# nLab enriched functor

enriched category theory

# Contents

## 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 $\left({C}_{0}×{C}_{0}\right)$-indexed collection of morphisms of $V$,

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

where $C\left(x,y\right)$ denotes the hom-object ${\mathrm{hom}}_{C}\left(x,y\right)$ 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:

$\begin{array}{ccc}C\left(b,c\right)\otimes C\left(a,b\right)& \stackrel{{\circ }_{a,b,c}}{\to }& C\left(a,c\right)\\ {↓}^{{F}_{b,c}\otimes {F}_{a,b}}& & {↓}^{{F}_{a,c}}\\ D\left({F}_{0}\left(b\right),{F}_{0}\left(c\right)\right)\otimes D\left({F}_{0}\left(a\right),{F}_{0}\left(b\right)\right)& \stackrel{{\circ }_{{F}_{0}\left(a\right),{F}_{0}\left(b\right),{F}_{0}\left(c\right)}}{\to }& D\left({F}_{0}\left(a\right),{F}_{0}\left(c\right)\right)\end{array}$\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:

$\begin{array}{ccc}& & I\\ & {}^{{j}_{a}}↙& & {↘}^{{j}_{{F}_{0}\left(a\right)}}\\ C\left(a,a\right)& & \stackrel{{F}_{a,a}}{\to }& & D\left({F}_{0}\left(a\right),{F}_{0}\left(a\right)\right)\end{array}$\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)) }

## References

The standard reference on enriched category theory is

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

Revised on June 18, 2010 08:17:10 by Urs Schreiber (87.212.203.135)