Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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). $\square$