Kerodon

Proof. Let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the projection map of Definition 5.3.3.1 and let $W'$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. Choose a functor of $\infty $-categories $T: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ with respect to $W'$. Let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the taut scaffold of Construction 5.3.4.11. Then $\lambda _{t}$ is a categorical equivalence of simplicial sets (Corollary 5.3.5.9). Moreover, a morphism of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ belongs to $W'$ if and only if it is isomorphic (as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$) to an element of $\lambda _{t}(W)$ (see Corollary 5.3.3.16). It follows that the composite map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \xrightarrow { \lambda _{t} } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \xrightarrow {T} \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ with respect to $W$. We conclude by observing that $\operatorname{\mathcal{D}}$ is a colimit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ (Corollary 7.4.5.5). $\square$

Proof. The equivalence of $(1)$ and $(2)$ follows by combining Example 7.5.2.11 with Corollary 7.5.5.15. The equivalence with $(3)$ follows by combining the same results with Proposition 6.3.1.13 and Example 7.5.2.8. $\square$

Proof. By virtue of Proposition 7.5.8.4 (and Proposition 4.5.3.8), it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{D}}$, any two of the following conditions imply the third:

$(1_{\operatorname{\mathcal{D}}} )$

The functor

\[ (\operatorname{\mathcal{C}}^{\triangleright })^{\operatorname{op}} \rightarrow \operatorname{QCat}\quad \quad C \mapsto \operatorname{Fun}( \overline{\mathscr {F}}(C), \operatorname{\mathcal{D}}) \]

is a categorical limit diagram.

$(2_{\operatorname{\mathcal{D}}})$

The functor

\[ (\operatorname{\mathcal{C}}^{\triangleright })^{\operatorname{op}} \rightarrow \operatorname{QCat}\quad \quad C \mapsto \operatorname{Fun}( \overline{\mathscr {G}}(C), \operatorname{\mathcal{D}}) \]

is a categorical limit diagram.

$(3_{\operatorname{\mathcal{D}}})$

The natural transformation $\alpha $ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \overline{\mathscr {G}}( {\bf 1} ), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \overline{\mathscr {F}}( {\bf 1} ), \operatorname{\mathcal{D}})$.

This follows from Remark 7.5.5.6. $\square$

Proof. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and define $\overline{\mathscr {G}}: (\operatorname{\mathcal{C}}^{\triangleright })^{\operatorname{op}} \rightarrow \operatorname{QCat}$ by the formula $\overline{\mathscr {G}}(C) = \operatorname{Fun}( \overline{\mathscr {F}}(C), \operatorname{\mathcal{D}})$. By virtue of Proposition 7.5.8.4, it will suffice to show that the diagram of Kan complexes $\overline{\mathscr {G}}^{\simeq }$ is a homotopy limit diagram. Setting $\mathscr {G} = \overline{\mathscr {G}}|_{ \operatorname{\mathcal{C}}^{\operatorname{op}} }$, our assumption that $\mathscr {F}$ is projectively cofibrant guarantees that the diagram $\mathscr {G}$ is isofibrant (Remark 7.5.6.6). It follows that the diagram of Kan complexes $\mathscr {G}^{\simeq }$ is also isofibrant, and that $\overline{\mathscr {G}}^{\simeq }$ is a limit diagram (Corollary 4.5.6.21). The desired result now follows from Example 7.5.4.2. $\square$

Proof. This is a restatement of Corollary 7.5.8.7. $\square$

Proof. Combine Proposition 7.5.8.4 with Corollary 7.5.4.6 (applied to the simplicial category $\operatorname{QCat}^{\operatorname{op}}$). $\square$

Proof. Combine Corollary 7.5.8.9, Corollary 7.5.7.7, and Remark 7.4.4.4. $\square$

Proof. Combine Proposition 7.5.8.4 with Corollary 7.5.5.15. $\square$

Proof. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$, let $U: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the projection map, and let $W$ be the collection of all horizontal edges of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$. The diagram $\overline{\mathscr {F}}$ then determines a morphism of simplicial sets $T: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow K$ which exhibits $K$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ with respect to $W$. It follows from assumption $(\ast )$ that for each object $C \in \operatorname{\mathcal{C}}$, the composite map

admits a colimit in $\operatorname{\mathcal{D}}$. Applying Corollary 7.3.5.3, we conclude that there is a functor $Q: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta : T \circ q \rightarrow Q \circ U$ which exhibits $Q$ as a left Kan extension of $T \circ q$ along $U$. We will complete the proof by showing that $Q$ satisfies conditions $(1)$, $(2)$, and $(3)$ of Proposition 7.5.8.12. Condition $(1)$ follows immediately from Remark 7.3.5.4.

We now prove $(2)$. Assume first that $X \in \operatorname{\mathcal{D}}$ is a colimit of the diagram $Q$. For every simplicial set $S$, we let $\underline{X}_{S}$ denote the image of $X$ in the $\infty $-category $\operatorname{Fun}(S, \operatorname{\mathcal{D}})$. Choose a natural transformation $\alpha : Q \rightarrow \underline{X}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}$ which exhibits $X \in \operatorname{\mathcal{D}}$ as a colimit of the diagram $Q$, let $\widetilde{\alpha }: Q \circ U \rightarrow \underline{X}_{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})}$ denote the image of $\alpha $ in $\operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}), \operatorname{\mathcal{D}})$, and let $\widetilde{\gamma }: q \circ T \rightarrow \underline{X}_{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})} = \underline{X}_{K} \circ T$ be a composition of $\beta $ with $\widetilde{\alpha }$ in $\operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}), \operatorname{\mathcal{D}})$. Since precomposition with $T$ induces a fully faithful functor $\operatorname{Fun}(K, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}), \operatorname{\mathcal{D}})$, we may assume without loss of generality that $\widetilde{\gamma }$ is the image of a natural transformation $\gamma : q \rightarrow \underline{X}_{K}$. Note that $\widetilde{\gamma }$ exhibits $X$ as a colimit of the diagram $q \circ T$ (Corollary 7.3.8.20). Since $T$ is right cofinal (Proposition 7.2.1.10), it follows that $\gamma $ exhibits $X$ as a colimit of the diagram $q$ (Corollary 7.2.2.7).

To prove the reverse implication, it will suffice to show that if the diagram $q: K \rightarrow \operatorname{\mathcal{D}}$ admits a colimit, then $Q$ also admits a colimit. Since $T$ is right cofinal, the diagram $q \circ T$ also admits a colimit in $\operatorname{\mathcal{D}}$ (Corollary 7.2.2.11), so the desired result is immediate from Corollary 7.3.8.20.

We now prove $(3)$. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories which preserves the colimit of the diagram $q_{C}$, for each object $C \in \operatorname{\mathcal{C}}$. Let $\alpha : Q \rightarrow \underline{X}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}$ and $\gamma : q \rightarrow \overline{X}_{K}$ be defined as above; we wish to show that $G(\alpha )$ exhibits $G(X)$ as a colimit of the diagram $G \circ Q$ if and only if $G(\gamma )$ exhibits $G(X)$ as a colimit of the diagram $G \circ q$. Using Corollary 7.2.2.7, we see that latter condition is equivalent to the requirement that $G( \widetilde{\gamma } )$ exhibits $G(X)$ as a colimit of the diagram $G \circ q \circ T$. By virtue of Corollary 7.3.8.20, we are reduced to showing that the natural transformation $G(\beta )$ exhibits $G \circ Q$ as a left Kan extension of $G \circ q \circ T$ along $U$. This follows from the criterion of Remark 7.3.5.4. $\square$