Quantum Sets and Finiteness Spaces

Richard Blute – 6 March 2024

We look at the possibility of extending the work of Coecke, Pavlovic, Vicary and others on the theory of quantum sets to infinite-dimensions. In previous work, those authors showed that the category of commutative dagger-Frobenius algebras in the category of finite-dimensional Hilbert spaces is equivalent to the category of finite sets. This allows one to have a notion of finite set in more general monoidal categories. But the extension of these ideas to infinite dimensions requires new structures. I'm going to suggest that Ehrhard's finiteness spaces are a sensible possibility. Rather than Hilbert spaces, one works in a category of Lefschetz topological spaces, which give a *-autonomous category of topological vector spaces. This work is a summary of ongoing discussions with J.S.Lemay.