The direction functor for Schreier extensions of monoids

TL;DR

This paper introduces a functorial approach to associating actions with Schreier extensions of monoids, generalizing the direction functor and revealing new categorical structures related to non-abelian cohomology.

Contribution

It defines a new functor that generalizes the direction functor for Schreier extensions, establishing its properties and connections to cohomology and monoidal structures.

Findings

The functor is conservative, product preserving, and a cofibration.

Fibres of the functor have a canonical symmetric monoidal structure.

Connected components of these structures relate to Patchkoria second cohomology monoids.

Abstract

We observe that the process of associating an action to any Schreier extension of monoids with commutative and cancellative kernel is functorial. We show that this functor is a generalisation of the direction functor, used to give a categorical description of non-abelian cohomology in terms of extensions. We further prove that our functor is a conservative, product preserving cofibration and from this we conclude that its fibres are endowed with a canonical symmetric monoidal structure. The commutative monoids obtained as connected components of these symmetric monoidal categories are isomorphic to Patchkoria second cohomology monoids of a monoid with coefficients in semimodules.