# nLab enriched functor

### Context

#### Enriched category theory

Could not include enriched category theory - contents

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

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

[where denotes the hom-object in ], 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:

$\array{ C(b,c) \otimes C(a,b) &\stackrel{\circ_{a,b,c}}{\to}& C(a,c) \\ \downarrow^{F_{b,c} \otimes F_{a,b}} && \downarrow^{F_{a,c}} \\ D(F_0(b), F_0(c)) \otimes D(F_0(a), F_0(b)) &\stackrel{\circ_{F_0(a),F_0(b), F_0(c)}}{\to}& D(F_0(a), F_0(c)) }$
• respect for units:

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

## References

The standard reference on enriched category theory is

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

Revised on March 11, 2014 02:53:21 by Urs Schreiber (89.204.155.115)