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.

  • $\begingroup$ 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? $\endgroup$ – David White Mar 4 at 15:43

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