Variant 9.1.4.4. Let $\kappa $, $\lambda $, and $\mu $ be infinite cardinals, let $K$ be a $\kappa $-small simplicial set, and let $L$ be a $\lambda $-small simplicial set. Assume that $\mu $ is uncountable of cofinality $\geq \kappa $ and exponential cofinality $\geq \lambda $, so that the $\infty $-category $\operatorname{\mathcal{S}}^{< \mu }$ admits $K$-indexed colimits (Example 7.6.6.8) and $L$-indexed limits (Example 7.6.6.4). Then $K$-indexed colimits commute with $L$-indexed limits in the $\infty $-category $\operatorname{\mathcal{S}}^{< \mu }$ if and only if the following condition is satisfied:
- $(\ast )$
Let $\operatorname{Fun}'(K, \operatorname{\mathcal{S}}^{< \mu })$ denote the full subcategory of $\operatorname{Fun}(K, \operatorname{\mathcal{S}}^{< \mu })$ spanned by those diagrams $\mathscr {F}: K \rightarrow \operatorname{\mathcal{S}}$ for which the colimit $\varinjlim (\mathscr {F}) \in \operatorname{\mathcal{S}}^{< \mu }$ is contractible. Then $\operatorname{Fun}'(K, \operatorname{\mathcal{S}}^{< \mu })$ is closed under $L$-indexed limits.
Proof.
Let $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{S}}^{< \mu }$ denote the full subcategory spanned by the contractible Kan complexes. Then $\operatorname{\mathcal{C}}$ is closed under the formation of $L$-indexed limits. By definition, $\operatorname{Fun}'( K, \operatorname{\mathcal{S}}^{< \mu } )$ is the inverse image of $\operatorname{\mathcal{C}}$ under the colimit functor $\varinjlim : \operatorname{Fun}(K, \operatorname{\mathcal{S}}^{< \mu } ) \rightarrow \operatorname{\mathcal{S}}^{< \mu }$. If $K$-indexed colimits commute with $L$-indexed limits in $\operatorname{\mathcal{S}}^{< \mu }$, then the functor $\varinjlim $ preserves $L$-indexed limits, so that $\operatorname{Fun}'( K, \operatorname{\mathcal{S}}^{< \mu } )$ is closed under the formation of $L$-indexed limits.
We now prove the converse. Assume that condition $(\ast )$ is satisfied; we wish to show that $K$-indexed colimits commute with $L$-indexed limits in the $\infty $-category $\operatorname{\mathcal{S}}^{< \mu }$. In what follows, if $Z$ is an object of $\operatorname{\mathcal{S}}^{< \mu }$, we let $\underline{Z}_{K}$ denote the constant diagram $K \rightarrow \operatorname{\mathcal{S}}^{\mu }$ taking the value $Z$, and $\underline{Z}_{L}$ for the constant diagram $L \rightarrow \operatorname{\mathcal{S}}^{< \mu }$ taking the value $Z$. Suppose we are given diagrams
\[ \mathscr {F}: L \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{S}}^{< \mu } )\quad \quad X: L \rightarrow \operatorname{\mathcal{S}}^{< \mu }, \]
having limits $\varprojlim (\mathscr {F} ) \in \operatorname{Fun}(K, \operatorname{\mathcal{S}}^{< \mu } )$ and $\varprojlim (X) \in \operatorname{\mathcal{S}}^{< \mu }$, respectovely. Write $\underline{X}_{K}$ for the $L$-indexed diagram in $\operatorname{Fun}(L, \operatorname{\mathcal{S}}^{< \mu }$ given by the construction $\ell \mapsto \underline{ X(\ell ) }$. Assume we are given a natural transformation $\gamma : \mathscr {F} \rightarrow \underline{X}_{K}$, carrying each vertex $\ell \in L$ to a natural transformation $\gamma (\ell ): \mathscr {F}(\ell ) \rightarrow \underline{ X(\ell ) }_{K}$. We wish to show that the induced map $\varprojlim (\gamma ): \varprojlim (\mathscr {F}) \rightarrow \underline{ \varprojlim (X) }_{K}$ exhibits the Kan complex $\varprojlim (X)$ as a colimit of the $K$-indexed diagram $\varprojlim (\mathscr {F})$. To prove this, we will show that $\varprojlim (\gamma )$ satisfies the criterion of Corollary 7.7.0.6. Let $f: Y \rightarrow \varprojlim (X)$ be a morphism of Kan complexes, where $Y$ is contractible, and suppose we are given a (levelwise) pullback square $\sigma :$
\[ \xymatrix { \mathscr {G} \ar [r]^{ \beta } \ar [d] & \underline{Y}_{K} \ar [d]^{ \underline{f}_{K} } \\ \varprojlim (\mathscr {F}) \ar [r]^{ \varprojlim (\gamma )} & \underline{ \varprojlim (X) }_{K} } \]
in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{S}}^{< \mu } )$; we wish to show that the diagram $\mathscr {G}$ has contractible colimit. Note that $f$ determines a natural transformation of $L$-indexed diagrams $\alpha : \underline{Y}_{L} \rightarrow X$. Since the limit functor $\varprojlim : \operatorname{Fun}(L, \operatorname{\mathcal{S}}^{< \mu } ) \rightarrow \operatorname{\mathcal{S}}^{< \mu }$ preserves pullback squares, we can identify $\sigma $ with the limit of an $L$-indexed diagram of pullback squares in $\operatorname{Fun}(K, \operatorname{\mathcal{S}}^{< \mu } )$, carrying each $\ell \in L$ to a diagram $\sigma _{\ell }:$
\[ \xymatrix { \mathscr {G}_{\ell } \ar [r]^{\beta _{\ell } } \ar [d] & \underline{Y}_{K} \ar [d]^{ \underline{ \alpha (\ell ) }_{K} } \\ \mathscr {F}(\ell ) \ar [r]^{ \gamma (\ell ) } & \underline{ X(\ell ) }_{K}. } \]
By virtue of condition $(\ast )$, it will suffice to show that each of the diagrams $\mathscr {G}_{\ell }: K \rightarrow \operatorname{\mathcal{S}}^{< \mu }$ has a contractible colimit. This follows from the criterion of Corollary 7.7.0.6, since $\gamma (\ell )$ exhibits $X(\ell )$ as a colimit of $\mathscr {F}(\ell )$.
$\square$