{"title":"Internal geometry and functors between sites","authors":"Konrad Waldorf","doi":"arxiv-2408.04989","DOIUrl":null,"url":null,"abstract":"Locality is implemented in an arbitrary category using Grothendieck\ntopologies. We explore how different Grothendieck topologies on one category\ncan be related, and, more general, how functors between categories can preserve\nthem. As applications of locality, we review geometric objects such as sheaves,\ngroupoids, functors, bibundles, and anafunctors internal to an arbitrary\nGrothendieck site. We give definitions such that all these objects are\ninvariant under equivalences of Grothendieck topologies and certain functors\nbetween sites. As examples of sites, we look at categories of smooth manifolds,\ndiffeological spaces, topological spaces, and sheaves, and we study properties\nof various functors between those.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Differential Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.04989","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Locality is implemented in an arbitrary category using Grothendieck
topologies. We explore how different Grothendieck topologies on one category
can be related, and, more general, how functors between categories can preserve
them. As applications of locality, we review geometric objects such as sheaves,
groupoids, functors, bibundles, and anafunctors internal to an arbitrary
Grothendieck site. We give definitions such that all these objects are
invariant under equivalences of Grothendieck topologies and certain functors
between sites. As examples of sites, we look at categories of smooth manifolds,
diffeological spaces, topological spaces, and sheaves, and we study properties
of various functors between those.