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 $\hom_C(c, d)$ to $C(c, d)$. An enriched natural transformation$\eta: F \to G$ is a family of morphisms of $V$

$\eta c: I \to D(F c, G c)$

indexed over $Ob(C)$, such that for any two objects $c$, $d$ of $C$ the following diagram commutes:

$\array{
C(c, d) & \stackrel{\rho}{\cong} & C(c, d) \otimes I & \stackrel{G_{c, d} \otimes \eta c}{\to} & D(G c, G d) \otimes D(F c, G c) \\
\stackrel{\lambda}{\cong} \downarrow & & & & \downarrow \circ_D \\
I \otimes C(c, d) & \underset{\eta d \otimes F_{c, d}}{\to} & D(F d, G d) \otimes D(F c, F d) & \underset{\circ_D}{\to} & D(F c, G d)
}$

(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)