Just a random thought: Are there any interesting "$\omega$-semiadditive" categories? Pointed categories admitting countable coproducts and products and s.t. the natural map between them is an equivalence? I suspect that such a category can't be additive unless its trivial since we (get an eilenberg swindle) can pick any non-trivial object X and consider the endomorphism given by T:= 1+1-1+... we then have that 1-T=-T so that 1=0. This does not rule out a semi-additive example, however.