Category Algebras and States on Categories

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