7.1.9 Colimits and Categorical Pullback Squares
Suppose we are given a categorical pullback diagram of $\infty $-categories
7.9
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-generic-diagram} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
In this section, we study assumptions which guarantee that colimits in the $\infty $-category $\operatorname{\mathcal{C}}_{01}$ can be computed “componentwise”. We can state our main result as follows:
Proposition 7.1.9.1. Let $K$ be a simplicial set, and suppose we are given a categorical pullback square of $\infty $-categories
7.10
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-relative-limit-all-existence} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r]^-{G_0} \ar [d]^{G_1} & \operatorname{\mathcal{C}}_0 \ar [d]^{F_0} \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{F_1} & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
Assume that the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ admit $K$-indexed colimits and that the functors $F_0$ and $F_1$ preserve $K$-indexed colimits. Then:
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits.
A morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{01}$ is a colimit diagram if and only if $G_0 \circ \overline{q}$ and $G_1 \circ \overline{q}$ are colimit diagrams (in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively).
In particular, the functors $G_0$ and $G_1$ preserve $K$-indexed colimits.
In the situation of Proposition 7.1.7.6, the hypothesis that (7.10) is a categorical pullback square is equivalent to the requirement that it induces an equivalence from $\operatorname{\mathcal{C}}_{01}$ to the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. Recall that $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ can be identified with a full subcategory of the oriented fiber product $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ introduced in Definition 4.6.4.1. Our starting point is the following:
Lemma 7.1.9.2. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, let $\operatorname{\mathcal{C}}_0 \vec{\times }_{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ denote the oriented fiber product of Definition 4.6.4.1, and consider the projection maps
\[ \operatorname{\mathcal{C}}_0 \xleftarrow { \operatorname{ev}_0 } \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \xrightarrow { \operatorname{ev}_1 } \operatorname{\mathcal{C}}_1. \]
Suppose we are given a diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ such that $F_0 \circ \operatorname{ev}_0 \circ \overline{q}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $\overline{q}$ is an $( \operatorname{ev}_0, \operatorname{ev}_1)$-colimit diagram in $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$.
Proof.
By construction, we have a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \ar [r]^-{G} \ar [d]^{ (\operatorname{ev}_0, \operatorname{ev}_1) } & \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \ar [d]^{U} \\ \operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) } \]
where the vertical maps are isofibrations. By virtue of Proposition 7.1.7.6, it will suffice to show that $G \circ \overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$, which is a special case of Example 7.1.8.12.
$\square$
Lemma 7.1.9.3 (Colimits in Oriented Fiber Products: Recognition). Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ be a diagram satisfying the following conditions:
The composition $(\operatorname{ev}_0 \circ \overline{q}): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0$ is a colimit diagram in $\operatorname{\mathcal{C}}_0$.
The composition $(\operatorname{ev}_1 \circ \overline{q}): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_1$ is a colimit diagram in $\operatorname{\mathcal{C}}_1$.
The composition $(F_0 \circ \operatorname{ev}_0 \circ \overline{q} ): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.
Then $\overline{q}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$.
Proof.
Combine Lemma 7.1.9.2, Corollary 7.1.6.11, and Example 7.1.3.11.
$\square$
Proposition 7.1.9.4 (Colimits in Oriented Fiber Products: Existence). Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $q: K \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ be a diagram satisfying the following conditions:
The diagram $(\operatorname{ev}_0 \circ q): K \rightarrow \operatorname{\mathcal{C}}_0$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0$ which is preserved by the functor $F_0$.
The diagram $(\operatorname{ev}_1 \circ q): K \rightarrow \operatorname{\mathcal{C}}_1$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_1$.
Then $q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. Moreover, an extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a colimit diagram if and only if $\operatorname{ev}_0 \circ \overline{q}$ and $\operatorname{ev}_1 \circ \overline{q}$ are colimit diagrams in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively.
Proof.
Let us identify $q$ with a triple $(q_0, q_1, \alpha )$, where $q_0 = \operatorname{ev}_0 \circ q$ is a diagram in $\operatorname{\mathcal{C}}_0$, $q_1 = \operatorname{ev}_1 \circ q$ is a diagram in $\operatorname{\mathcal{C}}_1$, and $\alpha : F_0 \circ q_0 \rightarrow F_1 \circ q_1$ is a natural transformation between $K$-indexed diagrams in $\operatorname{\mathcal{C}}$. By assumption, we can extend $q_0$ and $q_1$ to colimit diagrams $\overline{q}_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0$ and $\overline{q}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_1$, respectively. Moreover, the composition $F_0 \circ \overline{q}_0$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, so the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}) }( F_0 \circ \overline{q}_0, F_1 \circ \overline{q}_1 ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( K, \operatorname{\mathcal{C}}) }( F_0 \circ q_0, F_1 \circ q_1 ) \]
is a trivial Kan fibration (Corollary 7.1.6.20). It follows that $\alpha $ can be extended to a natural transformation $\overline{\alpha }: F_0 \circ \overline{q}_0 \rightarrow F_1 \circ \overline{q}_1$ between $K^{\triangleright }$-indexed diagrams in $\operatorname{\mathcal{C}}$. The triple $( \overline{q}_0, \overline{q}_1, \overline{\alpha } )$ can be identified with a diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ having the property that $\operatorname{ev}_0 \circ \overline{q}$ and $\operatorname{ev}_1 \circ \overline{q}$ are colimit diagrams in $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively. It follows from Lemma 7.1.9.3 that any such extension is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. Conversely, if $\overline{q}': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a colimit diagram satisfying $\overline{q}'|_{K} = q$, then $\overline{q}'$ is isomorphic to $\overline{q}$ (as an object of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$)), so that $\operatorname{ev}_0 \circ \overline{q}'$ and $\operatorname{ev}_1 \circ \overline{q}'$ are colimit diagrams in $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$.
$\square$
Corollary 7.1.9.5. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $K$ be a simplicial set satisfying the following conditions:
The $\infty $-category $\operatorname{\mathcal{C}}_0$ admits $K$-indexed colimits, which are preserved by the functor $F_0$.
The $\infty $-category $\operatorname{\mathcal{C}}_1$ admits $K$-indexed colimits.
Then the oriented fiber product $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ also admits $K$-indexed colimits. Moreover, a morphism of simplicial sets $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a colimit diagram if and only if its images in $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ are colimit diagrams.
See Proposition 7.1.3.14.
Corollary 7.1.9.7 (Colimits in Homotopy Fiber Products: Existence). Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $q: K \rightarrow \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ be a diagram satisfying the following conditions:
The diagram $(\operatorname{ev}_0 \circ q): K \rightarrow \operatorname{\mathcal{C}}_0$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0$ which is preserved by the functor $F_0$.
The diagram $(\operatorname{ev}_1 \circ q): K \rightarrow \operatorname{\mathcal{C}}_1$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_1$ which is preserved by the functor $F_1$.
Then $q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. Moreover, an extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a colimit diagram if and only if $\operatorname{ev}_0 \circ \overline{q}$ and $\operatorname{ev}_1 \circ \overline{q}$ are colimit diagrams in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively.
Proof.
Combine Proposition 7.1.9.4 with Remark 7.1.9.6.
$\square$
Proposition 7.1.9.1 is an immediate consequence of the following more precise assertion:
Proposition 7.1.9.8. Suppose we are given a categorical pullback square of $\infty $-categories
7.11
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-relative-limit-existence} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r]^-{G_0} \ar [d]^{G_1} & \operatorname{\mathcal{C}}_0 \ar [d]^{F_0} \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{F_1} & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}
and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram with the following properties:
The diagram $q_0 = G_0 \circ q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0$, which is preserved by the functor $F_0$.
The diagram $q_1 = G_1 \circ q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_1$, which is preserved by the functor $F_1$.
Then $q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_{01}$. Moreover, an extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if it satisfies the following condition:
- $(\ast )$
The compositions $G_0 \circ \overline{q}$ and $G_1 \circ \overline{q}$ are colimit diagrams (in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively).
Proof.
Since (7.11) is a categorical pullback square, it induces an equivalence from $\operatorname{\mathcal{C}}_{01}$ to the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. The desired result now follows from Corollary 7.1.9.7.
$\square$
The preceding results have counterparts for other inverse limit constructions. We consider here the case of sequential inverse limits (see Propositions 7.4.4.15 and 7.4.4.16 for a more systematic discussion).
Proposition 7.1.9.9. Suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_3 \xrightarrow { F_2 } \operatorname{\mathcal{C}}_2 \xrightarrow { F_1 } \operatorname{\mathcal{C}}_1 \xrightarrow { F_0 } \operatorname{\mathcal{C}}_0, \]
where each of the functors $F_{n}$ is an isofibration. Let $q: K \rightarrow \varprojlim _{n} \operatorname{\mathcal{C}}_{n}$ be a diagram, which we identify with a compatible sequence of diagrams $\{ q_ n: K \rightarrow \operatorname{\mathcal{C}}_ n \} _{n \geq 0}$. Assume that:
For each $n \geq 0$, the diagram $q_{n}$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_ n$.
For each $n \geq 0$, the functor $F_{n}: \operatorname{\mathcal{C}}_{n+1} \rightarrow \operatorname{\mathcal{C}}_ n$ preserves the colimit of the diagram $q_{n+1}$.
Then $q$ admits a colimit in the $\infty $-categoruy $\varprojlim _{n} \operatorname{\mathcal{C}}_ n$. Moreover, an extension $\overline{q}: K^{\triangleright } \rightarrow \varprojlim _{n} \operatorname{\mathcal{C}}_ n$ is a colimit diagram if and only if its image in each $\operatorname{\mathcal{C}}_{n}$ is a colimit diagram.
Proof.
Set $\operatorname{\mathcal{C}}= \prod _{n \geq 0} \operatorname{\mathcal{C}}_ n$ and let $S: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ be the shift functor (given on objects by the construction $\{ C_ n \} _{n \geq 0} \mapsto \{ F_ n( C_{n+1} ) \} _{n \geq 0}$). It follows from Corollary 4.5.6.19 that the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \varprojlim _{n} \operatorname{\mathcal{C}}_ n \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ (\operatorname{id}, \operatorname{id}) } \\ \operatorname{\mathcal{C}}\ar [r]^-{ (\operatorname{id}, S) } & \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}. } \]
is a categorical pullback square. The desired result now follows by combining Proposition 7.1.9.8 with Example 7.1.3.11.
$\square$
Corollary 7.1.9.10. Let $K$ be a simplicial set and suppose we are given a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{\mathcal{C}}_3 \xrightarrow { F_2 } \operatorname{\mathcal{C}}_2 \xrightarrow { F_1 } \operatorname{\mathcal{C}}_1 \xrightarrow { F_0 } \operatorname{\mathcal{C}}_0. \]
Assume that each of the $\infty $-categories $\operatorname{\mathcal{C}}_ n$ admits $K$-indexed colimits and that each of the functors $F_{n}$ is an isofibration which preserves $K$-indexed colimits. Then:
The $\infty $-category $\varprojlim _{n} \operatorname{\mathcal{C}}_ n$ admits $K$-indexed colimits.
A morphism $\overline{q}: K^{\triangleright } \rightarrow \varprojlim _{n} \operatorname{\mathcal{C}}_ n$ is a colimit diagram if and only if its image in each $\operatorname{\mathcal{C}}_{n}$ is a colimit diagram.