natural isomorphisms

Question

In the snippet below (taken from Hovey's MC book), I would like to understand in some detail the lemma 6.1.2 (what it says) and the proof of it. Its too sketchy for me. I do not even know what is Map$_{*\ell}(A,Y)$ there.

Answer 1

A bit on the definitions/background facts

As has been pointed out in the comments, $\newcommand\Map{\operatorname{Map}}\Map_{*\ell}$ is defined at the beginning of section 5.7.

You ask in the comments about what $\Map_{*\ell}$ is adjoint to, and the answer is that we have the following (Quillen) adjunction: $\Map_{*\ell}(A,-) : \newcommand\C{\mathcal{C}}\C\to\newcommand\sSet{\mathbf{sSet}}\sSet_*$ is right adjoint to $A\wedge - : \sSet_*\to\C$.

On the homotopical level, we have $R\Map_{*\ell}\cong R\Map_{*r}$, and the previous adjunction combines with the adjunction where $\newcommand\Hom{\operatorname{Hom}}\Hom(-,Y)$ is left adjoint to $\Map_{*r}(-,Y)$ adjunction to give an adjunction of 2-variables on the homotopical level (I won't need the adjunction of 2-variables part though).

Now as to how you should think of $\Map_{*\ell}(A,Y)$, that's another question entirely. I recommend the following: $\Map_{*\ell}(A,Y)$ is the pointed simplicial set $(\Map_\ell(A,Y),0)$.

The proof, using these facts

Then we have the following chain of natural isomorphisms $$ \newcommand\Ho{\operatorname{Ho}} \Ho\C(\Sigma^tA,Y)\simeq \Ho\C(A\wedge^L S^t,Y) \simeq \Ho\sSet^*(S^t,R\Map_{*\ell}(A,Y)). $$ The first natural isomorphism is associativity of $\wedge^L S^1$, since the $t$th smash power of $S^1$ is isomorphic to $S^t$ in the homotopy category. The next natural isomorphism is the derived adjunction between $A\wedge -$ and $\Map_{*\ell}(A,-)$. We also have the dual series of natural isomorphisms: $$ \newcommand\Ho{\operatorname{Ho}} \Ho\C(A,\Omega^tY)\simeq \Ho\C(A, R\Hom(S^t, Y)) \simeq \Ho\sSet^*(S^t,R\Map_{*r}(A,Y)), $$

Finally, as mentioned above, $R\Map_{*r}\simeq R\Map_{*\ell}$, so we can join these chains together to get the claimed series of isomorphisms, noting that for $M$ fibrant, we have $\pi_tM\simeq\Ho\sSet^*(S^t,M)$, and it turns out $\Map_{*\ell}(A,Y)$ and $\Map_{*r}(A,Y)$ are fibrant when $A$ is cofibrant and $Y$ is fibrant.

End note

For more details on $\Map_{*\ell}$, $\Map_{*r}$, $\Hom$, and $\wedge$, I recommend reading chapter 5 and make sure you understand the "nerve/realization" adjoint construction, which appears to be in chapter 3, as well as how to apply it when we are working with categories enriched over $\mathbf{Set}_*$ (pointed categories).