We consider certain groups of tree automorphisms as so-called diffeological groups. The notion of diffeology, due to Souriau, allows to endow non-manifold topological spaces, such as regular trees that we look at, with a kind of a differentiable structure that in many ways is close to that of a smooth manifold; a suita…
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The paper bridges diffeological bundle theory with higher topos theory.
problem Comparing Čech cohomology of diffeological spaces with existing notions.
method Using Čech model structure on simplicial presheaves and diffeological spaces as discrete simplicial presheaves.
result Nerve of diffeological principal -bundles is weak homotopy equivalent to -principal -bundles.
Lie algebras of quotient groups defined under specific conditions.
problem Conditions for Lie differentiation of quotient groups.
method Diffeological group theory, tangent structure, Lie functor instantiation.
result Lie algebra structure on quotient groups derived from Lie algebras of parent groups.
We define a diffeology on the Milnor classifying space of a diffeological group , constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any princip…
We show that every effective smooth action of a Lie group G on a manifold M is a diffeomorphism from G onto its image in Diff(M), where the image is equipped with the subset diffeology of the functional diffeology.
Defines sheaves and Čech cohomology for diffeological spaces and classifies principal bundles.
problem Defining sheaves and Čech cohomology for diffeological spaces.
method Defines sheaves for diffeological spaces and constructs Čech cohomology. Uses Čech cohomology to classify principal bundles.
result First degree Čech cohomology classes classify diffeological principal -bundles.
Study on Čech-de Rham obstruction in diffeological spaces.
problem Obstruction to Čech-de Rham map being an isomorphism in diffeological spaces.
method Higher topos theory, homotopy pullback diagrams, Čech-de Rham bicomplex, -stack cohomology.
result New exact sequences in all higher degrees and conceptual proof of cohomology agreement.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
New spherical Milnor spaces for diffeological groups with geometric and topological properties.
problem Understanding higher topological structures in diffeological spaces.
method Spherical Milnor construction based on quadratic normalization.
result Provides a natural setting for studying principal bundles with -twists and higher cohomology.
Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential…
The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
problem Understanding the structure of leaf spaces of Riemannian foliations.
method Analyzing the holonomy groupoid and using diffeological spaces.
result The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
On diffeological principal bundles of non-formal pseudo-differential operators over formal onesmath.DG
Study non-formal pseudo-differential operators over formal ones.
problem Understanding structure of non-formal pseudo-differential operators.
method Diffeological principal bundles, smoothing connections.
result Structure of diffeological bundle of non-formal pseudo-differential operators over formal ones.
New Morita equivalence for diffeological groupoids defined.
problem Defining Morita equivalence for diffeological groupoids.
method Developed diffeological groupoid actions, -bundles, and -bibundles; introduced principality; defined Hilsum-Skandalis tensor product.
result Biprincipal bibundles are weakly invertible in the bicategory DiffBiBund.
Local systems in diffeologymath.AT
Introduces a framework for rational homotopy theory in diffeological spaces.
problem Challenges in rational homotopy theory for smooth spaces with arbitrary fundamental groups.
method Utilizes local systems over simplicial sets and a model structure for diffeological spaces.
result Establishes an equivalence between fibrewise rational diffeological spaces and algebraic local systems.
Quasifold groupoids and diffeological quasifolds are studied, showing an equivalence of categories.
problem Understanding the structure and equivalence of quasifold groupoids and diffeological quasifolds.
method Examining the category of diffeological quasifolds and the bicategory of quasifold groupoids, proving an equivalence of categories under certain conditions.
result Restricting to locally invertible morphisms and effective quasifold groupoids, the orbit space functor is an equivalence of categories.
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth cur…
New method recovers differential cohomology from diffeological spaces.
problem Recovering differential cohomology from diffeological spaces.
method Introducing skeletal diffeologies and thin homotopies to recover differential cohomology.
result Ordinary differential cohomology can be recovered in terms of the homotopy theory of skeletal diffeological spaces.
Cohomology of Lie group quotient equals Lie algebra cohomology.
problem Cohomology of Lie group quotients.
method Diffeological de Rham cohomology and Lie algebra cohomology.
result Cohomology of equals Lie algebra cohomology of .
Elastic diffeological spacesmath.DG
New type of spaces with tangent structures for analysis.
problem Defining tangent structures for non-smooth spaces.
method Introducing elastic diffeological spaces and defining tangent structures.
result Elastic spaces have a natural tangent structure with graded commutation relations.
We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). Aft…
This note proves equivariant de Rham cohomology for quotient spaces.
problem Computing de Rham cohomology of quotient spaces under group actions.
method Equivariant identification of de Rham complexes using foliation theory.
result Canonical isomorphism of de Rham complexes for quotient spaces.
This paper aims to describe the behavior of diffeological differential forms under the operation of gluing of diffeological spaces along a smooth map. In the diffeological context, two ways of looking at diffeological forms are available, that of the vector space of all diffeological forms on a given space, and that of…
New cohomology theory for diffeological spaces developed.
problem Cohomology of diffeological spaces.
method Diffeological Čech cohomology theory.
result Established connections between diffeological Čech cohomology and de Rham cohomology.
Smooth classifying spacesmath.DG
We develop the theory of smooth principal bundles for a smooth group , using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define -numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling ba…
Why Diffeology?math.DG
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
This paper generalizes optimization techniques to diffeological spaces.
problem Challenges in applying optimization techniques to diffeological spaces due to various tangent space definitions.
method Suitable definition of tangent space, diffeological Riemannian space, diffeological gradient, and diffeological retraction.
result Formulation of an optimization algorithm on diffeological spaces.
This paper adapts submersions, immersions, and étale maps to diffeology.
problem Providing suitable analogs for submersions, immersions, and étale maps in diffeology.
method Nonlinear approach to diffeological submersions, immersions, and étale maps.
result Characterization and properties of diffeological embeddings and étale maps.
Universal connection constructed using diffeology theory.
problem Natural connection on bundles of paths on manifolds.
method Diffeological construction of Singer's universal connection.
result Functorial equivalence between holonomy categories and diffeological bundle-connection pairs.
Study tiling spaces over irrational tori using diffeological classification.
problem Understanding the structure of tiling spaces over irrational tori.
method Diffeological classification of irrational tori and analysis of fiber bundle structures.
result Inherited diffeological equivalence of one-dimensional tiling spaces over irrational tori.
Study of generalized vector bundles and their geometric tools.
problem Extension of differential geometric tools to infinite dimensional vector bundles.
method Analysis of automorphisms, frame bundle, connection 1-forms, and covariant derivatives in diffeological vector pseudo-bundles.
result Non-isomorphism between connection 1-forms and covariant derivatives in infinite dimensional cases.
Let be a foliated manifold. We prove that there is a canonical isomorphism between the complex of base-like forms of the foliation and the "De Rham complex" of the space of leaves when considered as a "diffeological" quotient. Consequently, the two corresponding …
New pushforward operation on vector pseudo-bundles creates new examples.
problem Creating new objects from vector bundle theory in diffeology.
method Introducing pushforward operation on diffeological vector pseudo-bundles.
result Pushforward operation produces new projective diffeological vector spaces.
If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the ident…
Analyzes geometric structures on profinite diffeological spaces.
problem Understanding geometric properties of spaces derived from finite-dimensional manifolds.
method Examines tangent and cotangent spaces, differential forms, metrics, connections, and cohomology.
result Unified geometric constructions across various contexts.
Introduces a new framework for Riemannian diffeology.
problem No specific problem stated; focuses on a new framework.
method Uses tangent functor and metric from Iglesias-Zemmour to establish weak Riemannian diffeological spaces.
result Establishes a category of weak Riemannian diffeological spaces and shows induced pseudodistance is a distance under technical conditions.
This work establishes properties on diffeological structures for set-valued maps and measures.
problem Establish rigorous properties on diffeological structures for set-valued maps and measures.
method Using diffeologies, the authors link various structures including set-valued maps, relations, gradients, measures, and shape analysis.
result Established rigorous properties on sample diffeologies.
Orbifolds as diffeologiesmath.DG
We consider orbifolds as diffeological spaces. This gives rise to a natural notion of differentiable maps between orbifolds, making them into a subcategory of diffeology. We prove that the diffeological approach to orbifolds is equivalent to Satake's notion of a V-manifold and to Haefliger's notion of an orbifold. This…
Develops orbibundle theory for complex hyperbolic geometry.
problem Calculating invariants of complex hyperbolic disc orbibundles.
method Using diffeology, calculates Euler number and Toledo invariant.
result New tools for invariants of complex hyperbolic orbibundles.
We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories o…
New bridge between diffeology and noncommutative geometry.
problem Connecting diffeology and noncommutative geometry.
method Embedding quasifolds into diffeology and associating C*-algebras.
result Morita classes of C*-algebras associated with diffeomorphic quasifolds.
The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It …
New derivations on diffeological spaces are not smooth, expanding tangent space definitions.
problem Lack of smoothness in derivations on diffeological spaces.
method Examined derivations satisfying the Leibniz rule but not smooth with respect to given diffeology.
result Tangent space defined via all derivations is larger than one defined using only smooth derivations.
A new method for optimization in diffeological spaces using linearizations.
problem Optimization in spaces with low regularity.
method Generalizing linearization to diffeological spaces and constructing smooth paths.
result Achieving weak convergence to minima or critical values in diffeological spaces.
Riemannian metrics on orbifolds are equivalent to diffeological ones.
problem Equivalence of Riemannian and diffeological orbifolds.
method Framework of Riemannian diffeology and analysis of 2-metrics.
result Riemannian metrics on orbifolds are equivalent to diffeological ones.
We study diffeologies on locally convex spaces and their application to smooth multiplication of distributions.
problem Constructing smooth multiplication of distributions on locally convex spaces.
method Using diffeological colimits and wavefront-set criterion.
result Proving smooth multiplication of microlocally multipliable distributions.
Integrates singular subalgebroids using diffeological groupoids.
problem Integration of singular subalgebroids.
method Definition of integration via diffeological groupoids with specific properties.
result Holonomy groupoids correspond to singular subalgebroids with submersive property.
Proves Mayer-Vietoris sequence for diffeological spaces using generating families.
problem No specific problem stated; focuses on extending a sequence.
method Uses generating families instead of coverings in diffeological spaces.
result Proves a Mayer-Vietoris sequence for diffeological spaces.
Solves open problem on Lie groupoids equivalence.
problem Whether Lie groupoids Morita equivalent are diffeologically Morita equivalent.
method Localisation of 2-categories, anafunctors, Lie groupoids, diffeological groupoids.
result Two Lie groupoids diffeologically Morita equivalent are Morita equivalent in the Lie sense.