Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier
{"title":"可构轴的同伦不变性","authors":"Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier","doi":"10.4310/hha.2023.v25.n2.a6","DOIUrl":null,"url":null,"abstract":"The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $\\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\\infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $\\mathcal{E}$ is a presentable $\\infty$-category, then we give a concrete formula for the constant hypersheaf functor $\\mathcal{E}\\to \\mathrm{Sh}^{\\mathrm{hyp}}(X;\\mathcal{E})$. This formula lets us show that the constant hypersheaf functor is a right adjoint, and is fully faithful if $X$ is also weakly contractible. It also lets us prove a general monodromy equivalence and categorical K\\\"unneth formula for locally constant hypersheaves.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"The homotopy-invariance of constructible sheaves\",\"authors\":\"Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier\",\"doi\":\"10.4310/hha.2023.v25.n2.a6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $\\\\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\\\\infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $\\\\mathcal{E}$ is a presentable $\\\\infty$-category, then we give a concrete formula for the constant hypersheaf functor $\\\\mathcal{E}\\\\to \\\\mathrm{Sh}^{\\\\mathrm{hyp}}(X;\\\\mathcal{E})$. This formula lets us show that the constant hypersheaf functor is a right adjoint, and is fully faithful if $X$ is also weakly contractible. It also lets us prove a general monodromy equivalence and categorical K\\\\\\\"unneth formula for locally constant hypersheaves.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4310/hha.2023.v25.n2.a6\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/hha.2023.v25.n2.a6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $\mathcal{E}$ is a presentable $\infty$-category, then we give a concrete formula for the constant hypersheaf functor $\mathcal{E}\to \mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$. This formula lets us show that the constant hypersheaf functor is a right adjoint, and is fully faithful if $X$ is also weakly contractible. It also lets us prove a general monodromy equivalence and categorical K\"unneth formula for locally constant hypersheaves.