Base change for category objects in topological spaces

Question

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 x_X Y x_X 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.)

Answer 1

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?

Answer 2

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).