Asymmetric Geometry as Irreversible Holonomy of an Oplax Functor-Correction (Version 4)

This paper develops a categorical framework for asymmetric geometry based on the accumulation of oplax functor-corrections along paths in a 2-category. Geometric settings are modeled as objects equipped with deforming monoids, while transport between settings is described by a functor whose global failure is corrected by path-dependent oplax transformations.

The work formalizes three distinct notions often conflated in informal discussions of geometric asymmetry: flatness, route-dependence, and irreversibility. Using a holonomy map from path composition into a deformation monoid, precise criteria are established linking route-dependence to non-commutativity and reversibility to the existence of suitable inverses within the underlying monoid.

The main results prove that:

• Route-dependence occurs if and only if the holonomy monoid is non-commutative.

• Reversibility occurs if and only if the holonomy monoid is left-invertible.

• Route-dependence and irreversibility are logically independent properties.

• The asymmetric regime, characterized by both route-dependence and irreversibility, is non-empty and is minimally realized by the bicyclic monoid.

The paper further identifies the coherence condition required for the accumulation of oplax corrections to define a well-behaved holonomy map and isolates the construction of a concrete geometric realization as an open problem. Consequently, the framework establishes a consistent and rigorous algebraic theory of asymmetric geometry while clearly distinguishing proved results from outstanding geometric questions.

Keywords: category theory, 2-categories, oplax natural transformations, holonomy, semigroup theory, monoids, bicyclic monoid, geometric deformation, asymmetric geometry, higher category theory.