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A categorical approach to inverse semigroups, étale groupoids, and their C*-algebras

Takuto Fujieda, Takeshi Katsura, Tomoki Uchimura

Abstract

We introduce a category of inverse semigroup actions and a category of étale groupoids. We show that there are three functors which send inverse semigroups to their spectral actions, inverse semigroup actions to their transformation groupoids, and étale groupoids to their groupoid C*-algebras, respectively. The composition of these functors is naturally equivalent to the already known functor which sends inverse semigroups to their C*-algebras. This result is a categorical version of the theorem by Paterson which states that the inverse semigroup C*-algebras coincide the groupoid C*-algebras of the universal groupoids. We also construct a functor from the category of étale groupoids to the category of inverse semigroup actions sending étale groupoids to their slice actions, and show that this functor is right adjoint to the functor sending inverse semigroup actions to their transformation groupoids.

A categorical approach to inverse semigroups, étale groupoids, and their C*-algebras

Abstract

We introduce a category of inverse semigroup actions and a category of étale groupoids. We show that there are three functors which send inverse semigroups to their spectral actions, inverse semigroup actions to their transformation groupoids, and étale groupoids to their groupoid C*-algebras, respectively. The composition of these functors is naturally equivalent to the already known functor which sends inverse semigroups to their C*-algebras. This result is a categorical version of the theorem by Paterson which states that the inverse semigroup C*-algebras coincide the groupoid C*-algebras of the universal groupoids. We also construct a functor from the category of étale groupoids to the category of inverse semigroup actions sending étale groupoids to their slice actions, and show that this functor is right adjoint to the functor sending inverse semigroup actions to their transformation groupoids.

Paper Structure

This paper contains 21 sections, 67 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial maps
  4. $C^*$-algebras
  5. Inverse semigroups and their $C^*$-algebras
  6. Inverse semigroups and their spectral actions
  7. Inverse semigroup $C^*$-algebras
  8. Étale groupoids and their $C^*$-algebras
  9. Étale groupoids and open bisections
  10. Groupoid homomorphisms
  11. Étale groupoid $C^*$-algebras
  12. Relationship between inverse semigroups and étale groupoids
  13. A category $\mathop{\mathrm{\mathbf{ISA}}}\nolimits$ and a functor $\mathop{\mathrm{SP}}\nolimits$
  14. A category $\mathop{\mathrm{\mathbf{EG}}}\nolimits$
  15. Two classes of groupoid homomorphisms
  16. ...and 6 more sections

Key Result

Proposition 1.11

The constructions $S \mapsto C^*(S)$ and $\theta\mapsto\sigma_\theta$ form a functor from $\mathop{\mathrm{\mathbf{IS}}}\nolimits$ to $\mathop{\mathrm{\mathbf{C^*_{alg}}}}\nolimits$. We denote the functor as $C^*$.

Theorems & Definitions (171)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 161 more