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7.5.9 Application: Filtered Colimits of $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ be a small filtered category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram, and let $\operatorname{\mathcal{E}}= \varinjlim ( \mathscr {F} )$ denote the colimit of $\mathscr {F}$ in the category of simplicial sets. If each of the simplicial sets $\mathscr {F}(C)$ is an $\infty $-category, then the simplicial set $\operatorname{\mathcal{E}}$ is also an $\infty $-category (Remark 1.3.0.9). Our goal in this section is to show that, in this case, we can also regard $\operatorname{\mathcal{E}}$ as a colimit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} ): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$. This is a consequence of the following more general result:

Proposition 7.5.9.1. 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. Then $\overline{\mathscr {F}}$ is a categorical colimit diagram.

Remark 7.5.9.2. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $W$ denote the collection of horizontal edges of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$. Proposition 7.5.9.1 asserts that, if the category $\operatorname{\mathcal{C}}$ is filtered, then the comparison map $\theta : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} )$ exhibits $\varinjlim ( \mathscr {F} )$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ with respect to $W$. In particular, $\theta $ is a weak homotopy equivalence.

Before giving the proof of Proposition 7.5.9.1, let us record some of its consequences.

Corollary 7.5.9.3. Let $\operatorname{\mathcal{C}}$ be a small filtered category. Then the inclusion map

\[ \operatorname{N}_{\bullet }( \operatorname{QCat}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]

preserves $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits.

Proof. We first observe that the full subcategory $\operatorname{QCat}\subseteq \operatorname{Set_{\Delta }}$ is closed under filtered colimits (Remark 1.3.0.9), so the category $\operatorname{QCat}$ admits $\operatorname{\mathcal{C}}$-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 7.5.9.1. $\square$

Variant 7.5.9.4. Let $\operatorname{\mathcal{C}}$ be a small filtered category. Then the inclusion map

\[ \operatorname{N}_{\bullet }( \operatorname{Kan}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}} \]

preserves $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits.

Corollary 7.5.9.5. Let $\operatorname{\mathcal{C}}$ be a small filtered category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets having colimit $K = \varinjlim (\mathscr {F} )$. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits and which admits $\mathscr {F}(C)$-indexed colimits, for each $C \in \operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{D}}$ also admits $K$-indexed colimits. Moreover, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a functor which preserves both $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits and $\mathscr {F}(C)$-indexed colimits for each $C \in \operatorname{\mathcal{C}}$, then $G$ also preserves $K$-indexed colimits.

Corollary 7.5.9.6. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits finite colimits and small filtered colimits. Then $\operatorname{\mathcal{D}}$ admits all small colimits. Moreover, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a functor of $\infty $-categories which preserves finite colimits and small filtered colimits, then $G$ preserves all small colimits.

Proof. This is a special case of Corollary 7.5.9.5, since every small simplicial set $K$ can be realized as a (small) filtered colimit of finite simplicial sets. For example, we can write $K$ as the union of all finite simplicial subsets of itself. $\square$

Our proof of Proposition 7.5.9.1 will require a brief digression. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. In ยง7.5.6, we showed that there exists a projectively cofibrant diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ equipped with a levelwise weak homotopy equivalence $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ (Proposition 7.5.6.9). Using a somewhat less explicit construction, we can obtain a better approximation to $\mathscr {G}$:

Proposition 7.5.9.7. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then there exists a projectively cofibrant diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ and a levelwise trivial Kan fibration $\alpha : \mathscr {F} \rightarrow \mathscr {G}$.

Proof of Proposition 7.5.9.1 from Proposition 7.5.9.7. 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 7.5.9.7, 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. $\square$

It will be useful to formulate a slightly stronger version of Proposition 7.5.9.7. First, we need some terminology.

Definition 7.5.9.8. Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha : \mathscr {F}' \rightarrow \mathscr {F}$ be a natural transformation between diagrams $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. We say that $\alpha $ is a projective cofibration if it has the left lifting property with respect to all levelwise trivial Kan fibrations (see Remark 4.5.6.2). That is, $\alpha $ is a projective cofibration if every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}' \ar [d]^{\alpha } \ar [r] & \mathscr {G}' \ar [d]^{ \beta } \\ \mathscr {F} \ar [r] \ar@ {-->}[ur] & \mathscr {G} } \]

admits a solution, under the assumption that $\beta $ is a levelwise trivial Kan fibration between diagrams $\mathscr {G}, \mathscr {G}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$.

Example 7.5.9.9. Let $\operatorname{\mathcal{C}}$ be a small category. Then a diagram of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant (in the sense of Definition 7.5.6.1) if and only if the unique natural transformation $\underline{\emptyset } \rightarrow \mathscr {F}$ is a projective cofibration (in the sense of Definition 7.5.9.8). Here $\underline{\emptyset }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denotes the initial object of the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$, which carries every object of $\operatorname{\mathcal{C}}$ to the empty simplicial set.

Example 7.5.9.10. Let $\operatorname{\mathcal{C}}$ be a small category. For each object $C \in \operatorname{\mathcal{C}}$, let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor corepresented by $C$ (given on objects by the formula $h^ C(D) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$). If $A \hookrightarrow B$ is a monomorphism of simplicial sets, then the natural transformation $\underline{A} \times h^{C} \hookrightarrow \underline{B} \times h^{C}$ is a projective cofibration in $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$; here $\underline{A}$ and $\underline{B}$ denote the constant simplicial sets taking the values $A$ and $B$, respectively.

Remark 7.5.9.11. Let $\operatorname{\mathcal{C}}$ be a small category. Then the collection of projective cofibrations in $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ is weakly saturated, in the sense of Definition 1.4.4.15. That is, it is closed under retracts, pushouts, and transfinite composition. See Proposition 1.4.4.16.

Proposition 7.5.9.12. Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha _0: \mathscr {F}_0 \rightarrow \mathscr {G}$ be a natural transformation between diagrams $\mathscr {F}_0, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then $\alpha _0$ factors as a composition

\[ \mathscr {F}_0 \xrightarrow {\beta } \mathscr {F} \xrightarrow {\alpha } \mathscr {G}, \]

where $\beta $ is a projective cofibration and $\alpha $ is a levelwise trivial Kan fibration.

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

\[ \mathscr {F}_{0} \rightarrow \mathscr {F}_{1} \rightarrow \mathscr {F}_{2} \rightarrow \mathscr {F}_{3} \rightarrow \cdots \]

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

\[ \mathscr {F}_{n}(C) \xrightarrow { \alpha '_{n,C} } \mathscr {F}'_{n}(C) \xrightarrow { \alpha ''_{n,C} } \mathscr {G}(C), \]

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

\[ \xymatrix@R =50pt@C=50pt{ \coprod _{C \in \operatorname{\mathcal{C}}} \underline{ \mathscr {F}_ n(C) } \times h^{C} \ar [r] \ar [d] & \coprod _{C \in \operatorname{\mathcal{C}}} \underline{ \mathscr {F}'_ n(C) } \times h^{C} \ar [d] \\ \mathscr {F}_{n} \ar [r] & \mathscr {F}_{n+1} } \]

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 7.5.9.10. Using Remark 7.5.9.11, we see that each of the maps

\[ \mathscr {F}_{0} \rightarrow \mathscr {F}_{1} \rightarrow \mathscr {F}_{2} \rightarrow \mathscr {F}_{3} \rightarrow \cdots \]

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 1.4.5.3). $\square$

Proof of Proposition 7.5.9.7. Apply Proposition 7.5.9.12 in the special case $\mathscr {F}_{0} = \underline{\emptyset }$ (see Example 7.5.9.9). $\square$

Corollary 7.5.9.13. Let $\operatorname{\mathcal{C}}$ be a small category, and let $S$ be the collection of all projective cofibrations in the category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$. Then $S$ is the smallest weakly saturated collection of morphisms which contains each of the inclusion maps $\iota _{n,C}: \underline{\operatorname{\partial \Delta }^{n}} \times h^{C} \hookrightarrow \underline{\Delta ^{n}} \times h^{C}$, for each $n \geq 0$ and each object $C \in \operatorname{\mathcal{C}}$.

Proof. It follows from Remark 7.5.9.11 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 7.5.9.10, 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 1.4.5.13 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 7.5.9.12 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

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}_{0} \ar [d]^{ \alpha _0 } \ar [r]^-{\beta } & \mathscr {F} \ar [d]^{ \alpha } \\ \mathscr {G} \ar@ {-->}[ur] \ar@ {=}[r] & \mathscr {G} } \]

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