Kerodon

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

Definition 2.1.1.1. Let $\operatorname{\mathcal{C}}$ be a category. A nonunital strict monoidal structure on $\operatorname{\mathcal{C}}$ is a functor

\[ \otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\quad \quad (X,Y) \mapsto X \otimes Y \]

which is strictly associative in the following sense:

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, we have an equality $X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z$ (as objects of $\operatorname{\mathcal{C}}$).

  • For every triple of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$, $h: Z \rightarrow Z'$, we have an equality

    \[ f \otimes (g \otimes h) = (f \otimes g) \otimes h \]

    of morphisms in $\operatorname{\mathcal{C}}$ from the object $X \otimes (Y \otimes Z) = (X \otimes Y) \otimes Z$ to the object $X' \otimes (Y' \otimes Z') = (X' \otimes Y') \otimes Z'$.

A nonunital strict monoidal category is a pair $(\operatorname{\mathcal{C}}, \otimes )$, where $\operatorname{\mathcal{C}}$ is a category and $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a nonunital strict monoidal structure on $\operatorname{\mathcal{C}}$.