ScalingStacks

Proof. The proposition is just a statement of the theory of localization with respect to a given map $f$, applied to the category of simplicial spaces. Although localization is now considered a standard technique, it seems that no treatment at the level of generality which we require has yet appeared in print. Goerss and Jardine [GJ, Ch. 9, Thm. 2.3] give a complete proof for localization of simplicial sets with respect to a map; the generalization to simplicial spaces is relatively straightforward. A complete proof is given by Hirschhorn [Hir].

We give a brief sketch of the proof here. Since the desired classes of cofibrations and $f$-local weak equivalences have been characterized, the class of $f$-local fibrations must be determined by these choices. To construct the localization model category structure, we must find a cofibration $j\colon A\rightarrow B$ which is also an $f$-local weak equivalence with the property that a map is an $f$-local fibration if and only if it has the right lifting property with respect to $j$. The proof of the model category structure follows using the “small object argument” to prove the factorization axiom. (That such a small object argument works here makes use of the fact that simplicial spaces is a left proper model category.)

It is still necessary to choose a $j$. Given an uncountable cardinal $\gamma$, take $j=\coprod_{\alpha}i_{\alpha}$, where $i_{\alpha}\colon A_{\alpha}\rightarrow B_{\alpha}$ ranges over isomorphism classes of maps which are cofibrations, $f$-local weak equivalences, and such that $B_{\alpha}$ has fewer than $\gamma$ simplices in each degree. That a sufficiently large $\gamma$ produces a map $j$ with the desired properties follows from the “Bousfield-Smith cardinality argument”. ∎

Proof. The proof proceeds in several stages. Suppose that $W$ is an $f$-local object. Then it follows by hypothesis that $W^{(F(1))^{k}}$ is $f$-local for all $k$, where $(F(1))^{k}$ denotes the $k$-fold product. Next one observes by elementary computation that $F(k)$ is a retract of $(F(1))^{k}$; thus it follows that $W^{F(k)}$ is a retract of $W^{(F(1))^{k}}$ and hence is also $f$-local.

Since the $f$-local model category is a simplicial model category, we see that for any $K\in{\operatorname{\mathcal{S}}}$ we have that $(W^{F(n)})^{K}=W^{F(n)\times K}$ is $f$-local (recall that we regard $K$ as a constant simplicial space). Since any simplicial space $X$ is a homotopy colimit (in the Reedy model category structure) of a diagram of simplicial spaces of the form $F(k)\times K$ where $K$ is a space, it follows that $W^{X}$ is a homotopy limit (again in the Reedy model category structure, assuming $W$ is Reedy fibrant) of a diagram of simplicial spaces of the form $W^{F(k)\times K}$. Since a homotopy limit of $f$-local objects is $f$-local, we see that $W^{X}$ is $f$-local for arbitrary $X$.

Now, to show that the $f$-local model category is compatible with the enrichment, it suffices to show that for a cofibration $i\colon X\rightarrow Y$ and an $f$-local trivial cofibration $j\colon U\rightarrow V$, the induced map

is an $f$-local equivalence. Equivalently, we must show that for every $f$-local object $W$ the square

is a homotopy pull-back of spaces. But this diagram is isomorphic to

and since $W^{X}$ and $W^{Y}$ are $f$-local, the columns are weak equivalences, whence the square is in fact a homotopy pull-back. ∎