A group seen as a category
Obsolete entry, go to this one.
Every group \(G\) is a category. It has only one object, \(G\) itself, and the morphisms corresponds to group elements.
Now, consider any set \(X\), and their bijections. We have other category.
A funtor from the former to the latter is nothing else that a group action of \(G\) over \(X\)! In this sense, we get a linear representation when \(X\) is a vector space and we shrink the image of the functor to their linear automorphisms (instead of all the bijections).
And a natural transformation between such two functors is a \(G\)-equivariant map.
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