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6.2.4 Local Existence Criterion

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

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}\quad \quad Z \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Z) ) \]

is corepresentable by an object of $\operatorname{\mathcal{D}}$, then the functor $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ (Corollary 6.2.4.4). Our goal in this section is to establish a counterpart of this criterion in the $\infty $-categorical setting. We begin with a simple observation.

Proposition 6.2.4.1. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Then $G$ admits a left adjoint if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the following condition is satisfied:

$(\ast _{X})$

There exists an object $Y \in \operatorname{\mathcal{D}}$ and a morphism $u: X \rightarrow G(Y)$ in $\operatorname{\mathcal{C}}$ such that, for every object $Z \in \operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(Y,Z) \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(Y), G(Z) ) \xrightarrow { \circ [u]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Z) ) \]

is a homotopy equivalence of Kan complexes.

Proof. We first prove necessity. Suppose that there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which exhibits $F$ as a left adjoint of $G$. Fix an object $X \in \operatorname{\mathcal{C}}$ and set $Y = F(X)$. Then $\eta $ determines a morphism $\eta _{X}: X \rightarrow G(Y)$ which satisfies the requirement of condition $(\ast _ X)$ (Proposition 6.2.1.17).

We now prove sufficiency. Let $\operatorname{\mathcal{E}}$ denote the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ be the cartesian fibration of Proposition 5.2.3.15. Let us abuse notation by identifying the fibers $\{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ with $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. Fix an object $X \in \operatorname{\mathcal{C}}$, and suppose that there exists an object $Y \in \operatorname{\mathcal{D}}$ together with a morphism $u: X \rightarrow G(Y)$ satisfying the requirement of condition $(\ast _ X)$. Then we can identify $u$ with a morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{E}}$. Our assumption on $u$ guarantees that the morphism $f$ is $U$-cocartesian (see Corollary 5.1.2.3). Consequently, if condition $(\ast _ X)$ is satisfied for every object $X \in \operatorname{\mathcal{C}}$, then $U$ is a cocartesian fibration. Applying Proposition 6.2.3.4, we conclude that $G$ admits a left adjoint. $\square$

Corollary 6.2.4.2. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. The following conditions are equivalent:

$(1)$

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

$(2)$

For every left fibration $\widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$, if the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ has an initial object, then the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}}$ also has an initial object.

$(3)$

For every object $X \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ has an initial object.

$(4)$

For every corepresentable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\lambda : \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$, the composite functor

\[ \mathrm{h} \mathit{\operatorname{\mathcal{D}}} \xrightarrow { \mathrm{h} \mathit{G} } \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \xrightarrow { \lambda } \mathrm{h} \mathit{\operatorname{Kan}} \]

is also corepresentable (in the sense of Definition 5.7.6.10).

$(5)$

For every corepresentable functor $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ of $\infty $-categories, the composite functor

\[ \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{C}}\xrightarrow { \lambda } \operatorname{\mathcal{S}} \]

is also corepresentable (in the sense of Definition 5.7.6.1).

Proof. The equivalence $(1) \Leftrightarrow (4)$ is a reformulation of Proposition 6.2.4.1. The implication $(2) \Rightarrow (3)$ is immediate. To see that $(3)$ implies $(4)$, we observe that if $\lambda : \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ be an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor which is corepresentable by an object $X \in \operatorname{\mathcal{C}}$. then $\lambda \circ \mathrm{h} \mathit{G}$ is isomorphic to the enriched homotopy transport representation of the left fibration $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{D}}$. If $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ has an initial object, then this functor is corepresentable by virtue of Proposition 5.7.6.21.

The implication $(4) \Rightarrow (5)$ follows from Remark 5.7.6.11. We will complete the proof by showing that $(5)$ implies $(2)$. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a left fibration, and let $\operatorname{Tr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $U$ (see Definition 5.7.5.1). If $\widetilde{\operatorname{\mathcal{C}}}$ has an initial object, then the functor $\operatorname{Tr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}}$ is corepresentable (Proposition 5.7.6.21). Assumption $(3)$ then guarantees that the functor $\operatorname{Tr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}} \circ G$ is also corepresentable. Identifying $\operatorname{Tr}_{ \widetilde{\operatorname{\mathcal{C}}} / \operatorname{\mathcal{C}}} \circ G$ with the covariant transport representation of the left fibration $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$, we see that the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}}$ also has an initial object (Proposition 5.7.6.21). $\square$

Remark 6.2.4.3. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories which satisfies the equivalent conditions of Corollary 6.2.4.2, so that $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. For each object $X \in \operatorname{\mathcal{C}}$, the value $F(X) \in \operatorname{\mathcal{D}}$ admits several characterizations:

  • The object $F(X)$ corepresents the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

    \[ \mathrm{h} \mathit{\operatorname{\mathcal{D}}} \xrightarrow { \mathrm{h} \mathit{G} } \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \xrightarrow { \underline{\operatorname{Hom}}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, \bullet ) } \mathrm{h} \mathit{\operatorname{Kan}}. \]
  • The object $F(X)$ corepresents the functor of $\infty $-categories

    \[ \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{C}}\xrightarrow { h^{X} } \operatorname{\mathcal{S}}, \]

    where $h^ X$ is the functor corepresented by $X$.

  • The object $F(X)$ is the image in $\operatorname{\mathcal{D}}$ of an initial object of the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$.

Corollary 6.2.4.4. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between ordinary categories. The following conditions are equivalent:

$(1)$

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

$(2)$

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

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}\quad \quad Z \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Z) ) \]

is corepresentable.

Corollary 6.2.4.5. 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 $X \in \operatorname{\mathcal{C}}$ and $Y \in \operatorname{\mathcal{D}}$, the composite map

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

is a homotopy equivalence of Kan complexes.

$(3)$

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

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

is a bijection.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 6.2.1.17, the implication $(2) \Rightarrow (3)$ follows from Proposition 6.2.4.1. We will 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 $G$. $\square$

Corollary 6.2.4.6. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories and let $u: K \rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, so that $G$ induces a functor of coslice $\infty $-categories $G': \operatorname{\mathcal{D}}_{/u} \rightarrow \operatorname{\mathcal{C}}_{ / (G \circ u)}$. If the functor $G$ admits a left adjoint, then the functor $G'$ also admits a left adjoint.

Proof. We will use the criterion of Corollary 6.2.4.2. Fix an object $\overline{X} \in \operatorname{\mathcal{C}}_{ / (G \circ u) }$; we wish to show that the $\infty $-category

\[ \operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}_{/u} \times _{ \operatorname{\mathcal{C}}_{ / G \circ u} } ( \operatorname{\mathcal{C}}_{ / (G \circ u) } )_{\overline{X} / } \]

has an initial object. Let $X$ denote the image of $\overline{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Unwinding the definitions, we can identify $\overline{X}$ with a morphism of simplicial sets $\overline{u}: K \rightarrow (\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/})$, and $\operatorname{\mathcal{E}}$ with the slice $\infty $-category $(\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/})_{ / \overline{u} }$. Since $G$ admits a left adjoint, the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$ has an initial object (Corollary 6.2.4.2). Applying Corollary 7.1.3.20, we conclude that $\operatorname{\mathcal{E}}$ also has an initial object. $\square$