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The Strict 2-Category Structure of Distorted Monoidal Categories

Joaquim Reizi Higuchi

TL;DR

The paper introduces distorted monoidal categories to model irreversible, direction-sensitive tensor interchange, formalizing non-invertible distortions $\sigma$ and unit distortions $\Lambda$ via typed hexagon axioms. It establishes that distorted monoidal categories, with $\sigma$-lax functors satisfying strict compatibility, assemble into a strict 2-category $\mathbf{DistMon}_\sigma$, where composition and interchange are on the nose. The work provides a constructive calculus with explicit composite laxators, existence schemes for non-invertible distortions (e.g., idempotent twists and graded unit distortions), and a 2-monad perspective that clarifies the algebraic structure and potential for mechanization. It connects to braided and skew-monoidal frameworks, while extending them to irreversible, resource-aware contexts and directing attention to coherence, centers, and enrichment in this broader setting. Overall, the framework offers a rigorous, verifiable foundation for directed tensorial phenomena relevant to directed homotopy theory, categorical quantum mechanics, and non-symmetric operadic structures.

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.

The Strict 2-Category Structure of Distorted Monoidal Categories

TL;DR

The paper introduces distorted monoidal categories to model irreversible, direction-sensitive tensor interchange, formalizing non-invertible distortions and unit distortions via typed hexagon axioms. It establishes that distorted monoidal categories, with -lax functors satisfying strict compatibility, assemble into a strict 2-category , where composition and interchange are on the nose. The work provides a constructive calculus with explicit composite laxators, existence schemes for non-invertible distortions (e.g., idempotent twists and graded unit distortions), and a 2-monad perspective that clarifies the algebraic structure and potential for mechanization. It connects to braided and skew-monoidal frameworks, while extending them to irreversible, resource-aware contexts and directing attention to coherence, centers, and enrichment in this broader setting. Overall, the framework offers a rigorous, verifiable foundation for directed tensorial phenomena relevant to directed homotopy theory, categorical quantum mechanics, and non-symmetric operadic structures.

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.

Paper Structure

This paper contains 20 sections, 19 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. From braidings to distortions.
  3. Conceptual position and novelty
  4. Main results
  5. Techniques and verification strategy
  6. Relationship to existing work
  7. Organization
  8. Preliminaries and Notation
  9. Auxiliary Lemmas
  10. Main Theorem: Strict 2-Category Structure
  11. Examples
  12. Discussion and Outlook
  13. A 2-monad presentation of distorted monoidal categories
  14. Adjoining the distortion operations.
  15. Strict $\Lambda$–compatibility: consequences and recognition principles
  16. ...and 5 more sections

Key Result

Lemma 3.1

With notation above, $(GF,\mu^{GF}_{-,-},\mu^{GF}_0)$ is a $\sigma$-lax monoidal functor.

Theorems & Definitions (50)

  • Definition 2.1: Distorted monoidal category
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: $\sigma$-lax monoidal functor
  • Remark 2.5: Scope of (S$\Lambda$) and why we keep it strict
  • Definition 2.6: Monoidal natural transformation
  • Remark 2.7
  • Lemma 3.1: Composite $\sigma$-lax data
  • proof
  • Lemma 3.2: Associativity for $\mu^{GF}_{-,-}$
  • ...and 40 more