Editing Natural transformation (section)

== Operations with natural transformations ==

[[File:Natural transformation composition.svg|thumb|Horizontal and vertical composition of natural transformations]]

=== Vertical composition ===

If <math>\eta : F \Rightarrow G</math> and <math>\epsilon: G \Rightarrow H</math> are natural transformations between functors <math>F, G, H: C \to D</math>, then we can compose them to get a natural transformation <math>\epsilon \circ \eta: F \Rightarrow H</math>. 
This is done component-wise:
:<math>(\epsilon \circ \eta)_X = \epsilon_X \circ \eta_X</math>. 

[[Image:Vertical composition of natural transformations.svg|center|300px]]

This vertical composition of natural transformations is [[associative]] and has an identity, and allows one to consider the collection of all functors <math>C \to D</math> itself as a category (see below under [[#Functor categories|Functor categories]]).
The identity natural transformation <math>\mathrm{id}_F</math> on functor <math>F</math> has components <math>(\mathrm{id}_F)_X = \mathrm{id}_{F(X)}</math>.<ref>{{cite web | url=https://ncatlab.org/nlab/show/identity+natural+transformation | title=Identity natural transformation in nLab }}</ref>
:For <math>\eta : F \Rightarrow G</math>, <math>\mathrm{id}_G \circ \eta = \eta = \eta \circ \mathrm{id}_F</math>.

=== Horizontal composition ===

If <math>\eta: F \Rightarrow G</math> is a natural transformation between functors <math>F, G: C \to D</math> and <math>\epsilon: J \Rightarrow K</math> is a natural transformation between functors <math>J, K: D \to E</math>, then the composition of functors allows a composition of natural transformations <math>\epsilon * \eta: J \circ F \Rightarrow K \circ G</math> with components
:<math>(\epsilon * \eta)_X = \epsilon_{G(X)} \circ J(\eta_X) = K(\eta_X) \circ \epsilon_{F(X)}</math>.
By using whiskering (see below), we can write
:<math>(\epsilon * \eta)_X = (\epsilon G)_X \circ (J \eta)_X = (K \eta)_X \circ (\epsilon F)_X</math>,
hence
:<math>\epsilon * \eta = \epsilon G \circ J \eta = K \eta \circ \epsilon F</math>.

[[Image:Horizontal composition of natural transformations.svg|center|400px|alt=This is a commutative diagram generated using LaTeX. The left hand square shows the result of applying J to the commutative diagram for eta:F to G on f:X to Y. The right had side shows the commutative diagram for epsilon:J to K  applied to G(f):G(X) to G(Y).]]

This horizontal composition of natural transformations is also associative with identity.
This identity is the identity natural transformation on the [[identity functor]], i.e., the natural transformation that associate to each object its [[identity morphism]]: for object <math>X</math> in category <math>C</math>, <math>(\mathrm{id}_{\mathrm{id}_C})_X = \mathrm{id}_{\mathrm{id}_C(X)} = \mathrm{id}_X</math>.
:For <math>\eta: F \Rightarrow G</math> with <math>F, G: C \to D</math>, <math>\mathrm{id}_{\mathrm{id}_D} * \eta = \eta = \eta * \mathrm{id}_{\mathrm{id}_C}</math>.
As identity functors <math>\mathrm{id}_C</math> and <math>\mathrm{id}_D</math> are functors, the identity for horizontal composition is also the identity for vertical composition, but not vice versa.<ref>{{cite web | url=https://bartoszmilewski.com/2015/04/07/natural-transformations/ | title=Natural Transformations | date=7 April 2015 }}</ref>

=== Whiskering ===

Whiskering is an [[external binary operation]] between a functor and a natural transformation.<ref>{{cite web | url=https://proofwiki.org/wiki/Definition:Whiskering | title=Definition:Whiskering - ProofWiki }}</ref><ref>{{cite web | url=https://ncatlab.org/nlab/show/whiskering | title=Whiskering in nLab }}</ref>

If <math>\eta: F \Rightarrow G</math> is a natural transformation between functors <math>F, G: C \to D</math>, and <math>H: D \to E</math> is another functor, then we can form the natural transformation <math>H \eta: H \circ F \Rightarrow H \circ G</math> by defining
:<math>(H \eta)_X = H(\eta_X)</math>.
If on the other hand <math>K: B \to C</math> is a functor, the natural transformation <math>\eta K: F \circ K \Rightarrow G \circ K</math> is defined by
:<math>(\eta K)_X = \eta_{K(X)}</math>.

It's also an horizontal composition where one of the natural transformations is the identity natural transformation:
:<math>H \eta = \mathrm{id}_H * \eta</math> and <math>\eta K = \eta * \mathrm{id}_K</math>.
Note that <math>\mathrm{id}_H</math> (resp. <math>\mathrm{id}_K</math>) is generally not the left (resp. right) identity of horizontal composition <math>*</math> (<math>H \eta \neq \eta</math> and <math>\eta K \neq \eta</math> in general), except if <math>H</math> (resp. <math>K</math>) is the [[identity functor]] of the category <math>D</math> (resp. <math>C</math>).

=== Interchange law ===

The two operations are related by an identity which exchanges vertical composition with horizontal composition: if we have four natural transformations <math>\alpha, \alpha', \beta, \beta'</math> as shown on the image to the right, then the following identity holds:
:<math> (\beta' \circ \alpha') * (\beta \circ \alpha) = (\beta' * \beta) \circ (\alpha' * \alpha)</math>.
Vertical and horizontal compositions are also linked through identity natural transformations:
:for <math>F: C \to D</math> and <math>G: D \to E</math>, <math>\mathrm{id}_G * \mathrm{id}_F = \mathrm{id}_{G \circ F}</math>.<ref>https://arxiv.org/pdf/1612.09375v1.pdf, p. 38</ref>

As whiskering is horizontal composition with an identity, the interchange law gives immediately the compact formulas of horizontal composition of <math>\eta: F \Rightarrow G</math> and <math>\epsilon: J \Rightarrow K</math> without having to analyze components and the commutative diagram:
:<math>\begin{align}
\epsilon * \eta & = (\epsilon \circ \mathrm{id}_J) * (\mathrm{id}_G \circ \eta) = (\epsilon * \mathrm{id}_G) \circ (\mathrm{id}_J * \eta) = \epsilon G \circ J \eta \\
                & = (\mathrm{id}_K \circ \epsilon) * (\eta \circ \mathrm{id}_F) = (\mathrm{id}_K * \eta) \circ (\epsilon * \mathrm{id}_F) = K \eta \circ \epsilon F
\end{align}</math>.