8.4.5 Adjoining Colimits to $\infty $-Categories
Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category and set $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. Theorem 8.4.0.3 asserts that the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$: that is, it is freely generated from $\operatorname{\mathcal{C}}$ by adjoining colimits of small diagrams. In this section, we consider a variant of this construction, where we adjoint colimits of an arbitrary collection of diagrams (and we drop the assumption that $\operatorname{\mathcal{C}}$ is essentially small).
Definition 8.4.5.1. Let $\mathbb {K}$ be a collection of simplicial sets. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete if it admits $K$-indexed colimits, for each $K \in \mathbb {K}$. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\mathbb {K}$-cocomplete $\infty $-categories, we let $\operatorname{Fun}^{\mathbb {K}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which preserve $K$-indexed colimits, for each $K \in \mathbb {K}$. We say that a functor of $\infty $-categories $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is $\mathbb {K}$-cocomplete.
For every $\mathbb {K}$-cocomplete $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $h$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
Example 8.4.5.2. Let $\kappa $ be an uncountable regular cardinal, and let $\mathbb {K}$ denote the collection of all $\kappa $-small simplicial sets. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete if and only if it is $\kappa $-cocomplete, in the sense of Variant 7.6.7.7. A functor of $\infty $-categories $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ if and only if it exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$, in the sense of Definition 8.4.3.2.
In particular, if $\mathbb {K}$ is the collection of all small simplicial sets, then an $\infty $-category $\operatorname{\mathcal{C}}$ is a $\mathbb {K}$-cocomplete if and only if it is cocomplete, and a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$ if and only if it exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$.
Our goal in this section is to prove the following existence result:
Proposition 8.4.5.3. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then there exists an $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ and a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Moreover, the functor $h$ is dense and fully faithful.
Warning 8.4.5.4. Let $\mathbb {K}$ be a collection of simplicial sets and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. In general, it is not true that every object of $\widehat{\operatorname{\mathcal{C}}}$ can be recovered as the colimit of a diagram
\[ K \rightarrow \operatorname{\mathcal{C}}\xrightarrow {h} \widehat{\operatorname{\mathcal{C}}} \]
for some $K \in \mathbb {K}$.
Let $\kappa $ be an uncountable regular cardinal. If $\mathbb {K}$ is the collection of all $\kappa $-small simplicial sets and the $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, then Proposition 8.4.5.3 follows from Theorem 8.4.3.3; in this case, we can take $\widehat{\operatorname{\mathcal{C}}}$ to be the $\infty $-category of functors $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })$. To prove Proposition 8.4.5.3 in general, we will build on this special case.
Construction 8.4.5.5. Let $\mathbb {K}$ be a collection of simplicial sets and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Choose an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small and every simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. We let $\widehat{\operatorname{\mathcal{C}}}$ denote the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ which contains all representable functors and is closed under the formation of $K$-indexed colimits, for each $K \in \mathbb {K}$. Note that covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ determines a functor $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$, which is dense (by virtue of Variant 8.4.2.4 and Remark 8.4.1.19) and fully faithful (by virtue of Theorem 8.3.3.13).
Proposition 8.4.5.3 is an immediate consequence of the following more precise result:
Proposition 8.4.5.7. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and define $\widehat{\operatorname{\mathcal{C}}}$ as in Construction 8.4.5.5. Then the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$.
The proof of Proposition 8.4.5.7 will require some preliminaries.
Lemma 8.4.5.8. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and define $\widehat{\operatorname{\mathcal{C}}}$ as in Construction 8.4.5.5. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete. Then there exists a functor $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ and an isomorphism $f \rightarrow F|_{\operatorname{\mathcal{C}}}$, where the functor $F$ is left Kan extended from the essential image of the Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$. Moreover, the functor $F$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.
Proof.
Fix an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and each $K \in \mathbb {K}$ is essentially $\kappa $-small. By virtue of Corollary 8.3.3.17, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a replete full subcategory of a $\kappa $-cocomplete $\infty $-category $\operatorname{\mathcal{D}}'$, and that the inclusion map $\operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{D}}'$ preserves all $\kappa $-small colimits which exist in $\operatorname{\mathcal{D}}$. By virtue of Theorem 8.4.3.3, we can also assume that $f$ factors as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {F'} \operatorname{\mathcal{D}}', \]
where $F'$ preserves $\kappa $-small colimits. The full subcategory $F'^{-1}( \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ contains all representable functors and is closed under $K$-indexed colimits for each $K \in \mathbb {K}$, and therefore contains $\widehat{\operatorname{\mathcal{C}}}$. It follows that $F'$ restricts to a functor $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ which preserves $K$-indexed colimits for each $K \in \mathbb {K}$. Theorem 8.4.3.6 implies that $F'$ is left Kan extended from the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$, so the functor $F = F'|_{ \widehat{\operatorname{\mathcal{C}}} }$ has the same property.
$\square$
Lemma 8.4.5.9. Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and define $\widehat{\operatorname{\mathcal{C}}}$ as in Construction 8.4.5.5. Let $\operatorname{\mathcal{D}}$ be a $\mathbb {K}$-cocomplete $\infty $-category and let $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $F$ is left Kan extended from the essential image of the Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$.
- $(2)$
The functor $F$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.
Proof.
Let $F_0$ denote the restriction of $F$ to the essential image of $h_{\bullet }$. Applying Lemma 8.4.5.8, we deduce that $F_0$ admits a left Kan extension $F': \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ which preserves $K$-indexed colimits for each $K \in \mathbb {K}$. Invoking the universal property of Kan extensions (Corollary 7.3.6.9), we see that there is an essentially unique natural transformation $\alpha : F' \rightarrow F$ which restricts to the identity transformation from $F_0$ to itself. We can then reformulate condition $(1)$ as follows:
- $(1')$
The natural transformation $\alpha $ is an isomorphism. That is, for each object $X \in \widehat{\operatorname{\mathcal{C}}}$, the induced map $\alpha _{X}: F'(X) \rightarrow F(X)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.
The implication $(1') \Rightarrow (2)$ follows from the fact that $F'$ preserves $K$-indexed colimits for each $K \in \mathbb {K}$. To prove the converse, let $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ denote the full subcategory spanned by those objects $X$ for which $\alpha _{X}$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$. By construction, $\widehat{\operatorname{\mathcal{C}}}'$ contains all representable functors $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$. If condition $(2)$ is satisfied, then $\widehat{\operatorname{\mathcal{C}}}'$ is closed under the formation of $K$-indexed colimits for each $K \in \mathbb {K}$, and therefore coincides with $\widehat{\operatorname{\mathcal{C}}}$.
$\square$
Proof of Proposition 8.4.5.7.
Let $\mathbb {K}$ be a collection of simplicial sets, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\widehat{\operatorname{\mathcal{C}}}$ be as in Construction 8.4.5.5. By construction, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is $\mathbb {K}$-cocomplete. To complete the proof, we must show that if $\operatorname{\mathcal{D}}$ is any $\mathbb {K}$-cocomplete $\infty $-category, then composition with the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ induces an equivalence of $\infty $-categories $\theta : \operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
Let $\operatorname{\mathcal{C}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ be the essential image of $h_{\bullet }$, so that $\theta $ factors as a composition
\[ \operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \xrightarrow {\theta '} \operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) \xrightarrow {\theta ''} \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]
where $\theta ''$ is an equivalence of $\infty $-categories (Theorem 8.3.3.13). Using Lemma 8.4.5.9, we see that $\operatorname{Fun}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ is the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by those functors which are left Kan extended from $\operatorname{\mathcal{C}}'$. It follows from Corollary 7.3.6.15 that $\theta '$ is a trivial Kan fibration onto a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}})$; in particular, it is fully faithful, so that $\theta $ is fully faithful. Lemma 8.4.5.8 implies that $\theta $ is essentially surjective, and therefore an equivalence of $\infty $-categories (Theorem 4.6.2.20).
$\square$