# Contents

## Idea

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

## Definition

Let $C$ and $D$ be categories enriched in a monoidal category $V$, and let $F,G:C\to D$ be enriched functors. We abbreviate hom-objects ${\mathrm{hom}}_{C}\left(c,d\right)$ to $C\left(c,d\right)$. An enriched natural transformation $\eta :F\to G$ is a family of morphisms of $V$

$\eta c:I\to D\left(Fc,Gc\right)$\eta c: I \to D(F c, G c)

indexed over $\mathrm{Ob}\left(C\right)$, such that for any two objects $c$, $d$ of $C$ the following diagram commutes:

$\begin{array}{ccccc}C\left(c,d\right)& \stackrel{\rho }{\cong }& C\left(c,d\right)\otimes I& \stackrel{{G}_{c,d}\otimes \eta c}{\to }& D\left(Gc,Gd\right)\otimes D\left(Fc,Gc\right)\\ \stackrel{\lambda }{\cong }↓& & & & ↓{\circ }_{D}\\ I\otimes C\left(c,d\right)& \underset{\eta d\otimes {F}_{c,d}}{\to }& D\left(Fd,Gd\right)\otimes D\left(Fc,Fd\right)& \underset{{\circ }_{D}}{\to }& D\left(Fc,Gd\right)\end{array}$\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)