7.1.7 Colimits Relative to a Fibration
In §7.1.6, we introduced the notion of a $U$-colimit diagram, where $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a functor of $\infty $-categories (Definition 7.1.6.1). In this section, we specialize to the case where $U$ is an inner fibration. Our starting point is the following:
Proposition 7.1.7.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets. Then $\overline{q}$ is a $U$-colimit diagram if and only if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ K \star \operatorname{\partial \Delta }^ n \ar [r]^-{\rho } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ K \star \Delta ^ n \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}} \]
admits a solution, provided that $n \geq 1$ and the restriction of $\rho $ to $K \star \{ 0\} \simeq K^{\triangleright }$ coincides with $\overline{q}$.
Proof.
Use the characterization of $U$-colimit diagrams given in Remark 7.1.6.8.
$\square$
Example 7.1.7.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{E}}$. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $U$-cocartesian (see Definition 5.1.1.1).
- $(2)$
The morphism $f$ is a $U$-colimit diagram when viewed as a map of simplicial sets $(\Delta ^0)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.
- $(3)$
The morphism $f$ is $U_{X/}$-initial when viewed as an object of the slice $\infty $-category $\operatorname{\mathcal{E}}_{X/}$, where $U_{X/}: \operatorname{\mathcal{E}}_{X/} \rightarrow \operatorname{\mathcal{C}}_{U(X)/ }$ is the functor induced by $U$.
- $(4)$
The morphism $f$ is $V$-initial when viewed as an object of the oriented fiber product $\{ X\} \vec{\times }_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}$, where $V: \{ X\} \vec{\times }_{ \operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \{ U(X) \} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$ is the functor induced by $U$.
The equivalence $(2) \Leftrightarrow (3)$ is immediate from the definition, the equivalence $(1) \Leftrightarrow (3)$ follows from Remark 7.1.6.8 and Proposition 5.1.1.14, and the equivalence $(3) \Leftrightarrow (4)$ follows from Corollary 4.6.4.20.
Example 7.1.7.3. Let $K$ be a weakly contractible simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories. Then every morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$ is a $U$-colimit diagram (see Proposition 4.3.7.6). Similarly, if $U$ is a right fibration, then every morphism $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{E}}$ is a $U$-limit diagram.
Corollary 7.1.7.4. Let $K$ be a weakly contractible simplicial set and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. If $U$ is a left fibration, then it creates $K$-indexed colimits. If $U$ is a right fibration, then it creates $K$-indexed limits.
Proof.
Assume $U$ is a left fibration; we will show that it creates $K$-indexed colimits (the analogous statement for right fibrations follows by a similar argument). Let $q: K \rightarrow \operatorname{\mathcal{E}}$ be a diagram and suppose that $U \circ q$ can be extended to a colimit diagram $g: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Since the inclusion $K \hookrightarrow K^{\triangleright }$ is left anodyne (Example 4.3.7.10), our assumption that $U$ is a left fibration guarantees that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ K \ar [d] \ar [r]^-{q} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ K^{\triangleright } \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{q} } & \operatorname{\mathcal{C}}} \]
has a solution. Since $K$ is weakly contractible, the morphism $\overline{q}$ is automatically a $U$-colimit diagram (Example 7.1.7.3). Applying Corollary 7.1.6.12, we see that $\overline{q}$ is also a colimit diagram.
$\square$
Corollary 7.1.7.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories and let $K$ be a weakly contractible simplicial set. Then:
If $U$ is a left fibration and the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, then the $\infty $-category $\operatorname{\mathcal{E}}$ also admits $K$-indexed colimits and the functor $U$ preserves $K$-indexed colimits.
If $U$ is a right fibration and the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, then the $\infty $-category $\operatorname{\mathcal{E}}$ also admits $K$-indexed limits and the functor $U$ preserves $K$-indexed limits.
Proof.
Combine Corollary 7.1.7.4 with Proposition 7.1.4.20.
$\square$
Proposition 7.1.7.6. Suppose we are given a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r]^-{F} \ar [d]^{U'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}. } \]
where the vertical maps are inner fibrations. Let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}'$ be a morphism of simplicial sets. If $F \circ \overline{q}$ is an $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$, then $\overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}'$.
Proof.
Use the characterization of relative colimit diagrams given in Proposition 7.1.7.1.
$\square$
Corollary 7.1.7.7. Suppose we are given a categorical pullback square of $\infty $-categories
7.3
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-relative-limit} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r]^-{F} \ar [d]^{U'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}'$ be a diagram. If $F \circ \overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$, then $\overline{q}$ is a $U'$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}'$.
Proof.
Using Corollary 4.5.2.24, we can factor $U$ as a composition $\operatorname{\mathcal{E}}\xrightarrow {E} \overline{\operatorname{\mathcal{D}}} \xrightarrow { V} \operatorname{\mathcal{D}}$, where $V$ is an isofibration and $E$ is an equivalence of $\infty $-categories. Applying Remark 7.1.6.6, we conclude that $(E \circ F \circ \overline{f}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $V$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. We may therefore replace $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{D}}$ and thereby reduce to proving Corollary 7.1.7.7 in the situation where $U$ is an isofibration. In this case, our assumption that (7.3) is a categorical pullback square guarantees that the induced map $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is an equivalence of $\infty $-categories. Using Remark 7.1.6.6 again, we can replace $\operatorname{\mathcal{E}}'$ by $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and thereby reduce to proving Corollary 7.1.7.7 in the situation where (7.3) is a pullback square (and the vertical maps are isofibrations). In this case, the desired result follows from Proposition 7.1.7.6.
$\square$
Corollary 7.1.7.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $–categories, let $C \in \operatorname{\mathcal{C}}$ be an object, and let
\[ \overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]
be a diagram. If $\overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$, then it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.
Proof.
Apply Proposition 7.1.7.6 in the special case $\operatorname{\mathcal{C}}' = \{ C\} $ (see Example 7.1.6.3).
$\square$
Beware that the converse of Corollary 7.1.7.8 is false in general: a colimit diagram in the fiber $\operatorname{\mathcal{E}}_{C}$ need not be a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$. To guarantee this, we need a slightly stronger condition.
Definition 7.1.7.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $C \in \operatorname{\mathcal{C}}$ be a vertex. We say that a morphism $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is an edgewise $U$-colimit diagram if it satisfies the following condition:
- $(\ast )$
For every edge $e: C \rightarrow C'$ of the simplicial set $\operatorname{\mathcal{C}}$, the composite map
\[ K^{\triangleright } \xrightarrow { \overline{f} } \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}} \]
is a colimit diagram in the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
Example 7.1.7.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets, let $C \in \operatorname{\mathcal{C}}$ be a vertex, and let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ be a morphism of simplicial sets. If $\overline{f}$ is an edgewise $U$-colimit diagram (in the sense of Definition 7.1.7.9), then it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. The converse holds if $U$ is a locally cartesian fibration. To see this, let $e: C \rightarrow C'$ be an edge of $\operatorname{\mathcal{C}}$ and let $\operatorname{\mathcal{E}}'$ denote the fiber product $\Delta ^1 \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. We wish to show that the inclusion map $\operatorname{\mathcal{E}}_{C} \simeq \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ carries colimit diagrams in $\operatorname{\mathcal{E}}_{C}$ to colimit diagrams in $\operatorname{\mathcal{E}}'$. This is a special case of Variant 7.1.4.26, since $\operatorname{\mathcal{E}}_{C}$ is a coreflective subcategory of $\operatorname{\mathcal{E}}'$ (see Corollary 6.2.5.2).
The implication $(\ast ) \Rightarrow (\ast ')$ follows from Corollary 7.1.6.20. To prove the converse, it suffices to show that the morphism $\theta $ is also a homotopy equivalence when $X$ belongs to $\operatorname{\mathcal{E}}_{C}$, which follows by applying condition $(\ast ')$ in the special case $e = \operatorname{id}_{C}$.
Example 7.1.7.12. In the situation of Remark 7.1.7.11, suppose that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a locally cocartesian fibration. Then condition $(\ast ')$ can be reformulated as follows:
- $(\ast '')$
For every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the composition
\[ K^{\triangleright } \xrightarrow { \overline{q} } \operatorname{\mathcal{E}}_{C} \xrightarrow { e_{!} } \operatorname{\mathcal{E}}_{C'} \]
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C'}$. Here $e_{!}$ denotes the covariant transport functor of Notation 5.2.2.9.
Proposition 7.1.7.14. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ be a diagram. Then $\overline{q}$ is a $U$-colimit diagram (in the sense of Definition 7.1.6.1) if and only if it is an edgewise $U$-colimit diagram (in the sense of Definition 7.1.7.9).
Proof.
Set $q = \overline{q}_{K}$. By virtue of Proposition 7.1.6.19, $\overline{q}$ is a $U$-colimit diagram if and only if for every object $X \in \operatorname{\mathcal{E}}$, the diagram of Kan complexes
7.4
\begin{equation} \begin{gathered}\label{equation:relative-colimit-by-fiber-preliminary} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{E}}) }( \overline{q}, \underline{X}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{E}}) }( q, \underline{X}|_{K} ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( U \circ \overline{q}, U \circ \underline{X}) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( U \circ q, U \circ \underline{X}|_{K}) } \end{gathered} \end{equation}
is a homotopy pullback square, where $\underline{X} \in \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{E}})$ denotes the constant diagram taking the value $X$. Since $U$ is an inner fibration, the vertical maps in (7.4) are Kan fibrations (Proposition 4.6.1.21 and Corollary 4.1.4.3). Using the criterion of Example 3.4.1.4, we see that (7.4) is a homotopy pullback square if and only if, for every vertex $u \in \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( U \circ \overline{q}, U \circ \underline{X})$, the induced map
\[ \xymatrix@R =50pt@C=50pt{ \{ u\} \times _{\operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( U \circ \overline{q}, U \circ \underline{X})} \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{E}}) }( \overline{q}, \underline{X}) \ar [d]^{\theta _ u} \\ \{ u|_{K} \} \times _{ \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( U \circ q, U \circ \underline{X}|_{K}) }\operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{E}}) }( q , \underline{X}|_{K}) } \]
is a homotopy equivalence of Kan complexes. Set $C' = U(X)$, so that $u$ can be identified with a morphism of simplicial sets $K^{\triangleleft } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, C' )$ and the condition that $\theta _ u$ is a homotopy equivalence depends only on the homotopy class of $u$. Since the simplicial set $K^{\triangleright }$ is weakly contractible (Example 4.3.7.11), it suffices to check that $\theta _ u$ is a homotopy equivalence in the special case where $u$ is the constant morphism taking the value $e$, for some morphism $e: C \rightarrow C'$ in the $\infty $-category $\operatorname{\mathcal{C}}$. The desired result is now a reformulation of Remark 7.1.7.11.
$\square$
Corollary 7.1.7.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of $\infty $-categories, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ be a morphism of simplicial sets. Then $\overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$ if and only if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.
Proof.
Combine Proposition 7.1.7.14 with Example 7.1.7.10.
$\square$
Corollary 7.1.7.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cocartesian fibration of $\infty $-categories, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$ be a morphism of simplicial sets. Then $\overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$ if and only if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ which is preserved by the covariant transport functor $e_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C'}$, for every morphism $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$.
Proof.
Combine Proposition 7.1.7.14 with Example 7.1.7.12.
$\square$
We close this section by recording a variant of Corollary 7.1.7.15 which will be useful later.
Proposition 7.1.7.18. Suppose we are given a commutative diagram of $\infty $-categories
7.5
\begin{equation} \begin{gathered}\label{equation:base-change-relative-limit5} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [rr]^{F'} \ar [dr]_{V'} \ar [dd]^{G} & & \operatorname{\mathcal{D}}' \ar [dl] \ar [dd] \\ & \operatorname{\mathcal{C}}' \ar [dd] & \\ \operatorname{\mathcal{E}}\ar [dr]_{ V } \ar [rr]^(.4){F} & & \operatorname{\mathcal{D}}\ar [dl]^{U} \\ & \operatorname{\mathcal{C}}& } \end{gathered} \end{equation}
where each square is a pullback, the diagonal maps are cartesian fibrations, and the functor $F$ carries $V$-cartesian morphisms of $\operatorname{\mathcal{E}}$ to $U$-cartesian morphisms of $\operatorname{\mathcal{D}}$. If $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{E}}'$ is an $F'$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}'$, then $G \circ \overline{f}$ is an $F$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$.
Proof.
Set $f = \overline{f}|_{K}$. By virtue of Corollary 4.3.6.10 and Proposition 5.1.4.20, we can replace (7.5) by the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}'_{f/} \ar [rr] \ar [dr] \ar [dd] & & \operatorname{\mathcal{D}}'_{ (F' \circ f)/} \ar [dl] \ar [dd] \\ & \operatorname{\mathcal{C}}'_{ (V' \circ f)/} \ar [dd] & \\ \operatorname{\mathcal{E}}_{ (G \circ f)/ } \ar [rr] \ar [dr] & & \operatorname{\mathcal{D}}_{ (F \circ G \circ f)/} \ar [dl] \\ & \operatorname{\mathcal{C}}_{ (V \circ G \circ f)/} & } \]
and thereby reduce to the special case $K = \emptyset $. In this case, the desired result follows from Proposition 7.1.5.19.
$\square$