Geometric realization of function complexes of simplicial sets

Let $\operatorname{sSet}$ be the category of simplicial sets and $\operatorname{Top}$ the category of (say, compactly generated) topological spaces. There is a pair of adjoint functors called geometric realization and singular set:

$$|\cdot |:\operatorname{sSet}\to \operatorname{Top}:S_\bullet$$

I am aware of some of the basic properties of this, but I am confused about the way it acts on function complexes, as I will explain.

Given two simplicail sets $X,Y$ we can form the simplicial set of maps between them, which we call the function complex and it is defined by $$\operatorname{Map}(X,Y)_n = \operatorname{Hom}(\Delta^n \times X,Y)$$

with the obvious face/degeneracy maps. This serves as an "internal hom" functor and it turns $\operatorname{sSet}$ into a cartesian closed category. In $\operatorname{Top}$ we also have an internal hom by endowing the set of maps between two topological spaces with the compact open topology. The general rather vague question is

What is the relationship between the space $|\operatorname{Map}(X,Y)|$ and $\operatorname{Map}(|X|,|Y|)$?

If I didn't make a mistake, I have the following. There is always a canonical map $$\phi:|\operatorname{Map}(X,Y)|\to \operatorname{Map}(|X|,|Y|)$$ which is adjoint to a map$$\psi:\operatorname{Map}(X,Y)\to S_{\bullet}(\operatorname{Map}(|X|,|Y|))$$

and we have a natural isomorphism $$S_{\bullet}(\operatorname{Map}(|X|,|Y|))\simeq \operatorname{Map}(X,S_\bullet|Y|)$$

The composition with $\psi$ is the map induced from the unit of the adjunction $Y\to S_\bullet|Y|$ which is a weak equivalence. Hence, I suspect that if $Y$ is a Kan complex then $\psi$, and hence $\phi$, is a weak equivalnce, but I lack the precise argument.

I do not expect $\phi$ to be weak homotopy equivalence in general, but I would like to know what can be said about it in this generality. For example, is $\phi$ injective on $\pi_0$? is it a cofibration?

It all seems very standard and well known, but I find it difficult to extract the answers from the standard sources like Goerss-Jardine for example.

• I am inclined to agree that you have a weak equivalence if $Y$ is a Kan complex. You need to use the fact that the standard model structure is compatible with exponential objects. Commented Mar 30, 2014 at 16:43
• Schwede theorem A.3.2 asserts your suspicion, though he doesn't really give either references or proofs. Commented Mar 22, 2016 at 8:35

For brevity I will write $[-, -]$ for exponential objects.
First things first: the comparison map $\newcommand{\rn}[1]{\left|{#1}\right|}$ $$\phi : \rn{[X, Y]} \to [\rn{X}, \rn{Y}]$$ is more easily described as the transpose of the map $$\rn{[X, Y]} \times \rn{X} \cong \rn{[X, Y] \times X} \to \rn{Y}$$ obtained by applying geometric realisation to the evaluation morphism $[X, Y] \times X \to Y$. (In particular, we need the fact that $\rn{-}$ preserves binary products.)
As you suspect, $\phi : \rn{[X, Y]} \to [\rn{X}, \rn{Y}]$ can fail to be a weak homotopy equivalence. For example, we may take $X$ to be $\{ 0, 1, 2, \ldots \}$ considered as a discrete simplicial set and $Y$ to be the simplicial set corresponding to the following directed graph (not category!): $$0 \to 1 \to 2 \to 3 \to \cdots$$ Clearly, $Y$ is connected. On the other hand, $[X, Y]$ is not: there is no path from the vertex $(0, 0, 0, \ldots)$ to $(0, 1, 2, \ldots)$. Thus $\phi : \rn{[X, Y]} \to [\rn{X}, \rn{Y}]$ even fails to be injective in $\pi_0$.
Also, as you suspect, $\phi : \rn{[X, Y]} \to [\rn{X}, \rn{Y}]$ is a weak homotopy equivalence if $Y$ is a Kan complex. It suffices to verify that $S \phi : S \rn{[X, Y]} \to S [\rn{X}, \rn{Y}]$ is a weak homotopy equivalence. There is a natural isomorphism $S [\rn{X}, T] \to [X, S T]$ and you can check that the composite $$[X, Y] \to S \rn{[X, Y]} \to S [\rn{X}, \rn{Y}] \to [X, S \rn{Y}]$$ is induced by the unit $Y \to S \rn{Y}$. But $[X, -]$ preserves simplicial homotopy equivalences, and $[X, Y] \to S \rn{[X, Y]}$ is a weak homotopy equivalence, so if $Y \to S \rn{Y}$ is a simplicial homotopy equivalence, then $S \phi : S \rn{[X, Y]} \to S [\rn{X}, \rn{Y}]$ is a weak homotopy equivalence as desired; in particular, this is true when $Y$ is a Kan complex.