Title: Dagger Frobenius relations for dagger linearly distributive categories
Abstract: Commutative dagger Frobenius algebras play a central role in categorical quantum mechanics due to their correspondence to orthogonal basis in the category of finite-dimensional Hilbert spaces. Subsequently, such algebras represent quantum observables. Measurement in a dagger monoidal category is an isometry, m: A \to X (i.e, m^\dagger m = 1_X) where A is any object, and X is a special commutative dagger Frobenius Algebra.
In this talk, I will generalize dagger Frobenius Algebras from dagger monoidal categories to dagger linearly distributive categories. We refer to the generalization as dagger linear monoids. I will provide the conditions under which a dagger linear monoid gives a dagger Frobenius algebra in a unitary category. We show the correspondence between dagger linear duals and dagger linear monoids. We find that in a complete and cocomplete category, limit of dagger linear monoids is a dagger linear monoid. Thus one can represent, possibly, infinite dimensional quantum observables using dagger linear monoids.
If time permits, I will discuss measurements for dagger linear monoids. A measurement for a dagger linear monoid is a retract from the linear monoid to a special commutative dagger Frobenius Algebra within the unitary core.