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Locally involutive semigroups

Clemens Berger, Jonathon Funk

TL;DR

This work introduces locally involutive semigroups and embeds them into ordered groupoids, establishing an ESN-correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the inverse semigroup theory. It develops an adjunction between the classifying topos ${oldsymbol{ extscr B}}(S)$ and the category ${oldsymbol{ extscr X}}(S)$ of semigroups over an inverse semigroup $S$, via the functors ${oldsymbol extLambda}$ and ${oldsymbol extGamma}$, and proves an equivalence on the appropriate subcategories, including left involutive and étale morphisms. The paper further analyzes representable left involutive semigroups $S(e)$, étale morphisms, and the fiber presheaf construction, linking presheaf data to involutive structures and topos-theoretic reflections. It connects these ideas to involutive $S$-sets, balanced $S$-sets, and balanced $S$-algebras, providing free and quotients constructions (free/balanced modules and algebras) and laying groundwork for $S$-modules and $S$-algebras in this generalized setting. Overall, the framework unifies semigroup-with-involution theory, category-theoretic adjunctions, and topos-theoretic perspectives to extend classical ESN correspondences and to realize desingularized embeddings into involutive algebras."

Abstract

We introduce locally involutive semigroups and embed them into the category of ordered groupoids. This embedding restricts to a correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the classical ESN-correspondence between inverse semigroups and inductive groupoids. An important subcategory of locally involutive semigroups is formed by left involutive semigroups because the classifying topos of an inverse semigroup S is equivalent to the category of left involutive semigroups étale over S [4]. We recover this equivalence from a general adjointness and use the latter to determine when a left involutive semigroup étale over S is actually an involutive semigroup. Any left involutive semigroup étale over S embeds into an involutive S-algebra as we call it. The underlying semigroup of this algebra is involutive.

Locally involutive semigroups

TL;DR

This work introduces locally involutive semigroups and embeds them into ordered groupoids, establishing an ESN-correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the inverse semigroup theory. It develops an adjunction between the classifying topos and the category of semigroups over an inverse semigroup , via the functors and , and proves an equivalence on the appropriate subcategories, including left involutive and étale morphisms. The paper further analyzes representable left involutive semigroups , étale morphisms, and the fiber presheaf construction, linking presheaf data to involutive structures and topos-theoretic reflections. It connects these ideas to involutive -sets, balanced -sets, and balanced -algebras, providing free and quotients constructions (free/balanced modules and algebras) and laying groundwork for -modules and -algebras in this generalized setting. Overall, the framework unifies semigroup-with-involution theory, category-theoretic adjunctions, and topos-theoretic perspectives to extend classical ESN correspondences and to realize desingularized embeddings into involutive algebras."

Abstract

We introduce locally involutive semigroups and embed them into the category of ordered groupoids. This embedding restricts to a correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the classical ESN-correspondence between inverse semigroups and inductive groupoids. An important subcategory of locally involutive semigroups is formed by left involutive semigroups because the classifying topos of an inverse semigroup S is equivalent to the category of left involutive semigroups étale over S [4]. We recover this equivalence from a general adjointness and use the latter to determine when a left involutive semigroup étale over S is actually an involutive semigroup. Any left involutive semigroup étale over S embeds into an involutive S-algebra as we call it. The underlying semigroup of this algebra is involutive.
Paper Structure (29 sections, 38 theorems, 176 equations)

This paper contains 29 sections, 38 theorems, 176 equations.

Table of Contents

  1. Locally involutive semigroups
  2. Terminology and basic properties
  3. Partial orderings of a $*$-semigroup
  4. Morphisms of $*$-semigroups
  5. ESN-correspondence for quasi-involutive semigroups
  6. Etale morphisms of $*$-semigroups
  7. Involutive $S$-sets
  8. Categories
  9. The category $\mathcal{X}(S)$ of semigroups over $S$
  10. The category $L(S)$
  11. Representable left involutive semigroups
  12. The classifying topos ${\mathscr{B}}\xspace(S)$
  13. The adjointness $\Lambda\dashv\Gamma$
  14. ${\mathscr{B}}\xspace(S)$ to $\mathcal{X}(S)$
  15. $\mathcal{X}(S)$ to ${\mathscr{B}}\xspace(S)$
  16. ...and 14 more sections

Key Result

Lemma 1.1

Any involutive semigroup is left and right involutive. Any left (or right) involutive semigroup is locally involutive.

Theorems & Definitions (117)

  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • Corollary 1.4
  • Remark 1.5
  • Lemma 1.6
  • proof
  • Remark 1.7
  • ...and 107 more