Kerodon

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

Example 2.1.7.14. Let $\operatorname{\mathcal{A}}$ be a monoidal category, let $A$ be an algebra object of $\operatorname{\mathcal{A}}$, which we can identify with an $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}_{A}$ having a single object $X$ (Example 2.1.7.3). For any $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{D}}$ containing an object $Y$, we have a canonical bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \text{$\operatorname{\mathcal{A}}$-Enriched Functors $F: \operatorname{\mathcal{C}}_{A} \rightarrow \operatorname{\mathcal{D}}$ with $F(X) = Y$} \} \ar [d]^{\sim } \\ \{ \text{Algebra homomorphisms $A \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(Y,Y)$} \} . } \]

In particular, if $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}_{B}$ for some other algebra object $B \in \operatorname{Alg}(\operatorname{\mathcal{D}})$, we obtain a bijection

\[ \operatorname{Hom}_{ \operatorname{Cat}(\operatorname{\mathcal{A}}) }( \operatorname{\mathcal{C}}_{A}, \operatorname{\mathcal{C}}_{B} ) \simeq \operatorname{Hom}_{ \operatorname{Alg}(\operatorname{\mathcal{A}})}( A, B). \]

In other words, the construction $A \mapsto \operatorname{\mathcal{C}}_{A}$ induces a fully faithful embedding $\operatorname{Alg}(\operatorname{\mathcal{A}}) \rightarrow \operatorname{Cat}(\operatorname{\mathcal{A}})$, whose essential image is spanned by those $\operatorname{\mathcal{A}}$-enriched categories having a single object.