[TYPES] [newbie] is {initial, terminal} object the identity for {coproduct, product}?

[Andreas Abel]

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 at seas.upenn.edu wrote:

> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list 
>  ]
>
> Quoting Larry Evans <cppljevans at 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 at ifi.lmu.de
http://www2.tcs.ifi.lmu.de/~abel/