Kerodon

Proof. By virtue of Variant 9.1.6.8, it will suffice to show that, for every small filtered category $\operatorname{\mathcal{C}}$, the functor $\iota $ preserves $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits. Fix 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 9.1.7.1. $\square$

Proof. Using Theorem 9.1.6.2, we can choose a directed partially ordered set $(A, \leq )$ and a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$. Using Corollary 7.2.2.3 we can replace $\operatorname{\mathcal{C}}$ by $\operatorname{N}_{\bullet }(A)$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is (the nerve of) a directed partially ordered set. Replacing $\mathscr {F}$ by an isomorphic functor if necessary, we can assume that it obtained from an $A$-indexed diagram in the ordinary category $\operatorname{QCat}$ (Corollary 5.6.5.18). In this case, the colimit $\varinjlim ( \mathscr {F} )$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ can be identified with its colimit in the ordinary category $\operatorname{QCat}\subset \operatorname{Set_{\Delta }}$ (Corollary 9.1.7.3), so the desired result follows from Corollary 8.5.8.10. $\square$

Proof. Combine Proposition 9.1.7.1 with Corollary 7.5.8.13. $\square$

Proof. This is a special case of Corollary 9.1.7.6, since every small simplicial set $K$ can be realized as a (small) $\kappa $-filtered colimit of $\kappa $-small simplicial sets. For example, we can write $K$ as the union of all $\kappa $-small simplicial subsets of itself. $\square$

Proof of Proposition 9.1.7.1 from Proposition 9.1.7.9. 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 9.1.7.9, 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 9.1.7.12. Using Remark 9.1.7.13, 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 9.1.7.9. Apply Proposition 9.1.7.14 in the special case $\mathscr {F}_{0} = \underline{\emptyset }$ (see Example 9.1.7.11). $\square$

Proof. It follows from Remark 9.1.7.13 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 9.1.7.12, 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 9.1.7.14 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 $S'$. $\square$