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9.1.10 Filtered Colimits of $\infty $-Categories

We now exploit the results of §9.1.6 and §9.1.8 to study filtered colimits in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Proposition 9.1.10.1. Let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a filtered diagram of $\infty $-categories. Assume that, for every object $J \in \operatorname{\mathcal{J}}$, the $\infty $-category $\mathscr {F}(J)$ is locally $n$-truncated, for some fixed integer $n$. Then the colimit $\varinjlim (\mathscr {F})$ is also locally $n$-truncated.

Proof. Using Theorem 9.1.8.1, we can choose a directed partially ordered set $(A, \leq )$ and a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{J}}$. Using Corollary 7.2.2.11, we can replace $\operatorname{\mathcal{J}}$ by $\operatorname{N}_{\bullet }(A)$ and thereby reduce to the case where $\operatorname{\mathcal{J}}$ is the nerve of a (directed) partially ordered set. Using Corollary 5.6.5.18, we can further reduce to the case where $\mathscr {F}$ is obtained from a strictly commutative diagram

\[ \mathscr {F}_0: (A, \leq ) \rightarrow \operatorname{QCat}\subset \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{E}}_{\alpha }. \]

It follows from Corollary 9.1.6.3 that we can identify the colimit $\varinjlim (\mathscr {F})$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$) with the colimit $\varinjlim (\mathscr {F}_0)$ (formed in the ordinary category of simplicial sets). The desired result now follows from Remark 4.8.2.4. $\square$

Remark 9.1.10.2. In the formulation of Proposition 9.1.10.1, we have implicitly assumed that the filtered $\infty $-category $\operatorname{\mathcal{J}}$ is small (so that the colimit $\varinjlim (\mathscr {F} )$ is well-defined). More generally, suppose that $\lambda $ is an uncountable regular cardinal and $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ is a $\lambda $-small filtered diagram. If each of the $\infty $-categories $\mathscr {F}(J)$ is locally $n$-truncated (for some fixed integer $n$), then the colimit $\varinjlim (\mathscr {F}) \in \operatorname{\mathcal{QC}}_{< \lambda }$ is also locally $n$-truncated. Similar remarks apply to the other results proved in this section.

Variant 9.1.10.3. Let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{S}}$ be a filtered diagram of spaces. Assume that, for every object $C \in \operatorname{\mathcal{C}}$, the Kan complex $\mathscr {F}(C)$ is $n$-truncated for some fixed integer $n$. Then the colimit $\varinjlim (\mathscr {F})$ is also $n$-truncated.

The proof of Proposition 9.1.10.1 can be adapted to prove many similar results.

Proposition 9.1.10.4. Let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a filtered diagram of $\infty $-categories. Suppose that each of the $\infty $-categories $\mathscr {F}(J)$ is idempotent complete. Then the colimit $\varinjlim (\mathscr {F})$ is also idempotent complete.

Proof. As in the proof of Proposition 9.1.10.1, we can assume that $\operatorname{\mathcal{J}}= \operatorname{N}_{\bullet }(A)$ is the nerve of a directed partially ordered set $(A, \leq )$ and that $\mathscr {F}$ is obtained from a strictly commutative diagram $\mathscr {F}_0: (A, \leq ) \rightarrow \operatorname{QCat}\subset \operatorname{Set_{\Delta }}$. Using Corollary 9.1.6.3 , we are reduced to proving that the colimit $\varinjlim (\mathscr {F}_0)$ (formed in the category of simplicial sets) is idempotent complete, which follows from Corollary 8.5.9.10. $\square$

Proposition 9.1.10.5. Let $\kappa $ be a regular cardinal, let $K$ be a $\kappa $-small simplicial set, and let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a $\kappa $-filtered diagram of $\infty $-categories. Assume that, for every morphism $u: J \rightarrow J'$ of $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}(u): \mathscr {F}(J) \rightarrow \mathscr {F}(J')$ preserves $K$-indexed colimits. Then, for every object $J \in \operatorname{\mathcal{J}}$, the natural map $\mathscr {F}(J) \rightarrow \varinjlim (\mathscr {F})$ preserves $K$-indexed colimits.

The proof of Proposition 9.1.10.5 will require some preliminaries. We begin with some simple observations.

Lemma 9.1.10.6. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be the colimit of a diagram of simplicial sets $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ indexed by a $\kappa $-directed partially ordered set $(A, \leq )$. Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is obtained as the colimit of a compatible system of morphisms $F_{\alpha }: K \rightarrow \operatorname{\mathcal{C}}_{\alpha }$. If $K$ is $\kappa $-small, then the canonical map $\theta _{\bullet }: \varinjlim _{\alpha } ( \operatorname{\mathcal{C}}_{\alpha } )_{ F_{\alpha } / } \rightarrow \operatorname{\mathcal{C}}_{F/}$ is an isomorphism of simplicial sets.

Proof. Fix an integer $n \geq 0$; we wish to show that $\theta _{\bullet }$ is bijective on simplices of dimension $n$. Unwinding the definitions, we see that $\theta _{n}$ is obtained from a commutative diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ \varinjlim _{\alpha } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K \star \Delta ^ n, \operatorname{\mathcal{C}}_{\alpha } ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K \star \Delta ^ n, \operatorname{\mathcal{C}}) \ar [d] \\ \varinjlim _{\alpha } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, \operatorname{\mathcal{C}}_{\alpha } ) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{\mathcal{C}}) } \]

by passing to fibers in the vertical direction. It will therefore suffice to show that the horizontal maps are bijective. Since $(A, \leq )$ is $\kappa $-directed, this follows from the fact that $K$ and $K \star \Delta ^ n$ are $\kappa $-small simplicial sets (see Variant 9.2.2.10). $\square$

Lemma 9.1.10.7. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be the colimit of a diagram of simplicial sets $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ indexed by a $\kappa $-directed partially ordered set $(A, \leq )$. Let $K$ be a $\kappa $-small simplicial set and let $\overline{F}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is obtained as the colimit of a compatible system of morphisms $\overline{F}_{\alpha }: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\alpha }$. Assume that each $\operatorname{\mathcal{C}}_{\alpha }$ is an $\infty $-category and that each $\overline{F}_{\alpha }$ is a colimit diagram in $\operatorname{\mathcal{C}}_{\alpha }$. Then $\operatorname{\mathcal{C}}$ is an $\infty $-category and $\overline{F}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Proof. The first assertion follows from Remark 1.4.0.9. To prove the second, set $F = \overline{F}|_{K}$ and $F_{\alpha } = ( \overline{F}_{\alpha } )|_{K}$ for each $\alpha \in A$. By virtue of Proposition 7.1.3.12, it will suffice to show that the restriction map $\theta : \operatorname{\mathcal{C}}_{ \overline{F} / } \rightarrow \operatorname{\mathcal{C}}_{F/}$ is a trivial Kan fibration. Using Lemma 9.1.10.6, we can realize $\theta $ as a filtered colimit of restriction maps $\theta _{\alpha }: (\operatorname{\mathcal{C}}_{\alpha })_{ \overline{F}_{\alpha } / } \rightarrow (\operatorname{\mathcal{C}}_{\alpha })_{ F_{\alpha } / }$. Our hypothesis guarantees that each $\theta _{\alpha }$ is a trivial Kan fibration (Proposition 7.1.3.12), so that $\theta $ is also a trivial Kan fibration (Remark 1.5.5.3). $\square$

Proof of Proposition 9.1.10.5. Let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a $\kappa $-filtered diagram of $\infty $-categories, let $K$ be a $\kappa $-small simplicial set, and suppose that each of the transition functors $\mathscr {F}(J) \rightarrow \mathscr {F}(J')$ preserves $K$-indexed colimits. As in the proof of Proposition 9.1.10.1, we can reduce to the case where $\operatorname{\mathcal{J}}= \operatorname{N}_{\bullet }(A)$ is the nerve of a $\kappa $-directed partially ordered set $(A, \leq )$ and $\mathscr {F}$ is obtained from a strictly commutative diagram of simplicial sets

\[ (A, \leq ) \rightarrow \operatorname{QCat}\subset \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{C}}_{\alpha }. \]

In this case, we are reduced to showing that each of the functors $\operatorname{\mathcal{C}}_{\beta } \rightarrow \varinjlim _{\alpha \in A} \operatorname{\mathcal{C}}_{\alpha }$ preserves $K$-indexed colimits (where the colimit is formed in the ordinary category of simplicial sets). Replacing $A$ by the subset $A_{\geq \beta } = \{ \alpha \in A: \alpha \geq \beta \} $, we can reduce to the case where $\beta $ is the least element of $A$. In this case, the result follows from Lemma 9.1.10.7. $\square$

Proposition 9.1.10.8. Let $\kappa $ be a regular cardinal, let $K$ be a $\kappa $-small simplicial set, and let $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}$ be a $\kappa $-filtered diagram of $\infty $-categories satisfying the following conditions:

$(1)$

For each $J \in \operatorname{\mathcal{J}}$, the $\infty $-category $\mathscr {F}(J)$ admits $K$-indexed colimits.

$(2)$

For each morphism $u: J \rightarrow J'$ in $\operatorname{\mathcal{J}}$, the functor $\mathscr {F}(u): \mathscr {F}(J) \rightarrow \mathscr {F}(J')$ preserves $K$-indexed colimits.

Then the colimit $\varinjlim (\mathscr {F})$ admits $K$-indexed colimits. Moreover, a functor of $\infty $-categories $T: \varinjlim (\mathscr {F}) \rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed colimits if and only if, for each $J \in \operatorname{\mathcal{J}}$, the composite functor $T_{J}: \mathscr {F}(J) \rightarrow \varinjlim (\mathscr {F}) \rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed colimits.

Proof. As in the proof of Proposition 9.1.10.1, we can reduce to the case where $\operatorname{\mathcal{J}}= \operatorname{N}_{\bullet }(A)$ is the nerve of a $\kappa $-directed partially ordered set $(A, \leq )$ and $\mathscr {F}$ is obtained from a strictly commutative diagram of simplicial sets

\[ (A, \leq ) \rightarrow \operatorname{QCat}\subset \operatorname{Set_{\Delta }}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{C}}_{\alpha }. \]

Set $\operatorname{\mathcal{C}}= \varinjlim _{\alpha } \operatorname{\mathcal{C}}_{\alpha }$, where the colimit is formed in the category of simplicial sets. Since $(A, \leq )$ is $\kappa $-directed and $K$ is $\kappa $-small, every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ can be lifted to a diagram $f_{\alpha }: K \rightarrow \operatorname{\mathcal{C}}_{\alpha }$ for some $\alpha \in A$ (see Variant 9.2.2.10). Using assumption $(1)$, we see that $f_{\alpha }$ can be extended to a colimit diagram $\overline{f}_{\alpha }: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\alpha }$. Using assumption $(2)$ and Proposition 9.1.10.5, we see that the image of $\overline{f}_{\alpha }$ in $\operatorname{\mathcal{C}}$ is a colimit diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ extending $f$. This proves the first assertion. Moreover, it shows that every colimit diagram in $\operatorname{\mathcal{C}}$ can be obtained (up to isomorphism) from a colimit diagram in some $\operatorname{\mathcal{C}}_{\alpha }$. Consequently, if $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories having the property that $T|_{\operatorname{\mathcal{C}}_{\alpha }}$ preserves $K$-indexed colimits for each $\alpha \in A$, then the functor $T$ itself preserves $K$-indexed colimits. The converse follows from Proposition 9.1.10.5. $\square$

Corollary 9.1.10.9. Let $\kappa $ be a small regular cardinal, let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets, and let $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}$ be the subcategory of $\operatorname{\mathcal{QC}}$ whose objects are $\mathbb {K}$-cocomplete $\infty $-categories and whose morphisms are $\mathbb {K}$-cocontinuous functors (see Notation 8.7.3.7). Then:

$(1)$

The $\infty $-category $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}$ admits small $\kappa $-filtered colimits.

$(2)$

The inclusion functor $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}} \hookrightarrow \operatorname{\mathcal{QC}}$ is $\kappa $-finitary: that is, it preserves small $\kappa $-filtered colimits.

Proof. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}$, where $\operatorname{\mathcal{J}}$ is a small $\kappa $-filtered $\infty $-category. It follows from Corollary 7.4.5.3 that $\mathscr {F}$ can be extended to a colimit diagram $\overline{\mathscr {F}}: \operatorname{\mathcal{J}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$. Using Propositions 9.1.10.5 and 9.1.10.8, we see that any such extension factors through the subcategory $\operatorname{\mathcal{QC}}^{ \mathbb {K}-\mathrm{cocont}} \subseteq \operatorname{\mathcal{QC}}$, where it is also a colimit diagram. $\square$

In the situation of Corollary 9.1.10.9, it is not necessary to work with small $\infty $-categories:

Variant 9.1.10.10. Let $\kappa $ be a regular cardinal and let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets. For every uncountable regular cardinal $\lambda \geq \kappa $, the $\infty $-category $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda }$ of Notation 8.7.3.7 admits $\lambda $-small $\kappa $-filtered colimits, which are preserved by the inclusion functor $\operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda }$.

Corollary 9.1.10.11. Let $\kappa < \lambda $ be regular cardinals, where $\lambda $ has exponential cofinality $\geq \kappa $. Let $\mathbb {K}$ be a collection of $\kappa $-small simplicial sets and let $T: \operatorname{\mathcal{QC}}_{< \lambda } \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ be a $\mathbb {K}$-cocompletion functor (see Definition 8.7.3.3 and Proposition 8.7.3.5). Then $T$ is $(\kappa , \lambda )$-finitary: that is, it preserves $\lambda $-small $\kappa $-filtered colimits.

Proof. Let $\iota : \operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda } \hookrightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ be the inclusion functor. By virtue of Proposition 8.7.3.9, the functor $T$ factors as a composition

\[ \operatorname{\mathcal{QC}}_{< \lambda } \xrightarrow {T_0} \operatorname{\mathcal{QC}}^{\mathbb {K}-\mathrm{cocont}}_{< \lambda } \xrightarrow {\iota } \operatorname{\mathcal{QC}}_{< \lambda }, \]

where $T_0$ is left adjoint to $\iota $ and therefore preserves arbitrary colimits (Corollary 7.1.4.22). It will therefore suffice to show that the functor $\iota $ is $(\kappa , \lambda )$-finitary, which follows from Variant 9.1.10.10. $\square$