The notion of $V$-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.
$V$-Enriched natural transformations constitute the 2-morphisms in the 2-category VCat.
For $V$ a cosmos for enrichment, let
$\mathbf{C}$, $\mathbf{D}$ be a pair of enriched categories
(we denote their hom-objects by $C(-, -)$ etc., instead of $\hom_C(-, -)$ or similar),
$F, G \,\colon\, \mathbf{C} \longrightarrow \mathbf{D}$ a pair of $V$-enriched functors between them.
Then:
An enriched natural transformation between these enriched functors
is a family of morphisms of $V$
(out of the tensor unit $I$ of $V$) indexed by $c\in Ob(C)$)
such that for any pair of objects $c, c' \,\in\, Obj(\mathbf{C})$ the following diagram commutes:
$\otimes$ denotes the tensor product in $V$,
$I$ denotes the tensor unit in $V$,
$l$,$r$ denote the left and right unitors of $V$,
$F_{c,c'}$, $G_{c,c'}$ denote the component maps of the enriched functors between hom-objects,
$\circ^{\mathbf{D}}$ denotes the composition-operation of $\mathbf{D}$.
The above diagram expresses the $V$-enriched version of commutativity of the plain naturality square:
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)
Given
where
$H$ is an enriched functor
then we obtain an enriched natural transformation of the form
with component maps given as the composition
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), $Cat$-enriched natural transformations are also known as strict 2-natural transformations.
Max Kelly, p. 9 of: Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982), Reprints in Theory and Applications of Categories, 10 (2005) 1-136 [tac:tr10, pdf]
Francis Borceux, def. 6.2.4 of: Handbook of Categorical Algebra Vol 2, Cambridge University Press (1994)
Emily Riehl, §3.5 in Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]
For more references see at enriched category.
Last revised on May 18, 2023 at 14:09:03. See the history of this page for a list of all contributions to it.