# Simplicial models for mapping spaces of filtered maps

Let $$S$$ and $$K$$ be simplicial sets with $$K$$ Kan. Given simplicial sets $$S$$ and $$K$$, we let $$SIMP(S,K)$$ be the internal hom in the category of simplicial sets. Hence $$SIMP(S,K)$$ is Kan.

Suppose that $${\cal S}=\{S_0 \subseteq S_1 \subseteq S_2 \subseteq S_3 \dots\}$$ and $${\cal K}=\{K_0 \subseteq K_1 \subseteq K_2 \subseteq K_3 \dots\}$$ are fitrations of $$S$$ and $$K$$, by sub-simplicial sets, where each $$K_i$$ is Kan. We can consider a filtered version $$SIMP({\cal S},{\cal K})$$ of $$SIMP(S,K)$$. The $$n$$-simplices are given by filtered simplical maps $${\cal S} \times \Delta(n) \to {\cal K}$$. (The case of fitrations of length two is treated in May's book "Simplical Objects in Algebraic Topology", page 17.)

It $$X$$ and $$Y$$ are topological spaces, we use $$TOP(X,Y)$$ to denote the space of functions $$X \to Y$$, with the k-ification of the compact open topology. We also have a simplicial mapping space $$TOP_{Simp}(X,Y)$$, which is essentially the singular complex $$Sing(TOP(X,Y))$$ of $$TOP(X,Y)$$.

If $${\cal X}=\{X_0 \subseteq X_1 \subseteq X_2\dots\}$$ and $${\cal Y}=\{Y_0 \subseteq Y_1\subseteq Y_e \dots\}$$ are fitrations of $$X$$ and $$Y$$ by subspaces, we let $$TOP({\cal X}, {\cal Y})$$ be the space of filtered functions $$\cal X \to \cal Y$$, with the $$k$$-ification of compact open topology.

We have a weak homotopy equivalence $$|Sing(X)| \to X$$ given any topological space $$X$$. There exists a well known weak homotopy equivalence $$|SIMP(S,K)| \to TOP(|S|,|K|)$$. It is essentially derived from the fact that $$K$$ is a strong deformation retract of the singular complex $$Sing(|K|)$$ if $$K$$ is Kan. This weak homotopy equivalence is the composition of the obvious maps: $$|SIMP(S,K)| \to |SIMP(S,Sing|K|)| \stackrel{\cong}{\to} |TOP_{Simp}(|S|,|K|)|\to TOP(|S|,|K|).$$

The question is the following: do we have a filtered version of the weak homotopy equivalence $$|SIMP(S,K)| \to TOP(|S|,|K|)$$? For instance, does it exist a weak homotopy equivalence $$|SIMP({\cal S}, {\cal K})| \to TOP(|\cal S|,|\cal K|)$$, where $$(|\cal S|$$ and $$|\cal K|)$$ are filtered by the geometric realisations of the $$S_i$$ and the $$K_i$$, respectively.

• Can you explain what goes wrong when trying to just use the answer to your previous question (with the obvious modifications) to answer this one too? – David White Mar 4 at 15:43