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...
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.)