Kerodon

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

Remark 6.1.6.4. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$ and morphisms

\[ \operatorname{coev}: {\bf 1} \rightarrow Y \otimes X \quad \quad \operatorname{ev}: X \otimes Y \rightarrow {\bf 1}. \]

Then:

  • The pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum in the monoidal category $\operatorname{\mathcal{C}}$ if and only if it is a duality datum in the reverse monoidal category $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ of Example 2.1.3.5. Note that passage to the reverse monoidal category reverses the roles of $X$ and $Y$: if $X$ is the left dual of $Y$ in the monoidal category $\operatorname{\mathcal{C}}$, then it is the right dual of $Y$ in the monoidal category $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ (and vice-versa).

  • The pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum in $\operatorname{\mathcal{C}}$ if and only if the pair $(\operatorname{ev}^{\operatorname{op}}, \operatorname{coev}^{\operatorname{op}})$ is a duality datum in the opposite monoidal category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (see Example 2.1.3.4). Note that passage to the opposite monoidal category reverses the roles of evaluation and coevaluation: $\operatorname{ev}^{\operatorname{op}}$ is the coevaluation morphism for the duality datum $(\operatorname{ev}^{\operatorname{op}}, \operatorname{coev}^{\operatorname{op}})$, while $\operatorname{coev}^{\operatorname{op}}$ is the evaluation morphism. Similarly, if $X$ is the left dual of $Y$ in the monoidal category $\operatorname{\mathcal{C}}$, then it is the right dual of $Y$ in the opposite monoidal category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (and vice-versa).