Yes, it is true. Let me use = to mean isomorphic. Begin by showing that if U + V = U + W = 1, then V = W. This is because U + V = 1 implies that U x V = 0. But then V = V x U + V x W = V x W and hence V is contained in W and vice versa. Now suppose that X + 1 = Y + 1. Then the 1 on the left can be written as 1 = U_0 + U_1, being the pullback with Y and 1. Similarly, X = V_0 + V_1. Moreover, Y = U_0 + V_0 and 1 = U_1 + V_1. But then U_0 = V_1 and then X = V_0 + V_1 = V_0 + U_0 = Y. On Wed, 13 Nov 2002, Michael Abbott wrote:
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 ?
So far I have neither a proof nor a counterexample. If this cancellation principle exists it'll be slightly subtle, since 0+N ~= 1+N, and so we'd only be able to cancel adding a restricted class of (finite?) objects.
I'd be grateful for either a proof or a counterexample!
Michael Abbott
13-Nov-2002 19:35:17 -0400,1945;000000000000-00000000