A group $G$ can be seen as 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 functor from the former to the latter is nothing else that a group action of $X$ over $G$! In this sense, we get a group 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.
Other example of category of the same flavour is a monoid. It is all the same than a group, but you don't need to have inverses. It is the simplest category at all.
If we shrink properties we arrive to a semigroup: neither inverses nor identity. But then we don't have a category.
On the other hand, a groupoid is a category like a group category, but with several objects, not only one. Es como si pegásemos varios grupos independientes...
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Author of the notes: Antonio J. Pan-Collantes
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