Categorical formulation of quantum algebras

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.

34 pages, to appear in Communications in Mathematical Physics

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• `What is a Thing?': Topos Theory in the Foundations of Physics
• Abstract Physical Traces
• State sum construction of two-dimensional open-closed Topological Quantum Field Theories
• Toy quantum categories
• Completeness of dagger-categories and the complex numbers

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