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I was prompted by this question, but the motivation is different.

Suppose we have an internal category object in topological spaces, i.e. an object space X and a morphism space Y, together with continuous domain, range, and composition maps satisfying the standard identities - these give a representable functor from spaces to categories. Associated to this category object (X,Y) we have an associated nerve N(X,Y), which is a simplicial space, and we can take geometric realization.

Suppose you have a (surjective) map Z -> X of topological spaces. You can then take the base change of this topological category, which is a new internal category (Z, Z xX Y xX Z) = (Z,W). There's a functor (Z,W) -> (X,Y) that represents the unique fully faithful functor from a category whose objects are represented by Z.

The question is: Under what conditions on the map Z -> X can we conclude that the map of geometric realizations |N(Z,W)| -> |N(X,Y)| is a (weak) homotopy equivalence?

In algebraic geometry the analogous questions are related to faithfully flat descent and stacks.

I know that sufficient conditions include:

  • Locally on X, the map Z -> X has sections
  • Z -> X and (X,Y) are realizations of corresponding diagrams of simplicial sets, with Z -> X surjective

I am interested in more general criteria. EDIT: The criteria I was hoping for were perhaps something like:

  • X is a CW complex and Z is a coproduct of subcomplexes (or cells) covering X
  • Z -> X is a proper map of real points of algebraic varieties
  • X has some kind of stratification over which the map Z behaves nicely, such as being a fibration...
Generally I want to replace "open" conditions by "closed" conditions.

Examples that do not work include: Z is the "discretification" of X, and Z is a coproduct of an open subset of X and the complemetary closed subset.

(The specific examples I'm interested in are categories of representations in unitary groups or general linear groups. The "domain" map Y -> X is projection from a cartesian product.)

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Just to be clear, Z x_X Y x_X Z is not a homotopy fiber product, correct? –  Reid Barton Oct 20 '09 at 16:52
    
That's correct, I'm still stuck in 20th century non-infinity-categories here. –  Tyler Lawson Oct 20 '09 at 18:09
    
@Tyler: From what you say, it seems like the paper MR0799449 (87d:55017) Zisman, M. Une remarque sur le classifiant d'une catégorie topologique. (French) [A remark on the classifying space of a topological category] Arch. Math. (Basel) 45 (1985), no. 1, 47–67. is relevant. The construction you mention is I think in the case of groupoids the universal construction_introduced by Higgins, see also my book _Topology and groupoids. –  Ronnie Brown Oct 6 '12 at 13:41
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2 Answers 2

There are a couple different versions of the geometric realization of simplicial spaces. There is the literal one, which is badly behaved in general. Then there are better realizations, e.g. the "fat" realization. All the good ones have the same (weak) homotopy type.

If you use the good realization, then it won't matter if you replace X, Y, and Z with the realizations of their singular simplicial sets, and this reduces the problem to the case that X, Y, and Z are simplicial sets. You've already mentioned one criteria for simplcial sets (which, btw, is implied by your first criteria). What more general criteria did you have in mind?

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I'm happy to take the "fat" realization - the point was not to worry about the homotopical properties of the nerve. One problem with reducing to the simplicial case is that Z -> X only induces a surjection Sing Z -> Sing X if the original map was a Serre fibration and a surjection. I'll add some more specific hopes above. –  Tyler Lawson Oct 20 '09 at 5:03
    
Also, the first criterion doesn't imply the second - a surjection of simplicial sets doesn't locally have sections on geometric realization, e.g. if Z is the disjoint union of two intervals of length 1 and X is formed by gluing these together at a point. –  Tyler Lawson Oct 20 '09 at 5:12
    
I meant it the other way. That if Z --> X has local sections, then Sing(Z) --> Sing(X) is surjective. But maybe this depends on what you mean by surjective. I'm guessing from your Serre fibration comment that you mean surjective on each level of the simplicial set. I was thinking surjective on pi_0 (which is a good notion when you view (Kan) simplicial sets as infinity-groupoids, but probably not what you want). –  Chris Schommer-Pries Oct 20 '09 at 13:42
    
Sorry, I misunderstood. Yes, I was only able to prove that you get an equivalence with an actual surjection. –  Tyler Lawson Oct 20 '09 at 18:11
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If $Z \to X$ admits local sections over a numerable open cover, then $|N(Z,W)| \to |N(X,Y)|$ is a homotopy equivalence (say if $X$ is paracompact), not just a weak equivalence. This boils down to a lemma by Segal that says the geometric realisation of the Cech nerve of a numerable open cover is homotopy equivalent to the original space. Note that if $p:Z \to X$ is a Hurewicz fibration, and $X$ is locally contractible, then $p$ admits local sections. Actually you could take $p$ to be a Dold fibration, which is strictly weaker than being a Hurewicz fibration, and get the same result.

If you are in the smooth category, $Z \to X$ could be a surjective submersion, but I"m sure you know this.

Actually all you really need is for $Z \times_X Y' \stackrel{pr_2}{\to} Y' \to X$ to admit local sections/Dold fibration over paracompact space/etc and you'll get the same results ($Y' \subset Y$ is the subobject of isomorphisms, the fibred product is taken over the source map $Y'\to X$ and the projection on the right is the target map).

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