ScalingStacks

Given a model category structure on $s{\operatorname{\mathcal{S}}}$, we say that it is compatible with the cartesian closure if for any cofibrations $i\colon A\rightarrow B$ and $j\colon C\rightarrow D$ and any fibration $k\colon X\rightarrow Y$, either (and hence both) of the following two equivalent assertions hold:

1. (1)

   The induced map $A\times D\amalg_{A\times C}B\times C\rightarrow B\times D$ is a cofibration, and additionally is a weak equivalence if either $i$ or $j$ is.

2. (2)

   The induced map $Y^{B}\rightarrow Y^{A}\times_{X^{A}}X^{B}$ is a fibration, and additionally is a weak equivalence if either $i$ or $k$ is.

(A closed symmetric monoidal category together with a Quillen closed model category structure which satisfies the above properties is sometimes also called a “Quillen ring”.) Assuming (as will always be the case for us) that a weak equivalence or a fibration $X\rightarrow Y$ in our model category structure induces a weak equivalence or a fibration $X_{0}\rightarrow Y_{0}$ on the degree $0$ spaces, then it follows that such a model category structure makes $s{\operatorname{\mathcal{S}}}$ into a simplicial model category in the sense of [Qui67], since $\Map_{s{\operatorname{\mathcal{S}}}}(X,Y)\approx(Y^{X})_{0}$ for any simplicial spaces $X$ and $Y$.

The Reedy model category structure on $s{\operatorname{\mathcal{S}}}$ is compatible with the cartesian closure; to prove (1) in this case, it suffices to recall that cofibrations are exactly inclusions, and that weak equivalences are degree-wise.