Topologically enriched homotopy colimits commuting with homotopy pullbacks

Question

Hi,

I am looking for an enriched analogon of Proposition 4.4 in https://www.google.de/url?q=http://hopf.math.purdue.edu/Rezk-Schwede-Shipley/simplicial.pdf

Concretely, I would like to prove the following statement:

Suppose $K$ is a topologically enriched category, i. e. the morphism sets carry a topology and composition is continuous. For a functor $X: K \to Top$, I can consider an enriched version of the homotopy colimit, namely $hocolim X$ is the realization of the simplicial space $srep X$, whose $n$-th level is given by

$ \coprod_{k_0,\ldots,k_n \in (ob K)^n} K(k_0, k_1) \times \ldots \times K(k_{n-1},k_n) \times X(k_0)$

Then, suppose there is a natural transformation between (enriched) functors $X,Y: K \to Top$, such that the diagram

$$ \begin{array}{ccc} X(k) & \to & Y(k) \end{array}$$

$$ \begin{array}{ccc} X(l) & \to &Y(l) \end{array} $$

is a homotopy pullback for all $k,l \in ob\ K$ and all morphisms $\alpha: k \to l$, which induce the (missing) vertical arrows.

Then the diagram

$$ \begin{array}{ccc} X(k) & \to & Y(k) \end{array}$$

$$ \begin{array}{ccc} hocolim X & \to &hocolim Y \end{array} $$

is a homotopy pullback for all $k \in ob K$.

Has anyone ever seen a statement like this or an idea on how to prove it? If it helps, one may assume that the natural transformations $X \to Y$ is levelwise a Serre fibration of topological spaces, since this is the only case, in which I need the statement to be true.

Thanks in advance, Alex

Answer 1

Rainer Vogt worked on this sort of problem originally back in the 1970s so check out his papers from that time. The theory involves homotopy coherence so you may need to check that out in his early paper (R. Vogt, Homotopy limits and colimits , Math. Z., 134, (1973), 11–52.)

Answer 2

I haven't thought about this hard (no time) but here are quick observations. Your homotopy colimit is the bar construction $B(\ast,K,X)$, the geometric realization of the simplicial space with $n$-simplices $B_n(\ast,K,X)$, as you state. The map $X(k) \to B(\ast,K,X)$ you are interested in is the geometric realization of the map from the constant simplicial space at $X(k)$ to $B_*(\ast, K, X)$ that identifies $X(k)$ with the subspace of $B_n(\ast,K,X)$ that sees only identity maps of the object $k$.

Homotopy pullbacks of diagrams one leg of which is a (Hurewicz) fibration are equivalent to actual pullbacks, so one approach might be to try to prove that $B(\ast,K,X) \to B(\ast,K,Y)$ is a fibration. It is standard that geometric realization of simplicial spaces preserves pullbacks (takes levelwise pullbacks to pullbacks). A variation on the theme of replacing maps by fibrations should convince you that geometric realization also preserves homotopy pullbacks (takes levelwise homotopy pullbacks to homotopy pullbacks).

So you would like your map to be the realization of a levelwise homotopy pullback. However, your stated hypothesis feels wrong to me, since it does not take the topology on the category K into account. Your hypothesis presumably should say that the evident square with upper left corner $K(\ell,k)$ and lower right corner $Map(X(k),Y(\ell))$ is a homotopy pullback. Assuming that, you should be able to prove that your map of simplicial spaces is a levelwise homotopy pullback, and then you would be done.

Hope that helps a bit.