The Strict 2-Category Structure of Distorted Monoidal Categories

TL;DR

This paper develops a strict 2-category framework for distorted monoidal categories, allowing non-invertible, direction-sensitive tensor structures, and provides a constructive calculus for formal reasoning about these generalized categories.

Contribution

It introduces distorted monoidal categories with non-invertible distortions, forming a strict 2-category, and offers a constructive, type-safe calculus for formal reasoning.

Findings

Established a strict 2-category structure for distorted monoidal categories.

Provided explicit construction schemes for non-invertible distortions.

Extended modeling capabilities to irreversible and resource-sensitive processes.

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.