nLab enriched natural transformation

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 $C$ and $D$ be categories enriched in a monoidal category $V$, and let $F, G \colon C \to D$ be enriched functors. We abbreviate hom-objects $\hom_C(c, d)$ to $C(c, d)$.

An enriched natural transformation $\eta \colon F \to G$ is a family of morphisms of $V$

$$\eta_c \;\colon\; I \longrightarrow D(F c, G c)$$

(out of the tensor unit $I$ of $V$) 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.)

Related concepts

• enriched adjunction

Reference

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

• Francis Borceux, Vol 2, def. 6.2.4 of Handbook of Categorical Algebra, Cambridge University Press (1994)

• Emily Riehl, chapter 3 Basics of enriched category theory in Categorical Homotopy Theory

For more references see at enriched category.