Kerodon

Proof. We will show that $(1) \Leftrightarrow (2)$ and that $(4) \Rightarrow (2)$; the implications $(3) \Leftrightarrow (4)$ and $(2) \Rightarrow (4)$ then follow by applying the same arguments to the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. In what follows, we let $u$ and $v$ denote the cone points of the simplicial sets $K^{\triangleright }$ and $L^{\triangleleft }$, respectively.

We first show that $(1) \Leftrightarrow (2)$. Let $\operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(L^{\triangleleft }, \operatorname{\mathcal{C}})$ spanned by the limit diagrams in $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ admits $L$-indexed limits, the restriction functor

is a trivial Kan fibration, and therefore admits a section $s: \operatorname{Fun}(L, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}})$ which is an equivalence of $\infty $-categories. In particular, the $\infty $-category $\operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}})$ admits $K$-indexed colimits, so condition $(2)$ is equivalent to the requirement that the inclusion functor $\operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}(L^{\triangleleft }, \operatorname{\mathcal{C}})$ preserves $K$-indexed colimits. Using the criterion of Proposition 7.1.7.2, we see that this is equivalent to the requirement that the composite functor

preserves $K$-indexed colimits. Since this composition is isomorphic to the limit functor $\varprojlim : \operatorname{Fun}(L, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ (Remark 7.1.7.8), this is reformulation of condition $(1)$.

We now show that $(2) \Rightarrow (4)$. Suppose we are given a colimit diagram $f: K \rightarrow \operatorname{Fun}( L^{\triangleleft }, \operatorname{\mathcal{C}})$ which carries each vertex $k \in K$ to an object of $\operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}})$; we wish to show that the colimit of $f$ has the same property. Choose an extension of $f$ to a colimit diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}( L^{\triangleleft }, \operatorname{\mathcal{C}})$, which we identify with a morphism of simplicial sets $\overline{F}: K^{\triangleright } \times L^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. The restriction $F|_{K^{\triangleright } \times L }$ can be identified with a diagram $g: L \rightarrow \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$. Our assumption that $\overline{f}$ is a colimit diagram guarantees that, for each vertex $\ell \in L$, the image $g_0( \ell )$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Since the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ admits $L$-indexed limits, we can extend $g$ to a limit diagram $\overline{g}: L^{\triangleleft } \rightarrow \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$. If condition $(4)$ is satisfied, then $\overline(v) \in \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ is also a colimit diagram in $\operatorname{\mathcal{C}}$. Let us identify $\overline{g}$ with a morphism of simplicial sets $\overline{G}: K^{\triangleright } \times L^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, which we further identify with a diagram $\overline{f}': K^{\triangleright } \rightarrow \operatorname{Fun}( L^{\triangleleft }, \operatorname{\mathcal{C}})$. Since $K$-indexed colimits in $\operatorname{Fun}( L^{\triangleleft }, \operatorname{\mathcal{C}})$ are computed levelwise (Proposition 7.1.7.2), $\overline{f}'$ is a colimit diagram. Moreover, we have $\overline{f}'|_{K} = f$ by construction. We are therefore reduced to showing that $\overline{f}'(u) \in \operatorname{Fun}(L^{\triangleleft }, \operatorname{\mathcal{C}})$ is a limit diagram. This follows from our assumption that $\overline{g}$ is a limit diagram, since the evaluation functor $\operatorname{ev}_{u}: \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ preserves $L$-indexed limits. $\square$

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

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 $\sigma :$

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 $\sigma _{\ell }:$

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 $\mathscr {F}(\ell )$. $\square$

Proof. By virtue of Corollary 7.2.3.9, it will suffice to show that for every diagram $f: L^{\operatorname{op}} \rightarrow \operatorname{\mathcal{K}}$, the coslice $\infty $-category $\operatorname{\mathcal{K}}_{f/}$ is weakly contractible. Let $h^{\bullet }: \operatorname{\mathcal{K}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{S}}^{< \mu } )$ be a contravariant Yoneda embedding for $\operatorname{\mathcal{K}}$. Since the $\infty $-category $\operatorname{\mathcal{S}}^{< \mu }$ admits $L$-indexed limits, the composite diagram

admits a limit $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{S}}^{< \mu } )$. It follows from Corollary 8.4.2.8 that $\mathscr {F}$ is a covariant transport representation for the left fibration $\operatorname{\mathcal{K}}_{f/} \rightarrow \operatorname{\mathcal{K}}$. Consequently, the $\infty $-category $\operatorname{\mathcal{K}}_{f/}$ is weakly contractible if and only if the colimit $\varinjlim (\mathscr {F} )$ is contractible (Proposition 7.4.3.1). Invoking our assumption that $\operatorname{\mathcal{K}}$-indexed colimits commute with $L$-indexed limits, we are reduced to proving that for each vertex $\ell \in L$, the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{K}}}( f(\ell ), \bullet )$ has contractible colimit, which follows from Example 7.4.3.7. $\square$