So I generally understand the idea of a monoid from set theory, but I'm trying to understand better the mapping to category theory.
I think there are a couple of things that are tripping me up, and I think it'd be instructive to have a concrete example, so let's do the monoid of integers. So the associative operator is +, and the zero is 0.
As a category, it has one object.. what is that object? And according to the wikipedia link, we have morphisms for every element... so we have a category whose object is Int, and the morphisms are 0, 1, 2, 3... (important question: can an object have multiple distinct morphisms? I didn't think that was defined, as I thought that morphisms were completely defined by the domain and codomain). So now we need our associative operator, ideally such that the map named 1 composed with the map named two is the same as the map named 3. I think this is where there is a breakdown in how I am thinking about these. How do we think about that last property?
Thank you very much for any help you have.
Update: the answer given gets me 95% of the way there, but there is one lingering question: The one thing I'm not sure about is how do we formalize equivalence? Like, we can definitely say that (a * b) * c = a * (b * c) (by the definition of a category), but how do we "define" 1 * 2 = 3 and think about it in a rigorous way when they are all morphisms with the same domain and codomain. Another way of asking this is if we have a morphism called 2, and a morphism called 3, and a morphism called 7, how do we know that 2*3=7?