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6.2.4 Existence of Adjoints

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Suppose that $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. For each object $D \in \operatorname{\mathcal{D}}$, the value $G(D) \in \operatorname{\mathcal{C}}$ is determined, up to canonical isomorphism, by the property that it represents the functor $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D)$: that is, there exists a bijection $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D ) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D))$ which depends functorially on $C$. This observation has a converse: if, for every object $D \in \operatorname{\mathcal{D}}$, the functor

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \]

is representable by an object of $\operatorname{\mathcal{C}}$, then the functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ (Corollary 6.2.4.6). Our goal in this section is to prove an analogous statement in the $\infty $-categorical setting. First, we need some terminology.

Definition 6.2.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the homotopy category of $\operatorname{\mathcal{C}}$, which we regard as enriched over the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$ (Construction 4.6.7.13). Let $U: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ be an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor. We will say that $U$ is representable if there exists an object $X \in \operatorname{\mathcal{C}}$ and a vertex $u \in U(X)$ with the following property: for every object $C \in \operatorname{\mathcal{C}}$, the composite map

\[ \{ u \} \times \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X ) \hookrightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X ) \times U(X) \rightarrow U(C) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. In this case, we say that the object $X \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ represents the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $U$.

We say that an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $V: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is corepresentable if there exists an object $Y \in \operatorname{\mathcal{C}}$ and a vertex $v \in V(Y)$ with the property that, for every object $C \in \operatorname{\mathcal{C}}$, the composite map

\[ \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, C ) \times \{ v \} \hookrightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, C ) \times V(Y) \rightarrow V(C) \]

is an isomorphism in $\mathrm{h} \mathit{\operatorname{Kan}}$. In this case, we will say that the object $Y \in \operatorname{\mathcal{C}}$ corepresents the functor $V$.

Remark 6.2.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $U: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ be a functor of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories. Then the construction $C \mapsto \pi _0( U(C) )$ determines a functor from $\mathrm{h} \mathit{\operatorname{Kan}}$ to the category of sets, which we will denote by $\pi _0(U)$. If $X \in \operatorname{\mathcal{C}}$ represents the functor $U$ (in the sense of Definition 6.2.4.1), then it also represents the set-valued functor $\pi _0(U)$: that is, there are bijections $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X) \simeq \pi _0( U(C) )$ which depend functorially on $C$. In particular, if the functor $U$ is representable (in the sense of Definition 6.2.4.1), then the functor $\pi _0(U)$ is also representable. Beware that the converse is false in general.

Remark 6.2.4.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Every object $X \in \operatorname{\mathcal{C}}$ determines an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $h_{X}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$, given on objects by the formula $h_{X}(C) = \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$. An arbitrary $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $U: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ is represented by $X$ (in the sense of Definition 6.2.4.1) if and only if it is isomorphic to $h_{X}$.

Example 6.2.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. For every object $D \in \operatorname{\mathcal{D}}$, the construction $(C \in \operatorname{\mathcal{C}}) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D)$ determines an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor from $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}}$ to the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. If the functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, then this functor is representable by the object $G(D) \in \operatorname{\mathcal{C}}$ (see Proposition 6.2.1.17).

Proposition 6.2.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:

$(1)$

The functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

$(2)$

For every object $D \in \operatorname{\mathcal{D}}$, the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \]

is representable by an object of $\operatorname{\mathcal{C}}$.

Proof. We first show that $(1)$ implies $(2)$. Suppose that there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which exhibits $G$ as a right adjoint of $F$. Let $D$ be an object of $\operatorname{\mathcal{D}}$, so that $\epsilon $ determines a morphism $\epsilon _ D: (F \circ G)(D) \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{D}}$. Proposition 6.2.1.17 guarantees that, for every object $C \in \operatorname{\mathcal{C}}$, the natural map

\begin{eqnarray*} \{ \epsilon _ D \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, G(D) ) & \hookrightarrow & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (F \circ G)(D), D) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(D) )\\ & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( (F \circ G)(D), D) \times \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), (F \circ G)(D) ) \\ & \xrightarrow {\circ } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \end{eqnarray*}

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. It follows that $G(D)$ represents the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D)$.

We now show that $(2)$ implies $(1)$. Choose a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ having fibers $\operatorname{\mathcal{C}}\simeq \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$, $\operatorname{\mathcal{D}}\simeq \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$, and for which the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is given by covariant transport along the nondegenerate edge of $\Delta ^1$ (for example, we can take $\operatorname{\mathcal{E}}$ to be the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$: see Proposition 5.2.4.15). Assume that condition $(2)$ is satisfied. We will show that $U$ is a cartesian fibration, so that $F$ admits a right adjoint by virtue of Proposition 6.2.3.4. Fix an object $D \in \operatorname{\mathcal{D}}$; we wish to show that there exists an object $X \in \operatorname{\mathcal{C}}$ and a $U$-cartesian morphism $f: X \rightarrow D$ in $\operatorname{\mathcal{E}}$. For each $C \in \operatorname{\mathcal{C}}$, we have a canonical isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \simeq \operatorname{Hom}_{\operatorname{\mathcal{E}}}(C,D)$ in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Consequently, condition $(2)$ guarantees that the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{E}}}(C,D) \]

is representable. In other words, there exists an object $X \in \operatorname{\mathcal{C}}$ and a morphism $f: X \rightarrow D$ with the property that, for every object $C \in \operatorname{\mathcal{C}}$, composition with the homotopy class $[f]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(C, D)$. Applying Corollary 5.1.2.3, we conclude that $f: X \rightarrow D$ is $U$-cartesian. $\square$

Corollary 6.2.4.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. The following conditions are equivalent:

$(1)$

The functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

$(2)$

For every object $D \in \operatorname{\mathcal{D}}$, the set-valued functor

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \]

is representable by an object of $\operatorname{\mathcal{C}}$.

Corollary 6.2.4.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between $\infty $-categories, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation. The following conditions are equivalent:

$(1)$

The natural transformation $\eta $ is the unit of an adjunction between $F$ and $G$.

$(2)$

For every pair of objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( (G \circ F)(C), G(D) ) \xrightarrow { \circ [ \eta _ C]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, G(D) ) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

$(3)$

The functor $F$ admits a right adjoint. Moreover, for every pair of objects $C \in \operatorname{\mathcal{C}}$ and $D \in \operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{D}}}}( F(C), D) \xrightarrow {G} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( (G \circ F)(C), G(D) ) \xrightarrow { \circ [ \eta _ C]} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( C, G(D) ) \]

is a bijection.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 6.2.1.17. Suppose that condition $(2)$ is satisfied. Then, for every object $D \in \operatorname{\mathcal{D}}$, the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D)$ is representable by the object $G(D) \in \operatorname{\mathcal{D}}$. Applying Proposition 6.2.4.5, we deduce that $F$ admits a right adjoint, so that condition $(3)$ is satisfied. We now complete the proof by showing that $(3) \Rightarrow (1)$. Note that, if condition $(3)$ is satisfied, then the natural transformation $\eta $ exhibits $\mathrm{h} \mathit{G}: \mathrm{h} \mathit{\operatorname{\mathcal{D}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as a right adjoint of the functor $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ (see Variant 6.1.2.11). Invoking Proposition 6.2.1.14, we deduce that $\eta $ is the unit of an adjunction between $F$ and $G$. $\square$