[ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ] In some categories 0 = 1, e.g. in the category of groups, so sometimes X * 0 = X (which bad math students believe to hold also in the natural numbers ;-). --Andreas On Sep 1, 2010, at 4:31 AM, wagnerdm@seas.upenn.edu wrote:
[ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
Quoting Larry Evans <cppljevans@suddenlink.net>:
Define + as the coproduct operator, IOW, X+Y is the coproduct of X and Y for some category C. Define * as the product operator, IOW X*Y is the product of X and Y for some category C. Define 0 as the initial object of C. Define 1 as the terminal object of C.
Being completely careful here, we must observe that if we view + as an operator, then X+Y is merely picking out one of many possible coproducts of X and Y. (Of course, any other coproducts that exist are isomorphic.) We must make a similar caveat for *, 0, and 1 (which are particular initial and terminal objects, though again unique up to isomorphism).
Is it true that, for all objects, X in C:
X+0 = X 0+X = X X*1 = X 1*X = X
Then, here, we must take equality as isomorphism, of course. It's pretty straightforward to show that X is *a* coproduct of X and 0 -- just take id : X -> X and the unique arrow i : 0 -> X as the injections. Therefore X is isomorphic to whatever object X+0 happens to be. The remaining equations follow by symmetry and duality.
Also, what's X*0 and X+1?
I wasn't able to come up with a more edifying description of these objects than simply expanding the definitions. Perhaps somebody else can come up with some further property of X*0/X+1 or show that there isn't anything additional we can assume...?
~d
Andreas Abel <>< Du bist der geliebte Mensch. Theoretical Computer Science, University of Munich Oettingenstr. 67, D-80538 Munich, GERMANY andreas.abel@ifi.lmu.de http://www2.tcs.ifi.lmu.de/~abel/