Classifying spaces and classifying topoi.

*(English)*Zbl 0838.55001
Lecture Notes in Mathematics. 1616. Berlin: Springer-Verlag. vii, 94 p. (1995).

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 [P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 44(1974), 5-77 (1975; Zbl 0237.14003)]. Then \({\mathcal D}C\) and the classifying space \(BC\) have the same weak homotopy type. From this the author obtains an answer to the question what \(BC\) classifies.

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 [P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 44(1974), 5-77 (1975; Zbl 0237.14003)]. Then \({\mathcal D}C\) and the classifying space \(BC\) have the same weak homotopy type. From this the author obtains an answer to the question what \(BC\) classifies.

Reviewer: M.Golasiński (Toruń)