nLab
enriched functor

Idea

Enriched functors are used in place of functors in enriched category theory: like functors they send objects to objects, but instead of mapping hom-sets to hom-sets they assign morphisms in the enriching category between hom-objects, while being compatible with composition and units in the obvious way.

Definition

Given two categories $C,D$ enriched in a monoidal category $V$, an enriched functor $F:C\to D$ consists of

• A function $F_0:C_0\to D_0$ between the underlying collections of objects;

• A $(C_0\times C_0)$-indexed collection of morphisms of $V$,

  $$F_{x,y}:C(x,y)\to D(F_{0}x,F_{0}y)$$F_{x, y}: C(x, y) \to D(F_0x, F_0y)

  where $C(x,y)$ denotes the hom-object ${\mathrm{hom}}_C(x,y)$ in $V$, compatible with the enriched identities and compositions of $C$ and $D$;

• such that the following diagrams commute for all $a,b,c\in C_0$:

  • respect for composition:

    $$\begin{array}{ccc}C(b,c)\otimes C(a,b)& \stackrel{{\circ }_{a,b,c}}{\to }& C(a,c)\\ {\downarrow }^{F_{b,c}\otimes F_{a,b}}& & {\downarrow }^{F_{a,c}}\\ D(F_0(b),F_0(c))\otimes D(F_0(a),F_0(b))& \stackrel{{\circ }_{F_0(a),F_0(b),F_0(c)}}{\to }& D(F_0(a),F_0(c))\end{array}$$\array{ C(b,c) \otimes C(a,b) &\stackrel{\circ_{a,b,c}}{\to}& C(a,c) \\ \downarrow^{F_{b,c} \otimes F_{a,b}} && \downarrow^{F_{a,c}} \\ D(F_0(b), F_0(c)) \otimes D(F_0(a), F_0(b)) &\stackrel{\circ_{F_0(a),F_0(b), F_0(c)}}{\to}& D(F_0(a), F_0(c)) }

  • respect for units:

    $$\begin{array}{ccc}& & I\\ & {}^{j_a}\swarrow & & {\searrow }^{j_{F_0(a)}}\\ C(a,a)& & \stackrel{F_{a,a}}{\to }& & D(F_0(a),F_0(a))\end{array}$$\array{ && I \\ & {}^{j_a}\swarrow && \searrow^{j_{F_0(a)}} \\ C(a,a) &&\stackrel{F_{a,a}}{\to}&& D(F_0(a), F_0(a)) }

Remarks

• Analogous to the functor category for ordinary functors between ordinary categories, enriched functors between two enriched categories form an enriched functor category.

References

The standard reference on enriched category theory is

• Max Kelly, Basic Concepts of Enriched Category Theory (web)