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,DC, D enriched in a monoidal category VV, an enriched functor F:CDF: C \to D consists of

  • A function F 0:C 0D 0F_0: C_0 \to D_0 between the underlying collections of objects;

  • A (C 0×C 0)(C_0 \times C_0)-indexed collection of morphisms of VV,

    F x,y:C(x,y)D(F 0x,F 0y)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 CC and DD;

  • such that the following diagrams commute for all a,b,cC 0a, b, c \in C_0:

    • respect for composition:

      C(b,c)C(a,b) a,b,c C(a,c) F b,cF a,b F a,c D(F 0(b),F 0(c))D(F 0(a),F 0(b)) F 0(a),F 0(b),F 0(c) D(F 0(a),F 0(c)) \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:

      I j a j F 0(a) C(a,a) F a,a D(F 0(a),F 0(a)) \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)) }

Properties

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)