Kerodon

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

Example 6.2.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between ordinary categories, and suppose we are given natural transformations $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$. Then the pair $(\eta , \epsilon )$ is an adjunction between $F$ and $G$ (in the sense of Definition 6.1.0.2) if and only if the induced maps

\[ \operatorname{N}_{\bullet }( \eta ): \operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \rightarrow \operatorname{N}_{\bullet }(G) \circ \operatorname{N}_{\bullet }(F) \quad \quad \operatorname{N}_{\bullet }( \epsilon ): \operatorname{N}_{\bullet }(F) \circ \operatorname{N}_{\bullet }(G) \rightarrow \operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) } \]

are compatible up to homotopy, in the sense of Definition 6.2.1.1. In particular:

  • A natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is the unit of an adjunction between functors of ordinary categories $F$ and $G$ if and only if $\operatorname{N}_{\bullet }( \eta ): \operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \rightarrow \operatorname{N}_{\bullet }(G) \circ \operatorname{N}_{\bullet }(F)$ is the unit of an adjunction between functors of $\infty $-categories $\operatorname{N}_{\bullet }(F)$ and $\operatorname{N}_{\bullet }(G)$.

  • A natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ is the counit of an adjunction between functors of ordinary categories $F$ and and $G$ if and only if $\operatorname{N}_{\bullet }( \epsilon ): \operatorname{N}_{\bullet }(F) \circ \operatorname{N}_{\bullet }(G ) \rightarrow \operatorname{id}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) }$ is the unit of an adjunction between functors of $\infty $-categories $\operatorname{N}_{\bullet }(F)$ and $\operatorname{N}_{\bullet }(G)$.

  • A functor of ordinary categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ admits a right adjoint $G$ if and only if the induced functor of $\infty $-categories $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ admits a right adjoint (in which case $\operatorname{N}_{\bullet }(G)$ is a right adjoint of $\operatorname{N}_{\bullet }(F)$).

  • A functor of ordinary categories $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $F$ if and only if the induced functor of $\infty $-categories $\operatorname{N}_{\bullet }(G): \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ admits a left adjoint (in which case $\operatorname{N}_{\bullet }(F)$ is a left adjoint of $\operatorname{N}_{\bullet }(G)$).