Classifying space of a colimit of topological categories

Question

Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. I would like to know when this induces a homotopy equivalence $$B(\text{colim}\, D)\stackrel{\sim}{\rightarrow}\text{hocolim}\, BD$$ Or more generally when there are known methods of computing the homotopy type of $B(\text{colim}\, D)$ from $BD$.

In the example I have in mind, $I$ is the poset of natural numbers, every space in sight is compactly generated and every functor $D(n)\rightarrow D(m)$ is a cofibration on both objects and morphisms. Futhermore, the indentity map in each $D(n)$ is a cofibration and the source and target maps are fibrations. In particular, in this case the above homotopy colimit is the ordinary colimit. However, I think the more general question is also of interest.

References to the literature are also welcome!

Answer 1

Espen, I would disagree with your description of the classifying space functor. Your question starts with a diagram in Cat(Top). The standard classifying space functor is the composite of the nerve functor $N$ from there to simplicial spaces and geometric realization. Here $N$ is defined in what should be an obvious way in terms of the space of objects and the spaces (defined by source target pullbacks) of composable morphisms. Geometric realization is generally understood in the usual, not the fat, sense. Since geometric realization commutes with colimits (it is a left adjoint), it is not a problem here. The problem is the nerve functor $N$. In your special case when I is the natural numbers, I see no problem: $N$ will take your cofibrations to levelwise cofibrations and will take unions to unions, those being the colimits in that special case.

However, the classifying space functor behaves quite badly with respect to colimits of general diagrams in Cat and therefore, more generally, with respect to general colimits in Cat(Top). Pushouts give a simple and central example of diagrams that behave badly: $N$ usually fails to preserve them. The key point of Thomason's paper in which he gives a model structure on Cat that is Quillen equivalent to the standard model structure on simplicial sets is to identify a class of maps, which he calls Dwyer maps, such that $N$ preserves pushouts in which one leg is a Dwyer map. Cisinski observed that a retract of a Dwyer map need not be a Dwyer map and identified an alternative notion of pseudo Dwyer maps that is closed under retracts. However, that is in Cat and I don't think that anyone has developed a theory of Dwyer maps in Cat(Top) that might help answer your general question.