Kerodon

Proof. Let $\operatorname{\mathcal{C}}$ be a small filtered category and suppose we are given a colimit diagram $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{QCat}$ in the ordinary category $\operatorname{QCat}$. We wish to show that the induced map $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} ): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$. By virtue of Corollary 7.5.8.9, this is equivalent to the requirement that $\overline{\mathscr {F}}$ is a categorical colimit diagram, which follows from Proposition 9.1.6.1. $\square$

Proof. Combine Proposition 9.1.6.1 with Corollary 7.5.8.13. $\square$

Proof of Proposition 9.1.6.1 from Proposition 9.1.6.7. Let $\operatorname{\mathcal{C}}$ be a small filtered category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a colimit diagram in the category of simplicial sets; we wish to show that $\overline{\mathscr {F}}$ is a categorical colimit diagram. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$. Using Proposition 9.1.6.7, we can choose a levelwise categorical equivalence $\alpha : \mathscr {E} \rightarrow \mathscr {F}$, where $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant. Let $\overline{\mathscr {E}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a colimit diagram extending $\mathscr {E}$, so that $\alpha $ extends uniquely to a natural transformation $\overline{\alpha }: \overline{\mathscr {E}} \rightarrow \overline{\mathscr {F}}$. Applying Corollary 4.5.7.2, we deduce that $\overline{\alpha }$ is also a levelwise categorical equivalence. Consequently, to show that $\overline{\mathscr {F}}$ is a categorical colimit diagram, it will suffice to show that $\overline{\mathscr {E}}$ is a categorical colimit diagram (Corollary 7.5.8.6). This follows from Corollary 7.5.8.7, since $\mathscr {E}$ is projectively cofibrant. $\square$

Proof. We will construct $\mathscr {F}$ as the colimit of a diagram of projective cofibrations

in the category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})_{ / \mathscr {G} }$. Fix $n \geq 0$, and suppose that we have constructed an object $\mathscr {F}_{n} \in \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})_{ / \mathscr {G} }$, which we identify with a natural transformation $\alpha _{n}: \mathscr {F}_{n} \rightarrow \mathscr {G}$. For each object $C \in \operatorname{\mathcal{C}}$, Exercise 3.1.7.11 guarantees that $\alpha _{n,C}$ factors as a composition

where $\alpha '_{n,C}$ is a monomorphism and $\alpha ''_{n,C}$ is a trivial Kan fibration (beware that $\mathscr {F}'_{n}(C)$ does not depend functorially on $C$). Form a pushout diagram

in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})_{ / \mathscr {G } }$, where the upper horizontal map is the coproduct of the projective cofibrations described in Example 9.1.6.10. Using Remark 9.1.6.11, we see that each of the maps

is a projective cofibration. Setting $\mathscr {F} = \varinjlim _{n} \mathscr {F}_{n}$, we obtain a factorization of $\alpha _{0}$ as a composition $\mathscr {F}_{0} \xrightarrow { \beta } \mathscr {F} \xrightarrow { \alpha } \mathscr {G}$, where $\beta $ is a projective cofibration. We complete the proof by observing that for each object $C \in \operatorname{\mathcal{C}}$, the morphism $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a trivial Kan fibration, since it can be written as a filtered colimit (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$) of the trivial cofibrations $\alpha ''_{n,C}: \mathscr {F}'_{n}(C) \rightarrow \mathscr {G}(C)$ (see Remark 1.5.5.3). $\square$

Proof of Proposition 9.1.6.7. Apply Proposition 9.1.6.12 in the special case $\mathscr {F}_{0} = \underline{\emptyset }$ (see Example 9.1.6.9). $\square$

Proof. It follows from Remark 9.1.6.11 that $S$ is weakly saturated. Let $S'$ be the smallest weakly saturated collection of morphisms of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ which contains each $\iota _{n,C}$. Using Example 9.1.6.10, we see that $S'$ is contained in $S$. For every monomorphism of simplicial sets $A \hookrightarrow B$ and every object $C \in \operatorname{\mathcal{C}}$, Proposition 1.5.5.14 guarantees that the projective cofibration $\underline{A} \times h^{C} \hookrightarrow \underline{B} \times h^{C}$ is contained in $S'$. It follows from the proof of Proposition 9.1.6.12 that every morphism $\alpha _0: \mathscr {F}_0 \rightarrow \mathscr {G}$ in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ factors as a composition $\mathscr {F}_0 \xrightarrow {\beta } \mathscr {F} \xrightarrow {\alpha } \mathscr {G}$, where $\beta $ belongs to $S'$ and $\alpha $ is a trivial Kan fibration. If $\alpha _0$ is projective cofibration, then the lifting problem

admits a solution. It follows that $\alpha _0$ is a retract of the morphism $\beta $, and therefore belongs to $S'$. $\square$