Authors: Richard Blute, Robin Cockett, Durgesh Kumar, J.S. Lemay

A result of Coecke, Pavlovic and Vicary states that a basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative dagger-Frobenius monoid in the category of finite-dimensional Hilbert spaces. This can be extended to an equivalence between such Frobenius algebras and the category of finite sets.

We describe an ongoing project to attempt to extend this result beyond finite sets. This

requires on the one hand replacing the category of sets with the category of Ehrhard’s

finiteness spaces, one of the motivating examples for the theory of differential linear logic. On

the other hand, Frobenius algebras must be replaced by linear monoids as defined by Priyaa

Srivinvasan.