nLab
enriched natural transformation

Contents

Idea

An enriched natural transformation is the appropriate notion of morphism between functors enriched in a monoidal category VV.

Definition

Let CC and DD be categories enriched in a monoidal category VV, and let F,G:CDF, G: 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: F \to G is a family of morphisms of VV

ηc:ID(Fc,Gc)\eta c: I \to D(F c, G c)

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,dηc D(Gc,Gd)D(Fc,Gc) λ D IC(c,d) ηdF 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

  • Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982) (pdf)

Revised on October 31, 2012 01:55:22 by Toby Bartels (64.89.53.173)