A monoid (weaker than group: doesn't need to have an inverse) is an example of a category, with a single object G (namely the underlying set) and a lot of arrow

A monoid (weaker than group: doesn't need to have an inverse) is an example of a category, with a single object G (namely the underlying set) and a lot of arrows from G -> G. If you take the natural numbers N = {0, 1, ...} as a set and + as a binary operator forming a monoid (N, +, 0), then the arrows would correspond to (. + 0), (. + 1), (. + 2) etc. Here (. + 0) can be seen as the identity arrow, and composition of arrows (. + a) and (. + b) yields a new arrow (. + (a+b)).