Groups, monoids, groupoids

This is an extension of this older entry.

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 endomaps. 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 linear representation when \(X\) is a vector space and we shrink the image of the functor to their linear endomorphisms (instead of all the endomaps). 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 cathegory ever.

On the other hand, a groupoid is a category like a group category, but with several objects, not only one.

 
If we start with a group, algebraically, and reduce properties to arrive to a monoid, and then continue shrinking properties we arrive to a semigroup: neither inverses nor identity. But then we don't have a category.

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