Adjunction of Cockett and Garner

• The adjunction establishes an equivalence between complete infinitely distributive inverse monoids and sober étale groupoids.
• The construction utilizes B-functor and G-functor to transfer properties between algebraic and topological structures.
• The duality extends classical Stone duality and supports practical applications in operator algebras, tiling theory, and noncommutative geometry.

The adjunction of Cockett and Garner describes a fundamental equivalence between the theory of pseudogroups—complete infinitely distributive inverse monoids—and the theory of sober étale topological groupoids. This adjunction provides a non-commutative generalization of Stone duality and underpins key applications in operator algebras, tiling theory, and topos theory. The construction generates synthetic accounts of étale groupoids as non-commutative spectra of pseudogroups, establishing a categorical framework in which properties and constructions in one setting transfer naturally to the other (Lawson et al., 2011, Taylor, 20 Dec 2025).

1. Definition of the Adjunction

The Cockett–Garner adjunction links two categories:

• Etale: The category of étale topological groupoids with covering functors.
• Inv: The category of pseudogroups (complete infinitely distributive inverse monoids), considered with morphisms opposite to those in Etale.

Two functors formalize this adjunction:

• The B-functor $$\mathcal{B}: \mathrm{Etale} \rightarrow \mathrm{Inv}^{\mathrm{op}}$$, assigning to an étale groupoid $G$ its pseudogroup of open bisections $\mathcal{B}(G)$.
• The G-functor $$\mathcal{G}: \mathrm{Inv}^{\mathrm{op}}\rightarrow \mathrm{Etale}$$, assigning to a pseudogroup $S$ its groupoid of completely prime filters $G(S)$.

The adjunction $\mathcal{B}\dashv \mathcal{G}$ is realized via unit and counit natural transformations—maps

$$\epsilon: S\rightarrow \mathcal{B}(G(S)) \qquad \eta: G\rightarrow G(\mathcal{B}(G))$$

that become isomorphisms when restricted to the subcategory of spatial pseudogroups and sober étale groupoids, thus inducing an equivalence of categories (Lawson et al., 2011, Taylor, 20 Dec 2025).

2. Construction of the Functors

From Étale Groupoids to Pseudogroups

Given an étale groupoid $G$, the set of open bisections $B(G)$ forms a pseudogroup under:

• Product $U \cdot V=\{gh: g\in U,\ h\in V,\ d(g)=r(h)\}$,
• Inversion $U\mapsto U^{-1}$,
• Unions of compatible bisections.

This structure functorially associates to every étale groupoid a pseudogroup, with covering functors inducing pullback of open bisections (Lawson et al., 2011).

From Pseudogroups to Étale Groupoids

Given a pseudogroup $S$, the spectrum $G(S)$ consists of completely prime filters on $S$. The groupoid structure is defined by:

• Objects: completely prime filters of idempotents,
• Arrows: completely prime filters in $S$,
• Composition: $F\cdot F'=(FF')^{\uparrow}$, if $d(F)=r(F')$,
• Topology: the subbasis $X_s = \{F: s\in F\}$ for $s\in S$ yields open bisections.

This realizes the germ groupoid (or spectrum) with desired properties (Taylor, 20 Dec 2025).

3. Spatiality, Sobriety, and the Equivalence Theorem

A pseudogroup is spatial if the natural map $\epsilon: S\to \mathcal{B}(G(S))$ is injective, ensuring that no two elements induce the same action on spectra. An étale groupoid is sober if its point-set space and arrow space are sober topological spaces, characterized by the homeomorphism $\eta: G\to G(\mathcal{B}(G))$.

Theorem: Restricting the adjunction to spatial pseudogroups and sober étale groupoids, the unit and counit are isomorphisms. This yields a categorical equivalence: $$\left\{\text{spatial pseudogroups}\right\}^{\mathrm{op}} \simeq \left\{\text{sober étale groupoids}\right\}$$

This provides a precise, algebraic-topological duality extending the frame–locale correspondence to the inverse semigroup and groupoid setting (Lawson et al., 2011, Taylor, 20 Dec 2025).

4. Classical and Non-commutative Stone Dualities

The adjunction subsumes classical Stone duality for frames as the commutative case where every element is idempotent. The non-commutative generalization provided by the adjunction yields several new correspondences:

• Boolean inverse semigroups $\leftrightarrow$ Boolean étale groupoids (Hausdorff, with compact-open bisections, unit space Boolean),
• Distributive inverse semigroups $\leftrightarrow$ Coherent étale groupoids,
• Weakly Boolean inverse semigroups $\leftrightarrow$ Étale groupoids with Boolean unit space.

These dualities support algebraic constructions of groupoid models for C*-algebras, notably the Cuntz–Krieger, graph, and tiling algebras, and unify various Stone-type theorems (Lawson et al., 2011).

5. Morphisms, Actors, and Functorial Properties

Morphisms in these categories are carefully matched:

• Actors are the morphisms between étale groupoids, preserving the étale and sobriety structures.
• Pseudogroup homomorphisms preserve joins of compatible families, essential for ensuring the functoriality of the spectrum and bisection constructions.

The adjoint equivalence naturally identifies the two as matching under the functors, with diagrammatic commutativity and satisfaction of triangle identities (Taylor, 20 Dec 2025).

6. Limits, Colimits, and Structural Completeness

The category of pseudogroups is complete and cocomplete, as the forgetful functor to sets creates all small limits and colimits. Via the Cockett–Garner equivalence, the category of sober étale groupoids and actors inherits these categorical properties:

• Products, equalizers, coequalizers, and general (small) limits are constructed by transporting the set-theoretic limits through the spectrum and bisection functors.
• For a diagram $D: I\to \mathrm{EtaleGrpd}_{\mathrm{sober}}$, the limit is given by

$$\underset{\longleftarrow}{\lim} D \cong \Gamma\left(\underset{\longleftarrow}{\lim}_i \mathcal{B}(D(i))\right)$$

where $\Gamma$ is the germ groupoid construction, and $\mathcal{B}$ is the pseudogroup of bisections (Taylor, 20 Dec 2025).

7. Applications and Conceptual Significance

The adjunction is instrumental in a range of areas:

• Noncommutative geometry: Models for noncommutative spaces arise from groupoid C*-algebras indexed by sober étale groupoids.
• Sheaf theory and topos theory: The categorical completeness enables gluing and descent arguments, as the classifying topos of an étale groupoid is naturally stable under limits.
• Tilings, dynamical systems, and inverse semigroup theory: The algebraic and topological techniques translate bidirectionally, offering tools for both model construction and classification.

A plausible implication is that this adjunction framework enables the algebraic characterization of geometric structures and their operator-algebraic invariants in a unified categorical setting (Lawson et al., 2011, Taylor, 20 Dec 2025).