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arXiv · 2602.20755

The direction functor for Schreier extensions of monoids

Stefano Ambra·Andrea Montoli·Diana Rodelo
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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.

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Stefano Ambra, Andrea Montoli, Diana Rodelo. 2026-02-24. The direction functor for Schreier extensions of monoids. https://arxiv.org/abs/2602.20755

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