On Nov 11, 10:44 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > Dan Christensen <Dan_Christen...@sympatico.ca> writes: > > It is an intriguing idea that something like a number can be seen as a > > kind of operator on a set, with say addition being the composition of > > those operators. > > > In category theory, how do you represent something like a ring with > > two kinds of composition corresponding to the + and * operations? > > Say, that's a good question, Dan. > > As far as I know, a ring is not a category in the same sense that a > monoid is. Categories come with only one notion of composition, not > two, so you couldn't have the arrows be elements of the ring in a simple > sense. > > Perhaps you could have two arrows for each element of the ring, with one > arrow corresponding to the left addition action, and the other > corresponding to the left multiplication action. But, no, it seems like > it would be more complicated than that, for composition of an addition > action with a multiplication action would lead to a new arrow, so we'd > have to have even more arrows. > > I'm just spitballing here, but I don't think there's a very elegant > ring-as-category picture analogous to the monoid-as-category. >
If several categories can somehow be combined into a single structure, how about combining 3 different categories: One analogous to + with identity 0, another to * with identity 1, and another to - (unary negation function) also with identity 0?
For +, comp1(x,y)=x+y For *, comp2(x,y)=x*y For -, ???? (don't know)