Natural Transformation

Suppose we are given functors A natural transformation is an operation assigning each a Morphism in such that for each in :

This is equivalent to a Commutative Diagram:

\usepackage{tikz-cd}
 
\begin{document}
\begin{tikzcd}
FA \arrow[r,"Ff"] \arrow[d,"\alpha_{A}"] & FB\arrow[d,"\alpha_{B}"]\\
GA\arrow[r,"Gf"] & GB
\end{tikzcd}
\end{document}

This is called a Naturality Square for at .

Natural transformations are Morphisms in the Category of Functors

Natural Isomorphism Equivalence

Example

Let be a category with only identity Morphisms. A Functor is just a sequence in indexed by . Given , a natural transformation between them is any assignment for . If there are no Morphisms for some , then there is no natural transformations.

Example

Given Group actions of a Group on and , a natural transformation between them is a -Equivariant A group action Functor is a functor sending the only element of to the set that its acting on, and sending each Morphism to a permutation of . Suppose and are such functors, representing Group actions of on sets and respectively. A natural transformation is then just a map such that for any , we have a Commutative Diagram:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \arrow[r,"F_{A}g"] \arrow[d,"\alpha"]
 & A \arrow[d,"\alpha"] \\
B \arrow[r,"F_{B}g"]
 & B
\end{tikzcd}
\end{document}

i.e. for any :

where represents the Group action in respective sets.