Cancelling coproducts?

Michael Abbott writes:

Am I being dim here? In an extensive category (coproducts are preserved by pullbacks and are disjoint), can I cancel thus: X+1 ~= Y+1 ?=>? X ~= Y ?

Several people have already pointed out that `1 is cancellable' in this sense. There is also a more obvious, but nonetheless usuful, kind of cancellation property which holds in extensive categories: for arbitrary Z, if X+Z is isomorphic to Y+Z via an isomorphism commuting with the injections of Z, then X is (canonically) isomorphic to Y. Steve Lack. 19-Nov-2002 21:14:18 -0400,6099;000000000001-00000000