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Categorical Formulation of Finite-Dimensional Quantum Algebras

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We describe how †-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 †-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.

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Correspondence to Jamie Vicary.

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Communicated by Y.Kawahigashi

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Vicary, J. Categorical Formulation of Finite-Dimensional Quantum Algebras. Commun. Math. Phys. 304, 765–796 (2011). https://doi.org/10.1007/s00220-010-1138-0

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