In Category Theory, a monoid is defined as "a category with one object [...] thus determined by the set of all its arrows, by the identity arrow and by the rule for the composition of arrows"[1].
If there is only one single object, then the domain and codomain assigns, to every arrow, the same object, a. Thus, every arrow has as domain the same object as codomain.
Given the definition of category, the identity operation assigns an arrow with dom f = cod f = a.
- Why can I say that the monoid has many arrows?? Why the different arrows are not the id?
- If I buy they are different arrows, then which axiom of the category definition asserts that the identity is the one that maps each element (inside the set) to itself? I thought the arrows were supposed 'not to see' the interns of the objects.
- What is the nature of the domain/codomain then? How f and g are diferent if the objects they point from/to are the same?
[1] Mac Lane, Category theory for the working mathematician, p.11