a topological category instead of a topological space\n\\cite{BaezSchreiber}.  These structures show up not only in\nmathematics, where they form a useful generalization of nonabelian\ngerbes \\cite{BreenMessing:2001}, but also in physics, where they can\nbe used to describe parallel transport of strings \\cite{SSS,Schreiber}.\n\nThe concepts of `\\v{C}ech cohomology' and `classifying space' play a\nwell-known and fundamental role in the theory of bundles.  For \nany topological group $G$, principal $G$-bundles over a space $M$ \nare classified by the first \\v{C}ech cohomology of $M$ with \ncoefficients in $G$.  Furthermore, under some mild conditions, \nthese \\v{C}ech cohomology classes are in 1-1 correspondence with \nhomotopy classes of maps from $M$ to the classifying space $BG$.  \nThis lets us define characteristic classes for bundles, coming from\ncohomology classes for $BG$.\n\nAll these concepts and results can be generalized from bundles to\n2-bundles.  Bartels \\cite{Bartels:2004} has defined principal\n$\\G$-2-bundles where $\\G$ is a `topological 2-group': roughly\nspeaking, a categorified version of a topological group.  Furthermore,\nhis work shows how principal $\\G$-2-bundles over $M$ are classified by\n$\\check{H}^1(M,\\G)$, the first \\v{C}ech cohomology of $M$ with\ncoefficients in $\\G$.  This form of cohomology, also known as\n`nonabelian cohomology', is familiar from work on nonabelian gerbes\n\\cite{Breen1,Giraud}.\n\nIn fact, under mild conditions on $\\G$ and $M$, there is a 1-1\ncorrespondence between $\\check{H}^1(M,\\G)$ and the set of homotopy \nclasses of maps from $M$ to a certain space $B|\\G|$: the classifying\nspace of the geometric realization of the nerve of $\\G$.   So, \n$B|\\G|$ serves as a classifying space for the topological 2-group\n$\\G$!  This paper seeks to provide an introduction to topological \n2-groups and nonabelian cohomology leading up to a self-contained \nproof of this fact.\n\nIn his pioneering work on this subject, Jur\\v{c}o \\cite{Jurco} \nasserted that a certain space homotopy equivalent to ours is a \nclassifying space for the first \\v{C}ech cohomology with \ncoefficients in $\\G$.  However, there are some gaps in his argument for \nthis assertion (see Section \\ref{remarks} for details).\n\nLater, Baas, B\\\"okstedt and Kro \\cite{BBK} gave the definitive\ntreatment of classifying spaces for 2-bundles.  For any `good'\ntopological 2-category $\\C$, they construct a classifying space \n$B\\C$.  They then show that for any space $M$ with the homotopy\ntype of a CW complex, concordance classes of `charted $\\C$-2-bundles' \ncorrespond to homotopy classes of maps from $M$ to $B\\C$.\nIn particular, a topological 2-group is just a topological \n2-category with one object and with all morphisms and 2-morphisms \ninvertible --- and in this special case, their result \\textit{almost} \nreduces to the fact mentioned above.\n\nThere are some subtleties, however.  \nMost importantly, while their `charted $\\C$-2-bundles' \nreduce precisely to our principal $\\G$-2-bundles, they classify\nthese 2-bundles up to concordance, while we classify them up to \na superficially different equivalence relation.  Two $\\G$-2-bundles \nover a space $X$ are `concordant' if they are restrictions of some \n$\\G$-2-bundle over $X \\times [0,1]$ to the two ends $X \\times \\{0\\}$ \nand $X \\times \\{1\\}$.  This makes it easy to see that homotopic maps \nfrom $X$ to the classifying space define concordant $\\G$-2-bundles.   \nWe instead consider two $\\G$-2-bundles to be equivalent if their \ndefining \\v{C}ech 1-cocycles are cohomologous.  In this approach, \nsome work is required to show that homotopic maps from $X$ to the \nclassifying space define equivalent $\\G$-2-bundles.  {\\em A priori}, \nit is not obvious that two $\\G$-2-bundles are equivalent\nin this \\v{C}ech sense if and only if they are concordant.  However, \nsince the classifying space of Baas, B\\\"okstedt and Kro is homotopy\nequivalent to the one we use, it follows from our work that these equivalence \nrelations are the same --- at least given $\\G$ and $M$ satisfying \nthe technical conditions of both their result and ours.\n\nWe also discuss an interesting example: the `string 2-group' \n$\\mathrm{String}(G)$ of a simply-connected compact simple Lie group $G$ \n\\cite{BCSS,Henriques}.  As its name suggests, this 2-group is of \nspecial interest in physics.  Mathematically, a key fact is that \n$|\\mathrm{String}(G)|$ --- the geometric realization of the nerve of \n$\\mathrm{String}(G)$ --- is the 3-connected cover of $G$.   Using this, \none can compute the rational cohomology of $B|\\mathrm{String}(G)|$.  \nThis is nice, because these cohomology classes give `characteristic \nclasses' for principal $\\G$-2-bundles, and when $M$ is a manifold\none can hope to compute these in terms of a connection and its \ncurvature, much as one does for ordinary principal bundles with a \nLie group as structure group.\n\nSection \\ref{summary} is an overview, starting with a review of\nthe classic results that people are now categorifying.  Section \n\\ref{2groups} reviews four viewpoints on topological 2-groups.  \nSection \\ref{nonabelian} explains nonabelian cohomology with\ncoefficients in a topological 2-group.  Finally, in Section \\ref{proofs} \nwe prove the results stated in Section \\ref{summary}, and comment\na bit further on the work of Jur\\v{c}o and Baas--B\\\"okstedt--Kro.\n\n\\section{Overview}\n\\label{summary}\n\nOnce one knows about `topological 2-groups', it is irresistibly tempting \nto generalize all ones favorite results about topological groups to \nthese new entities.  So, let us begin with a quick review of some \nclassic results about topological groups and their classifying spaces.\n\nSuppose that $G$ is a topological group.  The \\v{C}ech cohomology\n$\\check{H}^1(M,G)$ of a topological space $M$ with coefficients in $G$\nis a set carefully designed to be in 1-1 correspondence with the set\nof isomorphism classes of principal $G$-bundles on $M$.  Let us recall\nhow this works.\n\nFirst suppose $\\cU = \\{U_i\\}$ is an open cover of $M$ and $P$ is a principal\n$G$-bundle over $M$ that is trivial when restricted to each open set\n$U_i$.  Then by comparing local trivialisations of $P$ over $U_i$ and\n$U_j$ we can define maps $g_{ij}\\colon U_i\\cap U_j \\to G$: the\ntransition functions of the bundle.  On triple intersections \n$U_i\\cap U_j\\cap U_k$, these maps satisfy a cocycle condition:\n$$ \ng_{ij}(x)g_{jk}(x) = g_{ik}(x) \n$$ \nA collection of maps $g_{ij}\\colon U_i\\cap U_j \\to G$ satisfying \nthis condition is called a `\\v{C}ech 1-cocycle' subordinate to the\ncover $\\cU$.  Any such 1-cocycle defines a principal $G$-bundle \nover $M$ that is trivial over each set $U_i$.\n\nNext, suppose we have two principal $G$-bundles over $M$ that are\ntrivial over each set $U_i$, described by \\v{C}ech 1-cocycles $g_{ij}$ and\n$g'_{ij}$, respectively.  These bundles are isomorphic if and only if\nfor some maps $f_i \\colon U_i \\to G$ we have\n$$ \ng_{ij}(x) f_j(x) = f_i(x) g'_{ij}(x) \n$$\non every double intersection $U_i \\cap U_j$.  In this case\nwe say the \\v{C}ech 1-cocycles are `cohomologous'.  We\ndefine $\\check{H}^1(\\cU,G)$ to be the quotient of the set of \n\\v{C}ech 1-cocycles subordinate to $\\cU$ by this equivalence\nrelation.  \n\nRecall that a `good' cover of $M$ is an open cover $\\cU$ for which all\nthe non-empty finite intersections of open sets $U_i$ in $\\cU$ are\ncontractible.  We say a space $M$ \\textbf{admits good covers} if any\ncover of $M$ has a good cover that refines it.  For example, any\n(paracompact Hausdorff) smooth manifold admits good covers, as does any\nsimplicial complex.\n\nIf $M$ admits good covers, $\\check{H}^1(\\cU,G)$ is independent\nof the choice of good cover $\\cU$.  So, we can denote it simply by \n$\\check{H}^1(M,G)$.  Furthermore, this set $\\check{H}^1(M,G)$ is in \n1-1 correspondence with the set of isomorphism classes of \nprincipal $G$-bundles over $M$.  The reason is that we can always\ntrivialize any principal $G$-bundle over the open sets in a good\ncover.  \n\nFor more general spaces, we need to define the \\v{C}ech cohomology\nmore carefully.  If $M$ is a paracompact Hausdorff space, we can\ndefine it to be the limit \n\\[    \\check{H}^1(M,G) = \\varinjlim_{\\cU}\n\\check{H}^1(\\cU,G)  \\]\nover all open covers, partially ordered by refinement.  \n\nIt is a classic result in topology that $\\check{H}^1(M,G)$ can be \nunderstood using homotopy theory with the help of Milnor's construction\n\\cite{Dold,Milnor} of the classifying space $BG$: \n\n\\setcounter{theorem}{-1}\n\\begin{theorem} \n\\label{milnor_theorem}\nLet $G$ be a topological group.  Then there is a topological space \n$BG$ with the property that for any paracompact Hausdorff space \n$M$, there is a bijection \n$$ \n\\check{H}^1(M,G) \\cong [M,BG] \n$$ \n\\end{theorem}\n\n\\noindent\nHere $[X,Y]$ denotes the set of homotopy classes of maps from $X$ into\n$Y$.  The topological space $BG$ is called the \\textbf{classifying space} \nof $G$.  There is a canonical principal $G$-bundle on $BG$, called the\nuniversal $G$-bundle, and the theorem above is usually understood as\nthe assertion that every principal $G$-bundle $P$ on $M$ is obtained\nby pullback from the universal $G$-bundle under a certain map $M\\to\nBG$ (the classifying map of $P$).\n\nNow let us discuss how to generalize all these results to topological \n2-groups.  First of all, what is a `2-group'?  It is like a group, \nbut `categorified'. While a group is a \\emph{set}\nequipped with \\emph{functions} describing multiplication and inverses,\nand an identity \\emph{element}, a 2-group is a \\emph{category}\nequipped with \\emph{functors} describing multiplication and inverses,\nand an identity \\emph{object}.  Indeed, 2-groups are also known as\n`categorical groups'.\n\nA down-to-earth way to work with 2-groups involves treating them as\n`crossed modules'.  A crossed module consists of a pair of groups $H$\nand $G$, together with a homomorphism $t\\colon H\\to G$ and an action\n$\\alpha$ of $G$ on $H$ satisfying two conditions, equations\n\\eqref{equivariance} and \\eqref{peiffer} below.  Crossed modules were\nintroduced by J.\\ H.\\ C.\\ Whitehead \\cite{Whitehead} without the aid\nof category theory.  Mac Lane and Whitehead \\cite{MW} later proved\nthat just as the fundamental group captures all the homotopy-invariant\ninformation about a connected pointed homotopy 1-type, a crossed\nmodule captures all the homotopy-invariant information about a\nconnected pointed homotopy 2-type.  By the 1960s it was clear to\nVerdier and others that crossed modules are essentially the same as\ncategorical groups.  The first published proof of this may be due to Brown\nand Spencer \\cite{BS}.\n\nJust as one can define principal $G$-bundles over a space $M$ for any\ntopological group $G$, one can define `principal $\\G$-2-bundles' over\n$M$ for any topological 2-group $\\G$.  Just as a principal $G$-bundle\nhas a copy of $G$ as fiber, a principal $\\G$-2-bundle has a copy of\n$\\G$ as fiber.  Readers interested in more details are urged to read\nBartels' thesis, available online \\cite{Bartels:2004}.  We shall have\nnothing else to say about principal $\\G$-2-bundles except that they\nare classified by a categorified version of \\v{C}ech cohomology,\ndenoted $\\check{H}^1(M,\\cG)$.\n\nAs before, we can describe this categorified \\v{C}ech cohomology as a\nset of cocycles modulo an equivalence relation.  Let $\\cU$ be a cover\nof $M$.  If we think of the $2$-group $\\cG$ in terms of its associated\ncrossed module $(G,H,t,\\alpha)$, then a cocycle subordinate to $\\cU$\nconsists (in part) of maps $g_{ij}\\colon U_i\\cap U_j\\to G$ as before.\nHowever, we now `weaken' the cocycle condition and only require that\n\\begin{equation}\n\\label{triangle.eq}\nt(h_{ijk}) g_{ij}g_{jk} = g_{ik} \n\\end{equation}\nfor some maps $h_{ijk}\\colon U_i\\cap U_j\\cap U_k\\to H$.  These maps\nare in turn required to satisfy a cocycle condition of their own on\nquadruple intersections, namely\n\\begin{equation}\n\\label{tetrahedron.eq}\n\\a(g_{ij})(h_{jkl})h_{ijl} = h_{ijk}h_{ikl}\n\\end{equation}\nwhere $\\alpha$ is the action of $G$ on $H$.  This mildly\nintimidating equation will be easier to understand when we draw it as\na commuting tetrahedron --- see equation \\eqref{tetrahedron} in \nSection~\\ref{nonabelian}.  The pair $(g_{ij},h_{ijk})$ is called a \n$\\cG$-valued \\textbf{\\v{C}ech 1-cocycle} subordinate to $\\cU$.  \n\nSimilarly, we say two cocycles $(g_{ij},h_{ijk})$\nand $(g'_{ij},h'_{ijk})$ are \\textbf{cohomologous} if\n\\begin{equation}\n\\label{square.eq}\nt(k_{ij}) g_{ij} f_j = f_i g'_{ij} \n\\end{equation}\nfor some maps $f_i \\colon U_i \\to G$ and $k_{ij} \\colon U_i \\cap U_j \\to H$,\nwhich must make a certain prism commute --- see equation\n\\eqref{prism}.   We define $\\check{H}^1(\\cU,\\G)$ to be the set of \ncohomology classes of $\\G$-valued \\v{C}ech 1-cocycles.  \nTo capture the entire cohomology set $\\check{H}^1(M,\\cG)$, \nwe must next take a limit of the sets $\\check{H}^1(\\cU,\\G)$ \nas $\\cU$ ranges over all covers of $M$.  For more details we refer \nto Section~\\ref{nonabelian}. \n\nTheorem \\ref{milnor_theorem} generalizes nicely from topological \ngroups to topological 2-groups.  But, following the usual tradition\nin algebraic topology, we shall henceforth work in the category of \n$k$-spaces, i.e., compactly generated weak Hausdorff spaces.  So, \nby `topological space' we shall always mean a $k$-space,\nand by `topological group' we shall mean a group object in the\ncategory of $k$-spaces.  \n  \n\\begin{theorem} \n\\label{classifying}\nSuppose that $\\G$ is a well-pointed topological 2-group and\n$M$ is a paracompact Hausdorff space admitting good covers.  Then\nthere is a bijection\n$$ \\check{H}^1(M,\\G) \\cong [M,B|\\G|]\n$$ \nwhere the topological group $|\\G|$ is the geometric realization of\nthe nerve of $\\G$.\n\\end{theorem} \n\\noindent\nOne term here requires explanation.  A topological group $G$ is \nsaid to be `well pointed' if $(G,1)$ is an NDR pair, or in other words \nif the inclusion $\\{1\\}\\hookrightarrow G$ is a closed cofibration.  \nWe say that a topological 2-group $\\G$ is {\\bf well pointed} if \nthe topological groups $G$ and $H$ in its corresponding crossed \nmodule are well pointed.  \nFor example, any `Lie 2-group' is well pointed: a topological 2-group is \ncalled a {\\bf Lie 2-group} if $G$ and $H$ are Lie groups and the maps \n$t,\\alpha$ are smooth.\nMore generally, any `Fr\\'echet Lie 2-group' \\cite{BCSS} is well pointed.  \nWe explain the importance of this notion in Section\n\\ref{proof.thm.1}.\n\nBartels \\cite{Bartels:2004} has already considered two examples of principal\n$\\G$-2-bundles, corresponding to abelian gerbes and nonabelian gerbes.\nLet us discuss the classification of these before turning to a third,\nmore novel example.\n\nFor an abelian gerbe \\cite{Brylinski}, we first choose an abelian\ntopological group $H$ --- in practice, usually just $\\mathrm{U}(1)$.  Then, we\nform the crossed module with $G = 1$ and this choice of $H$, with $t$\nand $\\alpha$ trivial.  The corresponding topological 2-group deserves\nto be called $H[1]$, since it is a `shifted version' of $H$.  Bartels\nshows that the classification of abelian $H$-gerbes matches the\nclassification of $H[1]$-2-bundles.  It is well-known that\n$$  |H[1]| \\cong BH   $$\nso the classifying space for abelian $H$-gerbes is \n$$ B|H[1]| \\cong B(BH)  $$\nIn the case $H = \\mathrm{U}(1)$, this classifying space is just $K(\\ZZ,3)$.  \nSo, in this case, we recover the well-known fact that abelian \n$\\mathrm{U}(1)$-gerbes over $M$ are classified by\n\\[          [M, K(\\ZZ,3)] \\cong H^3(M,\\ZZ) \\]\njust as principal $\\mathrm{U}(1)$ bundles are classified by $H^2(M,\\ZZ)$.\n\nFor a nonabelian gerbe \\cite{Breen1,GS,Giraud}, we fix any topological\ngroup $H$.  Then we form the crossed module with $G = \\Aut(H)$ and\nthis choice of $H$, where $t \\colon H \\to G$ sends each element of $H$\nto the corresponding inner automorphism, and the action of $G$ on $H$\nis the tautologous one.  This gives a topological 2-group called\n$\\AUT(H)$.  Bartels shows that the classification of nonabelian\n$H$-gerbes matches the classification of $\\AUT(H)$-2-bundles.  It\nfollows that, under suitable conditions on $H$, nonabelian $H$-gerbes are classified by homotopy classes\nof maps into $B|\\AUT(H)|$.\n\nA third application of Theorem \\ref{classifying} arises when\n$G$ is a simply-connected compact simple Lie group.  For any such\ngroup there is an isomorphism $H^3(G,\\ZZ) \\cong \\ZZ$ and the generator\n$\\nu\\in H^3(G,\\ZZ)$ transgresses to a characteristic class $c\\in\nH^4(BG,\\ZZ) \\cong \\ZZ$.  Associated to $\\nu$ is a map $G\\to\nK(\\ZZ,3)$ and it can be shown that the homotopy fiber of this\ncan be given the structure of a topological group $\\hat{G}$.  \nThis group $\\hat{G}$ is the 3-connected cover of $G$.  \nWhen $G = \\Spin(n)$, this group $\\hat{G}$ is known as\n$\\mathrm{String}(n)$.  In general, we might call $\\hat{G}$ the {\\bf string\ngroup} of $G$.  Note that until one picks a specific construction for\nthe homotopy fiber, $\\hat{G}$ is only defined up to homotopy --- or\nmore precisely, up to equivalence of $A_\\infty$-spaces.\n\nIn \\cite{BCSS}, under the above hypotheses on $G$, a topological\n2-group subsequently dubbed the \\textbf{string 2-group} of $G$ was\nintroduced.  Let us denote this by $\\mathrm{String}(G)$.  A key\nresult about $\\mathrm{String}(G)$ is that the topological group $|\\mathrm{String}(G)|$\nis equivalent to $\\hat{G}$.  By construction $\\mathrm{String}(G)$ is a\nFr\\'echet Lie 2-group, hence well pointed.  So, from Theorem\n\\ref{classifying} we immediately conclude:\n\n\\begin{corollary}\n\\label{equivalence}\nSuppose that $G$ is a simply-connected compact simple Lie group.\nSuppose $M$ is a paracompact Hausdorff space admitting good\ncovers.  Then there are bijections between the following sets:\n\\begin{itemize}\n\\item\nthe set of equivalence classes of principal $\\mathrm{String}(G)$-2-bundles over $M$,\n\\item \nthe set of isomorphism classes of principal $\\hat{G}$-bundles over $M$,\n\\item\n$\\check{H}^1(M,\\mathrm{String}(G))$,\n\\item\n$\\check{H}^1(M,\\hat{G})$,\n\\item\n$[M, B\\hat{G}]$.\n\\end{itemize}\n\\end{corollary}\n\nOne can describe the rational cohomology of $B\\hat{G}$ in terms of the\nrational cohomology of $BG$, which is well-understood.  The following\nresult was pointed out to us by Matt Ando \\cite{Ando}, and later \ndiscussed by Greg Ginot \\cite{Ginot}:\n\n\\begin{theorem} \n\\label{string}\nSuppose that $G$ is a simply-connected compact simple Lie group, and\nlet $\\hat{G}$ be the string group of $G$.  Let $c\\in H^4(BG,\\QQ)\n= \\QQ$ denote the transgression of the generator $\\nu\\in H^3(G,\\QQ) =\n\\QQ$.  Then there is a ring isomorphism\n$$ \nH^*(B\\hat{G},\\QQ) \\cong H^*(BG,\\QQ)\/\\langle c\\rangle  \n$$ \nwhere $\\langle c \\rangle$ is the ideal generated by $c$.\n\\end{theorem} \n\nAs a result, we obtain characteristic classes for $\\mathrm{String}(G)$-2-bundles:\n\n\\begin{corollary} \n\\label{characteristic}\nSuppose that $G$ is a simply-connected compact simple Lie group and\n$M$ is a paracompact Hausdorff space admitting good covers.  Then an\nequivalence class of principal $\\mathrm{String}(G)$-2-bundles over $M$ \ndetermines a ring homomorphism\n$$ H^*(BG,\\QQ)\/\\langle c \\rangle \\to H^*(M,\\QQ) $$\n\\end{corollary} \n\nTo see this, we use Corollary \\ref{equivalence} to \nreinterpret an equivalence class of principal $\\G$-2-bundles over $M$ \nas a homotopy class of maps $f \\colon M \\to B|\\G|$.  Picking any \nrepresentative $f$, we obtain a ring homomorphism\n$$ f^\\ast \\colon H^*(B|\\G|,\\QQ) \\to H^*(M,\\QQ). $$\nThis is independent of the choice of representative.  Then,\nwe use Theorem \\ref{string}.\n\nIt is a nice problem to compute the rational characteristic classes \nof a principal $\\mathrm{String}(G)$-2-bundle over a manifold using de Rham\ncohomology.  It should be possible to do this using the curvature of\nan arbitrary connection on the 2-bundle, just as for ordinary principal\nbundles with a Lie group as structure group.  Sati, Schreiber and\nStasheff \\cite{SSS} have recently made excellent progress on solving\nthis problem and its generalizations to $n$-bundles for higher $n$.  \n\n\\section{Topological $2$-Groups} \n\\label{2groups}\n\nIn this section we recall four useful perspectives on topological\n$2$-groups.  For a more detailed account, we refer the reader to\n\\cite{HDA5}.  \n\nRecall that for us, a `topological space'\nreally means a $k$-space, and a `topological group' really means a\ngroup object in the category of $k$-spaces.\nA \\textbf{topological $2$-group} is a groupoid in the category of\ntopological groups.  In other words, it is a groupoid $\\G$ where\nthe set $\\Ob(\\G)$ of objects and the set $\\Mor(\\G)$ of morphisms \nare each equipped with the structure of a topological group such that \nthe source and target maps $s,t\\colon \\Mor(\\G)\\to \\Ob(\\G)$, the \nmap $i\\colon \\Ob(\\G)\\to \\Mor(\\G)$ assigning each object its identity\nmorphism, the composition map \n$\\circ\\colon \\Mor(\\G)\\times_{\\Ob(\\G)}\\Mor(\\G)\\to \\Mor(\\G)$,\nand the map sending each morphism to its inverse are all continuous \ngroup homomorphisms.   \n\nEquivalently, we can think of a topological 2-group as a group in\nthe category of topological groupoids.  A \\textbf{topological groupoid}\nis a groupoid $\\G$ where $\\Ob(\\G)$ and $\\Mor(\\G)$ are topological\nspaces (or more precisely, $k$-spaces) and all the groupoid\noperations just listed are continuous maps.  \nWe say that a functor $f\\colon \\G\\to\n\\G'$ between topological groupoids is \\textbf{continuous} if\nthe maps $f\\colon \\Ob(\\G)\\to \\Ob(\\G')$ and $f\\colon \\Mor(\\G)\\to\n\\Mor(\\G')$ are continous.\nA group in\nthe category of topological groupoids is such a thing equipped with\ncontinuous functors $m \\colon \\G \\times \\G \\to \\G$, $\\mathrm{inv} \\colon \\G \n\\to \\G$ and a unit object $1 \\in \\G$ satisfying the usual group axioms,\nwritten out as commutative diagrams.  \n\nThis second viewpoint is useful because\nany topological groupoid $\\G$ has a `nerve' $N\\G$, a simplicial \nspace where the space of $n$-simplices consists of composable strings of\nmorphisms\n$$  \nx_0 \\stackrel{f_1}{\\longrightarrow} x_1 \\stackrel{f_2}{\\longrightarrow}\n\\cdots \\stackrel{f_{n-1}}{\\longrightarrow} \nx_{n-1} \\stackrel{f_n}{\\longrightarrow} x_n \n$$\nTaking the geometric realization of this nerve, we obtain a \ntopological space which we denote as $|\\G|$ for short.\nIf $\\G$ is a topological 2-group, its nerve inherits a group\nstructure, so that $N\\G$ is a topological simplicial group.  This in\nturn makes $|\\G|$ into a topological group.  \n\nA third way to understand topological 2-groups is to view them as\ntopological crossed modules.  Recall that a {\\bf topological crossed module} \n$(G,H,t,\\a)$ consists of topological groups $G$ and $H$ together \nwith a continuous homomorphism \n$$ \nt\\colon H\\to G\\qquad \n$$ \nand a continuous action\n$$\n\\begin{array}{rcl}\n\\alpha \\colon G \\times H & \\to & H \\\\\n   (g,h) & \\mapsto & \\alpha(g) h\n\\end{array}\n$$\nof $G$ as automorphisms of $H$,\nsatisfying the following two identities: \n\\begin{align} \n\\label{equivariance}\n& t(\\a(g)(h)) = gt(h)g^{-1} \\\\ \n\\label{peiffer}\n& \\a(t(h))(h') = hh'h^{-1}. \n\\end{align} \nThe first equation above implies that the map $t\\colon H\\to G$ is equivariant \nfor the action of $G$ on $H$ defined by $\\a$ \nand the action of $G$ on itself by conjugation.  The second equation\nis called the \\textbf{Peiffer identity}.  When no confusion is likely to \nresult, we will sometimes denote the 2-group corresponding\nto a crossed module $(G,H,t,\\a)$ simply by $H \\to G$.  \n\nEvery topological crossed module determines a topological \n$2$-group and vice versa.  Since there are some choices of\nconvention involved in this construction, we briefly review it \nto fix our conventions.   Given a topological crossed module\n$(G,H,t,\\a)$, we define a topological 2-group $\\G$ as follows.\nFirst, define the group $\\Ob(\\G)$ of objects of $\\G$ and the group \n$\\Mor(\\G)$ of morphisms of $\\G$ by\n$$ \n\\Ob(\\G) = G,\\qquad \\Mor(\\G) = H \\rtimes G\n$$ \nwhere the semidirect product $H\\rtimes G$ is formed using the left \naction of $G$ on $H$ via $\\a$: \n$$ \n(h,g)\\cdot (h',g') = (h\\a(g)(h'), gg') \n$$ \nfor $g,g'\\in G$ and $h,h'\\in H$.  The source and target of \na morphism $(h,g) \\in \\Mor(\\G)$ are defined by\n$$ \ns(h,g) = g\\qquad \\text{and}\\qquad t(h,g) = t(h)g \n$$ \n(Denoting both the target map $t \\colon\n\\Mor(\\G) \\to \\Ob(\\G)$ and the homomorphism $t\\colon H\\to G$ by\nthe same letter should not cause any problems, since the first\nis the restriction of the second to $H \\subseteq \\Mor(\\G)$.)\nThe identity morphism of an object $g \\in \\Ob(G)$ is defined by\n$$i(g) = (1,g). $$\nFinally, the composite of the morphisms \n$$\\alpha = (h,g) \\colon g \\to t(h)g \\qquad \\text{and}\\qquad\n\\beta = (h',t(h)g) \\colon t(h)g \\to t(h'h)g'$$ \nis defined to be\n$$ \\beta \\circ \\alpha =  (h'h, g) \\colon g \\to t(h'h) g $$\nIt is easy to check that with these definitions, $\\G$ is a $2$-group.  \nConversely, given a topological $2$-group $\\G$, we define a crossed module \n$(G,H,t,\\a)$ by setting $G$ to be $\\Ob(\\G)$, $H$ to be \n$\\ker(s)\\subset \\Mor(\\G)$, \n$t$ to be the restriction of the target homomorphism \n$t\\colon \\Mor(\\G)\\to \\Ob(\\G)$ to the subgroup $H\\subset \\Mor(\\G)$, and\nsetting\n$$\\a(g)(h) = i(g) h i(g)^{-1}  $$\n\nIf $G$ is any topological group then there is a topological\ncrossed module $1\\to G$ where $t$ and $\\alpha$ are trivial.\nThe underlying groupoid of the corresponding topological 2-group \nhas $G$ as its space of objects, and only identity morphisms.  We\nsometimes call this 2-group the \\textbf{discrete} topological 2-group\nassociated to $G$ --- where `discrete' is used in the sense of\ncategory theory, not topology!  \n\nAt the other extreme, if $H$ is a topological group then it follows\nfrom the Peiffer identity that $H\\to 1$ can be made into topological\ncrossed module if and only if $H$ is abelian, and then in a unique\nway.  This is because a groupoid with one object and $H$ as\nmorphisms can be made into a 2-group precisely when $H$ is abelian.\nWe already mentioned this 2-group in the previous section, where we\ncalled it $H[1]$.\n\nWe will also need to talk about homomorphisms of $2$-groups.  \nWe shall understand these in the strictest possible sense.  So, \nwe say a \\textbf{homomorphism} of topological $2$-groups is a functor \nsuch that $f \\colon \\Ob(\\G)\\to \\Ob(\\G')$ and $f\\colon \\Mor(\\G)\\to \n\\Mor(\\G')$ are both continuous homomorphisms of topological groups.  We can also describe \n$f$ in terms of the crossed modules $(G,H,t,\\a)$ and $(G',H',t',\\a')$ \nassociated to $\\G$ and $\\G'$ respectively.  In these terms the data \nof the functor $f$ is described by the commutative diagram \n$$ \n\\xymatrix{ \nH \\ar[r]^-{f} \\ar[d]_-t & H' \\ar[d]^-{t'} \\\\ \nG \\ar[r]_-{f} & G' } \n$$ \nwhere the upper $f$ denotes the restriction of $f \\colon \\Mor(\\G) \n\\to \\Mor(\\G')$ to a map from $H$ to $H'$.  (We are using \n$f$ to mean several different things, but this makes the notation\nless cluttered, and should not cause any confusion.)\nThe maps $f \\colon G \\to G'$ and $f \\colon H \\to H'$\nmust both be continuous homomorphisms, and moreover must\nsatisfy an equivariance property with respect to the actions of $G$ on \n$H$ and $G'$ on $H'$: we have \n$$ \nf(\\a(g)(h)) = \\a(f(g))(f(h)) \n$$ \nfor all $g\\in G$ and $h\\in H$.  \n\nFinally, we will need to talk about short exact sequences of topological\ngroups and $2$-groups.   Here the topology is important.  \nIf $G$ is a topological group and $H$ is a normal\ntopological subgroup of $G$, then we can define an action of $H$ on $G\n$ by right translation.  In some circumstances, the projection \n$G\\to G\/H$ is a Hurewicz fibration.\nFor instance, this is the case if $G$ is a Lie group and $H$ is a closed \nnormal subgroup of $G$.  We define a \n{\\bf short exact sequence} of topological groups to be a sequence \n$$1 \\to H \\to G \\to K \\to 1$$ \nof topological groups and continuous homomorphisms \nsuch that the underlying sequence of groups is exact and \nthe map underlying the homomorphism $G\\to K$ is a Hurewicz fibration.\n\nSimilarly, we define a {\\bf short exact sequence} of topological\n2-groups to be a sequence\n$$1\\to \\G'\\to \\G\\to \\G''\\to 1$$\nof topological 2-groups and continuous homomorphisms between them\nsuch that both the resulting sequences\n\\[\n1\\to \\Ob(\\G')\\to \\Ob(\\G)\\to \\Ob(\\G'')\\to 1 \n\\]\n\\[\n1\\to \\Mor(\\G')\\to \\Mor(\\G)\\to \\Mor(\\G'')\\to 1\n\\]\nare short exact sequences of topological groups.\nAgain, we can interpret this in terms of the associated crossed modules: \nif $(G,H,t,\\a)$, $(G',H',t',\\a')$ and $(G'',H'',t'',\\a'')$ denote the \nassociated crossed modules, then it can be shown that the sequence of topological 2-groups\n$1\\to \\G'\\to \\G\\to \\G''\\to 1$\nis exact if and only if both rows in the commutative diagram \n$$ \n\\xymatrix{ \n1\\ar[r] & H' \\ar[r] \\ar[d] & H\\ar[r] \\ar[d] & H'' \\ar[d] \\ar[r] & 1 \\\\ \n1\\ar[r] & G' \\ar[r] & G\\ar[r] & G'' \\ar[r] &  1} \n$$ \nare short exact sequences of topological groups.  In this situation we\nalso say we have a short exact sequence of topological crossed\nmodules.  \n\nAt times we shall also need a fourth viewpoint on topological\n2-groups: they are strict topological 2-groupoids with a single\nobject, say $\\bullet$.  In this approach, what we had been calling `objects'\nare renamed `morphisms', and what we had been calling `morphisms'\nare renamed `2-morphisms'.  This verbal shift can be confusing, so we will \nnot engage in it!  However, the 2-groupoid viewpoint is very handy\nfor diagrammatic reasoning in nonabelian cohomology.  \nWe draw $g \\in \\Ob(\\G)$ as an arrow: \n\\[\n\\xymatrix{\n   \\bullet \\ar[rr]^{g}\n&& \\bullet\n}\n\\]\nand draw $(h,g) \\in \\Mor(\\G)$ as a bigon:\n\\[\n\\xymatrix{\n   \\bullet \\ar@\/^1pc\/[rr]^{g}_{}=\"0\"\n           \\ar@\/_1pc\/[rr]_{g'}_{}=\"1\"\n           \\ar@{=>}\"0\";\"1\"^{h}\n&& \\bullet\n}\n\\]\nwhere $g'$ is the target of $(h,g)$, namely $t(h) g$.\nWith our conventions, horizontal composition of\n2-morphisms is then given by:\n\\[ \n\\xymatrix{ \n\\bullet \\ar@\/^1pc\/[rr]^-{g_1}_{}=\"0\" \n            \\ar@\/_1pc\/[rr]_{g'_1}_{}=\"1\" \n            \\ar@{=>}\"0\";\"1\"^{h_1} \n& & \\bullet \n           \\ar@\/^1pc\/[rr]^-{g_2}_{}=\"2\" \n            \\ar@\/_1pc\/[rr]_{g'_2}_{}=\"3\" \n            \\ar@{=>}\"2\";\"3\"^{h_2} \n& & \\bullet \n}\n\\quad = \\quad \n\\xymatrix{\n\\bullet \\ar@\/^2pc\/[rrrr]^-{g_1g_2}_{}=\"0\" \n            \\ar@\/_2pc\/[rrrr]_{g'_1g'_2}_{}=\"1\" \n            \\ar@{=>}\"0\";\"1\"^{h_1\\a(g_1)(h_2)}\n& & & & \\bullet            \n}\n\\] \nwhile vertical composition is given by:\n\\[ \n\\xymatrix{ \n\\bullet \\ar@\/^2pc\/[rr]^-{g}_{}=\"0\" \n            \\ar[rr]_{}=\"1\" \n            \\ar@\/_2pc\/[rr]_{g'}_{}=\"2\" \n            \\ar@{=>} \"0\";\"1\"^{h} \n            \\ar@{=>} \"1\";\"2\"^{h'} \n& & \\bullet \n} \n\\quad = \\quad \n\\xymatrix{\n\\bullet \\ar@\/^1pc\/[rr]^-{g}_{}=\"0\" \n            \\ar@\/_1pc\/[rr]_{g'}_{}=\"1\" \n            \\ar@{=>}\"0\";\"1\"^{h'h}\n& & \\bullet            \n}\n\\]\n\n\\section{Nonabelian Cohomology} \n\\label{nonabelian}\n\nIn Section \\ref{summary} we gave a quick sketch of nonabelian\ncohomology.  The subject deserves a more thorough and more conceptual\nexplanation.  \n\nAs a warmup, consider the \\v{C}ech cohomology of a space $M$ with\ncoefficients in a topological group $G$.  In this case, Segal\n\\cite{Segal1} realized that we can reinterpret a \\v{C}ech 1-cocycle as\na \\textit{functor}.  Suppose $\\cU$ is an open cover of $M$.  Then there\nis a topological groupoid $\\hat{\\cU}$ whose objects are pairs $(x,i)$ with\n$x\\in U_i$, and with a single morphism from $(x,i)$ to\n$(x,j)$ when $x\\in U_i\\cap U_j$, and none otherwise.  \nWe can also think of $G$ as a topological groupoid with a single\nobject $\\bullet$.  Segal's key observation was that a continuous functor\n$$        g \\colon \\hat{\\cU} \\to G    $$\nis the same as a normalized \\v{C}ech 1-cocycle subordinate to $\\cU$.\n\nTo see this, note that a functor $g \\colon \\hat{\\cU} \\to G$ maps each\nobject of $\\hat{\\cU}$ to $\\bullet$, and each morphism $(x,i)\\to (x,j)$\nto some $g_{ij}(x) \\in G$.  For the functor to preserve composition,\nit is necessary and sufficient to have the cocycle equation\n$$ \ng_{ij}(x) g_{jk}(x) = g_{ik}(x) \n$$ \nWe can draw this suggestively as a commuting triangle in the groupoid\n$G$:\n\\[\n\\xy \n(0,0)*+{\\bullet}=\"1\"; \n(-12,-20.78)*+{\\bullet}=\"2\"; \n(12,-20.78)*+{\\bullet}=\"3\"; \n{\\ar^-{g_{ij}(x)} \"2\";\"1\"}; \n{\\ar^-{g_{jk}(x)} \"1\";\"3\"}; \n{\\ar_-{g_{ik}(x)} \"2\";\"3\"}; \n\\endxy\n\\]\nFor the functor to preserve identities, it is necessary and sufficient\nto have the normalization condition $g_{ii}(x) = 1$.\n\nIn fact, even more is true: two cocycles $g_{ij}$ and $g_{ij}'$\nsubordinate to $\\cU$ are cohomologous if and only if the corresponding\nfunctors $g$ and $g'$ from $\\hat{\\cU}$ to $G$ have a continuous\nnatural isomorphism between them.  To see this, note that $g_{ij}$ and\n$g'_{ij}$ are cohomologous precisely when there are maps $f_i \\colon\nU_i \\to G$ satisfying\n$$g_{ij}(x) f_j(x) = f_i(x) g'_{ij}(x)$$\nWe can draw this equation as a commuting square in the groupoid $G$:\n\\[ \n\\xymatrix{ \n\\bullet \\ar[r]^-{g_{ij}(x)} \\ar[d]_-{f_i(x)} & \\bullet \\ar[d]^-{f_j(x)} \\\\ \n\\bullet \\ar[r]_-{g_{ij}'(x)} & \\bullet }\n\\] \nThis is precisely the naturality square for a natural isomorphism\nbetween the functors $g$ and $g'$.\n\nOne can obtain \\v{C}ech cohomology with coefficients in a 2-group by\ncategorifying Segal's ideas.  Suppose $\\G$ is a topological 2-group\nand let $(G,H,t,\\alpha)$ be the corresponding topological crossed\nmodule.  Now $\\G$ is the same as a topological 2-groupoid with one\nobject $\\bullet$.  So, it is no longer appropriate to consider mere\n\\emph{functors} from $\\hat{\\cU}$ into $\\G$.  Instead, we should\nconsider \\emph{weak $2$-functors}, also known as `pseudofunctors'\n\\cite{KS}.  For this, we should think of $\\hat{\\cU}$ as a topological\n2-groupoid with only identity 2-morphisms.\n\nLet us sketch how this works.  A weak 2-functor $g \\colon \\hat{\\cU} \\to\n\\G$ sends each object of $\\hat{\\cU}$ to $\\bullet$, and each 1-morphism\n$(x,i)\\to (x,j)$ to some $g_{ij}(x) \\in G$.  However, composition of\n1-morphisms is only weakly preserved.  This means the above triangle\nwill now commute only up to isomorphism:\n\\[\n\\xy \n(0,0)*+{\\bullet}=\"1\"; \n(-12,-20.78)*+{\\bullet}=\"2\"; \n(12,-20.78)*+{\\bullet}=\"3\"; \n{\\ar^-{g_{ij}} \"2\";\"1\"}; \n{\\ar^-{g_{jk}} \"1\";\"3\"}; \n{\\ar_-{g_{ik}} \"2\";\"3\"}; \n{\\ar@2{->}^-{h_{ijk}} (0,-7)*{};(0,-18)*{}};\n\\endxy\n\\]\nwhere for readability we have omitted the dependence on $x\\in U_i\\cap\nU_j\\cap U_k$.  Translated into equations, this triangle says that we have\ncontinuous maps $h_{ijk} \\colon U_i \\cap U_j \\cap U_k \\to H$ satisfying\n\\[  \ng_{ik}(x) = t(h_{ijk}(x))g_{ij}(x)g_{jk}(x) \n\\]\nThis is precisely equation \\eqref{triangle.eq} from Section \\ref{summary}.   \n\nFor a weak 2-functor, it is not merely true that composition is preserved\nup to isomorphism: this isomorphism is also subject to a coherence law.\nNamely, the following tetrahedron must commute:  \n\\begin{equation}\n\\label{tetrahedron}\n\\begin{picture}(110,130)\n  \\includegraphics[scale = 0.85]{tetrahedron.eps}\n  \\put(-72,90){${}_{g_{jk}}$}\n  \\put(-40,35){${}_{g_{jl}}$}\n  \\put(-750,-5){$g_{il}$}\n  \\put(-125,60){$g_{ik}$}\n  \\put(-32.5,60){$g_{kl}$}\n  \\put(-100,35){${}_{g_{ij}}$}\n  \\put(-65,16){$h_{ijl}$}\n  \\put(-66.5,40){$h_{ikl}$}\n  \\put(-60,70){${}_{h_{jkl}}$}\n  \\put(-100,72.5){${}_{h_{ijk}}$}\n  \\put(-75,-5){$g_{il}$}\n\\end{picture}\n\\end{equation}\nwhere again we have omitted the dependence on $x$.  The commutativity\nof this tetrahedron is equivalent to the following equation:\n$$\n\\a(g_{ij})(h_{jkl})h_{ijl} = h_{ijk}h_{ikl}\n$$\nholding for all $x\\in U_i\\cap U_j\\cap U_k\\cap U_l$.  This is\nequation \\eqref{tetrahedron.eq}.\n\nA weak 2-functor may also preserve identity 1-morphisms only up to isomorphism.\nHowever, it turns out \\cite{Bartels:2004} that without loss of generality\nwe may assume that $g$ preserves identity 1-morphisms \\emph{strictly}.  \nThus we have $g_{ii}(x) = 1$ for all $x\\in U_i$.  We may also assume\n$h_{ijk}(x) = 1$ whenever two or more of the indices $i$, $j$ and $k$\nare equal.  Finally, just as for the case of an ordinary topological\ngroup, we require that $g$ is a \\textit{continuous} weak 2-functor\nWe shall not spell this out in detail; suffice it to say that the maps\n$g_{ij}\\colon U_i\\cap U_j\\to G$ and $h_{ijk}\\colon U_i\\cap U_j\\cap\nU_k\\to H$ should be continuous.   We say such continuous\nweak 2-functors $g\\colon \\hat{\\cU}\\to \\G$ are \\textbf{\\v{C}ech \n1-cocycles} valued in $\\G$, subordinate to the cover $\\cU$.\n\nWe now need to understand when two such cocycles should be considered \nequivalent.  In the case of cohomology with coefficients in an ordinary \ntopological group, we saw that two cocycles were cohomologous\nprecisely when there was a continuous natural isomorphism between \nthe corresponding functors.  In our categorified setting we should \ninstead use a `weak natural isomorphism', also called a\npseudonatural isomorphism \\cite{KS}.  So, we declare two cocycles to be \n{\\bf cohomologous} if there is a continuous weak natural isomorphism \n$f\\colon g \\Rightarrow g'$ between the corresponding weak 2-functors \n$g$ and $g'$.  \n\nIn a weak natural isomorphism, the usual naturality square commutes\nonly up to isomorphism.  So, $f \\colon g \\Rightarrow g'$ not only sends \nevery object $(x,i)$ of $\\hat{\\cU}$ to some $f_i(x) \\in G$, but also\nsends every morphism $(x,i)\\to(x,j)$ to some $k_{ij}(x) \\in H$ filling\nin this square:\n\\[ \n\\xy\n(-7.5,7.5)*+{\\bullet}=\"1\"; \n(-7.5,-7.5)*+{\\bullet}=\"2\"; \n(7.5,7.5)*+{\\bullet}=\"3\"; \n(7.5,-7.5)*+{\\bullet}=\"4\";\n{\\ar_-{f_{i}} \"1\";\"2\"}; \n{\\ar^-{g_{ij}} \"1\";\"3\"}; \n{\\ar^-{f_{j}} \"3\";\"4\"};\n{\\ar_-{g'_{ij}} \"2\";\"4\"}; \n{\\ar@2{->}^-{k_{ij}} (2,2)*{};(-2,-2)*{}};\n\\endxy \n\\]\nTranslated into equations, this square says that\n\\[ \nt(k_{ij}) g_{ij} f_j = f_i g'_{ij}\n\\]   \nThis is equation \\eqref{square.eq}.\n\nThere is also a coherence law that the $k_{ij}$ must satisfy: they\nmust make the following prism commute:\n\\begin{eqnarray}\n\\label{prism}\n\\begin{picture}(200,270)\n\\includegraphics{prism.eps}\n  \\put(-1,100){$f_k$}\n  \\put(-170,100){$f_i$}\n  \\put(-80,160){${}_{f_j}$}\n  \\put(-90,-7){$g'_{ik}$}\n  \\put(-100,187){$g_{ik}$}\n  \\put(-128,43){${}_{g'_{ij}}$}\n  \\put(-128,237){${}_{g_{ij}}$}\n  \\put(-47,43){${}_{g'_{jk}}$}\n  \\put(-47,237){${}_{g_{jk}}$}\n  \\put(-95,16){${}_{h'_{ijk}}$}\n  \\put(-95,212){${}_{h_{ijk}}$}\n  \\put(-95,115){$k_{ik}$}\n  \\put(-132,142){${}_{k_{ij}}$}\n  \\put(-35,121){${}_{k_{jk}}$}\n\\end{picture}\n\\end{eqnarray}\nAt this point, translating the diagrams\ninto equations becomes tiresome and unenlightening.\n \nIt can be shown that this notion of `cohomologousness' of \\v{C}ech\n1-cocycles $g\\colon \\hat{\\cU}\\to \\G$ is an equivalence relation.  We\ndenote by $\\check{H}^1(\\cU,\\G)$ the set of equivalence classes of\ncocycles obtained in this way.  In other words, we let\n$\\check{H}^1(\\cU,\\G)$ be the set of continuous weak natural\nisomorphism classes of continuous weak 2-functors $g \\colon \\hat{\\cU}\\to \\G$.\n\nFinally, to define $\\hat{H}^1(M,\\G)$, we need to take all covers into\naccount as follows.  The set of all open covers of $M$ is a directed\nset, partially ordered by refinement.  By restricting cocycles defined\nrelative to $\\cU$ to any finer cover $\\cV$, we obtain a map\n$\\check{H}^1(\\cU,\\G) \\to \\check{H}^1(\\cV,\\G)$.  This allows us \nto define the \\v{C}ech cohomology $\\check{H}^1(M,\\G)$ as a limit:\n\n\\begin{definition} Given a topological space $M$ and a topological\n2-group $\\G$, we define the \\textbf{first \\v{C}ech cohomology\nof} $M$ \\textbf{with coefficients in} $\\G$ to be\n$$ \n\\check{H}^1(M,\\G) = \\varinjlim_{\\cU}\\check{H}^1(\\cU,\\G)\n$$\n\\end{definition}\n\\noindent \nWhen we want to emphasize the crossed module, we will sometimes use\nthe notation $\\check{H}^1(M,H\\to G)$ instead of $\\check{H}^1(M,\\G)$.\nNote that $\\check{H}^1(M,\\G)$ is a pointed set, pointed by the trivial\ncocycle defined relative to any open cover $\\{U_i\\}$ by $g_{ij} = 1$,\n$h_{ijk} = 1$ for all indices $i$, $j$ and $k$.\n\nIn Theorem \\ref{classifying} we assume $M$ admits good covers, so that\nevery cover $\\cU$ of $M$ has a refinement by a good cover $\\cV$.  In\nother words, the directed set of good covers of $M$ is cofinal in the\nset of all covers of $M$.  As a result, in computing the limit\nabove, it is sufficient to only consider \\emph{good} covers $\\cU$.\n\nFinally, we remark that there is a more refined version of the set\n$\\check{H}^1(M,\\G)$ defined using the notion of `hypercover'\n\\cite{Breen1, Brown, Jardine}.  For a paracompact space $M$ this\nrefined cohomology set $H^1(M,G)$ is isomorphic to the set\n$\\check{H}^1(M,G)$ defined in terms of \\v{C}ech covers.  While the\ntechnology of hypercovers is certainly useful, and can simplify some\nproofs, our approach is sufficient for the applications we have in\nmind (see also the remark following the proof of Lemma~\\ref{lem2} in\nsub-section~\\ref{proof.lem.2}).\n\n\\section{Proofs}\n\\label{proofs}\n\n\\subsection{Proof of Theorem \\ref{classifying}} \n\\label{proof.thm.1}\n\nFirst, we need to distinguish between Milnor's \\cite{Milnor}\noriginal construction of a classifying space for a topological group and \na later construction introduced by Milgram, Segal and Steenrod\n\\cite{Milgram,Segal1,Steenrod} and further studied by May \n\\cite{May1}.  Milnor's construction is very powerful, \nas witnessed by the generality of Theorem \\ref{milnor_theorem}.  The\nlater construction is conceptually more beautiful: for any topological \ngroup $G$, it constructs $BG$ as the geometric realization of the nerve \nof the topological groupoid with one object associated to $G$.  But, \nhere we are performing this construction in the category of $k$-spaces,\nrather than the traditional category of topological spaces. \nIt also seems to give a slightly weaker result: to obtain a \nbijection \n$$\\check{H}^1(M,G) \\cong [M,BG]$$ \nall of the above cited works require some extra hypotheses on $G$: \nSegal \\cite{Segal2} requires that $G$ be locally contractible; May, \nMilgram and Steenrod require that $G$ be well pointed.  \nThis extra hypothesis on $G$ is required in the construction of \nthe universal principal $G$-bundle $EG$ over $BG$; to ensure \nthat the bundle is locally trivial we must make one of the above \nassumptions on $G$.  May's work goes further in this regard: he \nproves that if $G$ is well pointed then $EG$ is a \\emph{numerable} principal \n$G$-bundle over $BG$, and hence $EG\\to BG$ is a Hurewicz fibration.  \n\nAnother feature of this later construction is that $EG$ comes equipped\nwith the structure of a topological group.  In the work of May and Segal, \nthis arises from the fact that $EG$ is the geometric realization of the \nnerve of a topological 2-group.  We need the group structure on $EG$, \nso we will use this later construction rather than Milnor's.  For further \ncomparison of the constructions see tom Dieck \\cite{tomDieck}.\n\nWe prove Theorem \\ref{classifying} using three lemmas that are of \nsome interest in their own right.  The second, as far as we know, \nis due to Larry Breen:\n\n\\begin{lemma} \\label{lem1} \nLet $\\G$ be any well-pointed topological 2-group, and let\n$(G,H,t,\\alpha)$ be the corresponding topological crossed module.  \nThen:\n\\begin{enumerate} \n\\item $|\\G|$ is a well-pointed topological group.\n\\item There is a topological \n2-group $\\hat{\\G}$ such that $|\\hat{\\G}|$ fits into a short\nexact sequence of topological groups\n$$ \n1\\to H \\to |\\hat{\\G}| \\stackrel{p}{\\to} |\\G| \\to 1\n$$ \n\\item $G$ acts continuously via automorphisms on the topological group $EH$,\nand there is an isomorphism $|\\hat{\\G}| \\cong G\\ltimes EH$.  This exhibits   \n$|\\G|$ as $G \\ltimes_H EH$, the quotient of \n$G\\ltimes EH$ by the normal subgroup $H$. \n\\end{enumerate} \n\\end{lemma}\n\n\\begin{lemma} \\label{lem2}\nIf \n$$1\\to H \\stackrel{t}{\\to} G\\stackrel{p}{\\to} K\\to 1$$\nis a short exact sequence of topological groups, there is a bijection \n$$ \n\\check{H}^1(M,H\\to G)\\cong \\check{H}^1(M,K) \n$$ \nHere $H \\to G$ is our shorthand for the 2-group corresponding to the\ncrossed module $(G,H,t,\\alpha)$ where $t$ is the inclusion of the\nnormal subgroup $H$ in $G$ and $\\alpha$ is the action of $G$ by \nconjugation on $H$.\n\\end{lemma}\n\n\\begin{lemma} \\label{lem3}\nIf \n$$ 1 \\to \\G_0 \\stackrel{f}{\\to} \\G_1 \\stackrel{p}{\\to} \\G_2 \\to 1$$ \nis a short exact sequence of topological 2-groups, then \n$$ \n\\check{H}^1(M,\\G_0) \\stackrel{f_\\ast}{\\to} \\check{H}^1(M,\\G_1) \n\\stackrel{p_\\ast}{\\to} \\check{H}^1(M,\\G_2) \n$$ \nis an exact sequence of pointed sets.\n\\end{lemma}\n\nGiven these lemmas the proof of Theorem \\ref{classifying} goes as follows.  \nAssume that $\\G$ is a well-pointed topological 2-group.  \nFrom Lemma~\\ref{lem1} we see that $|\\G|$ is a well-pointed topological group.  \nIt follows that we have a bijection\n$$\\check{H}^1(M,|\\G|) \\cong [M,B|\\G|]$$ \nSo, to prove the theorem, it suffices to construct a bijection\n$$ \n\\check{H}^1(M,\\G) \\cong \\check{H}^1(M,|\\G|) \n$$ \nBy Lemma~\\ref{lem1}, $|\\G|$ fits into a short exact sequence \nof topological groups:\n$$1\\to H\\to G\\ltimes EH\\to |\\G|\\to 1$$\nWe can use Lemma~\\ref{lem2} to conclude that there is a bijection\n$$ \n\\check{H}^1(M,H\\to G\\ltimes EH)\n\\cong \\check{H}^1(M,|\\G|)\n$$ \nTo complete the proof it thus suffices to construct a bijection\n$$\n\\check{H}^1(M,H\\to G\\ltimes EH) \\cong\n\\check{H}^1(M,\\G) \n$$\nFor this, observe that we have a short exact sequence of topological \ncrossed modules:\n\\[\n\\xymatrix{ \n1 \\ar[r] & 1 \\ar[r] \\ar[d] & H \\ar[d] \\ar[r]^-1 & H \\ar[d]^-{t} \n\\ar[r] & 1 \\\\ \n1 \\ar[r] & EH \\ar[r] & G \\ltimes EH \\ar[r] & G \\ar[r] & 1 } \n\\]\nSo, by Lemma~\\ref{lem3}, we have an exact sequence of sets:\n$$ \n\\check{H}^1(M,EH) \\to \\check{H}^1(M, H\\to G\\ltimes EH) \n\\to \\check{H}^1(M, H\\to G) \n$$ \nSince $EH$ is contractible and $M$ is paracompact Hausdorff,\n$\\check{H}^1(M,EH)$ is easily seen to be trivial, so the \nmap $\\check{H}^1(M,H\\to G\\ltimes EH)\\to \\check{H}^1(M,H\\to G)$ \nis injective.  To see that this map is surjective, note that\nthere is a homomorphism of crossed modules going back:\n\\[\n\\xymatrix{ \n1 \\ar[r] & 1 \\ar[r] \\ar[d] & H \\ar[d] \\ar[r]^-1 & H \\ar[d]^-{t} \n\\ar@\/_1pc\/[l]_-1 \\ar[r] & 1 \\\\ \n1 \\ar[r] & EH \\ar[r] & G \\ltimes EH \\ar[r] & G \\ar[r] \n\\ar@\/_1pc\/[l]_-i & 1 } \n\\]\nwhere $i$ is the natural inclusion of $G$ in the\nsemidirect product $G\\ltimes EH$.  This homomorphism going\nback `splits' our exact sequence of crossed modules.\nIt follows that $\\check{H}^1(M,H\\to G \\ltimes EH)\\to\n\\check{H}^1(M, H\\to G)$ is onto, so we have a bijection\n$$ \n\\check{H}^1(M, H\\to G\\ltimes EH) \\cong \\check{H}^1(M,H\\to G) =\n\\check{H}^1(M, \\G)\n$$\ncompleting the proof.\n\n\\subsection{Remarks on Theorem \\ref{classifying}}\n\\label{remarks}\n\nTheorem \\ref{classifying}, asserting the existence of a classifying space \nfor first \\v{C}ech cohomology with coefficients in a topological 2-group, \nwas originally stated in a preprint by Jur\\v{c}o \\cite{Jurco}.  However,\nthe argument given there was missing some details.  In essence, the\nJur\\v{c}o's argument boils down to the following: he constructs\na map $\\check{H}^1(M,|\\G|)\\to \\check{H}^1(M,\\G)$ and sketches the\nconstruction of a map $\\check{H}^1(M,\\G)\\to \\check{H}^1(M,|\\G|)$.  The\nconstruction of the latter map however requires some further\njustification: for instance, it is not obvious that one can choose a\nclassifying map satisfying the cocycle property listed on the top of\npage 13 of \\cite{Jurco}.  Apart from this, it is not demonstrated that\nthese two maps are inverses of each other.\n\nAs mentioned earlier, Jur\\v{c}o and also Baas,\nB\\\"okstedt and Kro \\cite{BBK} use a different approach to construct\na classifying space for a topological 2-group $\\G$.  In their\napproach, $\\G$ is regarded as a topological 2-groupoid with one\nobject.  There is a well-known nerve construction that turns any\n2-groupoid (or even any 2-category) into a simplicial set \\cite{Duskin}.\nInternalizing this construction, these authors turn the topological\n2-groupoid $\\G$ into a simplicial space, and then take the geometric\nrealization of that to obtain a space.  Let us denote this space by $B\\G$.  \nThis is the classifying space used by Jur\\v{c}o and Baas--B\\\"okstedt--Kro.  \nIt should be noted that the assumption that $\\G$ is a well-pointed  \n2-group ensures that the nerve of the 2-groupoid $\\G$ is a `good' \nsimplicial space in the sense of Segal; this `goodness' condition is \nimportant in the work of Baas, B\\\"okstedt and Kro \\cite{BBK}.\n\nBaas, B\\\"okstedt and Kro also consider a third way to construct\na classifying space for $\\G$.  If we take the nerve $N\\G$ of $\\G$ we get a \nsimplicial group, as described in Section~\\ref{2groups} above.  \nBy thinking of each group of $p$-simplices $(N\\G)_p$ as a groupoid with \none object, we can think of $N\\G$ as a simplicial groupoid.  \nFrom $N\\G$ we can obtain a bisimplicial space $NN\\G$ by applying the \nnerve construction to each groupoid $(N\\G)_p$.  $NN\\G$ is sometimes\ncalled the `double nerve', since we apply the nerve construction twice.  \nFrom this bisimplicial space $NN\\G$ we can form an ordinary simplicial \nspace $dNN\\G$ by taking the diagonal.  Taking the geometric realization \nof this simplicial space, we obtain a space $|dNN\\G|$.\n\nIt turns out that this space $|dNN\\G|$ is homeomorphic to \n$B|\\G|$ \\cite{Braho, Quillen}.  It can also be shown that the spaces \n$|dNN\\G|$ and $B\\G$ are homotopy equivalent --- but although this fact \nseems well-known to experts, we have been unable to find a reference \nin the case of a \\emph{topological} 2-group $\\G$.  For ordinary 2-groups\n(without topology) the relation between all three nerves was worked out by \nMoerdijk and Svensson \\cite{MS} and Bullejos and Cegarra \\cite{BC}.  \nIn any case, since we do not use these facts in our arguments, we \nforgo providing the proofs here.\n\n\\subsection{Proof of Lemma \\ref{lem1}} \n\\label{proof.lem.1}\n\nSuppose $\\G$ is a well-pointed topological 2-group with \ntopological crossed module $(G,H,t,\\alpha)$, and let \n$|\\G|$ be the geometric realization of its nerve.\nWe shall prove that there is a topological 2-group $\\hat{\\G}$ \nfitting into a short exact sequence of topological 2-groups \n\\begin{equation} \n\\label{eq: exact seq of 2-groups} \n1\\to H\\to \\hat{\\G}\\to \\G\\to 1\n\\end{equation}\nwhere $H$ is the discrete topological $2$-group associated to the \ntopological group $H$.  On taking nerves and then geometric \nrealizations, this gives an exact sequence of groups: \n$$ \n1\\to H\\to |\\hat{\\G}| \\to |\\G|\\to 1\n$$ \nRedescribing the 2-group $\\hat{\\G}$ with the help of some work by Segal,\nwe shall show that $|\\hat{\\G}| \\cong G\\ltimes EH$ and thus $|\\G| \\cong\n(G \\ltimes EH)\/H$.  Then we prove that the above sequence is an exact\nsequence of {\\em topological} groups: this requires checking that\n$|\\hat{\\G}| \\to |\\G|$ is a Hurewicz fibration.  We conclude by showing that\n$|\\hat{\\G}|$ is well-pointed.\n\nTo build the exact sequence of $2$-groups in equation \n\\eqref{eq: exact seq of 2-groups}, we construct the corresponding\nexact sequence of topological crossed modules.  This\ntakes the following form:\n$$ \n\\xymatrix{ \n1 \\ar[r] & 1 \\ar[d] \\ar[r] & H \\ar[d]^-{t'} \\ar[r]^-1 & H \\ar[d]^-{t} \\ar[r] & 1  \\\\ \n1\\ar[r] & H \\ar[r]^-{f} & G\\ltimes H \\ar[r]^-{f'} & G \\ar[r] & 1} \n$$ \nHere the crossed module $(G\\ltimes H,H,t',\\alpha')$ is \ndefined as follows: \n\\begin{align*} \n& t'(h) = (1,h) \\\\ \n& \\a'(g,h)(h') = \\a(t(h)g)(h') \n\\end{align*} \nwhile $f$ and $f'$ are given by\n\\[\n\\begin{array}{rcl}\nf\\colon  H &\\to& G\\ltimes H \\\\ \n h &\\mapsto & (t(h),h^{-1}) \\\\ \n\\\\ \nf'\\colon G\\ltimes H &\\to& G \\\\ \n (g,h) &\\mapsto& t(h)g \n\\end{array} \n\\]\nIt is easy to check that these formulas define an exact sequence of\ntopological crossed modules.  The corresponding exact sequence \nof topological $2$-groups is\n$$ \n1\\to H\\to \\hat{\\G} \\to \\G\\to 1 \n$$ \nwhere $\\hat{\\G}$ \ndenotes the topological $2$-group associated to the\ntopological crossed module $(G\\ltimes H,H,t',\\alpha')$.  \n\nIn more detail, the $2$-group $\\hat{\\G}$ has \n\\begin{align*} \n& \\Ob(\\hat{\\G}) = G\\ltimes H \\\\ \n& \\Mor(\\hat{\\G}) = (G\\ltimes H)\\ltimes H \\\\ \n& s((g,h),h') = (g,h), \\qquad t((g,h),h') = (g,h'h) \\\\ \n& i(g,h) = ((g,h),1), \\qquad ((g,h'h),h'') \\circ ((g,h),h') = ((g,h),h''h')\n\\end{align*} \nNote that there is an isomorphism \n$(G\\ltimes H)\\ltimes H \\cong G\\ltimes H^2$ sending\n$((g,h),h')$ to $(g,(h,h'h))$.  Here by $G\\ltimes H^2$ we mean the \nsemidirect product \nformed with the diagonal action of $G$ on $H^2$, namely\n$g(h,h') = (\\a(g)(h),\\a(g)(h'))$.  Thus the group \n$\\Mor(\\hat{\\G})$ is isomorphic to $G\\ltimes H^2$.  \n\nWe can give a clearer description of the 2-group $\\hat{\\G}$ using the work \nof Segal \\cite{Segal1}.  Segal noted that for any topological group $H$,\nthere is a $2$-group $\\overline{H}$ with one object for each element of \n$H$, and one morphism from any object to any other.  In other words, \n$\\overline{H}$ is the 2-group with:\n\\begin{align*} \n& \\Ob(\\overline{H}) = H \\\\ \n& \\Mor(\\overline{H}) = H^2 \\\\ \n& s(h,h') = h, \\qquad t(h,h') = h' \\\\ \n& i(h) = (h,h), \\qquad (h',h'') \\circ (h,h') = (h,h'')\n\\end{align*} \nMoreover, Segal proved that the geometric realization $|\\overline{H}|$ \nof the nerve of $\\overline{H}$ is a model for $EH$.  Since \n$G$ acts on $H$ by automorphisms, we can define a `semidirect product' \n2-group $G\\ltimes \\overline{H}$ with \n\\begin{align*} \n& \\Ob(G\\ltimes \\overline{H}) = G\\ltimes H \\\\ \n& \\Mor(G\\ltimes \\overline{H}) = G\\ltimes H^2 \\\\ \n& s(g,(h,h')) = (g,h),\\qquad t(g,(h,h')) = (g,h') \\\\ \n& i(g,h) = (g,(h,h)),\\qquad (g,(h',h'')) \\circ (g,(h,h')) = (g,(h,h''))\n\\end{align*} \nThe isomorphism $(G\\ltimes H)\\ltimes H \\cong G\\ltimes H^2$ above can \nthen be interpreted as an isomorphism $\\Mor(\\hat{\\G})\\cong \\Mor(G\\ltimes \n\\overline{H})$.  It is easy to check that this isomorphism  \nis compatible with the structure maps for $\\hat{\\G}$ and $G\\ltimes \n\\overline{H}$, so we have an isomorphism of topological 2-groups:\n$$ \n\\hat{\\G} \\cong G\\ltimes \\overline{H}\n$$ \nIt follows that the nerve $N\\hat{\\G}$ of $\\hat{\\G}$ is isomorphic as a \nsimplicial topological group to the nerve of $G\\ltimes \\overline{H}$.  \nAs a simplicial space it is clear that $N(G\\ltimes \\overline{H}) = \nG\\times N\\overline{H}$. We need to identify the simplicial group structure \non $G\\times N\\overline{H}$.  \n\nFrom the definition of the products on $\\Ob(G\\ltimes \\overline{H})$ and \n$\\Mor(G\\ltimes \\overline{H})$, it is clear that the product on $N(G\\ltimes \n\\overline{H})$ is given by the simplicial map \n\\[   (G\\times N\\overline{H})\\times (G\\times N\\overline{H})\\to \nG\\times N\\overline{H} \\]\ndefined on $p$-simplices by \n$$ \n\\left((g,(h_1,\\ldots, h_p)),(g',(h_1',\\ldots, h'_p))\\right) \n\\mapsto (gg',(h_1\\a(g)(h'_1),\\ldots, h_p\\a(g)(h'_p))) \n$$\nThus one might well call $N(G\\ltimes \\overline{H})$ the `semidirect \nproduct' $G\\ltimes N\\overline{H}$.  Since geometric realization preserves \nproducts, it follows that there is an isomorphism of \ntopological groups \n$$ \n|\\hat{\\G}| \\cong G\\ltimes EH.  \n$$ \nHere the semidirect product is formed using the action of $G$ on $EH$ \ninduced from the action of $G$ on $H$. \nFinally note that $H$ is embedded as a normal subgroup of $G\\ltimes EH$ \nthrough \n\\[ \n\\begin{array}{ccc}\nH &\\to& G\\ltimes EH  \\\\\nh &\\mapsto& (t(h),h^{-1}) \n\\end{array}\n\\]\nIt follows that the exact sequence of groups \n$1\\to H\\to |\\hat{\\G}|\\to |\\G|\\to 1$ \ncan be identified with \n\\begin{equation} \n\\label{eq: fundamental extension} \n1\\to H\\to G\\ltimes EH\\to |\\G|\\to 1\n\\end{equation}\nIt follows that $|\\G|$ is isomorphic to the \nquotient $G\\ltimes_H EH$ of $G\\ltimes EH$ by the normal subgroup $H$.  \nThis amounts to factoring out by the action of $H$ on $G\\ltimes EH$ \ngiven by $h(g,x) = (t(h)g,xh^{-1})$.  \n\nNext we need to show that equation \\eqref{eq: fundamental extension} \nspecifies an exact sequence of {\\em topological} groups: in particular, \nthat the map $G\\ltimes EH \\to |\\G| = G\\ltimes_H EH$ is a Hurewicz fibration.   \nTo do this, we prove that the following diagram is a pullback: \n$$ \n\\xymatrix{ \nG\\ltimes EH\\ar[d] \\ar[r] & EH \\ar[d] \\\\ \nG\\ltimes_H EH \\ar[r] & BH } \n$$ \nSince $H$ is well pointed, $EH\\to BH$ is a numerable principal\nbundle (and hence a Hurewicz fibration) by the results of May\n\\cite{May1} referred to earlier.  The statement above now follows, as\nHurewicz fibrations are preserved under pullbacks.\n\nTo show the above diagram is a pullback, we construct a homeomorphism  \n$$ \n\\a\\colon (G\\ltimes_H EH) \\times_{BH} EH \\to G\\ltimes EH \n$$ \nwhose inverse is the canonical map $\\b\\colon G\\ltimes EH\\to (G\\ltimes_H EH) \n\\times_{BH} EH$.  To do this, suppose that \n$([g,x],y)\\in (G\\ltimes_H EH) \\times_{BH} EH$.  Then \n$x$ and $y$ belong to the same fiber of $EH$ over $BH$, so \n$y^{-1}x\\in H$.  We set \n$$ \n\\a([g,x],y) = ( t(y^{-1}x)g,y)\n$$ \nA straightforward calculation shows that $\\a$ is well defined\nand that $\\a$ and $\\b$ are inverse to one another.\n\nTo conclude, we need to show that $|\\G|$ is a well-pointed \ntopological group.  For this it is sufficient to show that $N\\G$ is a \n`proper' simplicial space in the sense of May \\cite{May2}\n(note that we can replace his `strong' NDR pairs with NDR pairs).\nFor, if we follow May and denote by $F_p|\\G|$ the image of \n$\\coprod^p_{i=0} \\Delta^i\\times N\\G_i$ in $|\\G|$, it then follows from \nhis Lemma 11.3 that $(|\\G|,F_p|\\G|)$ is an NDR pair for all $p$.  \nIn particular $(|\\G|,F_0|\\G|)$ is an NDR pair.  Since $F_0|\\G| = G$ \nand $(G,1)$ is an NDR pair, it follows that $(|\\G|,1)$ is an NDR pair:\nthat is, $|\\G|$ is well pointed.    \n\nWe still need to show that $N\\G$ is proper.  In fact it suffices to\nshow that $N\\G$ is a `good' simplicial space in the sense of Segal\n\\cite{Segal3}, meaning that all the degeneracies $s_i\\colon\nN\\G_n\\to N\\G_{n+1}$ are closed cofibrations.  The reason for this is\nthat every good simplicial space is automatically proper --- see the\nproof of Lewis' Corollary 2.4(b) \\cite{GaunceLewis}.  To see that\n$N\\G$ is good, note that every degeneracy homomorphism $s_i\\colon\nN\\G_n\\to N\\G_{n+1}$ is a section of the corresponding face\nhomomorphism $d_i$, so $N\\G_{n+1}$ splits as a semidirect\nproduct $N\\G_{n+1}\\cong N\\G_n\\ltimes \\ker(d_i)$.  Therefore, $s_i$ is\na closed cofibration provided that $\\ker(d_i)$ is well pointed.  But\n$\\ker(d_i)$ is a retract of $N\\G_{n+1}$, so $\\ker(d_i)$ will\nbe well pointed if $N\\G_{n+1}$ is well pointed.  For this, note that\n$N\\G_{n+1}$ is isomorphic as a space to $G\\times\nH^{n+1}$.  Since the groups $G$ and $H$ are well pointed by\nhypothesis, it follows that $N\\G_{n+1}$ is well pointed.  Here we have\nused the fact that if $X\\to Y$ and $X'\\to Y'$ are closed cofibrations\nthen $X\\times X'\\to Y\\times Y'$ is a closed cofibration.  \\qed\n  \n\\subsection{Proof of Lemma \\ref{lem2}} \n\\label{proof.lem.2}\n\nSuppose that $M$ is a topological space admitting good covers.\nAlso suppose that $1\\to H\\stackrel{t}{\\to} G\n\\stackrel{p}{\\to} K\\to 1$ is an exact sequence of topological groups.\n\nThis data gives rise to a topological crossed module \n$H \\stackrel{t}{\\to} G$ where $G$ acts on $H$ by conjugation.  \nFor short we denote this by $H \\to G$.  \nThe same data also gives a topological crossed module $1 \\to K$.  There is \na homomorphism of\ncrossed modules from $H \\to G$ to $1 \\to K$, arising from this commuting \nsquare:\n\\[ \n\\xymatrix{ \nH \\ar[r] \\ar[d]^{t} & 1 \\ar[d] \\\\ \nG \\ar[r]^{p} & K } \n\\]\nCall this homomorphism $\\alpha$.  It yields a map\n\\[ \\alpha_\\ast \\colon \\check{H}^1(M,H \\to G) \\to \\check{H}^1(M,1 \\to K). \\]\nNote that $\\check{H}^1(M,1 \\to K)$ is just the ordinary \\v{C}ech \ncohomology $\\check{H}^1(M,K)$.  \nTo prove Lemma \\ref{lem2}, we need to construct an inverse\n\\[ \\beta \\colon \\check{H}^1(M,K) \\to \\check{H}^1(M,H \\to G). \\]\nLet $\\cU = \\{U_i\\}$ be a good cover of $M$; then, as noted in\nSection~\\ref{nonabelian} there is a bijection\n\\[ \\check{H}^1(M,K) = \\check{H}^1(\\cU,K) \\]\nHence to define the map $\\beta$ it is sufficient to define a map\n$\\beta\\colon \\check{H}^1(\\cU,K)\\to \\check{H}^1(\\cU,H\\to G)$.  Let\n$k_{ij}$ be a $K$-valued \\v{C}ech 1-cocycle subordinate to $\\cU$.\nThen from it we construct a \\v{C}ech 1-cocycle $(g_{ij},h_{ijk})$\ntaking values in $H \\to G$ as follows.  Since the spaces $U_i\\cap U_j$\nare contractible and $p \\colon G \\to K$ is a Hurewicz fibration, \nwe can lift the maps $k_{ij} \\colon U_i\\cap U_j \\to K$ to maps\n$g_{ij}\\colon U_i\\cap U_j\\to G$.  The $g_{ij}$ need not satisfy the\ncocycle condition for ordinary \\v{C}ech cohomology, but instead we\nhave\n\\[\nt(h_{ijk}) g_{ij}g_{jk} = g_{ik}\n\\]\nfor some unique $h_{ijk}\\colon U_i\\cap U_j\\cap U_k\\to H$.   In terms\nof diagrams, this means we have triangles\n\\[\n\\xy \n(0,0)*+{\\bullet}=\"1\"; \n(-12,-20.78)*+{\\bullet}=\"2\"; \n(12,-20.78)*+{\\bullet}=\"3\"; \n{\\ar^-{g_{ij}} \"2\";\"1\"}; \n{\\ar^-{g_{jk}} \"1\";\"3\"}; \n{\\ar_-{g_{ik}} \"2\";\"3\"}; \n{\\ar@2{->}_-{h_{ijk}} (0,-7)*{};(0,-18)*{}};\n\\endxy\n\\]\nThe uniqueness of $h_{ijk}$ follows from the fact that the \nhomomorphism $t\\colon H\\to G$ is injective.\nTo show that the pair $(g_{ij},h_{ijk})$ defines a \\v{C}ech cocycle \nwe need to check that the tetrahedron~\\eqref{tetrahedron} commutes.  \nHowever, this follows from the commutativity of the corresponding\ntetrahedron built from triangles of this form:\n\\[\n\\xy \n(0,0)*+{\\bullet}=\"1\"; \n(-12,-20.78)*+{\\bullet}=\"2\"; \n(12,-20.78)*+{\\bullet}=\"3\"; \n{\\ar^-{k_{ij}} \"2\";\"1\"}; \n{\\ar^-{k_{jk}} \"1\";\"3\"}; \n{\\ar_-{k_{ik}} \"2\";\"3\"}; \n{\\ar@2{->}_-{1} (0,-7)*{};(0,-18)*{}};\n\\endxy\n\\]\nand the injectivity of $t$.  \n\nLet us show that this construction gives a well-defined map\n\\[  \\beta \\colon \\check{H}^1(M,K) = \n\\check{H}^1(\\cU,K) \\to \\check{H}^1(M,H\\to G) \\] \nsending $[k_{ij}]$ to $[g_{ij}, h_{ijk}]$.  \nSuppose that $k_{ij}'$ is another $K$-valued \\v{C}ech 1-cocycle subordinate to \n$\\cU$, such that $k'_{ij}$ and $k_{ij}$ are cohomologous.  Starting from the \ncocycle $k_{ij}'$ we can construct (in the same manner as above) a cocycle \n$(g'_{ij},h'_{ijk})$ taking values in $H\\to G$.  Our task is to show that \n$(g_{ij},h_{ijk})$ and $(g'_{ij},h'_{ijk})$ are cohomologous.  Since \n$k_{ij}$ and $k'_{ij}$ are cohomologous there exists a \nfamily of maps \n$\\kappa_i\\colon U_i\\to K$ \nfitting into the naturality square \n\\[ \n\\xymatrix{ \n\\bullet \\ar[r]^-{k_{ij}(x)} \\ar[d]_-{\\kappa_i(x)} & \\bullet \\ar[d]^-{\\kappa_j(x)} \\\\ \n\\bullet \\ar[r]_-{k_{ij}'(x)} & \\bullet }\n\\] \nChoose lifts $f_i\\colon U_i\\to G$ of the various $\\kappa_i$.  Since $p(g_{ij}) = p(g_{ij}') = k_{ij}$ \nand $p(f_i) = \\kappa_i$, $p(f_j) = \\kappa_j$ there is a unique map $\\eta_{ij}\\colon U_{i}\\cap U_{j} \\to H$ \n\\[ \nt(\\eta_{ij}) g_{ij}f_j = f_ig_{ij}' .\\]\nSo, in terms of diagrams, we have the following squares:\n\n\\[ \n\\xy \n(-7.5,0)*+{\\bullet} =\"1\"; \n(7.5,0)*+{\\bullet} = \"2\"; \n(-7.5,-15)*+{\\bullet} = \"3\"; \n(7.5,-15)*+{\\bullet}=\"4\"; \n{\\ar^-{g_{ij}} \"1\";\"2\"}; \n{\\ar_-{f_i} \"1\";\"3\"}; \n{\\ar^-{f_j} \"2\";\"4\"}; \n{\\ar_-{g'_{ij}} \"3\";\"4\"}; \n{\\ar@2{->}_-{\\eta_{ij}} (6,-2)*{};(-4,-13)*{}};\n\\endxy \n\\]\nThe triangles and squares defined so far fit together to form prisms:\n\\begin{center}\n\\begin{picture}(200,270)\n\\includegraphics{prism.eps}\n  \\put(-1,100){$f_k$}\n  \\put(-170,100){$f_i$}\n  \\put(-80,160){${}_{f_j}$}\n  \\put(-90,-7){$g'_{ik}$}\n  \\put(-100,187){$g_{ik}$}\n  \\put(-128,43){${}_{g'_{ij}}$}\n  \\put(-128,237){${}_{g_{ij}}$}\n  \\put(-47,43){${}_{g'_{jk}}$}\n  \\put(-47,237){${}_{g_{jk}}$}\n  \\put(-95,16){${}_{h'_{ijk}}$}\n  \\put(-95,212){${}_{h_{ijk}}$}\n  \\put(-95,115){$\\eta_{ik}$}\n  \\put(-132,142){${}_{\\eta_{ij}}$}\n  \\put(-35,121){${}_{\\eta_{jk}}$}\n\\end{picture}\n\\end{center}\nIt follows from the injectivity of the homomorphism $t$ that \nthese prisms commute, and therefore that $(g_{ij},h_{ijk})$\nand $(g'_{ij},h'_{ijk})$ are cohomologous. Therefore we have a \nwell-defined map $\\check{H}^1(\\cU,K)\\to \\check{H}^1(\\cU,H\\to G)$ \nand hence a well-defined map $\\beta\\colon \\check{H}^1(M,K)\\to \n\\check{H}^1(M,H\\to G)$.    \n\nFinally we need to check that $\\alpha$ and $\\beta$ are inverse to one\nanother.  It is obvious that $\\alpha\\circ \\beta$ is the identity on\n$\\check{H}^1(M,K)$.  To see that $\\beta \\circ \\alpha$ is the identity\non $\\check{H}^1(M,H\\to G)$ we argue as follows.  Choose a cocycle\n$(g_{ij},h_{ijk})$ subordinate to a good cover $\\cU = \\{U_i\\}$.  \nThen under $\\alpha$ the cocycle $[g_{ij},h_{ijk}]$ is sent to the \n$K$-valued cocycle $[p(g_{ij})]$.  But then we may take $g_{ij}$ as \nour lift of $p(g_{ij})$ in the definition of $\\beta(p(g_{ij}))$.  \nIt is then clear that $(\\beta \\circ \\alpha) [g_{ij},h_{ijk}] = \n[g_{ij},h_{ijk}]$.  \\qed\n\n\\vskip 1em\nAt this point a remark is in order.  The proof of the above lemma is \none place where the definition of $\\check{H}^1(M,H\\to G)$ in terms \nof hypercovers would lead to simplifications, and would allow us to \nreplace the hypothesis that the map underlying the homomorphism $G\\to K$ \nwas a fibration with a less restrictive condition.  The homomorphism \nof crossed modules \n\\[ \n\\xymatrix{ \nH \\ar[r] \\ar[d]^{t} & 1 \\ar[d] \\\\ \nG \\ar[r]^{p} & K } \n\\]\ngives a homomorphism between the associated 2-groups and\nhence a simplicial map between the nerves of the associated\n2-groupoids.  It turns out that this simplicial map belongs to a\ncertain class of simplicial maps with respect to which a subcategory\nof simplicial spaces is localized.  In the formalism of hypercovers, \nfor $M$ paracompact, \nthe nonabelian cohomology $\\check{H}^1(M,\\G)$ with coefficients in a 2-group\n$\\G$ is defined as a certain set of morphisms in this localized\nsubcategory.  It is then easy to see that the induced map\n$\\check{H}^1(M,H\\to G)\\to \\check{H}^1(M,K)$ is a bijection.\n\n\\subsection{Proof of Lemma \\ref{lem3}} \n\\label{proof.lem.3}\n\nSuppose that \n$$ 1 \\to \\G_0 \\stackrel{f}{\\to} \\G_1 \\stackrel{p}{\\to} \\G_2 \\to 1$$ \nis a short exact sequence of topological 2-groups, so that we have a short\nexact sequence of topological crossed modules:\n$$ \n\\xymatrix{ \n1 \\ar[d] \\ar[r] & H_0 \\ar[d]_-{t_0} \\ar[r]^-{f} & H_1 \\ar[d]_-{t_1} \\ar[r]^-{p}  \n& H_2 \\ar[d]_-{t_2} \\ar[r] & 1 \\ar[d] \\\\ \n1\\ar[r] & G_0 \\ar[r]^-f & G_1 \\ar[r]^-{p} & G_2 \\ar[r] & 1} \n$$ \nAlso suppose that $\\cU = \\{U_i\\}$ is a good cover of $M$, and that\n$(g_{ij},h_{ijk})$ is a cocycle representing a class in \n$\\check{H}^1(\\cU,\\G_1)$.  We claim that\nthe image of \n\\[ f_\\ast \\colon \\check{H}^1(M,\\G_0) \\to \\check{H}^1(M,\\G_1)  \\]\nequals the kernel of \n\\[ p_\\ast \\colon \\check{H}^1(M,\\G_1) \\to \\check{H}^1(M,\\G_2) . \\]\nIf the class $[g_{ij},h_{ijk}]$ is in the image of $f_\\ast$,\nit is clearly in the kernel of $p_\\ast$.  Conversely, \nsuppose it is in kernel of $p_\\ast$.  We need to show that it\nis in the image of $f_\\ast$.\n\nThe pair $(p(g_{ij}),p(h_{ijk}))$ is cohomologous to the trivial\ncocycle, at least after refining the cover $\\cU$, so there exist\n$x_i\\colon U_i\\to G_2$ and $\\xi_{ij} \\colon U_i\\cap U_j \\to H_2$ such\nthat this diagram commutes:\n\\begin{center}\n\\begin{picture}(180,270)\n\\includegraphics{prism.eps}\n  \\put(-1,100){$x_k$}\n  \\put(-170,100){$x_i$}\n  \\put(-80,160){${}_{x_j}$}\n  \\put(-85,-7){$1$}\n  \\put(-100,187){$p(g_{ik})$}\n  \\put(-128,43){${}_1$}\n  \\put(-129,237){${}_{p(g_{ij})}$}\n  \\put(-47,43){${}_1$}\n  \\put(-47,237){${}_{p(g_{jk})}$}\n  \\put(-95,16){${}_1$}\n  \\put(-108,212){${}_{p(h_{ijk})}$}\n  \\put(-95,115){$\\xi_{ik}$}\n  \\put(-132,142){${}_{\\xi_{ij}}$}\n  \\put(-35,121){${}_{\\xi_{jk}}$}\n\\end{picture}\n\\end{center}\n\nSince $p \\colon G_1 \\to G_2$ is a fibration and \n$U_i$ is contractible, we can lift $x_i$ to\na map $\\hat{x}_i \\colon U_i \\to G_1$.  Similarly,\nwe can lift $\\xi_{ij}$ to a map $\\hat{\\xi}_{ij} \\colon U_i\\cap U_j \\to H_1$.   \nThere are then unique maps $\\gamma_{ij} \\colon U_i\\cap U_j \\to G_1$ giving \nsquares like this:\n\\[\n\\xy\n(-7.5,0)*+{\\bullet} =\"1\";\n(7.5,0)*+{\\bullet} = \"2\";\n(-7.5,-15)*+{\\bullet} = \"3\";\n(7.5,-15)*+{\\bullet}=\"4\";\n{\\ar^-{g_{ij}} \"1\";\"2\"};\n{\\ar_-{\\hat{x}_i} \"1\";\"3\"};\n{\\ar^-{\\hat{x}_j} \"2\";\"4\"};\n{\\ar_-{\\c_{ij}} \"3\";\"4\"};\n{\\ar@2{->}_-{\\hat{\\xi}_{ij}} (6,-2)*{};(-4,-13)*{}};\n\\endxy\n\\]\nnamely \n\\[ \n \\c_{ij} = \\hat{x}_ig_{ij}\\hat{x}_j^{-1} t(\\hat{\\xi}_{ij}) \n\\]\nSimilarly, there are unique maps $c_{ijk} \\colon U_i\\cap U_j\\cap U_k \\to H_1$\nmaking this prism commute:\n\\begin{center}\n\\begin{picture}(180,270)\n\\includegraphics{prism.eps}\n  \\put(-1,100){$\\hat{x}_k$}\n  \\put(-170,100){$\\hat{x}_i$}\n  \\put(-80,160){${}_{\\hat{x}_j}$}\n  \\put(-90,-7){$\\c_{ik}$}\n  \\put(-100,187){$g_{ik}$}\n  \\put(-128,43){${}_{\\c_{ij}}$}\n  \\put(-129,237){${}_{g_{ij}}$}\n  \\put(-47,43){${}_{\\c_{jk}}$}\n  \\put(-47,237){${}_{g_{jk}}$}\n  \\put(-95,16){${}_{c_{ijk}}$}\n  \\put(-95,212){${}_{h_{ijk}}$}\n  \\put(-95,115){$\\hat{\\xi}_{ik}$}\n  \\put(-132,142){${}_{\\hat{\\xi}_{ij}}$}\n  \\put(-35,121){${}_{\\hat{\\xi}_{jk}}$}\n\\end{picture}\n\\end{center}\nTo define $c_{ijk}$, we simply compose the 2-morphisms on the\nsides and top of the prism.\n\nApplying $p$ to the prism above we obtain the previous prism.\nSo, $\\c_{ij}$ and $c_{ijk}$ must take values in the kernel of \n$p\\colon G_1 \\to G_2$ and $p \\colon H_1 \\to H_2$, respectively.\nIt follows that $\\c_{ij}$ and $c_{ijk}$ take values in the image\nof $f$. \n\nThe above prism says that $(\\c_{ij},c_{ijk})$ is cohomologous\nto $(g_{ij}, h_{ijk})$, and therefore a cocycle in its own\nright.  Since $\\c_{ij}$ and $c_{ijk}$ take values in the image\nof $f$, they represent a class in the image of \n\\[ f_\\ast \\colon \\check{H}^1(M,\\G_0) \\to \\check{H}^1(M,\\G_1) . \\]\nSo, $[g_{ij}, h_{ijk}] = [\\c_{ij},c_{ijk}]$ is in the image of\n$f_\\ast$, as was to be shown.   \\qed\n\n\\subsection*{Proof of Theorem \\ref{string}}\n\\label{proof.thm.2}\n\nThe following proof was first described to us by Matt Ando \\cite{Ando}, and\nlater discussed by Greg Ginot \\cite{Ginot}.\n\nSuppose that $G$ is a simply-connected, compact, simple Lie group.  \nThen the string group $\\hat{G}$ of $G$ fits into a short exact sequence \nof topological groups \n$$ \n1\\to K(\\ZZ,2)\\to \\hat{G}\\to G\\to 1 \n$$ \nfor some realization of the Eilenberg-Mac Lane space $K(\\ZZ,2)$ \nas a topological group.  Applying the classifying space functor $B$ \nto this short exact sequence gives rise to a fibration \n$$ \nK(\\ZZ,3)\\to B\\hat{G}\\stackrel{p}{\\to} BG. \n$$ \nWe want to compute the rational cohomology of $B\\hat{G}$.  \n\n\n\n\nWe can use the Serre spectral sequence to compute $H^*(B\\hat{G},\\QQ)$.  \nSince $BG$ is simply connected the $E_2$ term of this spectral sequence is \n$$ \nE_2^{p,q} = H^p(BG,\\QQ)\\otimes H^q(K(\\ZZ,3),\\QQ). \n$$ \nBecause $K(\\ZZ,3)$ is rationally indistinguishable from $S^3$, the first \nnonzero differential is $d_4$.  Furthermore, the differentials of this \nspectral sequence are all derivations.  It follows that $d_4(y\\otimes x_3) = \n(-1)^{p}y\\otimes d_4(x_3)$ if $y\\in H^p(BG,\\QQ)$.  It is not hard to identify \n$d_4(x_3)$ with $c$, the class in $H^4(BG,\\QQ)$ which is the transgression \nof the generator $\\nu$ of $H^3(G,\\QQ) = \\QQ$.  It follows that the spectral \nsequence collapses at the $E_5$ stage with \n$$ \nE_5^{p,q} = E_{\\infty}^{p,q} = \\begin{cases} \n0 & \\text{if}\\ q > 0 \\\\ \nH^p(BG,\\QQ)\/\\langle c\\rangle & \\text{if}\\ q = 0. \n\\end{cases} \n$$ \nOne checks that all the subcomplexes $F^iH^*(B\\hat{G},\\QQ)$ in the \nfiltration of $H^*(B\\hat{G},\\QQ)$ are zero for $i\\geq 1$.  Hence \n$H^p(B\\hat{G},\\QQ) = E^{p,0}_{\\infty} = H^p(BG,\\QQ)\/\\langle c \\rangle$ \nand so Theorem \\ref{string} is proved.  \n\n\n\\subsubsection*{Acknowledgements}  We would like to thank Matthew\nAndo and Greg Ginot for showing us the proof of Theorem \\ref{string}, \nPeter May for helpful remarks on classifying spaces, and Toby Bartels,\nBranislav Jur\\v{c}o, and Urs Schreiber for many discussions on higher \ngauge theory.  We also thank Nils Baas, Bj\\o rn Dundas, Eric Friedlander, \nBj\\o rn Jahren, John Rognes, Stefan Schwede, and Graeme Segal for\norganizing the 2007 Abel Symposium and for useful discussions.\nFinally, DS has received support from Collaborative Research \nCenter 676: `Particles, Strings, and the Early Universe'.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\n\n\nIn recent years, session types have \nbeen integrated into several mainstream programming languages\n(see, e.g.,~\\cite{HuY16,Padovani17,SY2016,LindleyM16,OrchardY16,AnconaEtAl16,NHYA2018})\nwhere they specify the pattern of interactions that each endpoint must\nfollow, i.e., a communication protocol.\nAll of these practical applications show \na good level of maturity of the session type theory, \nbut there are still some limitations. In particular, the notion of\nsubtyping considered in such tools usually assumes \nsynchronous communication channels,\nwhile, in many cases, communication\ntakes place over asynchronous point-to-point \\textsc{fifo} channels\n(where outputs are non-blocking). In this setting, the emitted messages are stored \ninside channels, and there may be an arbitrary delay between \nan output (on an endpoint) and the corresponding input (on the opposite\nendpoint).\nThe impact on session subtyping of these aspects related with \nasynchronous communication has been \ninitially studied in \\cite{MostrousESOP09,MostrousIC15,LMCSasync}, but the\nnotions of subtyping proposed therein were subsequently proved to be\nundecidable \\cite{BravettiCZ17,LangeY17}. Only recently, sound (but not complete)\nalgorithms for asynchronous session subtyping have been\nproposed~\\cite{BCZ18,BCLYZ21,BLZ21}.\nHowever, the theory behind asynchronous session types (see \\cite{BravettiZ21} for a gentle introduction) and \nrelated algorithms is rather intricate and this  \ncould limit their dissemination in the research community, \nas well as their adoption in practical applications.\n\nThe aim of this paper, and of the tool that we introduce, \nis to make the growing body of knowledge about asynchronous session subtyping more accessible.\nMore precisely, we present in an uniform and intuitive way various \nnotions of (a)synchronous session subtyping that were presented in the literature\nfollowing different formalisms, e.g., types in \\cite{BLZ21} or \ncommunicating finite-state machines in \\cite{BCLYZ21}. Our tool \nintegrates several algorithms for checking subtyping that can \nbe invoked from an easy-to-use Python GUI. This interface\nallows the user to input, using standard session type\nsyntax, two types:\nthe candidate subtype and supertype. The tool\nautomatically generates the \ngraphical representation of these session types \nas communicating finite-state\nmachines~\\cite{cfsm83}. It is also possible\nto execute on them the desired subtyping algorithm(s).\nThe tool has been implemented in a modular way, and it \nis possible to easily include several subtyping algorithms, \nsimply by customizing a JSON configuration file. In the current\nversion, we consider: two algorithms from~\\cite{LangeY16} for synchronous \nsession subtyping (based on Gay and Hole's~\\cite{GH05}  and  Kozen et al.'s~\\cite{KozenPS95} algorithms), a sound algorithm \nfor checking (orphan message free) asynchronous\nsession subtyping \\cite{BCLYZ21}, and a sound algorithm for\nchecking fair asynchronous session subtyping \\cite{BLZ21}.\nThe implementations of these algorithms, besides returning \na verdict about subtyping of the two types, \nalso return a graphical representation of the so-called\n\\emph{subtyping simulation game}: i.e., \nthe procedure to check that each relevant input\/output action that can be performed\nby the candidate subtype has a corresponding matching action in the candidate supertype.\nThis graphical representation is helpful to \nunderstand the reason behind the given verdict.\nThe original command line Haskell implementations of the algorithms in \\cite{LangeY16,BCLYZ21,BLZ21} have been adapted and integrated by: $(i)$ uniformizing their graphical notation\/colors (e.g.,  {\\it inner\/outer} states represented as rectangles, with the initial one being thicker, error ones being red, content of outer ones being blue, etc\\dots),  $(ii)$ reimplementing the synchronous algorithm so to also generate the simulation graph, $(iii)$ completely rewriting, in the fair asynchronous algorithm, the controllability check (existance of a compliant peer, see Section~\\ref{examples}) and $(iv)$ error detection with generation of red states for all algorithms, $(v)$ pre-transforming inputted types with a Python ANTLR4 parser that produces a common raw syntax.\n\n\\textit{Synopsis.}\nSection \\ref{SEC:subtyping} recalls basic notions about session\nsubtyping using tool-simulated examples\nand Section \\ref{SEC:tool} describes the functionalities of the tool. Finally, in Section \\ref{SEC:conc} we conclude the paper. \n\nThe tool sources\/binaries are available at~\\cite{toolrepo}.\n\n\n\n\n\n\n\\section{Session Subtyping} \\label{SEC:subtyping}\n\n\n\\begin{figure}[t]\n\\vspace{-.3cm}\n\\begin{minipage}[t]{0.44\\linewidth}\n\\centering\n\\vspace{.3cm}\n\\subfloat[$\\mathsf{LTS}(T_{HS})$]{\\includegraphics[scale=.56, keepaspectratio, valign=t]{pics\/thc_dual.pdf}}\n\\hspace*{0.15cm}\n\\vspace*{1.75cm}\n\\caption{Hospital server.}\n\\label{fig:THSSAut}\n\\end{minipage}%\n\\begin{minipage}[t]{0.56\\linewidth}\n\\centering\n\\subfloat[$\\mathsf{LTS}(T''_{HC})$]{\\includegraphics[scale=.56,  keepaspectratio, valign=t]{pics\/thc2.pdf}}\n\\subfloat[$\\mathsf{LTS}(T'_{HC})$]{\\includegraphics[scale=.56, keepaspectratio, valign=t]{pics\/thc1.pdf}}\n\\hspace*{0.01cm}\n\\subfloat[$\\mathsf{LTS}(T_{HC})$]{\\includegraphics[scale=.56, keepaspectratio, valign=t]{pics\/thc.pdf}}\n\\caption{Hospital clients.}\n\\label{fig:AsyncAut}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}\n\\begin{minipage}[t]{0.49\\linewidth}\n\\centering\n\\vspace*{.3cm}\n\\subfloat[$\\mathsf{LTS}(T_{SS})$]{\\includegraphics[scale=.56, keepaspectratio, valign=t]{pics\/tsc_dual.pdf}}\n\\vspace*{.05cm}\n\\caption{Satellite protocol server.}\n\\label{fig:SFAsyncAut}\n\\end{minipage}%\n\\begin{minipage}[t]{0.49\\linewidth}\n\\centering\n\\subfloat[$\\mathsf{LTS}(T'_{SC})$]{\\includegraphics[scale=.56, keepaspectratio, valign=t]{pics\/tsc.pdf}}\n\\hspace*{.5cm}\n\\subfloat[$\\mathsf{LTS}(T_{SC})$]{\\includegraphics[scale=.56, keepaspectratio, valign=t]{pics\/tsc1.pdf}}\n\\caption{Satellite protocol clients.}\n\\label{fig:FAsyncAut}\n\\end{minipage}\n\\end{figure}\n\n\n\nWe first recall the syntax of session types\nand their automata representation in the style of \ncommunicating finite-state machines (CFSM) \\cite{cfsm83}. We then show how our tool can generate simulation\ngraphs for supported session subtyping relations:\nsynchronous \\cite{LangeY16}, asynchronous \\cite{BCLYZ21} and fair asynchronous subtyping~\\cite{BLZ21}.\nIn the asynchronous cases automata are assumed to communicate over unbounded \\textsc{fifo} channels as for CFSMs.\n\n\n\\subsection{Session Types and their Automata Representation}\\label{examples}\n\nThe formal syntax of two-party session types is given below. Notice that we follow\nthe simplified notation used in, e.g., \\cite{DenielouY13,BravettiCZ17,BLZ21}, \nwhich abstracts away \nfrom data carried by messages (payloads). This is done in order to focus on the key aspects of the session subtyping problem (as we will see co\/contra-variance of output\/input and output anticipation): passing data or channels (delegation) are features that we deem orthogonal to such a problem.\n\n\\begin{definition}[Session Types]\\label{def:sessiontypes}\n  Given a set of label names $\\mathcal{L}$, ranged over by $l$, the syntax\n  of two-party session types is given by the following grammar: \\\\[.05cm]\n \n\\centerline{$    \\begin{array}{lrl}\n      T\\ \\ &::=&\\ \\  \n     \n     \n                 \\Tselect{l}{T} \n                 \\quad \\mid\\quad  \\Tbranch{l}{T} \n                 \\quad \\mid\\quad  \\Trec X.T\n                 \\quad \\mid\\quad \\Tvar X\n                 \\quad \\mid\\quad \\mathbf{end}\n                \n                \n    \\end{array}$} \\\\[.05cm]\nwhere $I \\!\\neq\\! \\emptyset$ and $\\forall i \\!\\neq\\! j \\!\\in\\! I .\\  l_i\\! \\neq\\! l_j$.\n\\end{definition}\n\n\nType $\\Tselect{l}{T}$ represents an internal \nchoice among {\\it outputs}, specifying that the chosen label name $l_i\\! \\in\\! {\\cal L}$ is sent and, then,\ncontinuation $T_i$  is executed.\n$\\Tbranch{l}{T}$ represents, instead, an external \nchoice among {\\it inputs},\nspecifying that, once a\nlabel name $l_i\\! \\in\\! {\\cal L}$ is\nreceived, continuation $T_i$ takes place.\nTypes $\\Trec X.T$ and $\\Tvar X$ denote standard recursion\nconstructs.\nWe assume recursion to be guarded,\ni.e., in $\\Trec X.T$ the recursion variable $\\Tvar X$ \noccurs only after receving or sending a label.\nType $\\mathbf{end}$ denotes the end of the interaction.\nSession types are closed,\ni.e.,\nall recursion variables $\\Tvar X$ occur under the scope of a\ncorresponding binder $\\Trec X.T$.  \n\n\nIn the tool we graphically represent the behaviour of a session type $T$ as a Labeled Transition System (LTS), see, e.g., Figures \\ref{fig:THSSAut}, \\ref{fig:AsyncAut},\n\\ref{fig:SFAsyncAut} and \\ref{fig:FAsyncAut}.\nFollowing the notation of CFSMs, we denote a LTS by $(Q, q_0, \\Trans)$, with $Q$ being a \nset of states, $q_0$ the initial state and $\\Trans$ a transition relation over $Q \\!\\times\\! (\\{!,\\!?\\} \\!\\times\\! {\\cal L}) \\!\\times\\! Q$,\nwith label ``$! \\, l$'' representing output on $l$ and label ``$? \\, l$'' representing input on $l$.\n\nWe use $\\mathsf{LTS}(T)$ to denote the LTS of type $T$. Let ${\\cal T}$ be the set of all session types $T$.\nWe define transition relation\n$\\longrightarrow \\; \\subseteq {\\cal T} \\!\\times\\! (\\{!,\\!?\\} \\!\\times\\! {\\cal L}) \\!\\times\\!\n{\\cal T}$, as the least transition set satisfying the following rules\\\\\n$\n\\!\\Tselect{l}{T} \\!\\arrow{ \\! ! \\, l_i}\\! T_i \\hspace{.4cm} i \\!\\in\\! I \\hspace{.8cm}\n \\Tbranch{l}{T} \\!\\arrow{ \\! ? \\, l_i}\\! T_i \\hspace{.4cm} i \\!\\in\\! I \\hspace{.8cm}\n\\infr{ T\\{\\Trec X.{T}\/\\Tvar{X}\\} \\!\\arrow{\\ell}\\! T'}{\\Trec X.{T} \\!\\arrow{\\ell}\\! T' }\n$\\\\\nwith label $\\ell$ ranging over $\\{!,\\!?\\} \\!\\times\\! {\\cal L}$.\nNotice that a state $\\mathbf{end}$, called {\\it termination state}, has no outgoing transitions.\nGiven a session type $T$ we define $\\mathsf{LTS}(T)$ as being $(Q_T,T,\n\\Trans_T)$, where:\n$Q_T$ is the set of terms $T'$ which are reachable from $T$ according to $\\longrightarrow$ relation and $\\Trans_T$ is defined as the restriction of $\\longrightarrow$ to  $Q_T \\!\\times\\! (\\{!,\\!?\\} \\!\\times\\! {\\cal L}) \\!\\times\\! Q_T$.\n\n\nNotice that in general an LTS may express more\nbehaviours than the ones described by session types: \nit can include non-deterministic and mixed choices, i.e.\\ choices including both inputs and outputs. Here we only consider LTSs $(Q, q_0, \\Trans)$ such that $\\exists T \\!\\in\\! {\\cal T} \\ldotp \\mathsf{LTS}(T)=(Q, q_0, \\Trans)$.\n\n\n\n\\begin{example}\nAs an example of session types we consider the Hospital server from~\\cite{BCLYZ21}:\\\\\n\\centerline{$\n    T_{HS} = \\Trec{X}.\n                     \\mysum \\!\n                     \\left\\{\n                     \\msg{ti} ; \\inchoicetop \\{ko; \\Tvar{X}, \\ ok; \\Tvar{X} \\}, \\\n                     \\msg{to} ; \\inchoicetop \\{ko; \\Tvar{X}, \\ ok; \\Tvar{X} \\}\n                     \\right\\}\n$}\\\\[.05cm]\nFigure \\ref{fig:THSSAut} shows $\\mathsf{LTS}(T_{HS})$ as produced by our tool.\nThe server $T_{HS}$ expects to receive two types of\nmessages: $\\msg{ti}$ (next patient data) or $\\msg{to}$ (patient report).\nThen it may send either\n$\\msg{ok}$ or $\\exKO$, indicating whether the evaluation of received data\nwas successful or not, and it loops.\n\\end{example}\n\nWe now define the \\textit{dual} of a session type \n$T$, written $\\dual{T}$. $\\dual{T}$ is inductively\nobtained from $T$ as follows:\n$\\dual{\\Tselect{l}{T}} = \\Tbranch{l}{\\dual{T}}$,\n$\\dual{\\Tbranch{l}{T}} = \\Tselect{l}{\\dual{T}}$,\n$\\dual{\\mathbf{end}} = \\mathbf{end}$, $\\dual{\\Tvar{X}} = \\Tvar{X}$, and\n$\\dual{\\Trec{X}.T} = \\Trec{X}.\\dual{T}$. \nFor example, \nthe dual of the Hospital server $T_{HS}$ is:\\\\[.05cm]\n\\centerline{$\\dual{T_{HS}} =  \\Trec{X}.\n                     \\inchoicetop \\!\n                     \\left\\{\n                     \\msg{ti} ; \\mysum \\{ko; \\Tvar{X}, \\ ok; \\Tvar{X} \\}, \\\n                     \\msg{to} ; \\mysum \\{ko; \\Tvar{X}, \\ ok; \\Tvar{X} \\}\n                     \\right\\}\n$}\n\n\\begin{example}\\label{ExAsync}\nWe now consider examples of session types that are clients of the Hospital service: an ``ideal'' client $T_{HC}$ and two specific ones $T'_{HC}$ and  $T''_{HC}$, respectively. \\\\%[.05cm]\n\\centerline{$\n  \\begin{array}{lcl}\n    T_{HC} & = & \\dual{T_{HS}} = \\Trec{X}.\n                     \\inchoicetop \\!\n                     \\left\\{\n                     \\msg{ti} ; \\mysum \\{ko; \\Tvar{X}, \\ ok; \\Tvar{X} \\}, \\\n                     \\msg{to} ; \\mysum \\{ko; \\Tvar{X}, \\ ok; \\Tvar{X} \\}\n                     \\right\\}\n                         \\\\\n    T'_{HC} & = & \\Trec{X}.\n                     \\inchoicetop \\!\n                     \\left\\{\n                     \\msg{ti} ; \\mysum \\{ko; \\Tvar{X}, \\ ok; \\Tvar{X}, \\ dk; \\Tvar{X} \\}\n                     \\right\\}\n                         \\\\\n    T''_{HC} & = & \\Trec{X}.  \\inchoicetop \\! \\left\\{\\msg{ti}; \\mysum \\{ ko;  \\Tvar{X}, \\ ok; \\inchoicetop \\! \\{\\msg{to}; \\Tvar{X} \\}\\}\\right\\}\n    \\\\\n  \\end{array}\n$}\\\\[.05cm]\nFigure \\ref{fig:AsyncAut} shows $\\mathsf{LTS}(T''_{HC})$, $\\mathsf{LTS}(T'_{HC})$ and $\\mathsf{LTS}(T_{HC})$,\\footnote{As we will see, the order in which the LTSs are presented reflects the subtyping relation (we will show that $T''_{HC}$ and $T'_{HC}$ are subtypes of $T_{HC}$) and the positions in which types are inputed in the tool.} as produced by our tool.\n\nThe ``ideal'' client  $T_{HC}$ is simply the dual of the Hospital server: first it may send two types of\nmessages $\\msg{ti}$ or $\\msg{to}$, then it expects to receive either $\\msg{ok}$ or $\\exKO$. In general, a client that is {\\it compliant} with the Hospital server is a type such that: $(i)$ each message sent by the client (resp.\\ server) can be received by the server (resp.\\ client), and $(ii)$ neither the server nor the client blocks in a receive.\nFor example, the client $T'_{HC}$, a slightly modified version of $T_{HC}$ that may send $\\msg{ti}$ only\nand expects to receive also $dk$ (don't know) besides $\\msg{ok}$ and $\\exKO$ (i.e., it applies covariance of outputs and contravariance of inputs, see \\cite{LangeY16}) is still compliant with the Hospital server. \nHence we say that $T'_{HC}$ is a subtype of $T_{HC}$.\n\nUnder asynchronous communication client compliance is relaxed by requiring that all messages that are sent are {\\it eventually} received. For example, in this setting,\nclient $T''_{HC}$ (that may anticipate output $\\msg{ti}$ w.r.t. inputs) is also a compliant client, see~\\cite{BCLYZ21}, hence $T''_{HC}$ is an asynchronous subtype of $T_{HC}$.\n\nNotice that, both for synchronous and asynchronous communication (see~\\cite{BCZ18}),  it holds, for any session type $T,T'$: $T'$ subtype of $T$ implies $\\dual{T}$ subtype of $\\dual{T'}$ (closure under duality). As we will see, our tool automatically handles the generation of the {\\it dual subtyping problem} ($\\dual{T}$ subtype of $\\dual{T'}$) from $T'$ subtype of $T$ by exchanging and dualizing inputted types.\n\n\\end{example}\n\\begin{example}\\label{SatEx}\nAs another example, \nwe consider clients of the Satellite protocol from~\\cite{BLZ21}: an ``ideal'' client (the dual of the Satellite protocol server $T_{SS}$ whose LTS is depicted in Figure \\ref{fig:SFAsyncAut}) and a specific one; here denoted with $T_{SC}$ and $T'_{SC}$, respectively.\\\\%[.05cm]\n\\centerline{$\n  \\begin{array}{lcl}\n    T_{SC} & = & \\dual{T_{SS}} = \n                 \\Trec{X}.~\\Tbra{ \\msg{tm} ; \\Tvar{X} ,  \\,\\msg{over} ; \\Trec{Y}\n                 . \\Tsel{ \\msg{tc} ; \\Tvar{Y}  , \\,\\msg{done} ; \\mathbf{end} } } \n    \\\\\n    T'_{SC} & = & \\Trec  X .\\Tsel{ \\msg{tc} ; \\Tvar X  , \\, \\msg{done} ;\n                  \\Trec{Y} . \n                  ~\\Tbra{ \\msg{tm} ; \\Tvar{Y} ,  \\,\\msg{over} ; \\mathbf{end} } } \n  \\end{array}\n$}\\\\[.05cm]\nFigure \\ref{fig:FAsyncAut} shows $\\mathsf{LTS}(T'_{SC})$ and $\\mathsf{LTS}(T_{SC})$, as produced by our tool. \nThe ``ideal'' client  $T_{SC}$ may receive\na\nnumber of telemetries ($\\msg{tm}$), followed by a\nmessage $\\msg{over}$.\nIn the second phase, the client\nsends a number of telecommands\n($\\msg{tc}$), followed by a message $\\msg{done}$. Under {\\it fair} asynchronous communication  client $T'_{SC}$ (with phases exchanged) is also compliant with the server, i.e.\\ $T'_{SC}$ a fair asynchronous subtype of $T_{SC}$, see \\cite{BLZ21}. Compared to asynchronous communication considered in Example \\ref{ExAsync}, here client compliance entails that,\nunder {\\it fairness assumption} (i.e.\\ communication loops with some exit are assumed to be eventually escaped), both the client and the server must reach successfull termination with no messages left to be consumed in the \\textsc{fifo}\nchannels.\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\begin{minipage}[t]{0.49\\linewidth}\n\\centering\n\\subfloat{\\includegraphics[scale=.3,  keepaspectratio, valign=t]{pics\/async_sim.pdf}}\n\\caption{Asynchronous simulation.}\n\\label{fig:AsyncSim}\n\\end{minipage}\n\\begin{minipage}[t]{0.49\\linewidth}\n\\centering\n\\subfloat{\\includegraphics[scale=.7,  keepaspectratio]{pics\/sync_sim.pdf}}\n\\caption{Synchronous simulation.} \n\\label{fig:SyncSim}\n\\subfloat{\\includegraphics[scale=.7,  keepaspectratio]{pics\/sync_failure.pdf}}\n\\caption{Failed synchronous simulation.} \n\\label{fig:FailSyncSim}\n\\end{minipage}\n\\end{figure}\n\n\\subsection{Synchronous Session Subtyping}\\label{sub:syncsub}\nIn order to establish whether type $T'$ is a synchronous subtype of a type $T$ \\cite{LangeY16} we can perform synchronous simulation of the (ordered) pair of LTSs $\\mathsf{LTS}(T')=(Q',q'_0,\\Trans')$ and $\\mathsf{LTS}(T)=(Q,q_0,\\Trans)$. Simulation states are pairs $(q',q)$, with $q' \\!\\in\\! Q'$ and $q \\!\\in\\! Q$. The simulation proceeds by starting from state $(q'_0,q_0)$ and by synchronously matching transitions of $\\mathsf{LTS}(T')$ and $\\mathsf{LTS}(T)$ having the {\\it same} labels (both\n``$! \\, l$'' or both ``$? \\, l$'').\nFor each reached simulation state $(q',q)$\nwe must have: $(i)$ the set of outputs (resp.\\ inputs) fireable by $q'$ is subset (resp.\\ superset) or equal to the set of outputs (resp.\\ inputs) fireable by $q$; this enacts covariance (resp.\\ contravariance) of outputs (resp.\\ inputs), $(ii)$ if $(q',q)$ performs no transitions then both $q'$ and $q$ must perform no transitions (successfully terminate). \n\nOn the contrary, simulation states $(q',q)$ for which the above constraints are not satisfied are called \\textit{failure simulation states} (depicted in red in our tool) and cause synchronous subtyping not to hold. \n\\begin{example}\nFigure \\ref{fig:SyncSim} shows the synchronous simulation graph, as produced from our tool, for the pair $\\mathsf{LTS}(T'_{HC})$ and $\\mathsf{LTS}(T_{HC})$. Notice that our tool builds the simulation graph as a tree: when a pair $(q',q)$ is reached, which was previously traversed (as e.g.\\ for the $(1,1)$ pair), simulation does not proceed further in that branch and a dashed line is depicted connecting the two copies of $(q',q)$. Notice that, if in $T'_{HC}$ we turn $\\rcv{ko}$ into $\\rcv{ko1}$ (creating a mismatch with the server), $T'_{HC}$ is no longer a synchronous subtype of $T_{HC}$. This can be seen in Figure \\ref{fig:FailSyncSim} where the originated failure simulation state is depicted in red.\n\\end{example}\nWe now give the formal definition of synchronous subtyping. We first define set of inputs and set of outputs fireable by a state $q$ as follows:  $\\inTrans{q}=\\{l \\mid \\exists q'.q\\trans{?\\, l} q'\\}$ and\n  $\\outTrans{q}=\\{l \\mid \\exists q'.q\\trans{!\\, l} q'\\}$.\n\n\\begin{example}\n  Consider $\\mathsf{LTS}(T_{HC})$ (Figure~\\ref{fig:AsyncAut}), we have the following:\n \n  \\[\n    \\begin{array}{lllll}\n      \\inTrans{1} = \\emptyset\n      &\n        \\hspace{.3cm}\\inTrans{2}=  \\{ \\exKO, \\msg{ok} \\}\n      \\\\\n       \\outTrans{1} = \\{ \\msg{ti}, \\msg{ti}\\}\n      &\n        \\hspace{.3cm}\\outTrans{2} = \\emptyset\n   \n    \\end{array}\n  \\]\n\\end{example}\n\n\\begin{definition}[Synchronous Simulation]\\label{def:syncsimtree}\nGiven set of label names ${\\cal L}$ and two\nLTSs $(P,p_0,\\Trans_1)$ and $(Q,q_0,\\Trans_2)$,\nsynchronous simulation is defined as a labeled transition system over states of $P \\!\\times\\! Q$, i.e. pairs denoted by $\\simtreepair pq$, with $p \\!\\in\\! P$ and $q \\!\\in\\! Q$. In particular, the initial state is $\\simtreepair{p_0}{q_0}$ and the transition relation $\\simtreetrans{}$, labeled over  $\\{!,\\!?\\} \\!\\times\\! {\\cal L}$,\nis defined as the minimal relation satisfying rules:\\\\[.1cm]\n \n$    \\begin{array}{c}\n      \\infer[\\!\\mathsf{(In)}]\n     \n      {\\simtreepair pq\\ \\simtreetrans{? \\,l}\\ \\simtreepair{p'}{q'}}\n     \n      {p\\trans{? \\,l}_1 p' \\;\\; q\\trans{? \\,l}_2 q' \\;\\; \\inTrans{p} \\!\\supseteq\\! \\inTrans{q}}      \n     \n      \\quad\\quad\n      \\infer[\\!\\mathsf{(Out)}]\n     \n                                         { \\simtreepair pq\\ \\simtreetrans{!\\,l}\\ \\simtreepair{p'}{q'} }\n     \n      {p\\trans{! \\,l}_1 p' \\;\\; q\\trans{! \\,l}_2 q' \\;\\; \\outTrans{p} \\!\\subseteq\\! \\outTrans{q}}\n      \\\\\n    \\end{array}\n$\n\\end{definition}\n\nFormally, a type $T'$ is a synchronous subtype of a type $T$ if the LTS obtained as the synchronous simulation of the pair $\\mathsf{LTS}(T')$ and $\\mathsf{LTS}(T)$ is such that, for every state $\\simtreepair{q'}{q}$ reachable from the initial simulation state, we have: if $\\simtreepair{q'}{q}$ performs no transitions then both $q'$ and $q$ perform no transitions. \n\n\n\n\\subsection{Asynchronous Session Subtyping}\\label{sub:asyncsub}\n\nIn contrast to synchronous simulation, the asynchronous one gives the possibility of ``anticipating'', in the right-hand LTS, output transitions w.r.t.\\ input transitions that precede them. This can be shown, using our tool, via an example. \n\\begin{example}\\label{exAsync}\nFigure~\\ref{fig:AsyncSim} shows the asynchronous simulation tree, as produced from our tool, for the pair $\\mathsf{LTS}(T''_{HC})$ and $\\mathsf{LTS}(T_{HC})$. Simulation of Figure \\ref{fig:AsyncSim} proceeds as follows.\nFor instance, after transitions $\\snd{\\msg{ti}}$, $\\rcv{ko}$ and $\\snd{\\msg{to}}$ (i.e.\\ path ``$\\snd{\\msg{ti}} \\, \\rcv{ko} \\, \\snd{\\msg{to}}$'') are synchronously performed by $\\mathsf{LTS}(T''_{HC})$ and $\\mathsf{LTS}(T_{HC})$, they reach states $1$ and $2$, respectively. Now, $\\mathsf{LTS}(T''_{HC})$ in state $1$ can only do output $\\snd{\\msg{ti}}$, while $\\mathsf{LTS}(T_{HC})$ in state $2$ can only do inputs. Being asynchronous, the simulation can proceed by calculating the so-called state $2$\n{\\it input tree} $\\intree{2} = \\outchoicetop{\\exKO \\!:\\! 1 , \\, \\msg{ok} \\!:\\! 1}{}$, i.e.\\ the spanning tree from state $2$ (constructed considering input transitions only), which has two leaves, both being state $1$. \nProvided that all leaves \nof\n$\\intree{2}$ can perform $\\snd{\\msg{ti}}$, the simulation can proceed by considering $\\outchoicetop{\\exKO \\!:\\! [\\,] , \\, \\msg{ok} \\!:\\! [\\,]}{}$ as an ``accumulated'' input for the right-hand LTS and by making all states in its leaves evolve by performing the $\\snd{\\msg{ti}}$ transition. Therefore, after simulation performs\n$\\snd{\\msg{ti}}$, $\\mathsf{LTS}(T''_{HC})$ and $\\mathsf{LTS}(T_{HC})$ reach states $2$ and $\\outchoicetop{\\exKO \\!:\\! 2 , \\, \\msg{ok} \\!:\\! 2}{}$, respectively (the state reached by the right-hand LTS\nis actually an input tree). \n\\end{example}\n\nIn general, input trees (e.g. $\\outchoicetop{\\exKO : 2 , \\, \\msg{ok} : 2}{}$ in the example above) are defined in \\cite{BCLYZ21} \nas {\\it input contexts} ${\\mathcal A}$ (representing ``accumulated'' input, e.g. $\\outchoicetop{\\exKO : [] , \\, \\msg{ok} : []}{}$ in the example above) with {\\it holes} ``$[\\,]$'' replaced by LTS states. Their syntax is:\n\\\\%[.02cm]\n\\centerline{$ {\\mathcal A}\\ \\ ::=\\ \\ [\\,] \\ \\mid\\ \\outchoicetop{l_i:{\\mathcal\n      A}_i}{i\\in I} $}\\\\%[.02cm] \nIn the tool we represent input trees by nested boxes. For instance the input tree\n$\\outchoicetop{\\exKO : \\outchoicetop{\\exKO : 1 , \\, \\msg{ok} : 1}{} , \\, \\msg{ok} : \\outchoicetop{\\exKO : 1 , \\, \\msg{ok} : 1}{} }{}$ is represented as:\\\\[.2cm]\n\\centerline{\\includegraphics[scale=.8, keepaspectratio]{pics\/intree_ext.PNG}}\\\\\n\\vspace*{-.4cm}\n\nIn general, due to input accumulation, represented by an input tree,  even if two types are in an asynchronous subtyping relation, the simulation could proceed infinitely without meeting failure simulation states (as it would happen for the pair $\\mathsf{LTS}(T''_{HC})$ and $\\mathsf{LTS}(T_{HC})$ of Example \\ref{exAsync}).\nIn our tool we use the algorithm of \\cite{BCLYZ21} for checking asynchronous subtyping, which is sound but not complete (in some cases it terminates without returning a decisive verdict). In a nutshell, such an algorithm proceeds as follows. The subtyping simulation terminates when we encounter a failure state (depicted in red in our tool), meaning that the two types are not in the subtyping relation, or when we detect a repetitive behaviour in the simulation (which, we show, can always be found, in case of infinite simulation).  In the latter case, we check whether this repetitive behaviour satisfies sufficient conditions (see \\cite{BCLYZ21} for details) that guarantee that the subtyping simulation will never encounter failures. If the conditions are satisfied the algorithm concludes that the two types are in the subtyping relation, otherwise a \\textit{maybe} verdict is returned. \n\nTherefore, the tool always produces a finite simulation tree: for types that are detected to be subtypes, simulation can stop in a state that is identified (via a dashed transition in our tool) to a previously encountered state, even if they are not identical; see bottommost state of Figure \\ref{fig:AsyncSim} (outgoing dashed transition). The sufficient conditions checked by the algorithm guarantees that the behaviour beyond such a simulation state is a repetition of the behaviour already observed.\n\nWe now give the formal definition of asynchronous subtyping. Given a LTS $(Q, q_0, \\Trans)$,\nwe write $q_0 \\trans{\\ell_1 \\cdots \\ell_k} q_k$ iff there\nare $q_1, \\ldots , q_{k-1} \\in Q$ such that\n$q_{i-1} \\trans{\\ell_i} q_i$ for $1 \\leq i \\leq k$.\nGiven a list of messages $\\omega = \\msg{l}_1 \\cdots \\msg{l}_k$ ($k \\geq\n0$), we write $\\rcvL{\\omega}$ for the list $\\rcv{l}_1 \\cdots \\rcv{l}_k$\nand $\\sndL{\\omega}$ for $\\snd{l}_1 \\cdots \\snd{l}_k$.\n\n\n\n\\begin{definition}[Input Context]\n  An input\ncontext is a term of the grammar\\\\[.1cm]\n\\centerline{$ {\\mathcal A}\\ \\ ::=\\ \\ [\\,]_j \\ \\mid\\ \\outchoicetop{l_i:{\\mathcal\n      A}_i}{i\\in I} $}\\\\[.1cm] \nwhere: All indices $j$, denoted by\n  $I(\\mathcal A)$, are distinct and are associated to holes. Moreover, $I \\!\\neq\\! \\emptyset$ and $\\forall i \\!\\neq\\! j \\!\\in\\! I .\\  l_i\\! \\neq\\! l_j$.\n\\end{definition}\nHoles are, thus, actually indexed so to make it possible to individually replace them. In this way ${\\mathcal A}[q_i]^{i \\in I({\\cal A})}$ denotes the {\\it input tree} obtained by syntactically replacing each hole $[\\,]_i$ in\n$\\ctx A$ by a specific state $q_i \\in Q$. \nIn the sequel, we use ${\\cal I}{\\cal T}_Q$ to denote the set of input trees over states\n$q \\in Q$. \n\n\n\n\n\n\\subsubsection{Auxiliary functions}\nGiven a CSFM $(Q, q_0, \\Trans)$ and a state\n$q \\in Q$, we define:\n\\vspace{-.3cm}\n\\begin{itemize}\n\\item\n  $\\loopio{\\star}{q} \\iff \\exists \\omega \\in {\\cal L}^{*}, \\omega' \\in\n  {\\cal L}^{+}, q' \\in Q .\\  q \\trans{\\star {\\omega}} q' \\trans{\\star\n    {\\omega'}} q'$ (with $\\star \\in \\{!,?\\}$),\n\n \n\\item the \\emph{partial} function $\\intree{\\cdot}$ as\\\\\n \n$\\intree{q} =\n\\begin{cases}\n  \\perp & \\text{if }  \\loopio{?}{q}\n  \\\\\n  q & \\text{if } \\inTrans{q} = \\varnothing\n  \\\\\n\\outchoicetop{l_i: \\intree{q'_i}}{i\\in I} & \\text{if }   \\inTrans{q} = \\{l_i \\, \\mid \\, i\\! \\in\\! I\\} \\neq  \\varnothing\n\\end{cases}$\\\\\nwith $q'_i$ being the state such that $q\\trans{?\\, l_i} q'_i$.\n\\end{itemize}\nPredicate $\\loopio{\\star}{q}$ says that, from $q$, we\ncan\nreach a cycle with only sends (resp.\\ receives), depending on whether\n$\\star=!$ or $\\star=?$.\nThe partial function $\\intree{q}$, when defined,  returns the tree containing all sequences\nof messages which can be received from $q$ until a final or sending\nstate is reached.\nIntuitively, $\\intree{q}$ is undefined when $\\loopio{?}{q}$ as it\nwould return an infinite tree.\n\n\\begin{example}\n  Consider $\\mathsf{LTS}(T_{HC})$ (Figure~\\ref{fig:AsyncAut}), we have the following:\n \n \\vspace*{-0.5cm}\\begin{figure}[h]\n \\begin{minipage}[t]{0.65\\linewidth}\n  \\[\n    \\begin{array}{lllll} \n      \\\\\n      \\intree{1} = 1\n      &   \\hspace{.3cm}\\intree{2} = \\outchoicetop{\\exKO : 1 , \\,\n                          \\msg{ok} : 1}{}\n\n   \n    \\end{array}\n  \\]\n\\end{minipage}\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=0.4\\linewidth, valign=t]{pics\/intree.PNG}\n\\\\[.1cm]{\\!\\!$\\intree{2}$ tool representation}\n\\label{intree}\n\\end{minipage}%\n\\end{figure}\n\\end{example}\n\n\\begin{figure}[t]\n\\begin{minipage}[t]{0.4\\linewidth}\n\\centering\n\\subfloat{\\includegraphics[scale=.62, keepaspectratio]{pics\/example2.pdf}}\n\\caption{Output cycle example.} \n\\label{fig:CycleAut}\n\\end{minipage}\n\\begin{minipage}[t]{0.6\\linewidth}\n\\centering\n\\subfloat{\\includegraphics[scale=.4, keepaspectratio, valign=t]{pics\/fair_sim.pdf}}\n\\caption{Fair asynchronous simulation fragment.}\n\\label{fig:FAsyncSim}\n\\end{minipage}\n\\end{figure}\n\n\\begin{example}\\label{example:newexample}\n  Consider the LTS of Figure \\ref{fig:CycleAut}.\n  From state $1$ we can reach state $2$ with an output. The latter can loop with an output into itself. Hence, we have both $\\loopio{!}{1}$ and\n  $\\loopio{!}{2}$.\n\\end{example}\n\n\\begin{definition}[Asynchronous Simulation]\\label{def:simtree}\nGiven set of label names ${\\cal L}$ and two\nLTSs: LTS$_1\\!=\\!(P,p_0,\\Trans_1)$ and LTS$_2\\!=\\!(Q,q_0,\\Trans_2)$,\nasynchronous simulation is defined as a labeled transition system over states of $P \\!\\times\\! {\\cal I}{\\cal T}_Q$, i.e. pairs denoted by $\\simtreepair{p}{\\ctx A[q_j]^{j\\in J}}$, with $p \\!\\in\\! P$ and $\\ctx A[q_j]^{j\\in J} \\!\\in\\! {\\cal I}{\\cal T}_Q$. In particular, the initial state is $\\simtreepair{p_0}{q_0}$ and the transition relation $\\simtreetrans{}$, labeled over  $\\{!,\\!?\\} \\!\\times\\! {\\cal L}$,\nis defined as the minimal relation satisfying rules in Definition \\ref{def:syncsimtree}, plus the following ones:\\\\[.1cm]\n \n$    \\begin{array}{c}\n\t\\infer[\\!\\mathsf{(InCtx)}]\n     \n      {\n       \n      \\simtreepair{p}{ \\outchoicetop{l_i:\\ctx A_i[q_{i,j}]^{j\\in J_i}}}{i\\in I} \n     \n      \\ \\simtreetrans{?\\, l_k}\\ \n     \n      \\simtreepair{p'}{\\ctx A_k[q_{k,j}]^{j\\in J_k}}} \n     \n      {p \\trans{?\\, l_k}_1 p' &  k\\in I &\n                                      \\inTrans{p} \\supseteq \\{l_i \\mid\n                                      i\\in I\\, \\}}\n     \n      \\\\[4mm]\n      \\infer[\\!\\!\\mathsf{(OutA)}]\n     \n      {\n      \\simtreepair p{\\ctx A[q_j]^{j\\in J}}\n      \\simtreetrans{!\\, l}\\simtreepair{p'}\n      {\\ctx A[\\ctx A_j[q'_{j,h}]^{h\\in H_j}]^{j\\in J}}\n      }\n     \n      {\n      \\begin{array}{c@{}}\n        \\!\n        p\\trans{!\\, l}_1 p'\n        \\qquad\\qquad\n        \\neg\\loopio{!}{p}\n        \\\\\n        \\!\\forall j \\!\\!\\in\\!\\! J. \\big(\n        \\intree{q_{j\\!}}\\! =\\! \\ctx A_{j\\!}[q_{j,h}]^{h\\in H_j} \\!\\!\\land\\!\n        \\forall h \\!\\in\\!\\! H_j.\n        ( \\outTrans{p} \\!\\subseteq\\! \\outTrans{q_{j,h}} \\!\\land\\!\n        q_{j,h\\!} \\!\\trans{!\\, l}_2 \\! q'_{j,h}) \\big)\n      \\end{array}\n      }\n    \\end{array}\n$\n\n\\end{definition}\nThe two additional rules express how inputs are accumulated and consumed by means of input trees in the LTS$_2$.\nThe first one is applicable when the input tree state of the LTS$_2$ is\nnon-empty and the state $p$ of the LTS$_1$ is able to perform a\nreceive action corresponding to any message located at the root of the\ninput tree (contra-variance of receive actions).\nThe second rule allows the LTS$_1$ to execute some send\nactions by matching them with send actions that, in the LTS$_2$, occur after receives.\nIntuitively, each send action outgoing from state $p$ of the LTS$_1$ must also be\nexecutable from each of the states $q_{j,h}$ of $\\intree{q_{j\\!}}\\! =\\! \\ctx A_{j\\!}[q_{j,h}]^{h\\in H_j}$, with\n$q_{j\\!}$ being a leaf of the input tree state ${\\ctx A[q_j]^{j\\in J}}$ of the LTS$_2$ (covariance of send actions).\nThe constraint $\\neg\\loopio{!}{p}$ guarantees that accumulated receive actions will be eventually executed.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Fair Asynchronous Session Subtyping}\\label{sub:fairasyncsimtree}\n\n\n\n\n\n\nConsider again the satellite protocol of Example \\ref{SatEx}. The asynchronous subtyping of previous Section \\ref{sub:asyncsub} rejects $T'_{SC}$ as a\nsubtype of $T_{SC}$.\nIndeed that notion of subtyping allows for anticipation of outputs\nonly when they are preceded by a \\emph{bounded} number of inputs.\nHowever the outputs of $T_{SC}$ occur after an arbitrary\nnumber of inputs.\nThat notion of subtyping requires that all sent messages are consumed\nalong \\emph{all} possible computations of the receiver.\nWhile in $T'_{SC}$ there is a degenerate execution where\nthe candidate subtype sends an infinite number of $\\mathsf{tc}$\nmessages and thus never performs the required inputs.\n\nIn contrast, {\\it fair} asynchronous session subtyping~\\cite{BLZ21} relies on the\nassumption that such degenerate executions cannot occur under the\nnatural assumption the loop of outputs eventually terminates, i.e.,\nonly a finite (but unspecified) amount of messages can be emitted.\n\nConcretely, the fair subtyping uses a more expressive notion of input\ncontexts ${\\mathcal A}$ that also include recursive constructs.\nTheir syntax becomes:\n\\\\%[.05cm]\n\\centerline{$ {\\mathcal A}\\ \\ ::=\\ \\ [\\,] \\ \\mid\\ \\outchoicetop{l_i:{\\mathcal\n      A}_i}{i\\in I} \\ \\mid\\ \\Trec X.{\\mathcal A}  \\ \\mid \\ \n    \\Tvar X \n    $}\\\\%[.05cm] %\n  %\n  %\n  These input context can encode the recursive reception of\n  messages in the satellite example and thus identify\n  $T'_{SC}$ as a fair asynchronous subtype of\n  $T_{SC}$.\n \n \n\nFigure~\\ref{fig:FAsyncSim} shows a fragment of the resulting fair\nasynchronous simulation tree, as produced from our tool: due to the\nmore complex syntax of input contexts, states now contain a (possibly\nlooping) automaton instead of an input tree.\n\n\n\n\n\n\\section{Main Functionalities of the Tool} \\label{SEC:tool}\n\n\nBesides the classic operations that a text editor allows (e.g.\\ edit, load, save), users can compute the dual of: either a single session type or the entire subtyping problem. To facilitate understanding of session types, the tool offers the possibility to view\/save the graphical representation of a given type by means of \"Show Image\" and \"Save Image\" respectively.  In our tool types are inputted by means of {\\it two text areas}: the leftmost one is used for the candidate subtype and the rightmost one for the candidate supertype. Input types must be expressed with the syntax presented in Definition \\ref{def:sessiontypes},  with ``$+$'' standing for ``$\\oplus$'' and ``$rec$'' standing for ``$\\mu$''. In addition the tool accepts: $(i)$ the alternative ``raw'' syntax $[! \\, a \\, ; T , ! \\, b \\, ; T' , \\dots]$ standing for $\\oplus \\{a \\, ; T , b \\, ; T' , \\dots \\}$\nand $[? \\, a \\, ; T , ? \\, b \\, ; T' , \\dots]$\nfor $\\& \\{a \\, ; T , b \\, ; T' , \\dots \\}$ $(ii)$ the abbreviations $! \\, a \\, ; T$ and $? \\, a \\, ; T$ standing for $\\oplus \\{a \\, ; T \\}$ and $\\& \\{a \\, ; T \\}$.  A Python parser checks that the inputted types fit the above syntax using the following \\textit{EBNF} syntax: \\\\[.1cm]\n$\n  \\begin{array}{lcl}\n    S &::=&  \\,OP\\, \\textbf{\\{}\\,\\textit{\\textbf{id}}\\, \\textbf{;} \\,S\\, (\\textbf{, } \\,\\textit{\\textbf{id}}\\, \\textbf{;} \\,S\\,)^*\\textbf{\\} } \\ \\mid\\ \\ \\textbf{rec} \\,\\textit{\\textbf{id}}\\,\\textbf{. }\\,S\\,\\ \\mid\\ \\ \\,\\textit{\\textbf{id}}\\, \\ \\mid\\ \\ \\textbf{end}\n     \\\\\n    && \\textbf{!} \\,\\textit{\\textbf{id}}\\, \\textbf{;} \\,S\\,  \\ \\mid\\ \\  \\textbf{[ !} \\,\\textit{\\textbf{id}}\\, \\textbf{;} \\,S\\, (\\textbf{, !} \\,\\textit{\\textbf{id}}\\, \\textbf{;} \\,S\\,)^* \\textbf{]}  \\ \\mid \\ \\ \\textbf{?} \\,\\textit{\\textbf{id}}\\, \\textbf{;} \\,S\\, \\ \\mid\\ \\ \\textbf{[ ?} \\,\\textit{\\textbf{id}}\\, \\textbf{;} \\,S\\, (\\textbf{, ?} \\,\\textit{\\textbf{id}}\\, \\textbf{;} \\,S\\,)^* \\textbf{]}\n  \\\\\n  OP& ::=& \\textbf{+} \\ \\mid\\ \\ \\textbf{\\&}\n  \\end{array}\n$\\\\[.2cm]\nwhere \\textit{\\textbf{id}} is a non-empty sequence of uppercase and lowercase letters possibly followed by trailing numbers.\n\nThe core of the tool is the algorithm menu.  Users can choose between different subtyping algorithms and possibly set a maximum number of  execution steps. The algorithm response can be: \"true\", \"false\" or \"maybe\" (for asynchronous algorithms, due to undecidability, or when the specified number of steps\nis not enough to determine the subtyping relation), along with the time needed.  In addition,  it is possible to run all the algorithms to have an overview\nof the types of relationship that hold.\nFinally, the \"Simulation Result\" menu, which is initially disabled, makes it possible to show or save the graphical output of the last performed algorithm. \n\n\n\\subsection{Extensibility of the Tool}\\label{extens}\n\nOur tool (and its GUI) is automatically extensible with new subtyping algorithms by simply modifying its json configuration file that we will detail in the next section. Such a file can also be modified, directly from the GUI, by using the ``Algorithm configuration\" menu under ``Settings\".  Configuration of a new algorithm is done by providing: its displayed name and the path and calling pattern of its execution command, in the form \\\\\n\\centerline{\\texttt{ExecutableName [flag] [t1] [t2] [steps]}}\\\\%[.1cm]\nThe tool will replace [flag][t1][t2][steps] with: the user-selected flags, the pair of session types the user wants to check and the number of steps the algorithm is requested to do. The above order of bracketed elements ([steps] is optional) may change according to the algorithm. The json file also maps algorithm-dependent flag names into tool functionalities by categorizing them. For instance, the default \\textit{flag} category includes flags that simply modify the behaviour of algorithms: e.g.\\ the asynchronous one has the -\\;\\!-\\;\\!$\\mathsf{nofallback}$ flag that prevents the algorithm from trying to fall back to the dual subtyping problem in case of an initial maybe verdict. Moreover,\nthe \\textit{execution flag} category is useful when an executable encloses different (alternative) algorithms, e.g.\\ Gay and Hole (-\\;\\!-\\;\\!$\\mathsf{gayhole}$) and Kozen et al.'s (-\\;\\!-\\;\\!$\\mathsf{kozen}$) algorithms for synchronous subtyping (with one indicated as being the default). Morever, the \\textit{visual flag} category includes just the name of the flag causing the algorithm to produce the graphical simulation. \n\nWhen adding an algorithm to the tool, the following requirements have to be satisfied: they have to support command line execution (with the possibility of taking .txt files as input) and have to fit the ``raw\" syntax described above. Regarding the algorithm response, the only requirement is that it is printed on the standard output. Finally, to generate the graphical output, it is mandatory that the algorithm creates a .dot file no matter what its name is (since it is specified in dedicated section of the json configuration). It is important to observe that our tool is agnostic to the implementation language of algorithms, since it makes use of their executable version. \n\n\\subsection{Configuration of Tool Algorithms}\nThe json file presented below is an example of the ``algorithms\\_config.json\" currently used by the tool. The \\textit{standard\\_exec} field specifies the default\nexecution flag, e.g.\\ Gay and Hole or Kozen.  Moreover, the \\textit{simulation\\_file} field indicates the relative path to the algorithm generated \\textit{Graphviz} \".dot\" simulation file. Similarly \\textit{win, osx} and \\textit{linux} point at the folder in which the tool looks for the algorithm binaries for that specific os. \n\\begin{lstlisting}[language=json,firstnumber=1]\n[{\n   \"alg_name\": \"Async Subtyping\",\n   \"flag\": \"--nofallback\",\n   \"execution_flag\": \"\",\n   \"standard_exec\": \"\",\n   \"visual_flag\": \"--pics\",\n   \"simulation_file\": \"tmp\/simulation_tree\",\n   \"win\": \"asynchronous-subtyping\\\\win\\\\\",\n   \"osx\": \"asynchronous-subtyping\/osx\/\",\n   \"linux\": \"asynchronous-subtyping\/linux\/\",\n   \"exec_comm\": \"Checker [flags] [t1] [t2]\"\n },\n {\n   \"alg_name\": \"Fair Async Subtyping\",\n   \"flag\": \"\",\n   \"execution_flag\": \"\",\n   \"standard_exec\": \"\",\n   \"visual_flag\": \"--debug\",\n   \"simulation_file\": \"tmp\/simulation_tree\",\n   \"win\": \"fair-asynchronous-subtyping\\\\win\\\\\",\n   \"osx\": \"fair-asynchronous-subtyping\/osx\/\",\n   \"linux\": \"fair-asynchronous-subtyping\/linux\/\",\n   \"exec_comm\": \"Checker [flags] [t1] [t2] [steps]\"\n },\n {\n   \"alg_name\": \"Sync Subtyping\",\n   \"flag\": \"\",\n   \"execution_flag\": \"--gayhole,--kozen\",\n   \"standard_exec\": \"--gayhole\",\n   \"visual_flag\": \"--pics\",\n   \"simulation_file\": \"tmp\/simulation_tree\",\n   \"osx\": \"sync_subtyping\/osx\/\",\n   \"win\": \"sync_subtyping\\\\win\\\\\",\n   \"linux\": \"sync_subtyping\/linux\/\",\n   \"exec_comm\": \"Checker [flags] [t1] [t2]\"\n }]\n\\end{lstlisting}\n\n\\section{Conclusion}\\label{SEC:conc}\nIn this paper we introduced an integrated extensible GUI-based tool which: applies algorithms for \nsynchronous and (fair) asynchronous session subtyping, and\ngenerates graphical simulations showing how underlying algorithms work. \n\nConcerning future work, we plan to use our synchronous subtyping simulation algorithm (with error detection) in the context of type checking for object oriented programming languages where classes are endowed with usage protocols~\\cite{BravettiFGHJKR20}. Indeed, extending the theory of \\cite{BravettiFGHJKR20} with protocol subtyping, would make it possible to verify correctness also for class inheritance.\nIn particular, we have started integrating our algorithm (see Appendix \\ref{integration}) into the Java checker \\cite{MGR21}, which is based on~\\cite{BravettiFGHJKR20}.\nFinally, we plan to extend the syntax of session types managed by our tool, e.g.\\ by including passing of data\/channels and, possibly, by also encompassing pre-emption mechanisms~\\cite{BravettiZ09,Bravetti21}, which are often used in\ncommunication protocols. \n\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}