TITLE: Monoid as a single object category
QUESTION [25 upvotes]: I'm struggling with comprehending what monoids are in terms of category theory.
In examples they view integer numbers as a monoid. I think I get the set theoretic definition. We have a set and a associative binary operator (addition) and the neutral element (zero).
Then they are saying something like that - view the whole set as a single object and binary operator as bunch of morphisms for every element of the set.
Like add0 is an identity morphism. Which would really give us the same object i.e. the same set of all integer numbers. I think I understand this.
But let's view the morphism add1. After applying it to the our single object (the set of all integers) we would have a set {1,2,3…} not the {0,1,2,3…}. Aren't domain and codomain different in that case?
That's what's bothering me. Can someone clarify that to me?
Here is the text that gives me problems.
The text

REPLY [8 votes]: In addition to Henning Makholm's crisp and clear answer, you might find the opening six pages of my Notes on Category Theory helpful. They too give the example of a monoid as a category, but also give some other examples of categories where the arrows are not functions in any ordinary sense. Another important illustration is the case of a posets treated as a category.
In fact these examples suggest why we might well prefer to talk of 'arrows' rather than 'morphisms' (because the very term 'morphism' comes with baggage, and almost inevitably makes us think of a function -- but to repeat, arrows need not be functions).