Diffeological Groupoids: Extending Lie Theory

• Diffeological groupoids are groupoid objects in diffeological spaces that generalize Lie groupoids to include singular, quotient, and infinite-dimensional settings.
• They utilize concepts such as quotient diffeology, subduction, and concrete sheaf methods to extend smooth manifold techniques to broader smooth structures.
• They provide a unifying framework for Morita theory, principal bundles, and Lie algebroid integrations, impacting both smooth and singular geometry.

A diffeological groupoid is, in the broadest and now standard sense, a groupoid object in the category of diffeological spaces: the object space and arrow space carry diffeologies, and the source, target, unit, inversion, and composition maps are smooth in the diffeological sense. This enlarges the Lie-groupoid framework from manifolds to quotient spaces, singular orbit spaces, quasifolds, and various infinite-dimensional or non-Hausdorff smooth objects. In current literature the term also appears in narrower forms, notably Villatoro’s $QUED$-groupoids and singular Lie groupoids, and in an orbit-space-centered usage where Lie groupoids are studied through the quotient diffeology on their coarse moduli spaces (Villatoro, 2023).

1. Diffeological foundations

A diffeological space is a set $X$ equipped with a specified family of plots $p:U\to X$, where $U\subset \mathbb{R}^n$ is open, satisfying three axioms: constants are plots, precomposition with ordinary smooth maps preserves plots, and plots satisfy a locality or gluing condition. A map $f:X\to Y$ between diffeological spaces is smooth when $f\circ p$ is a plot of $Y$ for every plot $p$ of $X$. Smooth manifolds embed fully faithfully into this category by taking the standard diffeology whose plots are the ordinary smooth maps from Euclidean domains.

Two constructions are fundamental for groupoid theory in diffeology. The first is quotient diffeology: if $X\to X/\!\sim$ is a set-theoretic quotient, a parametrization into the quotient is a plot precisely when it locally lifts through the quotient map. The second is the notion of subduction, a smooth surjection $X$0 such that every plot of $X$1 locally lifts to a plot of $X$2. Villatoro also uses the sharper notion of local subduction, where the lift is required through a prescribed point of the fiber; for smooth maps between manifolds, local subductions are exactly the possibly non-surjective submersions (Villatoro, 2023).

Diffeological spaces admit a sheaf-theoretic reformulation. Baez–Hoffnung identify the category of diffeological spaces with the category of concrete sheaves on the site $X$3 of Cartesian spaces, and the same concrete-sheaf viewpoint underlies the stack-theoretic treatment of diffeological coarse moduli spaces. This makes diffeology simultaneously a plot-based and a sheaf-based smooth category, which is one reason it interfaces naturally with groupoids, stacks, and classifying objects (Minichiello, 2024).

2. Definitions and competing usages

In a direct internal-language formulation, a diffeological groupoid $X$4 consists of diffeological spaces $X$5 and $X$6 together with smooth source and target maps

$X$7

a smooth inversion map, a smooth composition

$X$8

and a unit map $X$9; in one expository formulation the unit map is required to be an induction, meaning a smooth map that is a diffeomorphism onto its image with the subset diffeology. In the more categorical formulation of diffeological Morita theory, a diffeological groupoid is simply a groupoid object in $p:U\to X$0, and in that setting source and target are automatically subductions (Iglesias-Zemmour, 24 Aug 2025, Schaaf, 2020).

This definition contains Lie groupoids as the manifold case. If both object and arrow spaces are manifolds and the source and target maps are ordinary submersions, one recovers the usual Lie-groupoid notion. Villatoro refines the picture by replacing the ambient category $p:U\to X$1 with the subcategory $p:U\to X$2 of quasi-étale diffeological spaces and requiring source and target to be $p:U\to X$3-submersions. A $p:U\to X$4-groupoid whose object space is a smooth manifold is called a singular Lie groupoid; this is the class for which the paper constructs a Lie functor to Lie algebroids (Villatoro, 2023).

The terminology is not entirely uniform across the literature. In work on quasifolds, lift-complete groupoids, and orbit-space geometry, “diffeological groupoid” may function more as a viewpoint than as a formal definition: one starts with a Lie groupoid and then studies its orbit space $p:U\to X$5 as a diffeological space, or treats the groupoid through bibundles and the quotient diffeology on its coarse orbit space. That broader usage is central in the study of quasifold groupoids and in results asserting that certain Lie groupoids are determined up to Morita equivalence by their diffeological orbit spaces (Karshon et al., 2022, Miyamoto, 2023, Watts, 2013).

3. Orbit spaces, coarse moduli, and basic forms

For a Lie groupoid $p:U\to X$6, the orbit space $p:U\to X$7 is generally singular, but it always carries a canonical quotient diffeology. The basic theorem in this direction states that for a proper Lie groupoid the quotient map

$p:U\to X$8

induces an isomorphism

$p:U\to X$9

where a form on $U\subset \mathbb{R}^n$0 is basic when $U\subset \mathbb{R}^n$1. As a consequence, the diffeological de Rham cohomology of the orbit space agrees with singular cohomology: $U\subset \mathbb{R}^n$2 This identifies the de Rham complex of the orbit space with the intrinsic basic-form complex of the groupoid (Watts, 2013).

The same correspondence persists for many action groupoids. If a Lie group $U\subset \mathbb{R}^n$3 acts on a manifold $U\subset \mathbb{R}^n$4, then the pullback by the quotient map

$U\subset \mathbb{R}^n$5

is always injective on diffeological forms, and when the identity component $U\subset \mathbb{R}^n$6 acts properly it yields an isomorphism from diffeological forms on $U\subset \mathbb{R}^n$7 to basic forms on $U\subset \mathbb{R}^n$8. The quotient may fail to be a manifold and may even have trivial topology, but the quotient diffeology still carries the differential forms detected by basic invariants upstairs (Karshon et al., 2014).

From the stack-theoretic side, the coarse-moduli construction makes this orbit-space picture functorial. The paper on diffeological coarse moduli spaces defines a functor $U\subset \mathbb{R}^n$9 from stacks over manifolds to concrete sheaves, hence to diffeological spaces, and proves that if a differentiable stack is presented by a Lie groupoid $f:X\to Y$0, then its diffeological coarse moduli space is the quotient diffeology on $f:X\to Y$1. In the same framework, basic differential forms on the stack agree with basic forms on a presenting Lie groupoid, and under properness or gerbe-type hypotheses they agree with diffeological forms on the orbit space (Watts et al., 2014).

4. Actions, bibundles, and Morita equivalence

A full Morita theory for diffeological groupoids replaces manifold-theoretic surjective submersions by subductions. A left action of a diffeological groupoid $f:X\to Y$2 on a diffeological space $f:X\to Y$3 is given by a smooth moment map $f:X\to Y$4 and a smooth map

$f:X\to Y$5

satisfying the usual unit and associativity axioms; right actions are defined similarly. The quotient of an action carries the quotient diffeology, and equivariant maps organize actions into a category $f:X\to Y$6.

A principal bundle for a diffeological groupoid is defined by combining a subduction condition on the bundle projection with a pre-principality condition that the action map be a diffeomorphism onto the fiber product $f:X\to Y$7. For a $f:X\to Y$8-bibundle $f:X\to Y$9, one has left and right underlying bundles, left and right principality, and the notion of biprincipal bibundle when both sides are principal. Two diffeological groupoids are Morita equivalent precisely when there exists a biprincipal bibundle between them. The Hilsum–Skandalis tensor product

$f\circ p$0

provides the composition law on bibundles and yields a bicategory $f\circ p$1 (Schaaf, 2020).

The decisive Morita theorem states that a diffeological bibundle is biprincipal if and only if it is weakly invertible in $f\circ p$2. This directly generalizes the corresponding theorem for Lie groupoids. Several structural consequences then follow: orbit spaces of Morita equivalent diffeological groupoids are diffeomorphic, the fibrating property of a diffeological groupoid is Morita invariant, and the action categories $f\circ p$3 and $f\circ p$4 are equivalent whenever $f\circ p$5 and $f\circ p$6 are Morita equivalent (Schaaf, 2020).

5. Quasi-étale geometry, singular Lie groupoids, and integrability

Villatoro’s construction isolates a subcategory $f\circ p$7 of quasi-étale diffeological spaces. A smooth map $f\circ p$8 is quasi-étale when it is a local subduction, its fibers are totally disconnected, and it satisfies a rigidity condition: if $f\circ p$9 is open and $Y$0 is smooth with $Y$1, then $Y$2 is a local diffeomorphism. A diffeological space is quasi-étale if every point has a quasi-étale chart from a manifold. Examples include manifolds, quotients $Y$3 for countable affine groups $Y$4, and homogeneous spaces $Y$5 with $Y$6 a totally disconnected normal subgroup (Villatoro, 2023).

A $Y$7-groupoid is then a groupoid object in $Y$8 with source and target $Y$9-submersions. When the object space is a smooth manifold, one obtains a singular Lie groupoid. This class is large enough to contain orbifolds, quasifolds, and the Ševera–Weinstein groupoids of arbitrary Lie algebroids, yet rigid enough to support differentiation. Villatoro proves the existence of a functor

$p$0

extending the classical Lie functor, and shows that the Ševera–Weinstein groupoid $p$1 of any Lie algebroid $p$2 is a singular Lie groupoid with

$p$3

This establishes a diffeological version of Lie’s third theorem: every Lie algebroid is integrable by a singular Lie groupoid, even when no smooth Lie groupoid exists (Villatoro, 2023).

A complementary orbit-space theorem is provided by lift-complete Lie groupoids. Miyamoto defines a lift-complete Lie groupoid as an étale effective Lie groupoid with totally disconnected orbits satisfying a lifting condition for orbit-preserving smooth maps. The quotient functor from Lie groupoids to diffeological orbit spaces then restricts to an equivalence between lift-complete Lie groupoids with isomorphism classes of surjective submersive bibundles and quasi-étale diffeological spaces with surjective local subductions. In particular, the Morita equivalence class of a lift-complete Lie groupoid is determined by its diffeological orbit space (Miyamoto, 2023).

6. Higher, bundle-theoretic, and infinitesimal developments

Diffeological groupoids appear naturally as classifying objects for bundles and connections. For a diffeological group $p$4, the presheaf of groupoids

$p$5

is an $p$6-stack that classifies diffeological principal $p$7-bundles, and the presheaf of groupoids

$p$8

classifies principal $p$9-bundles with connection. In that latter groupoid, objects are $X$0-valued connection $X$1-forms and morphisms are gauge transformations with source and target determined by the Maurer–Cartan form. The resulting cocycle groupoid of bundles with connection is equivalent to Waldorf’s diffeological groupoid of diffeological principal $X$2-bundles with connection (Minichiello, 2024). A related higher-topos result identifies the nerve of the category of diffeological principal $X$3-bundles with the nerve of the category of $X$4-principal $X$5-bundles on a diffeological space $X$6, so the diffeological groupoid of bundles and the stack-theoretic groupoid present the same weak homotopy type (Minichiello, 2022).

The same pattern persists for principal Lie groupoid bundles. Given a principal Lie groupoid bundle $X$7 and a connection on the structure Lie groupoid, one obtains diffeological quotients that play the roles of adjoint and Atiyah objects. The resulting diffeological groupoids $X$8, $X$9, and $X\to X/\!\sim$0 fit into a short exact sequence of diffeological groupoids over the discrete category $X\to X/\!\sim$1, and a connection on the bundle splits this sequence. In this setting diffeological groupoids are the natural home for quotient constructions defined only up to homotopy, where manifold quotients are generally unavailable (Chatterjee et al., 4 Feb 2025).

At the infinitesimal end, elastic diffeological spaces supply a tangent-category environment in which diffeology supports a Cartan calculus. Blohmann’s elastic spaces are precisely those on which the left Kan extension of the classical tangent functor yields a tangent structure with vector fields, differential forms, Lie bracket, de Rham differential, inner derivative, and Lie derivative (Blohmann, 2023). Building on that framework, differentiable groupoid objects in a tangent category admit abstract Lie algebroids: the source-vertical tangent bundle restricted to the identity section carries an anchor into the tangent of the object space, sections correspond to invariant vector fields, and these are closed under the tangent-category Lie bracket. The stated examples include diffeological symmetry groupoids of general relativity, symmetry groupoids in Lagrangian field theory, holonomy groupoids of singular foliations, elastic diffeological groupoids, and groupoid objects in differentiable stacks (Aintablian et al., 2024).

An expository synthesis places these constructions in a longer development running from fibrating groupoids and universal coverings to orbifold and quasifold structure groupoids, the Klein groupoid of germs of local diffeomorphisms, groupoid $X\to X/\!\sim$2-algebras, and prequantum groupoids built from path spaces. In that broader sense, diffeological groupoids function as a unifying language for singular smooth geometry, orbit-space stratification, noncommutative geometry, and geometric quantization (Iglesias-Zemmour, 24 Aug 2025).