nLab enriched natural transformation




The notion of VV-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.

VV-Enriched natural transformations constitute the 2-morphisms in the 2-category VCat.


For VV a cosmos for enrichment, let

  • C\mathbf{C}, D\mathbf{D} be a pair of VV-enriched categories

    (we denote their hom-objects by C(,)C(-, -) etc., instead of hom C(,)\hom_C(-, -) or similar),

  • F,G:CDF, G \,\colon\, \mathbf{C} \longrightarrow \mathbf{D} a pair of VV-enriched functors between them.



An enriched natural transformation between these enriched functors

η:FG \eta \colon F \longrightarrow 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 by cOb(C)c\in Ob(C))

such that for any pair of objects c,cObj(C)c, c' \,\in\, Obj(\mathbf{C}) the following diagram commutes:



The above diagram expresses the VV-enriched version of commutativity of the plain naturality square:

c F(c) η c G(c) f F(f) G(f) c F(c) η c G(c) \array{ c && F(c) &\overset{ \eta_c }{\longrightarrow}& G(c) \\ \mathllap{{}^{f}} \Big\downarrow &\mapsto\;\;\;& \mathllap{{}^{F(f)}} \Big\downarrow && \Big\downarrow\mathrlap{{}^{G(f)}} \\ c' && F(c') &\underset{\eta_{c'}}{\longrightarrow}& G(c') }


As 2-morphisms

In generalization of how for plain natural transformations there is a notion of horizontal composition (“whiskering”) and vertical composition, so for enriched natural transformations:


(horizontal composition)


  1. η\eta is an enriched natural transformation (Def. )

  2. HH is an enriched functor

then we obtain an enriched natural transformation of the form

with component maps given as the composition

Iη cD(F(c),G(c))H F(c),G(c)E(HF(c),HG(c)). I \overset{\eta_c}{\longrightarrow} \mathbf{D}\big(F(c),\,G(c)\big) \overset{H_{F(c), G(c)}}{\longrightarrow} \mathbf{E}\big(H \circ F(c),\,H \circ G(c)\big) \,.

Relation to strong natural transformations

For closed monoidal categories VV there is a close relation between VV-enriched natural transformations and VV-strong natural transformations.

For the moment see at enriched monad – relation to strong monads for more.



With Set denoting the category of sets and functions equipped with its cartesian monoidal-structure (via Cartesian product of sets), Set-enriched natural transformations are just plain natural transformations between functors between locally small categories.


With Cat denoting the 1-category of small strict categories equipped with its cartesian monoidal structure (via forming product categories), CatCat-enriched natural transformations are also known as strict 2-natural transformations.


For more references see at enriched category.

Last revised on December 20, 2023 at 08:53:29. See the history of this page for a list of all contributions to it.