nLab enriched natural transformation

Idea

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.

Definition

For $V$ a cosmos for enrichment, let

• $\mathbf{C}$, $\mathbf{D}$ be a pair of $V$-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:

Definition

An enriched natural transformation between these enriched functors

$$\eta \colon F \longrightarrow G$$

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 by $c\in Ob(C)$)

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

(1)

Here

• $\otimes$ 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:

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

Properties

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:

Proposition

(horizontal composition)
Given

where

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

2. $H$ is an enriched functor

then we obtain an enriched natural transformation of the form

with component maps given as the composition

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

Relation to strong natural transformations

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

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

Examples

Example

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.

Example

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.

Related concepts

• strong natural transformation

• enriched functor

• enriched functor category

• enriched adjoint functor

References

• 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.