The Strict 2-Category Structure of Distorted Monoidal Categories

Abstract

This paper introduces the concept of distorted monoidal categories, a generalization of monoidal and braided monoidal categories that supports non-reversible and direction-sensitive tensor structures. Unlike the classical setting, where the braiding symmetry is required to be invertible, distorted monoidal categories admit non-invertible binary distortions and unit distortions while preserving coherent tensorial reasoning. We show that these structures naturally assemble into a strict 2-category whose composition and interchange laws hold on the nose, not merely up to isomorphism. Beyond the abstract 2-monad justification, our contribution is a fully constructive and type-safe calculus that enables formal reasoning about non-invertible interchange. We provide explicit construction schemes for such distortions, including idempotent twists of classical braidings and graded unit distortions arising from characters on monoidal gradings. This framework extends the expressive power of monoidal categories to model irreversible, resource-sensitive, and direction-dependent processes --such as those in directed homotopy theory, categorical quantum mechanics, and non-symmetric operadic structures--while remaining amenable to mechanization and formal verification.