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
For closed monoidal categories $V$ there is a close relation between $V$-senriched natural transformations and $V$-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), $Cat$-enriched natural transformations are also known as strict 2-natural transformations.
Samuel Eilenberg, G. Max Kelly, §I.10 of: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]
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 August 23, 2023 at 10:07:15. See the history of this page for a list of all contributions to it.