Enriched category

Definition

Suppose ${\mathcal {M}}$ is a monoidal category. A category enriched in ${\mathcal {M}}$ is defined as a ${\mathcal {C}}$ equipped with the following data:

• A collection of objects $\operatorname {Ob} {\mathcal {C}}$.
• For any objects $A,B\in \operatorname {Ob} {\mathcal {C}}$, an object ${\mathcal {C}}(A,B)\in \operatorname {Ob} {\mathcal {M}}$.
• For any object $A\in \operatorname {Ob} {\mathcal {C}}$, a unit map $u_{A}$ from the unit object in ${\mathcal {M}}$ to ${\mathcal {C}}(A,A)$.
• For any objects $A,B,C\in \operatorname {Ob} {\mathcal {C}}$, a morphism in ${\mathcal {M}}$:

$\circ :{\mathcal {C}}(B,C)\otimes {\mathcal {C}}(A,B)\to {\mathcal {C}}(A,C)$

where $\otimes$ denotes the monoidal operation on ${\mathcal {M}}$.

satisfying the following conditions:

• For any $A,B,C,D\in \operatorname {Ob} {\mathcal {C}}$, with $f\in {\mathcal {C}}(A,B),g\in {\mathcal {C}}(B,C),h\in {\mathcal {C}}(C,D)$, we have $h\circ (g\circ f)=(h\circ g)\circ f$.
• For any $A,B\in \operatorname {Ob} {\mathcal {C}}$, the map $\circ :{\mathcal {C}}(B,B)\otimes {\mathcal {C}}(A,B)\to {\mathcal {C}}(A,B)$, composed with $u_{B}$ on the first coordinate, gives the identity map on the second coordinate. Similarly, the other way around. Need to make more precise -- Fill this in later