nLab
enriched natural transformation

Context

Enriched category theory

Could not include enriched category theory - contents

Contents

Idea

An enriched natural transformation is the appropriate notion of homomorphism between enriched functors, the analog in enriched category theory of the ordinary notion of natural transformation in ordinary category theory.

Definition

Let CC and DD be categories enriched in a monoidal category VV, and let F,G:CDF, G \colon C \to D be enriched functors. We abbreviate hom-objects hom C(c,d)\hom_C(c, d) to C(c,d)C(c, d).

An enriched natural transformation η:FG\eta \colon F \to G is a family of morphisms of VV

η c:ID(Fc,Gc) \eta_c \;\colon\; I \longrightarrow D(F c, G c)

(out of the tensor unit II of VV) indexed over Ob(C)Ob(C), such that for any two objects cc, dd of CC the following diagram commutes:

C(c,d) ρ C(c,d)I G c,dη c D(Gc,Gd)D(Fc,Gc) λ D IC(c,d) η dF c,d D(Fd,Gd)D(Fc,Fd) D D(Fc,Gd)\array{ C(c, d) & \stackrel{\rho}{\cong} & C(c, d) \otimes I & \stackrel{G_{c, d} \otimes \eta_c}{\to} & D(G c, G d) \otimes D(F c, G c) \\ \stackrel{\lambda}{\cong} \downarrow & & & & \downarrow \circ_D \\ I \otimes C(c, d) & \underset{\eta_d \otimes F_{c, d}}{\to} & D(F d, G d) \otimes D(F c, F d) & \underset{\circ_D}{\to} & D(F c, G d) }

(Should expand to include other notions of enriched category.)

Reference

For more references see at enriched category.

Revised on May 3, 2016 12:10:45 by Urs Schreiber (131.220.184.222)