Enriched category over a monoidal base

Let $(\mathcal{V},\otimes ,\mathbf{1},\alpha ,\lambda ,\rho )$ be a monoidal category (Monoidal category).

A $\mathcal{V}$-enriched category, or $\mathcal{V}$-category, consists of

• a set $\mathrm{Ob}(\mathcal{A})$ of objects;
• for every ordered pair $A,B\in \mathrm{Ob}(\mathcal{A})$, an object $\mathcal{A}(A,B)$ of $\mathcal{V}$;
• for every triple $A,B,C$, a composition morphism $M_{A,B,C}:\mathcal{A}(B,C)\otimes \mathcal{A}(A,B)⟶\mathcal{A}(A,C);$
• for every object $A$, an identity morphism $j_A:\mathbf{1}⟶\mathcal{A}(A,A);$

such that the following diagrams commute.

For every $A,B,C,D$, the two composites

$((\mathcal{A}(C,D)\otimes \mathcal{A}(B,C))\otimes \mathcal{A}(A,B))\rightrightarrows \mathcal{A}(A,D)$

agree, namely

$M_{A,B,D}\circ (M_{B,C,D}\otimes 1)=M_{A,C,D}\circ (1\otimes M_{A,B,C})\circ {\alpha }_{\mathcal{A}(C,D),\mathcal{A}(B,C),\mathcal{A}(A,B)}\mathrm{.}$

For every $A,B$, the left and right unit laws hold:

$M_{A,B,B}\circ (j_B\otimes 1)\circ {\lambda }_{\mathcal{A}(A,B)}^{-1}=1_{\mathcal{A}(A,B)},$

$M_{A,A,B}\circ (1\otimes j_A)\circ {\rho }_{\mathcal{A}(A,B)}^{-1}=1_{\mathcal{A}(A,B)}\mathrm{.}$

The order of the two hom-objects in the tensor product is part of the definition: the factor $\mathcal{A}(B,C)$ stands on the left and $\mathcal{A}(A,B)$ on the right because composition is "first $A\to B$, then $B\to C$".

The object collection is required here to be a set. This is the standing size convention for the present page, and it is what makes later statements about $\mathcal{V}\text{-}\mathbf{C}\mathbf{a}\mathbf{t}$ honest inside the library's current foundations.