# What does the classifying space of a category classify?

A finite group $G$ can be considered as a category with one object. Taking its nerve $NG$, and then geometrically realizing we get $BG$ the classifying space of $G$, which classifies principle $G$ bundles.

Instead starting with any category $C$, what does $NC$ classify? (Either before or after taking realization.) Does it classify something reasonable?

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Have sometimes wondered! Don't know; but note that another way to say "principle G-bundle" is "Sheaf of G-torsors"; unless C is a groupoid (and even then) it's difficult to imagine what's the right notion for Torsor; one day should read HTT and see how the multi-object picture works. BG also comes with a natural principle bundle $EG\to BG$ with $EG$ contractible as a space --- it's easy to build a contractible space mimicking the group construction, but describing a good map to $BG$ is again difficult without significant assumptions. –  some guy on the street May 7 '10 at 14:38
I suppose you already know the paper by Michael Weiss (MR2175298, Homology Homotopy Appl. 7 (2005), no. 1, 185--195) whose title is exactly the title of your question. –  villemoes May 7 '10 at 15:31
I just took a look at the MathReview and am astonished - the result as stated there is completely contained in the book by Moerdijk which is 10 years older... –  Peter Arndt May 7 '10 at 16:50
I don't know the paper of Michael Weiss, however when I was writing the title of this question I had the feeling I have read this title before. So probably I at least read the title of that paper. –  Don Stanley May 7 '10 at 21:43
@PeterArndt: Michael Weiss gives a reference to Moerdijk's book in his paper and discusses the relationship between his result and Moerdijk's (they are not identical), so I don't think that the existence of Weiss's paper is astonishing. –  Dmitri Pavlov Nov 11 '13 at 11:19

Ieke Moerdijk has written a small Springer Lecture Notes tome addressing this question:"Classifying Spaces and Classifying Topoi" SLNM 1616.

Roughly the answer is: A $G$-bundle is a map whose fibers have a $G$-action, i.e. are $G$-sets (if they are discrete), i.e. they are functors from $G$ seen as a category to $\mathsf{Sets}$. Likewise a $\mathcal C$-bundle for a category $\mathcal C$ is a map whose fibers are functors from $\mathcal C$ to $\mathsf{Sets}$, or, if you want, a disjoint union of sets (one for each object of $\mathcal C$) and an action by the morphisms of $\mathcal C$ — a morphism $A \to B$ in $\mathcal C$ takes elements of the set corresponding to $A$ to elements of the set corresponding to $B$.

There is a completely analogous version for topological categories also.

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Thank you. I will look at this. –  Don Stanley May 7 '10 at 15:23
While recently reading V. Srinvas book "Algebraic K-theory", I learned the following (lemma 6.1), which is contained in what Peter says: The category of covering spaces of $BC$ is naturally equivalent to the category of functors $F:C\to\mbox{Sets}$ for which $F(f)$ is an isomorphism for every morphism $f$ in $C$. (Via the usual fibre construction: Fix a covering space. Then each object $X$ in $C$ gives a point of $BC$, and taking the fibre in the covering space gives a set; this gives the functor $C\to \mbox{Sets}$.) –  Matthew Morrow May 7 '10 at 22:39

It's one level up the categorical ladder, but you may find this paper interesting:

http://arxiv.org/abs/math/0612549
Two-Categorical Bundles and Their Classifying Spaces
Authors: Nils. A. Baas, Marcel Bokstedt, Tore August Kro

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I will look at this too. –  Don Stanley May 7 '10 at 15:23