The paper examines how differential forms behave under space gluing in diffeological spaces.
problem Behavior of differential forms under diffeological space gluing.
method Description of differential forms behavior under gluing of diffeological spaces.
result Describes the behavior of differential forms under diffeological space gluing.
The paper explores diffeological Clifford algebras and pseudo-bundles.
problem Constructing pseudo-bundles of diffeological Clifford algebras and modules.
method Using diffeological gluing to construct pseudo-bundles.
result Construction of pseudo-bundles of diffeological Clifford algebras and modules.
Study pseudo-bundles of exterior algebras and their Clifford modules, addressing compatibility issues.
problem Compatibility issues in diffeological pseudo-bundles and their duals.
method Analysis of pseudo-bundles of exterior algebras, Clifford actions, and gluing conditions.
result A natural map ensuring commutativity of duals under gluing, also an isometry.
Diffeological vector pseudo-bundles can be non-trivial and require new gluing methods.
problem Non-triviality of diffeological vector pseudo-bundles.
method Introducing diffeological gluing and pseudo-metrics.
result Diffeological gluing provides a substitute for local trivializations.
The paper defines Dirac operators in diffeological spaces and describes gluing procedures.
problem Defining Dirac operators in non-smooth spaces.
method Develops diffeological Dirac operators and describes gluing procedures.
result A procedure for gluing diffeological Dirac operators is described.
Study of pseudo-bundles and pseudo-metrics on them.
problem Developing an analog of Riemannian metrics for diffeological vector pseudo-bundles.
method Detailed study of gluing operation for pseudo-bundles, construction of pseudo-metrics, and analysis of induced pseudo-metrics.
result Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them.
A simpler definition of diffeological connections on vector pseudo-bundles.
problem Defining connections on diffeological vector pseudo-bundles.
method Adapting standard connection definition to diffeological context.
result Simplified definition of connections is straightforward and uses dual pseudo-bundle as tangent space.
Geometric framework for Milnor classifying spaces in diffeological spaces.
problem Milnor classifying spaces in diffeological spaces.
method Developed spherical and projective models with natural diffeological structures, constructed Riemannian metrics, defined differential forms, and introduced Clifford structures.
result Established a coherent geometric setting combining classifying spaces, diffeology, and higher geometric structures.
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.
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 G-bundles is weak homotopy equivalent to G-principal ∞-bundles. Study of tree automorphisms as diffeological groups.
problem Understanding automorphisms of trees with a diffeological structure.
method Introducing diffeology to non-manifold spaces, specifically trees, and applying it to groups of automorphisms.
result The group of tree automorphisms has a discrete diffeology.
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.
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 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.
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 G-bundles. 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.
The paper extends tangent space theory for diffeological spaces and bundles.
problem Understanding tangent spaces of diffeological bundles and spaces.
method Introducing weakly filtered and filtered diffeological spaces, extending exact sequences, and defining Hector's tangent bundle.
result Tangent bundles of filtered diffeological spaces are diffeological vector spaces.
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.
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.
New cochain algebra for diffeological spaces connects de Rham and singular cohomologies.
problem Incompatibility of de Rham and singular cohomologies in diffeology.
method Introduces a new singular de Rham complex and proves it quasi-isomorphic to the original de Rham complex for manifolds and spaces with singularities.
result The new cochain complex resolves the incompatibility issue in diffeology.
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.
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…
We define a diffeology on Milnor's classifying space and prove existence of connections.
problem Classifying spaces and connections in diffeology.
method Definition of diffeology on Milnor's classifying space and proof of connection existence.
result Existence of diffeological connections on principal bundles.
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.
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.
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.
Study the conditions on diffeological vector spaces and their implications.
problem Understanding the conditions and implications on diffeological vector spaces.
method Analyzing various conditions and their relationships using a total order.
result Majority of conditions fit into a total order, with examples showing which implications do not hold.
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.
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.
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.
The paper explores algebraic structures on diffeological vector spaces.
problem Defining and understanding algebraic structures on diffeological vector spaces.
method Formal checks, verification of properties, and analysis of decompositions.
result The maximal isotropic subspace is invariant and unique, while characteristic subspaces are not.
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.
Study on diffeologies on locally convex spaces and smooth multiplication of distributions.
problem Geometric characterization and smoothness of distribution multiplication.
method Investigation of canonical and c∞-diffeologies on locally convex spaces, proving geometric characterizations, and comparing diffeologies. result Established a framework for nonlinear distribution theory beyond manifolds, realizing microlocally multipliable distributions as a diffeological colimit.
A new pseudo-metric approach allows smooth vector spaces to have scalar products.
problem Finite-dimensional diffeological vector spaces lack a standard scalar product.
method Introducing pseudo-metrics to circumvent the standard scalar product issue.
result Pseudo-metrics on finite-dimensional diffeological vector spaces induce smooth scalar products on their duals.
Survey and extend work on singular foliations in diffeology.
problem Understanding singular foliations and their properties in diffeological settings.
method Survey Stefan and Sussmann's work, introduce transverse equivalence, and define basic cohomology.
result Transverse equivalence of singular foliations preserves leaf spaces diffeologically but not conversely.
Paper constructs infinitely many tangent functors on diffeological spaces.
problem Tangent spaces in diffeological spaces are not uniquely defined.
method Introduced and constructed infinitely many non-isomorphic tangent functors.
result The choice of tangent functor is not unique outside smooth manifolds.
Diffeological submanifolds are a new type of submanifold in manifold theory.
problem Defining and understanding different types of submanifolds in manifold theory.
method Introducing diffeological submanifolds and comparing them with other types of submanifolds.
result A diffeological submanifold can be included in a manifold without being an immersion.
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.
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.
Finite spaces can be or not coproducts of subspaces.
problem When finite-dimensional diffeological vector spaces are coproducts of their subspaces.
method Reviewing the question in diffeological vector spaces and comparing with other categories.
result Finite-dimensional spaces can be coproducts, but not always.
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.
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 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.
Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.
problem Combining infinite-dimensional geometry and PDEs for optimization problems.
method Review and extension of classical function spaces and mapping spaces.
result Diffeologies provide a unified framework for evolution equations and optimization problems.