ScalingStacks

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

$$U\times Y\coprod_{U\times X}V\times X\rightarrow V\times Y$$

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. ∎