I don’t think you lose much by focusing on finite Cartesian frames. I have mostly only been imagining finite cases.
I think there is some potential for later extending the theory to encompass game theory and probabilistic strategies, and then we might want to think of the infinite space of mixed strategies as the agent, but it wouldn’t surprise me if in doing this, we also put continuity into the system and want to assume compactness.
I don’t think you lose much by focusing on finite Cartesian frames. I have mostly only been imagining finite cases.
I think there is some potential for later extending the theory to encompass game theory and probabilistic strategies, and then we might want to think of the infinite space of mixed strategies as the agent, but it wouldn’t surprise me if in doing this, we also put continuity into the system and want to assume compactness.