nLab enriched natural transformation

Contents

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:

\begin{definition} \label{EnrichedNaturalTransformation} 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:

\begin{tikzcd} & \mathbf{C}(c,c’) \otimes I \ar[r, { G_{c,c} \otimes \eta_c }] & \mathbf{D}\big(G(c),\, G(c’)\big) \otimes \mathbf{D}\big(F(c),\, G(c)\big) \ar[dr, { \circ^{\mathbf{D}} }] &[-40pt] \ \mathbf{C}(c, c’) \ar[ur, { r^{-1} }] \ar[dr, { l^{-1} }{swap}] & & & \mathbf{D}\big( F(c) ,\,G(c’) \big) \ & I \otimes \mathbf{C}(c, c’) \ar[r, { \eta_{c} \otimes F_{c,c} }{swap}] & \mathbf{D}\big(F(c’),\, G(c’) \big) \otimes \mathbf{D}\big(F(c),\, F(c’)\big) \ar[ur, { \circ^{\mathbf{D}} }{swap}] \end{tikzcd}

\end{definition} 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:

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:

\begin{proposition}\label{RightWhiskering} (horizontal composition) \linebreak Given

\begin{tikzcd} \mathbf{C} \ar[rr, bend left=30, { F }, {\ }{name=s, swap}] \ar[rr, bend right=30, { G }{swap}, {\ }{name=t}] \ar[from=s, to=t, Rightarrow, shorten=10pt, { \eta }] && \mathbf{D} \ar[rr, { H }] && \mathbf{E} \end{tikzcd}

where

1. $\eta$ is an enriched natural transformation (Def. \ref{EnrichedNaturalTransformation})

2. $H$ is an enriched functor

then we obtain an enriched natural transformation of the form

\begin{tikzcd}
\mathbf{C} \ar[rr, bend left=30, { H \circ F }, {\ }{name=s, swap}] \ar[rr, bend right=30, { H \circ G }{swap}, {\ }{name=t}] \ar[from=s, to=t, Rightarrow, shorten=10pt, { H(\eta) }] && \mathbf{E} \end{tikzcd}

with component maps given as the composition

\end{proposition}

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

\begin{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. \end{example}

\begin{example} \label{StrictTwoNaturalTransformations} 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. \end{example}

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](https://doi.org/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](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), 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](https://doi.org/10.1017/CBO9781107261457), pdf]

For more references see at enriched category.