$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

8.3.4 Example: Extensions as Adjoints

Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is small and $\operatorname{\mathcal{D}}$ admits small colimits. It follows from Theorem that, up to isomorphism, the functor $f$ factors as a composition

\[ \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow {F} \operatorname{\mathcal{D}}, \]

where $h_{\bullet }$ denotes a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$ and $F$ is a functor which preserves small colimits. The functor $F$ is uniquely determined up to isomorphism: by virtue of Theorem, it can be characterized as a left Kan extension of $f$ along $h_{\bullet }$. Our goal in this section is to show that, if the $\infty $-category $\operatorname{\mathcal{D}}$ is locally small, then we can give another characterization of the functor $F$: it is left adjoint to the functor

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(\bullet ), \operatorname{\mathcal{D}}). \]

Proposition Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is essentially small and that $\operatorname{\mathcal{D}}$ is locally small, and let

\[ h_{\bullet }^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad h_{\bullet }^{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \]

be covariant Yoneda embeddings for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. Let $G$ denote the composite functor

\[ \operatorname{\mathcal{D}}\xrightarrow { h_{\bullet }^{\operatorname{\mathcal{D}}} } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow { \circ f^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}). \]

If $\operatorname{\mathcal{D}}$ admits small colimits, then the functor $G$ admits a left adjoint $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$. Moreover, the composition $F \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$ is isomorphic to $f$.

Following the convention of Remark, we will deduce Proposition from the following more general assertion:

Variant Let $\kappa $ be an uncountable regular cardinal and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, that $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits, and that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D)$ is essentially $\kappa $-small for every pair of objects $C \in \operatorname{\mathcal{C}}$, $D \in \operatorname{\mathcal{D}}$. Then the functor

\[ G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(\bullet ), D ) \]

admits a left adjoint $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ which preserves $\kappa $-small colimits. Moreover, the composite functor $\operatorname{\mathcal{C}}\xrightarrow { h^{\operatorname{\mathcal{C}}}_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {F} \operatorname{\mathcal{D}}$ is isomorphic to $f$.

Proof. We first prove the existence of the functor $F$. Fix a cardinal $\lambda $ of exponential cofinality $\geq \kappa $, so that the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda $-small (see Corollary By virtue of Proposition, it will suffice to show that for every functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, the composite functor

\[ \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow { \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }) }( \mathscr {F}, \bullet ) } \operatorname{\mathcal{S}}^{< \lambda } \]

is corepresentable by an object of $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits, the collection of functors $\mathscr {F}$ which satisfy this condition is closed under $\kappa $-small colimits (Remark Using Corollary, we can reduce to the case where the functor $\mathscr {F}$ is representable by an object $C \in \operatorname{\mathcal{C}}$. In this case, the object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })$ corepresents the evaluation functor $\operatorname{ev}_{C}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ (Remark It now follows from the definition of the functor $G$ that the composition $\operatorname{ev}_{C} \circ G$ is corepresentable by the object $f(C) \in \operatorname{\mathcal{D}}$.

Choose functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\epsilon : (F \circ G) \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which exhibits $F$ as a left adjoint to the functor $G$. It follows from Corollary that the functor $F$ preserves all colimits which exist in $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$; in particular, it preserves $\kappa $-small colimits. We will complete the proof by showing that $F \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$ is isomorphic to $f$.

For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, let $\alpha _{X,Y}$ denote the morphism of Kan complexes

\[ h_{Y}^{\operatorname{\mathcal{C}}}(X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(X), f(Y) ) = G( f(Y) )(X). \]

By virtue of Proposition, we can promote the construction $(X,Y) \mapsto \alpha _{X,Y}$ to a natural transformation of functors $\alpha : h_{\bullet }^{\operatorname{\mathcal{C}}} \rightarrow G \circ f$. Let $\beta $ denote a composition of the natural transformations

\[ F \circ h_{\bullet }^{\operatorname{\mathcal{C}}} \xrightarrow { F(\alpha ) } F \circ G \circ f \xrightarrow {\epsilon } \operatorname{id}_{\operatorname{\mathcal{D}}} \circ f = f. \]

We claim that $\beta $ is an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. By virtue of Theorem, it will suffice to show that $\beta $ induces an isomorphism $\beta _{X}: F( h_{X}^{\operatorname{\mathcal{C}}} ) \rightarrow f(X)$ for each object $X \in \operatorname{\mathcal{C}}$. Fix an object $D \in \operatorname{\mathcal{D}}$; we wish to show that precomposition with $\beta _{X}$ induces a homotopy equivalence of Kan complexes

\[ \beta _{X,D}: \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(X), D) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F( h_{X}^{\operatorname{\mathcal{C}}} ),D ) \simeq \operatorname{Hom}_{\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })}( h_{X}^{\operatorname{\mathcal{C}}}, G(D) ) \]

We conclude by observing that $\beta _{X,D}$ is left homotopy inverse to the morphism

\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })}( h_{X}^{\operatorname{\mathcal{C}}}, G(D) ) \rightarrow G(D)(X) \simeq \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(X), D) \]

given by evaluation at $\operatorname{id}_{X} \in h_{X}^{\operatorname{\mathcal{C}}}(X)$, which is a homotopy equivalence by virtue of Proposition $\square$

Example (Functoriality of the Presheaf Construction). Let $\kappa $ be an uncountable regular cardinal, let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and that $\operatorname{\mathcal{D}}$ is locally $\kappa $-small, and fix covariant Yoneda embeddings

\[ h_{\bullet }^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \quad \quad h_{\bullet }^{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ). \]

Let $G: \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa } )$ be given by precomposition with $f$. Then the functor $G$ admits a left adjoint $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa })$. Moreover, the diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{C}}} } \ar [d]^{f} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [d]^{F} \\ \operatorname{\mathcal{D}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{D}}} } & \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ). } \]

commutes up to isomorphism. This follows by applying Variant to the composite functor $(h_{\bullet }^{\operatorname{\mathcal{D}}} \circ f): \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.