Simplicial models for fibrations between mapping spaces

Question

Let $S,T$ 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. We let $|S|$ denote the geometric realisation of a simplicial set $S$.

Given $X$ and $Y$, (compactly generated) topological spaces, we use $TOP(X,Y)$ to denote the function space, 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 $TOP(X,Y)$.

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 is the composition of the obvious maps: $$ |SIMP(S,K)| \to |SIMP(S,Sing|K|)| \stackrel{\cong}{\to} |TOP_{Simp}(|S|,|K|)|\to TOP(|S|,|K|). $$

Suppose that we have a cofibration (meaning inclusion) $i\colon T \to S$ of simplicial sets. We have an induced fibration $i^*\colon SIMP(S,K) \to SIMP(T,K)$ of simplicial sets, hence a Serre fibration $|i^*|\colon |SIMP(S,K)| \to |SIMP(T,K)|$.

We also have a cofibration $|i|\colon |T| \to |S|$, thus a (Hurewicz, hence Serre) fibration $|i|^*\colon TOP(|S|,|K|) \to TOP(|T|,|K|)$. (We just restrict a map $|S| \to |K|$ to $|T|\subset |S|$.)

It is also clear that we have a commutative diagram: $\require{AMScd}$ \begin{CD} |SIMP(S,K)| @>>> TOP(|S|,|K|)\\ @V |i^*| V V @VV |i|^* V\\ |SIMP(T,K)| @>>> TOP(|T|,|K|)\\ \end{CD}

The question I would like to ask is the following. Does $i^*\colon SIMP(S,K) \to SIMP(T,K)$ give a 'faithful' simplicial model of the fibration $|i|^*\colon TOP(|S|,|K|) \to TOP(|T|,|K|)$ (whatever this means). For instance (this would suffice), does the homotopy long exact sequence of $|i|^*$ coincide with that of $|i^*|$?

Answer 1

Yes, these agree.

The usual model structures on $C = sSet$ and on $C = Top$ are both cartesian monoidal. So the functor $[-,X] : C^{op} \to C$ is a right Quillen functor when $X$ is fibrant (where $[-,-]$ denotes the internal hom). Thus this functor is "already derived" when evaluated on cofibrant objects.

Moreover, geometric realization is a left Quillen functor. Because every object of $sSet$ is cofibrant, it preserves weak equivalences, and is "already derived".

So when $K$ is Kan, $|i^\ast|: |[S,K]| \to |[T,K]|$ computes "the correct" map because it is a a component of a transformation between functors which are already derived. And $|i|^\ast: [|S|,|K|] \to [|T|,|K|]$ also computes "the correct" map because it is a component of a transformation between functors which are already derived. Because these two maps agree in the homotopy category, they agree up to homotopy. Unwinding what this means, the horizontal maps in the square you've written are weak equivalences.

The remarkable fact that geometric realization preserves fibrations means that you don't even have to worry about taking homotopy fibers to construct the homotopy long exact sequence!