Pseudogroups and Their Morphisms

• Pseudogroups are algebraic structures capturing partial symmetries through complete inverse monoids enriched with lattice operations.
• Pseudogroup morphisms preserve both inverse-monoid and lattice-theoretic operations, establishing an equivalence with étale groupoids.
• Universal constructions in pseudogroups extend applications in topology, operator algebras, and foliation theory, supporting robust classification results.

A pseudogroup is an algebraic structure capturing symmetries described by partially defined transformations, with applications across topology, operator algebras, foliation theory, and dynamical systems. Formally, a pseudogroup is a complete inverse monoid with a rich order and lattice structure, and morphisms between pseudogroups are homomorphisms that respect both the inverse-monoid and lattice-theoretic operations. The category of pseudogroups with these morphisms supports an equivalence with sober étale groupoids, provides a framework for both universal constructions and rigid classification results, and serves as the setting for generalizations to higher categorical and internalized contexts.

1. Formal Structure of Pseudogroups

A pseudogroup $S$ is defined as a tuple $$(S, \le, \wedge, \bigvee, \cdot, {}^{-1}, 0_S, 1_S)$$ where:

• $(S, \cdot, 1_S)$ is a monoid with an involution $s \mapsto s^{-1}$ satisfying $(st)^{-1} = t^{-1}s^{-1}$ and $(s^{-1})^{-1} = s$.
• $(S, \le)$ is a complete meet-semiltattice: for all $s, t$, the meet $s \wedge t$ exists; for every compatible subset $X \subseteq S$, the join $\bigvee X$ exists.
• The set $E(S) = \{ e \in S : e^2 = e = e^{-1} \}$ is a frame: a complete lattice with distributivity of finite meets over arbitrary joins.
• Multiplication and inversion preserve all lattice operations as follows:
  • $(s \wedge t)u = (su) \wedge (tu)$ and $u(s \wedge t) = (us) \wedge (ut)$,
  • $s(\bigvee t_i) = \bigvee (st_i)$ and $(\bigvee t_i)s = \bigvee (t_i s)$ for compatible families,
  • $(s \wedge t)^{-1} = s^{-1} \wedge t^{-1}$ and $(\bigvee t_i)^{-1} = \bigvee t_i^{-1}$.
• The inverse semigroup law $ss^{-1}s = s$, $s^{-1} ss^{-1} = s^{-1}$ holds.

This renders $S$ an inverse monoid "completed" by all joins of compatible families, ensuring both algebraic and topological fit for describing partial symmetries (Taylor, 20 Dec 2025, Lawson et al., 2011).

2. Pseudogroup Morphisms and the Category PsGrp

Given pseudogroups $S$ and $T$, a pseudogroup morphism $f: S \to T$ is a function satisfying:

• $f(1_S) = 1_T$, $f(0_S) = 0_T$,
• $f(s \cdot s') = f(s) \cdot f(s')$, $f(s^{-1}) = f(s)^{-1}$,
• $f(s \wedge t) = f(s) \wedge f(t)$,
• $$f(\bigvee_{i \in I} s_i) = \bigvee_{i \in I} f(s_i)$$ for any compatible family,
• $s \le t \implies f(s) \le f(t)$.

Pseudogroup morphisms preserve both the inverse-monoid structure and the lattice-theoretic operations. This defines the category $\mathbf{PsGrp}$ of pseudogroups and their morphisms, which supports all small limits: limits are computed on underlying sets and equipped pointwise with the unique compatible pseudogroup structure, as ensured by the forgetful functor to $\mathbf{Set}$ creating limits (Taylor, 20 Dec 2025).

3. Universal Constructions, Limits, and the Cockett–Garner Adjunction

The category $\mathbf{PsGrp}$ admits standard categorical limits—products, equalizers, pullbacks—constructed via their set-theoretic analogues, with all structure given pointwise. For a diagram $D: J \to \mathbf{PsGrp}$, the limit $L = \lim_{j \in J} U(D_j)$ is the subset of $\prod_j D_j$ of tuples $(x_j)$ compatible with the connecting morphisms, equipped pointwise with pseudogroup operations. Key lemmas establish closure under operations and universal lifting of cones (Taylor, 20 Dec 2025).

A central structural result is the adjunction (Cockett–Garner) between pseudogroups and étale groupoids:

$$\mathbf{Pseudogroups} \overset{\mathcal G}{\longrightarrow} \mathbf{ÉtaleGroupoids} \overset{\mathcal B}{\longrightarrow} \mathbf{Pseudogroups}$$

with $\mathcal{G}$ assigning to a pseudogroup $S$ its germ groupoid, and $\mathcal{B}$ assigning to $G$ its pseudogroup of open local bisections (Cockett et al., 2020, Lawson et al., 2011). In the sober/ spatial setting, this adjunction is an equivalence, allowing translation of limit and colimit computations between categories.

4. Key Examples and Representations

Classical instances realize pseudogroups as partial homeomorphisms or local diffeomorphisms:

• Topological pseudogroups: Submonoids of the inverse monoid of partial homeomorphisms of a topological space $X$ closed under composition, inversion, joins of compatible families. The associated groupoid is the “germ groupoid” (Haefliger).
• Localic pseudogroups: Inverse monoid of partial automorphisms of a locale $L$ (Cockett et al., 2020).
• Full groups and dynamical applications: The full group $[[\mathcal{G}]]$ associated to a pseudogroup $\mathcal{G}$ of partial homeomorphisms is critical in topological dynamics, rigidity phenomena, and classification of transformation groups. Rigidity results show that group homomorphisms between full groups extend to continuous morphisms of pseudogroups, imposing strong restrictions on embeddings and yielding invariants tied to orbital graph geometry, dynamical homology, and complexity (Bon, 2018).

5. Generalized Correspondence: Groupoids, Inverse Categories, and Internalization

The classical étale groupoid–pseudogroup correspondence, originating with Haefliger, Resende, and Lawson–Lenz, has been systematically generalized through several axes:

1. Enriched morphisms: Arbitrary monoid homomorphisms (with compatible maps of representing spaces) replace restrictive “callitic” morphisms; on groupoid side, strict functors are replaced by cofunctors (opfibrations).
2. Multi-object generalization: Inverse monoids/inverse categories correspond to groupoids/partite étale groupoids, allowing for families of objects.
3. Non-invertible generalization: Étale groupoids vs. join-inverse categories broadens to source-étale categories vs. complete restriction categories by allowing non-invertible arrows.
4. Internalization: All these correspondences hold, not just for topological spaces, but for any join restriction category with local glueings (e.g., smooth manifolds, locales, schemes). Internal pseudogroups correspond to groupoids internal to such categories, yielding “complete $\mathcal{C}$-pseudogroups” (Cockett et al., 2020).

The resulting adjunction generalizes the classical spatial–sober duality and manifests as an equivalence of categories under appropriate fixpoints (hyperconnected pseudogroups, source-étale groupoids).

6. Pseudogroup Morphisms in Foliation Theory and Dynamical Rigidity

In the geometric context, notably for Riemannian foliations, Álvarez-López and Masa define morphisms between pseudogroups as maximal families of continuous maps closed under patching with elements of both source and target pseudogroups. They prove completeness and closure results: any morphism between complete Riemannian pseudogroups is itself complete, possesses a closure, and its local maps are smooth along orbit closures (López et al., 2013). This underpins homotopy invariance of the foliation spectral sequence and the density of smooth maps in the adapted $C^\infty$ topology.

Rigidity phenomena in topological full groups associated to minimal pseudogroups on the Cantor set further connect pseudogroup morphisms, invariants, and classification of subgroups, revealing new non-Kazhdan ($T$) sources of combinatorial rigidity (Bon, 2018).

7. Universal and Presentational Aspects

Pseudogroups admit universal constructions via coverages—systems of subsets specifying the join operations enforced. The universal $C$-pseudogroup $P_C(S)$ (for a coverage $C$) expresses the completion of an inverse semigroup $S$ by compatible-join order ideals that are $C$-closed (Lawson et al., 2011). This leads to tight completions, Booleanizations (constructing Boolean inverse monoids), and dualities with groupoids equipped with appropriate covering structures—essential in non-commutative Stone duality and applications to $C^*$-algebras and aperiodic tilings.

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References

• "Limits in categories of étale groupoids and pseudogroups" (Taylor, 20 Dec 2025)
• "Pseudogroups and their etale groupoids" (Lawson et al., 2011)
• "Generalising the étale groupoid--complete pseudogroup correspondence" (Cockett et al., 2020)
• "Morphisms between complete Riemannian pseudogroups" (López et al., 2013)
• "Rigidity properties of full groups of pseudogroups over the Cantor set" (Bon, 2018)