Enriching Diagrams with Algebraic Operations

Abstract: In this paper, we extend diagrammatic reasoning in monoidal categories with algebraic operations and equations. We achieve this by considering monoidal categories that are enriched in the category of Eilenberg-Moore algebras for a monad. Under the condition that this monad is monoidal and affine, we construct an adjunction between symmetric monoidal categories and symmetric monoidal categories enriched over algebras for the monad. This allows us to devise an extension, and its semantics, of the ZX-calculus with probabilistic choices by freely enriching over convex algebras, which are the algebras of the finite distribution monad. We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.