Kerodon

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

Example 2.1.3.4 (The Opposite of a Monoidal Category). Let $\operatorname{\mathcal{C}}$ be a category equipped with a nonunital monoidal structure $( \otimes , \{ \alpha _{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$. Then the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ inherits a nonunital monoidal structure, which can be described concretely as follows:

  • The tensor product on $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is obtained from the tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ by passing to opposite categories.

  • Let $X$, $Y$, and $Z$ be objects of $\operatorname{\mathcal{C}}$, and let us write $X^{\operatorname{op}}$, $Y^{\operatorname{op}}$, and $Z^{\operatorname{op}}$ for the corresponding objects of $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then the associativity constraint $\alpha _{ X^{\operatorname{op}}, Y^{\operatorname{op}}, Z^{\operatorname{op}}}$ for $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is the inverse of the associativity constraint $\alpha _{X,Y,Z}$ for $\operatorname{\mathcal{C}}$.

If the nonunital monoidal category $\operatorname{\mathcal{C}}$ is equipped with a unit structure $( \mathbf{1}, \upsilon )$, then we can regard $( \mathbf{1}^{\operatorname{op}}, \upsilon ^{-1} )$ as a unit structure for the nonunital monoidal category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. In particular, every monoidal structure on a category $\operatorname{\mathcal{C}}$ determines a monoidal structure on the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.