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”. ∎