8.4 Cocompletion
Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category. It is very rare for $\operatorname{\mathcal{C}}$ to admit small colimits: this is possible only if $\operatorname{\mathcal{C}}$ is (equivalent to the nerve of) a partially ordered set (Proposition 7.6.2.16). However, it is always possible to embed $\operatorname{\mathcal{C}}$ into a larger $\infty $-category which admits small colimits. Our goal in this section is to study the universal example of such an enlargement.
Definition 8.4.0.1. Let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. We will say that $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
- $(1)$
The $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ admits small colimits.
- $(2)$
Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits small colimits and let $\operatorname{Fun}'( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}})$ spanned by those functors which preserve small colimits. Then precomposition with $h$ induces an equivalence of $\infty $-categories $\operatorname{Fun}'( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
It follows immediately from the definition that if an $\infty $-category $\operatorname{\mathcal{C}}$ admits a cocompletion $\widehat{\operatorname{\mathcal{C}}}$, then $\widehat{\operatorname{\mathcal{C}}}$ is determined uniquely up to equivalence. Our primary goal in this section is to prove the following existence result:
Theorem 8.4.0.3. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ (Definition 8.3.3.9). Then $h_{\bullet }$ exhibit $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ as a cocompletion of $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.4.0.1).
Example 8.4.0.4. Let $X$ be a contractible Kan complex, which we identify with a vertex $x$ of the simplicial set $\operatorname{\mathcal{S}}$. Applying Theorem 8.4.0.3 in the special case where $\operatorname{\mathcal{C}}= \Delta ^0$, we deduce that the map $x: \Delta ^0 \rightarrow \operatorname{\mathcal{S}}$ exhibits $\operatorname{\mathcal{S}}$ as a cocompletion of the $0$-simplex $\Delta ^0$. That is, for every $\infty $-category $\operatorname{\mathcal{D}}$ which admits small colimits, the evaluation map
\[ \operatorname{Fun}'( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}\quad \quad F \mapsto F( X ) \]
is an equivalence of $\infty $-categories, where $\operatorname{Fun}'( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}})$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}})$ spanned by the colimit-preserving functors. Note that this property characterizes the $\infty $-category $\operatorname{\mathcal{S}}$ up to equivalence: it is “freely generated” under small colimits by the object $\Delta ^0$.
Warning 8.4.0.5. In §8.4.5, we will show that every $\infty $-category $\operatorname{\mathcal{C}}$ admits a cocompletion $\widehat{\operatorname{\mathcal{C}}}$ (Proposition 8.4.5.3). Beware that, if $\operatorname{\mathcal{C}}$ is not essentially small, then $\widehat{\operatorname{\mathcal{C}}}$ cannot necessarily be identified with the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (Warning 8.4.3.4). However, if $\operatorname{\mathcal{C}}$ is locally small, then it can be identified with a full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (see Construction 8.4.5.5).
Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is essentially small and $\operatorname{\mathcal{D}}$ admits small colimits. Using the $\infty $-categorical version of Yoneda's lemma (Theorem 8.3.3.13), we see that $f$ factors (up to isomorphism) as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow {F_0} \operatorname{\mathcal{D}}, \]
where $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by the representable functors. Theorem 8.4.0.3 asserts that $F_0$ admits an essentially unique extension to a functor $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ which preserves small colimits. To prove this, it will be useful to characterize this extension in a different way. For any functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$, we will show that the following conditions are equivalent:
- $(a)$
The functor $F$ preserves small colimits.
- $(b)$
The functor $F$ is left Kan extended from the subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.
Granting this equivalence, the proof of Theorem 8.4.0.3 is reduced to showing that every functor $F_0: \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ admits an essentially unique left Kan extension $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$, which follows from the general results of §7.3.
To establish the equivalence of $(a)$ and $(b)$, we will proceed by reduction to an important special case. Suppose that $\operatorname{\mathcal{D}}= \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ and that $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ is the identity functor. In this case, condition $(a)$ is automatically satisfied. Condition $(b)$ then asserts that the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is dense: that is, every object $\mathscr {G} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ can be recovered as the colimit of the diagram
\[ \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}{\operatorname{op}}, \operatorname{\mathcal{S}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{ / \mathscr {F} } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \]
of representable functors over $\mathscr {G}$ (see Definition 8.4.1.5). In §8.4.1, we discuss dense subcategories in general and provide a concrete criterion can be used to show that a subcategory is dense (Proposition 8.4.1.8). In §8.4.2, we apply this criterion to establish the density of the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (Corollary 8.4.2.2). In §8.4.3, we use this result to establish the equivalence of $(a)$ and $(b)$ in general (Theorem 8.4.3.6), and deduce Theorem 8.4.0.3 as an easy consequence.
Let us now specialize the preceding discussion to the situation where the $\infty $-category $\operatorname{\mathcal{D}}$ is locally small. In this case, we will show that conditions $(a)$ and $(b)$ above are equivalent to the following:
- $(c)$
The functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.
The implication $(c) \Rightarrow (a)$ is formal (by virtue of Corollary 7.1.3.21, every left adjoint preserves colimits). In §8.4.4, we prove the reverse implication by giving an explicit construction of the right adjoint $G$: it carries each object $D \in \operatorname{\mathcal{D}}$ to the functor
\[ \mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}\quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D) \]
where $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(\bullet , \bullet )$ is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{\mathcal{D}}$ (see Proposition 8.4.4.1). The equivalence of $(a)$ and $(c)$ is a special case of the $\infty $-categorical adjoint functor theorem (Theorem 
), which we discuss in § .
If $\operatorname{\mathcal{C}}$ is an essentially small $\infty $-category, then its cocompletion $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is a drastic enlargement of $\operatorname{\mathcal{C}}$, obtained by (freely) adjoining a colimit for every small diagram. In practice, it will be useful to consider a variant of Definition 8.4.0.1, where we restrict our attention to diagrams indexed by some collection of simplicial sets $\mathbb {K}$. 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 $\widehat{\operatorname{\mathcal{C}}}$ admits $K$-indexed colimits for each $K \in \mathbb {K}$, and is universal with respect to this property (see Definition 8.4.5.1). In §8.4.5, we show that every $\infty $-category $\operatorname{\mathcal{C}}$ admits a $\mathbb {K}$-cocompletion $\widehat{\operatorname{\mathcal{C}}}$ (Proposition 8.4.5.3). Our proof proceeds by explicit construction. Assume for simplicity that the $\infty $-category $\operatorname{\mathcal{C}}$ and each of the simplicial sets $K \in \mathbb {K}$ is essentially small; in this case, we show that we can take $\widehat{\operatorname{\mathcal{C}}}$ to be the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ which contains all representable functors and is closed under $K$-indexed colimits, for each $K \in \mathbb {K}$ (Construction 8.4.5.5 and Proposition 8.4.5.7).
Let us isolate another important feature of the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ associated to an essentially small $\infty $-category $\operatorname{\mathcal{C}}$. For every pair of objects $X \in \operatorname{\mathcal{C}}$ and $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, the $\infty $-categorical analogue of Yoneda's lemma supplies a homotopy equivalence
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})}( h_{X}, \mathscr {F} ) \xrightarrow {\sim } \mathscr {F}(X) \]
(Proposition 8.3.1.3). It follows that $h_{X}$ is an atomic object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$: that is, it corepresents a functor which preserves small colimits (Definition 8.4.6.1). The Yoneda embedding is essentially characterized by this property, together with the fact that it is dense and fully faithful. More precisely, suppose that we are given a functor of $\infty $-categories $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$, where $\operatorname{\mathcal{C}}$ is small and $\widehat{\operatorname{\mathcal{C}}}$ admits small colimits. In §8.4.6, we show that $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$ if and only if it is dense, fully faithful, and carries each object of $\operatorname{\mathcal{C}}$ to an atomic object of $\widehat{\operatorname{\mathcal{C}}}$ (Proposition 8.4.6.6). In §8.4.7, we apply this characterization to show that the formation of cocompletions is compatible with the formation of slice $\infty $-categories (Proposition 8.4.7.1). In particular, if $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration between essentially small $\infty $-categories, we show that there is an equivalence of $\infty $-categories $\operatorname{Fun}( \widetilde{\operatorname{\mathcal{C}}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \simeq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})_{/ \mathscr {F} }$, where $\mathscr {F}$ denotes a covariant transport representation for the left fibration $U^{\operatorname{op}}$ (Corollary 8.4.7.2).
Structure
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Subsection 8.4.1: Dense Functors
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Subsection 8.4.2: Density of Yoneda Embeddings
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Subsection 8.4.3: Cocompletion via the Yoneda Embedding
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Subsection 8.4.4: Example: Extensions as Adjoints
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Subsection 8.4.5: Adjoining Colimits to $\infty $-Categories
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Subsection 8.4.6: Recognition of Cocompletions
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Subsection 8.4.7: Slices of Cocompletions