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7.3.4 Colimits of $\infty $-Categories

Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.4.4.1). Our goal in this section is to show that the $\infty $-category $\operatorname{\mathcal{QC}}$ admits small colimits (Corollary 7.3.4.13). In fact, we will prove something more precise: if $\operatorname{\mathcal{C}}$ is a small $\infty $-category, then the colimit of any diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ can be described explicitly as the localization $(\int _{\operatorname{\mathcal{C}}} \mathscr {F})[W^{-1} ]$, where $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denotes the $\infty $-category of elements of $\mathscr {F}$ (Definition 5.5.4.4) and $W$ is the collection of all morphisms of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which are cocartesian with respect to the forgetful functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ (Corollary 7.3.4.12).

We begin with some general remarks. Let $\operatorname{\mathcal{C}}^{\triangleright }$ denote the right cone on on a simplicial set $\operatorname{\mathcal{C}}$ (Construction 4.3.3.25), and let ${\bf 1} \in \operatorname{\mathcal{C}}^{\triangleright }$ denote the cone point. For every vertex $C \in \operatorname{\mathcal{C}}$, there is a unique edge $e_{C}: C \rightarrow {\bf 1}$ in $\operatorname{\mathcal{C}}^{\triangleright }$. If $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ is a cocartesian fibration of simplicial sets, then covariant transport along $e_{C}$ determines a functor

\[ e_{C!}: \overline{\operatorname{\mathcal{E}}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}} \rightarrow \{ {\bf 1} \} \times _{ \operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}} = \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }. \]

In what follows, it will be convenient to amalgamate the functors $\{ e_{C!} \} _{C \in \operatorname{\mathcal{C}}}$ into a single morphism $\mathrm{Rf}: \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}} \rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$, which we will refer to as the covariant refraction diagram.

Definition 7.3.4.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and let ${\bf 1}$ denote the cone point of the simplicial set $\operatorname{\mathcal{C}}^{\triangleright } \simeq \operatorname{\mathcal{C}}\star \{ {\bf 1} \} $. Suppose that we are given a cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$, and set

\[ \operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}} \quad \quad \overline{\operatorname{\mathcal{E}}}_{{\bf 1}} = \{ {\bf 1} \} \times _{\operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}}. \]

We will say that a morphism $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is a covariant refraction diagram if there exists a morphism of simplicial sets $H: \Delta ^1 \times \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}$ satisfying the following conditions:

  • The restriction $H|_{ \{ 0\} \times \operatorname{\mathcal{E}}}$ is the identity morphism from $\operatorname{\mathcal{E}}$ to itself.

  • The restriction $H|_{ \{ 1\} \times \operatorname{\mathcal{E}}}$ is equal to $\mathrm{Rf}$.

  • For every vertex $X \in \operatorname{\mathcal{E}}$, the restriction $H|_{ \Delta ^1 \times \{ X\} }$ is a $\overline{U}$-cocartesian edge of $\overline{\operatorname{\mathcal{E}}}$.

Remark 7.3.4.2. In the situation of Definition 7.3.4.1, suppose that $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is a covariant refraction diagram. Then, for every vertex $C \in \operatorname{\mathcal{C}}$, the restriction $\mathrm{Rf}|_{ \operatorname{\mathcal{E}}_{C} }: \operatorname{\mathcal{E}}_{C} \rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is given by covariant transport along the unique edge $e_{C}: C \rightarrow {\bf 1}$ of $\operatorname{\mathcal{C}}^{\triangleright }$, in the sense of Definition 5.2.2.1.

Proposition 7.3.4.3. Let $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ be a cocartesian fibration of simplicial sets, set $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}}$, and let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. Then:

$(1)$

There exists a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ (Definition 7.3.4.1).

$(2)$

Let $F: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{{\bf 1}}$ be any morphism of simplicial sets. Then $F$ is a covariant refraction diagram if and only if it is isomorphic to $\mathrm{Rf}$ as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \overline{\operatorname{\mathcal{E}}}_{{\bf 1}} )$.

Proof. This is a special case of Lemma 5.2.2.9. $\square$

Example 7.3.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. Using Example 5.2.4.18, we see that the tautological map $V: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow (\Delta ^{0})^{\triangleright } \simeq \Delta ^1$ is a cocartesian fibration. If $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ is another cocartesian fibration, then the $\infty $-categories $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{C}}^{\triangleright }} \overline{\operatorname{\mathcal{E}}}$ and $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} } = \{ {\bf 1} \} \times _{ \operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}}$ can be identified with the fibers of the composite map

\[ (V \circ \overline{U} ): \overline{\operatorname{\mathcal{E}}} \rightarrow \Delta ^1, \]

which is also a cocartesian fibration (Proposition 5.1.4.13). In this case, the covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ of Proposition 7.3.4.3 is given by covariant transport for the cocartesian fibration $V \circ \overline{U}$ (along the nondegenerate edge of $\Delta ^1$).

Remark 7.3.4.5. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]

where $U$ and $\overline{U}$ are cocartesian fibrations. Let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$ and let $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ be a covariant refraction diagram. For every $U$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$, the image $U(e)$ is an isomorphism in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$. To prove this, we observe that there is a morphism $\Delta ^1 \times \Delta ^1 \rightarrow \overline{\operatorname{\mathcal{E}}}$ as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d]^{e} & \mathrm{Rf}(X) \ar [d]^{\mathrm{Rf}(e) } \\ Y \ar [r] & \mathrm{Rf}(Y), } \]

where the horizontal maps are $\overline{U}$-cocartesian. Applying Proposition 5.1.4.12, we deduce that $\mathrm{Rf}(e)$ is an $\overline{U}$-cocartesian edge of $\overline{\operatorname{\mathcal{E}}}$, and therefore an isomorphism in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ (Proposition 5.1.4.11).

Our study of colimits in the $\infty $-category $\operatorname{\mathcal{QC}}$ will make use of the following recognition principle for colimits in the $\infty $-category $\operatorname{\mathcal{QC}}$:

Theorem 7.3.4.6 (Refraction Criterion). Suppose we are given a pullback diagram of small simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]

where $U$ and $\overline{U}$ are cocartesian fibrations. Let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$ and let $W$ be the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. The following conditions are equivalent:

$(1)$

The covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ of Proposition 7.3.4.3 exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect $W$.

$(2)$

The covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$ of Notation 5.6.2.12 is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Remark 7.3.4.7. In the statement of Theorem 7.3.4.6, the covariant refraction diagram $F: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ and the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$ are only well-defined up to isomorphism (as objects of the $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{E}}, \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} })$ and $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{QC}})$, respectively). However, conditions $(1)$ and $(2)$ depend only on their isomorphism classes (see Exercise 6.3.1.11 and Corollary 7.1.5.7).

Exercise 7.3.4.8. Let $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ and $\overline{U}': \overline{\operatorname{\mathcal{E}}}' \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ be cocartesian fibrations of simplicial sets which are equivalent as inner fibrations over $\operatorname{\mathcal{C}}^{\triangleright }$ (in the sense of Definition 5.1.6.1). Show that $\overline{U}$ satisfies condition $(1)$ of Theorem 7.3.4.6 if and only if $\overline{U}'$ satisfies condition $(1)$ of Theorem 7.3.4.6.

We will prove Theorem 7.3.4.6 in ยง7.3.5. The remainder of this section is devoted to explaining some of its consequences. We begin by showing that there is a good supply of cocartesian fibrations which satisfy the assumptions of Theorem 7.3.4.6.

Proposition 7.3.4.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. Then there exists a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{\overline{U}} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]

where $\overline{U}$ is a cocartesian fibration and a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ which exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization fo $\operatorname{\mathcal{E}}$ with respect to the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$.

Proof. Let $W$ be the collection of all $U$-cocartesian edges of $\operatorname{\mathcal{E}}$. Applying Proposition 6.3.2.1, we deduce that there exists an $\infty $-category $\operatorname{\mathcal{E}}[W^{-1}]$ and a diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$ which exhibits $\operatorname{\mathcal{E}}[W^{-1}]$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. In particular, the diagram $\mathrm{Rf}$ carries each $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ to an isomorphism in $\operatorname{\mathcal{E}}[W^{-1}]$. Let $\overline{\operatorname{\mathcal{E}}}$ denote the relative join $\operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}]$ (Construction 5.2.4.1). Applying Lemma 5.2.4.17 to the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \mathrm{Rf}} \ar [d]^{U} & \operatorname{\mathcal{E}}[W^{-1} ] \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \Delta ^0, } \]

we deduce that vertical maps induce a cocartesian fibration

\[ \overline{U}: \overline{\operatorname{\mathcal{E}}} = \operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}] \rightarrow \operatorname{\mathcal{C}}\star _{\Delta ^0} \Delta ^0 \simeq \operatorname{\mathcal{C}}^{\triangleright }. \]

By construction, we have a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{\overline{U}} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright }, } \]

and the fiber of $\overline{U}$ over the cone point ${\bf 1} \in \operatorname{\mathcal{C}}^{\triangleright }$ can be identified with the $\infty $-category $\operatorname{\mathcal{E}}[W^{-1}]$. Moreover, $\mathrm{Rf}$ induces a morphism of simplicial sets

\[ H: \Delta ^1 \times \operatorname{\mathcal{E}}\simeq \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star _{ \operatorname{\mathcal{E}}[W^{-1}] } \operatorname{\mathcal{E}}[W^{-1}] = \overline{\operatorname{\mathcal{E}}} \]

for which $H|_{ \{ 0\} \times \operatorname{\mathcal{E}}}$ is the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$, and $H|_{ \{ 1\} \times \operatorname{\mathcal{E}}}$ is the diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$. For every vertex $X \in \operatorname{\mathcal{E}}$, the criterion of Lemma 5.2.4.17 guarantees that $H|_{ \Delta ^1 \times \{ X\} }$ is a $\overline{U}$-cocartesian edge of $\overline{\operatorname{\mathcal{E}}}$, so that $H$ exhibits $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}[W^{-1}]$ as a covariant refraction diagram. $\square$

Remark 7.3.4.10. In the situation of Proposition 7.3.4.9, suppose that the simplicial sets $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{C}}$ are small. Then the localization $\operatorname{\mathcal{E}}[W^{-1}]$ supplied by Proposition 6.3.2.1 can also be chosen to be small. It follows that the simplicial set $\overline{\operatorname{\mathcal{E}}}$ constructed in the proof is also small.

Corollary 7.3.4.11. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration between small simplicial sets, and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a covariant transport representation of $U$. Then the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ admits a colimit in $\operatorname{\mathcal{QC}}$. Moreover, an object $\operatorname{\mathcal{D}}\in \operatorname{\mathcal{QC}}$ is a colimit of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ if and only if it is equivalent to the localization $\operatorname{\mathcal{E}}[W^{-1}]$, where $W$ is the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.

Proof. Let ${\bf 1}$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. By virtue of Proposition 7.3.4.9 (and Remark 7.3.4.10), there exists a pullback diagram of small simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright } } \]

where $\overline{U}$ is a cocartesian fibration, and a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ which exhibits $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W$. Applying Corollary 5.6.2.9, we see that $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ extends to a covariant transport representation $ \operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$. By virtue of Theorem 7.3.4.6, this extension is a colimit diagram carrying ${\bf 0}$ to the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ {\bf 1} } \simeq \operatorname{\mathcal{E}}[W^{-1}]$. $\square$

Corollary 7.3.4.12. Let $\operatorname{\mathcal{C}}$ be a small simplicial set, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram, let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, and let $W$ be the collection of all $U$-cocartesian morphisms of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Then the localization $( \int _{\operatorname{\mathcal{C}}} \mathscr {F})[W^{-1} ]$ is a colimit of the diagram $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Proof. Apply Corollary 7.3.4.11 to the cocartesian fibration $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$. $\square$

Corollary 7.3.4.13. The $\infty $-category $\operatorname{\mathcal{QC}}$ admits small colimits.