TITLE: Trying to understand significance of monoid as a one object category
QUESTION [10 upvotes]: So I generally understand the idea of a monoid from set theory, but I'm trying to understand better the mapping to category theory.
http://en.wikipedia.org/wiki/Monoid#Relation_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?

REPLY [2 votes]: You already know what the integers with addition are.
When you construct this category, you define composition to be addition. There's not a problem doing that, because you already know what addition is.

important question: can an object have multiple distinct morphisms

Yes, if I understand you: in general, the set of homomorphisms from $X$ to $Y$ may have many elements. One of the typical examples of a category is $\mathbf{Set}$, the category whose objects are sets and the whose morphisms are functions.
I this category, the domain and codomain of a function in the sense of the category is its given by its domain and codomain in the sense of set theory. (I'm assuming a definition of 'function' that includes the domain and codomain as part of the definition. If you pick a different definition: e.g. that a function is simply the graph of a relation, you'll have to modify the definition of morphism in $\mathbf{Set}$ appropriately)
And in $\mathbf{Set}$, the product of two morphisms is their composite.
In this category, we can easily see that $\hom(X,Y)$ generally has lots and lots of elements, except in special cases. (e.g. $X = \varnothing$ or $|Y| = 1$)