Working Paper Article Version 1 This version is not peer-reviewed

Category Algebras and States on Categories

Version 1 : Received: 28 May 2021 / Approved: 31 May 2021 / Online: 31 May 2021 (08:31:29 CEST)

A peer-reviewed article of this Preprint also exists.

Saigo, H. Category Algebras and States on Categories. Symmetry 2021, 13, 1172. Saigo, H. Category Algebras and States on Categories. Symmetry 2021, 13, 1172.


The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand-Naimark-Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs.


Category; Algebra; State; Category Algebra; State on Category; Noncommutative Probability; Quantum Probability; GNS representation


Computer Science and Mathematics, Mathematics

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