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Re: The Monoid Category in DC Proof
Posted:
Nov 12, 2012 9:15 AM
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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)
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
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