Kerodon

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

Example 2.1.3.5 (The Reverse of a Monoidal Structure). 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 we can equip $\operatorname{\mathcal{C}}$ with another nonunital monoidal structure $(\otimes ^{\operatorname{rev}}, \{ \alpha ^{\operatorname{rev}}_{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$, defined as follows:

  • The tensor product functor $\otimes ^{\operatorname{rev}}: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is given on objects by the formula $X \otimes ^{\operatorname{rev}} Y = Y \otimes X$ (and similarly on morphisms).

  • The associativity constraint on $\otimes ^{\operatorname{rev}}$ is given by the formula $\alpha ^{\operatorname{rev}}_{X,Y,Z} = \alpha _{Z,Y,X}^{-1}$.

We will refer to the nonunital monoidal structure $( \otimes ^{\operatorname{rev}}, \{ \alpha ^{\operatorname{rev}}_{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$ as the reverse of the nonunital monoidal structure $(\otimes , \{ \alpha _{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$. In this case, we will write $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ to denote the nonunital monoidal category whose underlying category is $\operatorname{\mathcal{C}}$, equipped with the nonunital monoidal structure $(\otimes ^{\operatorname{rev}}, \{ \alpha ^{\operatorname{rev}}_{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$.

If the nonunital monoidal category $\operatorname{\mathcal{C}}$ is equipped with a unit structure $( \mathbf{1}, \upsilon )$, then we can also regard $( \mathbf{1}, \upsilon )$ as a unit structure for the nonunital monoidal category $\operatorname{\mathcal{C}}^{\operatorname{rev}}$. In other words, if $\operatorname{\mathcal{C}}$ is a monoidal category, then we can regard $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ as a monoidal category (having the same underlying category and unit object, but “reversed” tensor product).