Proposition 8.4.5.13. 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 $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete.
- $(2)$
The functor $h$ admits a left adjoint $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$.
Moreover, if these conditions are satisfied, then $F$ is the $\mathbb {K}$-cocontinuous extension of the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (see Remark 8.4.5.5).
Proof.
We first show that $(2)$ implies $(1)$. Assume that $h$ admits a left adjoint $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$. Since $h$ is fully faithful (Proposition 8.4.5.3), it induces an equivalence from $\operatorname{\mathcal{C}}$ to a reflective subcategory of $\widehat{\operatorname{\mathcal{C}}}$ (Remark 6.3.3.4). For each $K \in \mathbb {K}$, our assumption the existence of $K$-indexed colimits in $\widehat{\operatorname{\mathcal{C}}}$ then guarantees the existence of $K$-indexed colimits in $\operatorname{\mathcal{C}}$ (Corollary 7.1.4.23).
We now prove the converse. Assume that $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-cocomplete. Let $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a $\mathbb {K}$-cocontinuous extension of the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$, so that there exists an isomorphism of functors $\epsilon : F \circ h \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$. We will complete the proof by showing that $\epsilon $ is the counit of an adjunction between $F$ and $h$. By virtue of Corollary 6.2.6.5, it will suffice to show that for every pair of objects $\widehat{X} \in \widehat{\operatorname{\mathcal{C}}}$, $Y \in \operatorname{\mathcal{C}}$, the composite map
\[ \theta _{X,Y}: \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}} }( \widehat{X}, h(Y) ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F( \widehat{X} ), (F \circ h)(Y) ) \xrightarrow { [ \epsilon _{Y} ] \circ } \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( F( \widehat{X} ), Y) \]
is a homotopy equivalence of Kan complexes. Let us regard the object $Y$ as fixed. Since the functor $F$ is $\mathbb {K}$-cocontinuous, the collection of objects $\widehat{X} \in \widehat{\operatorname{\mathcal{C}}}$ which satisfy this condition is closed under the formation of $K$-indexed colimits for each $K \in \mathbb {K}$ (see Proposition 7.4.1.18). We may therefore assume without loss of generality that $\widehat{X} = h(X)$ for some $X \in \operatorname{\mathcal{C}}$. In this case, we can identify $\theta _{X,Y}$ with a left homotopy inverse of the natural map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( h(X), h(Y) )$, which is a homotopy equivalence because $h$ is fully faithful (Proposition 8.4.5.3).
$\square$