# Kerodon

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

Example 2.1.7.3. Let $\operatorname{\mathcal{A}}$ be a monoidal category. If $\operatorname{\mathcal{C}}$ is a category enriched over $\operatorname{\mathcal{A}}$ and $X$ is an object of $\operatorname{\mathcal{C}}$, then the composition law

$c_{X,X,X}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$

exhibits $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ as an algebra object of $\operatorname{\mathcal{A}}$, in the sense of Example 2.1.5.21. Moreover, this construction induces a bijection

$\{ \text{\operatorname{\mathcal{A}}-Enriched Categories \operatorname{\mathcal{C}} with \operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X \} } \} \simeq \{ \text{Algebra objects of \operatorname{\mathcal{A}}} \} .$

Consequently, the theory of enriched categories can be regarded as a generalization of the theory of associative algebras (See Example 2.1.7.14 for a more precise statement).