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Thanks for these remarks! So the definition of algebraic space that I was taught only requires that the diagonal is representable, not necessarily quasi-compact. I'm not an algebraic geometer so I'm a little rusty. Does this example have a representable diagonal? Is there an easy criteria that would guarantee this?
I just thought of another possibility, which seems likely after reading Anton's post again. Is $\mathbb{G}_M / \mathbb{Z}$ a scheme? if so what scheme? It seems like it would have to be a single point.
In certain contexts "biproduct" can have a different meaning (e.g. in the context of bicategories). But arguing over terminology is a little ridiculous. The main point is that this is a beautiful answer.
Two questions: (1) is this equivalent to asking whether the homomorphism $\Gamma \to SO(8)$ lifts to Spin(8)? (2) What is the counter example in the case of the five-sphere $S^5$. I have a sketch of a proof, but it seems to work too generally so something must be wrong.
I thought of this as I was writing the question, but wanted to see what other people would say, anyway. When you say "infinite direct sum" you mean coproduct, right? When I hear "direct sum" I usually think something which is simultaneously a product and a coproduct. Even for the category of abelian groups infinite direct sums in that sense don't exists.
