nLab enriched natural transformation




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.


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.)


For more references see at enriched category.

Last revised on July 30, 2018 at 11:39:00. See the history of this page for a list of all contributions to it.