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Étale inverse semigroupoids: elementary properties, universal constructions and duality

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Abstract

The main goal of this article is to unite the theories of groupoids, semigroups and their dynamical aspects under a unique umbrella theory, which is attained with semigroupoids. We start by dealing with a small point of contention, and show that two axiomatizations of semigroupoids which have appeared in the literature, akin to the “graph-theoretic” and “algebraic” descriptions of categories, are not equivalent (to the contrary of the analogous category-theoretical result). The remainder of the paper is specialized to the study of (étale) inverse semigroupoids. Elementary results and notions from the theories of discrete inverse semigroups and étale groupoids are generalized—such as the Vagner–Preston Theorem, (partial and pre-)actions, and semidirect products (which include groupoids of germs by inverse semigroup actions). As our main result, we finish this paper with a non-commutative Stone-type duality for ample inverse semigroupoids.

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Acknowledgements

I sincerely thank the reviewer for the careful reading of this paper and the many comments which helped polish this material and make the results clearer.

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Authors and Affiliations

  1. Departamento de Engenharias da Mobilidade, Universidade Federal de Santa Catarina, Joinville, SC, Brazil

    Luiz Gustavo Cordeiro

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  1. Luiz Gustavo Cordeiro
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Correspondence to Luiz Gustavo Cordeiro.

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Communicated by Mark V. Lawson.

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The author was partly supported by the ANR project GAMME (ANR-14-CE25-0004), during his postdoctoral fellowship at UMPA – ENS Lyon.

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Cordeiro, L.G. Étale inverse semigroupoids: elementary properties, universal constructions and duality. Semigroup Forum 106, 67–127 (2023). https://doi.org/10.1007/s00233-022-10329-8

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