#coproducts

[isomorphismes]

Direct sum is not the coproduct in the category of quadratic vector spaces. The absence of terminal objects and (co)products in QVec is due to the fact that projections do not generally preserve norms.

José Figueroa O'Farrill, Spin Geometry

[eka-mark]

what is it about the binary coproducts of finite sets that makes A+B = A+C imply B = C? (note: “=“ means “is isomorphic to” in this post)

it’s easy to see that this isn’t the case for general coproducts, and indeed this isn’t even true for infinite sets, since strict inequalities of infinite sets A>B>C imply A+B = A+C = A (maybe you need choice for that?)

so apparently the property of interest isn’t a consequence of FinSet being a topos, nor the fact that its coproducts are freely generated

what’s so special about coproducts in FinSet, and where else does A+B = A+C imply B = C?