The point i wanted to make is that Soule and C-C (and Deitmar) use affine schemes in their set-up, but require only a tiny fraction of their power, as they restrict to functors from finite dimensional algebras (over C, finite and flat over Z). All that info is contained in the dual coalgebra, and that thing can be generalized to the noncommutative world, whereas generalizing affine schemes would cause more problems than it will solve…

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