Sober Étale Groupoids and Pseudogroup Duality

• Sober étale groupoids are étale topological groupoids with sober spaces of objects and arrows, ensuring each irreducible closed set uniquely corresponds to a point.
• They establish a duality with pseudogroups via the groupoid of germs, offering categorical equivalence and clear mappings between structure-preserving morphisms.
• Their completeness and limit constructions facilitate advanced applications in noncommutative topology and C*-algebra theory.

A sober étale groupoid is an étale topological groupoid in which both the space of objects and the space of morphisms are sober topological spaces. Sobriety is a separation property essential in spatial dualities, ensuring correspondence between irreducible closed subsets and points. The theory of sober étale groupoids features canonical dualities with pseudogroups, categorical completeness, and applications to operator algebras and noncommutative topology.

1. Definition and Basic Structure

An étale groupoid comprises a groupoid object $G$ where the spaces of units $G^{(0)}$ and arrows $G^{(1)}$ are topological spaces, all structure maps (source $s$, target $t$, multiplication $m$, unit $u$, inversion $i$) are continuous, and the source and target maps are local homeomorphisms. An open bisection is an open subset $U \subseteq G^{(1)}$ such that both $s|_U$ and $t|_U$ restrict to homeomorphisms onto open subsets of $G^{(0)}$. A topological space $X$ is sober if every irreducible closed subset is the closure of a unique point. A sober étale groupoid is then defined as a topological groupoid where both $G^{(0)}$ and $G^{(1)}$ are sober and the groupoid data is étale (Taylor, 20 Dec 2025, Lawson et al., 2011).

2. Pseudogroups and Groupoid Correspondence

A pseudogroup $\mathcal{P}$ is a set of partial homeomorphisms between open subsets of a sober space $X$, closed under composition, inversion, arbitrary unions of pairwise compatible families, and all open-set identities. The open bisections of a sober étale groupoid $G$ form a pseudogroup $\Sigma(G)$ under composition and unions (Taylor, 20 Dec 2025).

Conversely, from a pseudogroup $\mathcal{P}$, one constructs the groupoid of germs $\Gamma(\mathcal{P})$, whose objects are points in the base sober space, and morphisms are equivalence classes $[x, f]$ with $f \in \mathcal{P}$ and $x$ in the domain of $f$, with the equivalence given by local agreement. This groupoid inherits a topology from the pseudogroup and is itself a sober étale groupoid. The functoriality extends to morphisms: actors (functors preserving open bisections) of groupoids correspond to pseudogroup morphisms, i.e., structure-preserving maps of partial homeomorphisms (Taylor, 20 Dec 2025, Lawson et al., 2011).

3. Duality and Categorical Equivalence

There is an equivalence of categories between sober étale groupoids (with actors) and pseudogroups (with pseudogroup morphisms). Two functors realize this equivalence:

• $\Sigma: G \mapsto \Sigma(G)$, assigning a groupoid to its pseudogroup of open bisections.
• $\Gamma: \mathcal{P} \mapsto \Gamma(\mathcal{P})$, assigning a pseudogroup to its groupoid of germs.

Restricting the Cockett–Garner adjunction to sober groupoids and spatial pseudogroups gives an equivalence: $\mathbf{SoberEtaleGpd} \simeq \mathbf{Pseudo}$ (Taylor, 20 Dec 2025, Lawson et al., 2011). This correspondence underpin noncommutative generalizations of Stone duality and enables transfer of categorical properties between groupoids and pseudogroups.

4. Limits, Products, and Categorical Completeness

The category of sober étale groupoids admits all small limits, including products, equalizers, pullbacks, and inverse limits. Limits in sober étale groupoids are computed via the equivalence with pseudogroups: form the diagram of pseudogroups, compute its limit in the category of pseudogroups (using the forgetful functor to sets, which creates all small limits), and then reconstruct the groupoid limit as the germ groupoid of the resulting limit pseudogroup (Taylor, 20 Dec 2025).

Typical Constructions

• Products: Given $G$ and $H$, the product groupoid's space of objects is $G^{(0)} \times H^{(0)}$, arrows $G^{(1)} \times H^{(1)}$, and open bisections correspond on the nose to products of bisections.
• Equalizers: For actors $\alpha, \beta: G \to H$, their equalizer is the groupoid of germs of the sub-pseudogroup $$E = \{ U \mid \Sigma(\alpha)(U) = \Sigma(\beta)(U) \}$$.
• Fibered Constructions: The completeness ensures the existence of fiber products and inverse systems (Taylor, 20 Dec 2025).

5. Noncommutative Stone Duality and Booleanization

The equivalence described provides a noncommutative generalization of Stone duality. Boolean pseudogroups (complete infinitely distributive inverse monoids with Boolean idempotent frame) correspond to Hausdorff étale groupoids whose unit space is a Boolean space and open bisections form a basis of compact opens (Lawson et al., 2011). This extends classical Stone duality (Boolean algebras and Stone spaces) and Kellendonk–Paterson duality (tiling groupoids and inverse semigroups).

Table: Levels of Dualities

Pseudogroup Type	Dual Étale Groupoid	Remarks
Spatial pseudogroup	Sober étale groupoid	General case
Boolean pseudogroup	Boolean étale groupoid	Hausdorff, Boolean unit space
Distributive inverse semigroup	Distributive étale gpd	Idl-completion, finite joins
Tight completion	Tight groupoid	Ultrafilter groupoids (Paterson)

6. Applications and Further Structure

Sober étale groupoids are structurally central in groupoid $C^*$-algebra theory. The categorical completeness ensures that projective/inverse systems of groupoids give rise to direct systems of $C^*$-algebras, relevant for the construction of AF-algebras and analysis of group actions (Taylor, 20 Dec 2025). The spectrum construction provides a spatial realization of the ‘locale of points’ of a pseudogroup.

Further, the duality framework accommodates refinements:

• The Idl-completion of inverse semigroups and the theory of tight filters yield more specialized dualities, including for weakly Boolean, distributive, and tight semigroups, recovering ultrafilter and Paterson groupoids (Lawson et al., 2011).

7. Examples and Classification

Transformation pseudogroups on a sober space $X$ (partial homeomorphisms with open domains/ranges) form pseudogroups, whose groupoid of germs recovers classical transformation or foliation groupoids. In these cases, sobriety of $X$ ensures sobriety of the groupoid. When the pseudogroup is ‘spatial’ (germ topology separates arrows), the construction recovers the original structure up to equivalence (Lawson et al., 2011).

• J. Taylor, "Limits in categories of étale groupoids and pseudogroups" (Taylor, 20 Dec 2025).
• M. V. Lawson and D. H. Lenz, "Pseudogroups and their étale groupoids" (Lawson et al., 2011).