Inner hom and geometric realization.

Question

I would like to prove the following fact, which I learned from a previous MO question.

Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a weak equivalence $$ |\underline{\mathbf{sSET}} (S_\cdot,T_\cdot)|\simeq \underline{\mathbf{TOP}} (|S_\cdot|,|T_\cdot|) $$

Here is what I have so far: By a Quillen adjunction, $$ \mathbf{TOP} (|\underline{\mathbf{sSET}} (S_\cdot,T_\cdot)|, \underline{\mathbf{TOP}} (|S_\cdot|,|T_\cdot|)) \cong \mathbf{sSET}(\underline{\mathbf{sSET}} (S_\cdot,T_\cdot),\textrm{Sing}\ \underline{\mathbf{TOP}} (|S_\cdot|,|T_\cdot|)) $$ So we need to find a weak equivalence in the second set. Notice $$ \textrm{Sing}\ \underline{\mathbf{TOP}} (|S_\cdot|,|T_\cdot|))_k \cong \mathbf{TOP}(\Delta^k\times |S_\cdot|,|T_\cdot|) $$ And by the universal property of coends, there's a map (*) $$ |\Delta(k)|\rightarrow \Delta^k $$
($\Delta(k)$ is the simplicial set corepresenting $[k]\in\mathbb{\Delta}$), so we get a map to $\mathbf{TOP}(|\Delta(k) \times S_\cdot|,|T_\cdot|)$, which is in turn in bijection with $\mathbf{sSET}(\Delta(k) \times S_\cdot,\textrm{Sing} |T_\cdot|)$, and that is the $k^{\textrm{th}}$ space of $\underline{\mathbf{sSET}}(S_\cdot,\textrm{Sing}\ |T_\cdot|)$. So unraveling, we need to find a weak equivalence $$ \underline{\mathbf{sSET}} (S_\cdot,T_\cdot) \rightarrow \underline{\mathbf{sSET}} (S_\cdot, \textrm{Sing} |T_\cdot|) $$ We do have the unit map $T_\cdot\rightarrow \textrm{Sing} |T_\cdot|$ of the adjunction in the target, and since all simplicial sets are cofibrant, the result would follow if this unit map is a trivial fibration when $T_\cdot$ is fibrant (by compatibility of inner-hom with the model structure in $\mathbf{sSET}$). Here's where I'm stuck; it seems like I'm missing a key ingredient to finish.

(*) also here I need to show that these set maps $$ \mathbf{TOP}(\Delta^k\times |S_\cdot|,|T_\cdot|) \rightarrow \mathbf{TOP}(|\Delta(k) \times S_\cdot|,|T_\cdot|) $$ assemble to a weak equivalence of simplicial sets.

Answer 1

The unit map $T\to \mathrm{Sing}|T|$ is far from being a trivial fibration; it's actually injective. Did you mean to say it's a trivial cofibration?

It is a weak equivalence for any simplicial set $T$ (so the map is actually a trivial cofibration), and this would complete your proof. Unfortunately, this seems to take some hard work. Goerss & Jardine state this as Proposition 11.1 in chapter 1 of their book; their proof relies on much of the material in the previous 40 pages or so of the chapter.

The main idea in their proof (which goes back to Quillen) is to show that geometric realization $|-|:\mathbf{sSet}\to \mathbf{Top}$ preserves fibrations. It's clear from the definitions that $\mathrm{Sing}$ preserves fibrations, so therefore the composite functor $\mathrm{Sing}|-|$ takes Kan fibrations to Kan fibrations. In particular, it preserves the path-loop fibration, so you can reduce the problem of comparing homotopy groups $\pi_n(T,t)\to \pi_n(\mathrm{S}|T|,t)$ to the case $n=0$.

Showing that $|-|$ preserves fibrations involves a detour through the theory of minimal Kan fibrations, which is charming; but it would be nice if there was a more direct proof. I don't know one.

Answer 2

As another shameless advertisement for the forthcoming book ``More concise algebraic topology: localization, completion, and model categories'', by Kate Ponto and myself, the book will contain a proof of the model axioms for simplicial sets that avoids the theory of minimal fibrations. It is due to Pete Bousfield and myself, mainly Pete. In particular, Corollary 17.5.13 in the book is the statement that the unit map T --> S|T| is a weak equivalence for any simplicial set T, and no result about the behavior of |-| on fibrations is required in the proof. Actually though, this much is or at least should be classical. It can be deduced directly from the easily checked fact that the homotopy groups of a space X are isomorphic to the homotopy groups of the Kan complex SX, Milnor's 1957 result that the unit map is a weak equivalence when T is a Kan complex, and the two triangle identities for the (|-|,S) adjunction. The deduction does use that a map of Kan complexes induces an isomorphism of homotopy groups iff it is a homotopy equivalence, but that is also an old result. (It's in my 1967 book "Simplicial objects in algebraic topology'', but I don't remember the original source).