{"title":"Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations","authors":"Fernando Lucatelli Nunes, Rui Prezado, L. Sousa","doi":"10.36045/j.bbms.221021","DOIUrl":null,"url":null,"abstract":"For any suitable base category $\\mathcal{V} $, we find that $\\mathcal{V} $-fully faithful lax epimorphisms in $\\mathcal{V} $-$\\mathsf{Cat} $ are precisely those $\\mathcal{V}$-functors $F \\colon \\mathcal{A} \\to \\mathcal{B}$ whose induced $\\mathcal{V} $-functors $\\mathsf{Cauchy} F \\colon \\mathsf{Cauchy} \\mathcal{A} \\to \\mathsf{Cauchy} \\mathcal{B} $ between the Cauchy completions are equivalences. For the case $\\mathcal{V} = \\mathsf{Set} $, this is equivalent to requiring that the induced functor $\\mathsf{CAT} \\left( F,\\mathsf{Cat}\\right) $ between the categories of split (op)fibrations is an equivalence. By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations $\\mathcal{F}$ to the study of the codescent factorization, we find that these observations on fully faithful lax epimorphisms provide us with a characterization of (effective) $\\mathcal{F}$-descent morphisms in the category of small categories $\\mathcal{Cat}$; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of discrete opfibrations -- previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"10 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Belgian Mathematical Society-Simon Stevin","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.36045/j.bbms.221021","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For any suitable base category $\mathcal{V} $, we find that $\mathcal{V} $-fully faithful lax epimorphisms in $\mathcal{V} $-$\mathsf{Cat} $ are precisely those $\mathcal{V}$-functors $F \colon \mathcal{A} \to \mathcal{B}$ whose induced $\mathcal{V} $-functors $\mathsf{Cauchy} F \colon \mathsf{Cauchy} \mathcal{A} \to \mathsf{Cauchy} \mathcal{B} $ between the Cauchy completions are equivalences. For the case $\mathcal{V} = \mathsf{Set} $, this is equivalent to requiring that the induced functor $\mathsf{CAT} \left( F,\mathsf{Cat}\right) $ between the categories of split (op)fibrations is an equivalence. By reducing the study of effective descent functors with respect to the indexed category of split (op)fibrations $\mathcal{F}$ to the study of the codescent factorization, we find that these observations on fully faithful lax epimorphisms provide us with a characterization of (effective) $\mathcal{F}$-descent morphisms in the category of small categories $\mathcal{Cat}$; namely, we find that they are precisely the (effective) descent morphisms with respect to the indexed categories of discrete opfibrations -- previously studied by Sobral. We include some comments on the Beck-Chevalley condition and future work.
期刊介绍:
The Bulletin of the Belgian Mathematical Society - Simon Stevin (BBMS) is a peer-reviewed journal devoted to recent developments in all areas in pure and applied mathematics. It is published as one yearly volume, containing five issues.
The main focus lies on high level original research papers. They should aim to a broader mathematical audience in the sense that a well-written introduction is attractive to mathematicians outside the circle of experts in the subject, bringing motivation, background information, history and philosophy. The content has to be substantial enough: short one-small-result papers will not be taken into account in general, unless there are some particular arguments motivating publication, like an original point of view, a new short proof of a famous result etc.
The BBMS also publishes expository papers that bring the state of the art of a current mainstream topic in mathematics. Here it is even more important that at leat a substantial part of the paper is accessible to a broader audience of mathematicians.
The BBMS publishes papers in English, Dutch, French and German. All papers should have an abstract in English.