# Kerodon

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

Example 2.1.4.8 (The Left Regular Representation). Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category and let $\operatorname{End}(\operatorname{\mathcal{C}}) = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ be the category of functors from $\operatorname{\mathcal{C}}$ to itself, endowed with the strict monoidal structure of Example 2.1.1.4. For each object $X \in \operatorname{\mathcal{C}}$, let $\ell _{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denote the functor given on objects by the formula $\ell _{X}(Y) = X \otimes Y$. The construction $X \mapsto \ell _{X}$ then determines a functor $\ell : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there is a natural isomorphism $\mu _{X,Y}: \ell _{X} \circ \ell _{Y} \xrightarrow {\sim } \ell _{X \otimes Y}$, whose value on an object $Z \in \operatorname{\mathcal{C}}$ is given by the associativity constraint

$(\ell _{X} \circ \ell _{Y})(Z) = X \otimes (Y \otimes Z) \xrightarrow { \alpha _{X,Y,Z} } (X \otimes Y) \otimes Z = \ell _{X \otimes Y}(Z).$

Then $\mu = \{ \mu _{X,Y} \} _{X,Y}$ is a nonunital monoidal structure on the functor $X \mapsto \ell _{X}$: property $(a)$ of Definition 2.1.4.3 follows from the naturality of the associativity constraint on $\operatorname{\mathcal{C}}$, and property $(b)$ is a reformulation of the pentagon identity.