Categorical formulation of quantum algebras

TL;DR

This paper develops a categorical framework using dagger-Frobenius monoids to describe quantum algebras, linking finite-dimensional C*-algebras with categorical structures and extending spectral theorems.

Contribution

It introduces involution monoids and establishes a correspondence between C*-algebras and dagger-Frobenius monoids in Hilbert spaces, recasting spectral theorems categorically.

Findings

Categorical description of quantum algebras via dagger-Frobenius monoids

Reformulation of spectral theorems in categorical language

Application to measurement objects in Boolean toposes

Abstract

We describe how dagger-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional 'quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of dagger-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory.