Classifying spaces and classifying topoi
type: Item
instance of: scholarly article
title: Classifying spaces and classifying topoi
zbMATH Open document ID: 0838.55001
published in: Lecture Notes in Mathematics
publication date: 1995-11-13T00:00:00Z
review text: This notes arose out of two related questions: ``what does the so-called classifying space of a small category actually classify'' and ``what is the relation between classifying spaces and classifying topoi?'' The first purpose is to extend the relation between classifying space and classifying topos from the well-known case of a group \(G\) to that of a small category \(C\). In Chapter I, the classifying topos \({\mathcal B} C\) of \(C\) is constructed as the topos of all presheaves on \(C\) and the classifying space \(BC\) as the geometric realization of the nerve of \(C\). Replacing the space \(C\) by its topos \(\text{Sh} (BC)\) (of all sheaves on \(BC)\), in Chapter IV is constructed a nice weak homotopy equivalence \(p : \text{Sh} (BC) \to {\mathcal B} C\) which contains a lot of more information. For example, for a CW-space \(X\) one concludes a bijective correspondence \([X,BC] \approx [\text{Sh} (X), {\mathcal B}C]\) between homotopy classes of maps \(X \to BC\) and homotopy classes of maps of topoi \(\text{Sh} (X) \to {\mathcal B} C\). Using the notion of a principal \(C\)-bundle in Section IV the following theorem of Diaconescu is proved: Homotopy classes of maps \(X \to BC\) are in bijective correspondence with concordance classes of principal \(C\)-bundles over \(X\). Then the author moves to the problem of extending these results to topological categories. In the case (including that of discrete categories) of topological categories \(C\) with the property that the source map \(s : C_1 \to C_0\) is étale the map \(p : \text{Sh} (BC) \to {\mathcal B} C\) is a weak homotopy equivalence. The theorem just stated holds for étale topological categories as well. To obtain a suitable comparison with the classifying space \(BC\) in that case the author considers \({\mathcal D} C\) the Deligne classifying topos [\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 44(1974), 5-77 (1975; Zbl 0237.14003)]. Then \({\mathcal D}C\) and the classifying space \(BC\…
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 55N30
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 18F10
zbMATH DE Number: 815350
zbMATH Keywords: classifying space
zbMATH Keywords: classifying topos
zbMATH Keywords: étale map
zbMATH Keywords: sheaf
zbMATH Keywords: presheaf
author: Ieke Moerdijk
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0094441
OpenAlex ID: W243555764
DOI: 10.1007/BFB0094441
description: scientific article; zbMATH DE number 815350