Enriched Categories

Let \( (\mathcal V, \otimes, 1) \) be a symmetric monoidal category. An \( \mathcal V \)-enriched category \( \mathcal C \) consists of

• a collection \( \Ob \mathcal C \) of objects,
• an hom-object \( \mathcal C(x,y) \) for all objects \( x, y \in \Ob \mathcal C \),
• an identity morphism \( 1_x \colon 1 \rightarrow \mathcal C(x,x) \) for every object \( x in \Ob \mathcal C \),
• a composition morphism \( \circ _{x,y,z} \colon \mathcal C(y,z) \otimes \mathcal C(x,y) \rightarrow \mathcal C(x,z) \) in \( \mathcal V \) for all objects \( x,y,z \in \Ob \mathcal C \)

such that

• composition is associative:

  \begin{equation*} \xymatrix{ \mathcal C(y,z) \otimes \mathcal C(x,y) \otimes \mathcal C(w,x) \ar@{->}[d]_{\circ \otimes 1} \ar@{->}[r]^{1 \otimes \circ} & \mathcal C(y,z) \otimes \mathcal C(w,y) \ar@{->}[d]^{\circ} \\ \mathcal C(x,z) \otimes \mathcal C(w,x) \ar@{->}[r]^{\circ} & \mathcal C(w,z)} \end{equation*}

  commutes in \(\mathcal V \).

• composition is unital:

  \begin{equation*} \xymatrix{ \mathcal C(x,y) \otimes I \ar@{->}[r]^{1 \otimes 1_x} \ar@{->}[rd]_{r} & \mathcal C(x,y) \otimes \mathcal C(x,x) \ar@{->}[d]^{\circ} \\ & \mathcal C(x,y)} \xymatrix{ I \otimes \mathcal C(x,y) \ar@{->}[r]^{1_y \otimes 1} \ar@{->}[rd]_{l} & \mathcal C(y,y) \otimes \mathcal C(x,y) \ar@{->}[d]^{\circ} \\ & \mathcal C(x,y)} \end{equation*}

  commute in \( \mathcal V \) where \( l \) and \( r \) are the left and right unitors of \( \mathcal V \).