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The direction functor for Schreier extensions of monoids

Stefano Ambra, Andrea Montoli, Diana Rodelo

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.

The direction functor for Schreier extensions of 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.
Paper Structure (8 sections, 40 theorems, 85 equations)

This paper contains 8 sections, 40 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Definition and first properties of Schreier extensions of monoids
  3. Action induced by a Schreier extension
  4. Interlude on Schreier points and $S$-reflexive relations
  5. The Chasles relation of a cc-Schreier extension
  6. Definition of the direction functor for cc-Schreier extensions
  7. Properties of the direction functor
  8. Monomorphisms and regular epimorphisms in $cc\text{-}SExt_M$

Key Result

Lemma 2.1

In the above notation:

Theorems & Definitions (80)

  • Lemma 2.1
  • Definition 2.2: Redei
  • Corollary 2.3
  • proof
  • Lemma 2.4: cf. schreier_book
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6: Cf. P-II
  • proof
  • ...and 70 more