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