Kerodon

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

Definition 2.1.7.1. Let $\operatorname{\mathcal{A}}$ be a monoidal category with unit object $\mathbf{1}$. An $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ consists of the following data:

$(1)$

A collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$, whose elements we refer to as objects of $\operatorname{\mathcal{C}}$. We will often abuse notation by writing $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an element of $\operatorname{Ob}(\operatorname{\mathcal{C}})$.

$(2)$

For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, an object $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ of the monoidal category $\operatorname{\mathcal{A}}$.

$(3)$

For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism

\[ c_{Z,Y,X}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \]

in the category $\operatorname{\mathcal{A}}$, which we will refer to as the composition law.

$(4)$

For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism $e_{X}: \mathbf{1} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ in the category $\operatorname{\mathcal{A}}$, which we refer to as the identity of $X$.

These data are required to satisfy the following conditions:

$(A)$

For every quadruple of objects $W,X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the diagram

\[ \xymatrix { & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,Y) \ar [dd]^-{ c_{Z,Y,W} } \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes ( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,X) ) \ar [dd]^-{\alpha } \ar [ur]^{ \operatorname{id}\otimes c_{Y,X,W} } & \\ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,Z ) \\ ( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X,Y) ) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,X) \ar [dr]_{ c_{Z,Y,X} \otimes \operatorname{id}} & \\ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,X) \ar [uu]_-{ c_{Z,X,W} } } \]

commutes. Here $\alpha $ denotes the associativity constraint on the monoidal category $\operatorname{\mathcal{A}}$.

$(U)$

For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the diagrams

\[ \xymatrix@C =50pt{ \mathbf{1} \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [rr]^-{e_{Y} \otimes \operatorname{id}} \ar [dr]_-{\lambda } & & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Y) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [dl]^-{ c_{Y,Y,X} } \\ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) & } \]
\[ \xymatrix@C =50pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \otimes \mathbf{1} \ar [rr]^-{\operatorname{id}\otimes e_ X} \ar [dr]_-{\rho } & & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \ar [dl]^-{ c_{Y,X,X} } \\ & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) & } \]

commute, where $\lambda $ and $\rho $ denote the left and right unit constraints on $\operatorname{\mathcal{A}}$ (see Construction 2.1.2.17).