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Monoids abound in mathematics; they are in a sense the most primitive interesting algebraic structures.http://math.ucr.edu/home/baez/week74.html
A monoid is a generalisation of group, in that the requirement that each element has an inverse is dropped.
what is a category with one object? It is a "monoid". The usual definition of a monoid is this: a set M with an associative binary product and a unit element 1 such that a1 = 1a = a for all a in S. […]
We tend to think of this ability to "undo" any process as a key aspect of symmetry. I.e., if we rotate a beer bottle, we can rotate it back so it was just as it was before. We don't tend to think of SMASHING the beer bottle as a symmetry, because it can't be undone. But while processes that can be undone are especially interesting, it's also nice to consider other ones… so for a full understanding of symmetry we should really study monoids as well as groups.
But we also should be interested in "partially defined" processes, processes that can be done only if the initial conditions are right. This is where categories come in! Suppose that we have a bunch of boxes, and a bunch of processes we can do to a bottle in one box to turn it into a bottle in another box: for example, "take the bottle out of box x, rotate it 90 degrees clockwise, and put it in box y". We can then think of the boxes as objects and the processes as morphisms: a process that turns a bottle in box x to a bottle in box y is a morphism f: x → y. We can only do a morphism f: x → y to a bottle in box x, not to a bottle in any other box, so f is a "partially defined" process. This implies we can only compose f: x → y and g: u → v to get fg: x → v if y = u.
[A] monoid is like a group, but the "symmetries" no longer need be invertible; a category is like a monoid, but the "symmetries" no longer need to be composable!
Note for physicists: the operation of "evolving initial data from one spacelike slice to another" is a good example of a "partially defined" process: it only applies to initial data on that particular spacelike slice. So dynamics in special relativity is most naturally described using groupoids. Only after pretending that all the spacelike slices are the same can we pretend we are using a group. It is very common to pretend that groupoids are groups, since groups are more familiar, but often insight is lost in the process. Also, one can only pretend a groupoid is a group if all its objects are isomorphic. Groupoids really are more general.