8.4.4 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 8.4.0.3 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 8.4.3.6, 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
\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(\bullet ), D ). \]
Proposition 8.4.4.1. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is essentially small, that $\operatorname{\mathcal{D}}$ is cocomplete and 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
\[ \operatorname{\mathcal{D}}\xrightarrow { h_{\bullet }^{\operatorname{\mathcal{D}}} } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow { \circ f^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}). \]
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 $f$.
Corollary 8.4.4.2. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is cocomplete and locally small, and let $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $F$ preserves small colimits.
- $(2)$
The functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.
Proof.
Assume that $F$ preserves small colimits; we will show that it admits a right adjoint (the reverse implication follows from Corollary 7.1.3.21). Choose covariant Yoneda embeddings
\[ h_{\bullet }^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad h_{\bullet }^{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \]
set $f = F \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$, and let $G$ denote the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow { h_{\bullet }^{\operatorname{\mathcal{D}}} } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow { \circ f^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}). \]
It follows from Proposition 8.4.4.1 that $G$ admits a left adjoint $F': \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ such that $F' \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$ is isomorphic to $f = F \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$. Since the functor $F'$ also preserves small colimits (Corollary 7.1.3.21), Theorem 8.4.0.3 implies that it is isomorphic to $F$. It follows that $G$ is also a right adjoint of $F$.
$\square$
Corollary 8.4.4.3. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category and let $\mathscr {F}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $\mathscr {F}$ admits a left adjoint.
- $(2)$
The functor $\mathscr {F}$ is representable by an object of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$.
- $(3)$
The functor $\mathscr {F}$ preserves small limits.
Proof.
Since the identity functor $\operatorname{id}: \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable (by the object $\Delta ^0 \in \operatorname{\mathcal{S}}$), the implication $(1) \Rightarrow (2)$ follows from Corollary 6.2.4.2. The implication $(2) \Rightarrow (3)$ is a special case of Corollary 7.4.5.17, and the implication $(3) \Rightarrow (1)$ follows by applying Corollary 8.4.4.2 to to the opposite functor $\mathscr {F}^{\operatorname{op}}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}^{\operatorname{op}}$.
$\square$
Following the convention of Remark 4.7.0.5, we will deduce Proposition 8.4.4.1 from the following more general assertion:
Variant 8.4.4.4. 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 4.7.8.8). By virtue of Proposition 6.2.4.1, 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 8.3.3.16). Using Corollary 8.4.3.9, 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 8.3.1.5). 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 7.1.3.21 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 Corollary 8.3.5.8, 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 4.4.4.4, 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 8.3.1.3
$\square$
Example 8.4.4.5 (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 8.4.4.4 to the composite functor $(h_{\bullet }^{\operatorname{\mathcal{D}}} \circ f): \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.