The direction functor for Schreier extensions of monoids
Category Theory
2026-02-25 v1
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.
Cite
@article{arxiv.2602.20755,
title = {The direction functor for Schreier extensions of monoids},
author = {Stefano Ambra and Andrea Montoli and Diana Rodelo},
journal= {arXiv preprint arXiv:2602.20755},
year = {2026}
}