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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).

Remark 8.4.5.6. In the situation of Construction 8.4.5.5, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ is independent of the choice of $\kappa $ (provided that $\kappa $ is chosen large enough that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small and each $K \in \mathbb {K}$ is essentially $\kappa $-small).

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). $\square$

Corollary 8.4.5.10. 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 $\operatorname{\mathcal{C}}$. Let $U: \operatorname{\mathcal{E}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a left fibration of $\infty $-categories. The following conditions are equivalent:

$(1)$

The covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\widehat{\operatorname{\mathcal{C}}}}: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.

$(2)$

The projection map $\operatorname{\mathcal{E}}\times _{ \widehat{\operatorname{\mathcal{C}}} } \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is left cofinal.

Proof. It follows from Proposition 8.4.5.3 that the functor $h$ is fully faithful. We can therefore replace $\operatorname{\mathcal{C}}$ by its essential image and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is a replete full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ (and $h$ is the inclusion functor). In this case, condition $(1)$ is equivalent to the requirement that the functor $\operatorname{Tr}_{\operatorname{\mathcal{E}}/ \widehat{\operatorname{\mathcal{C}}} }$ is left Kan extended from $\operatorname{\mathcal{C}}$ (Lemma 8.4.5.9). The equivalence $(1) \Leftrightarrow (2)$ is now a special case of Corollary 7.4.5.16. $\square$

Remark 8.4.5.11. The $\operatorname{\mathcal{K}}$-cocompletion construction of this section has been studied in more detail by Rezk; we refer the reader to [rezk2022free] for more details.