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…
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48 results for “diffeological differential forms”
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.
Diffeology explores -forms and bundles with more information than traditional differential forms.
problem Understanding -forms and bundles in diffeological spaces.
method Developed theory of diffeological vector pseudo-bundles, including limits and colimits, and various operations.
result Sections of bundles of -forms contain more information than differential forms.
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…
In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left…
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.
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…
Differential forms and symmetric tensors show contrasting singular behaviors in a specific geometric setting.
problem Exploring differential forms and symmetric tensors on a specific geometric setting.
method Analyzing differential forms and symmetric tensors on the quadrant with subset diffeology.
result Symmetric tensors exhibit singularities that accumulate, while differential forms are smooth.
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.
The paper proves isomorphisms between two complexes related to singular foliations.
problem Understanding the isomorphisms between two complexes associated with singular foliations.
method Analyzing the quotient map and proving isomorphisms in specific cases.
result Isomorphisms between the complexes of differential forms on the leaf space and basic differential forms on the manifold.
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.
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.
Projective model structures on diffeological spaces and smooth sets and the smooth Oka principlemath.AT
Smooth structures on diffeological spaces and sheaves, resolving conjectures.
problem Model structures on diffeological spaces and sheaves of sets.
method Embedding diffeological spaces into sheaves of sets, studying combinatorial model structures, and proving equivalence to simplicial sets.
result Established a model structure on sheaves of sets that is Quillen equivalent to simplicial sets and has properties like cartesian and cofibrant smooth manifolds.
We consider the notion of the De Rham operator on finite-dimensional diffeological spaces such that the diffeological counterpart Λ^1(X) of the cotangent bundle, the so-called pseudo-bundle of values of differential 1-forms, has bounded dimension. The operator is defined as the composition of the Levi-Civita connection…
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.
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.
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.
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.
Lie groupoids and their orbit spaces are linked through equivalence classes.
problem Understanding the relationship between Lie groupoids and their orbit spaces.
method Introducing lift-complete Lie groupoids and showing equivalence between categories.
result Morita equivalence class of a lift-complete Lie groupoid is determined by its orbit space.
We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections…
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…
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.
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.
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…
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.
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.
A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is general…
Differential calculus on Euclidean spaces has many generalisations. In particular, on a set , a diffeological structure is given by maps from open subsets of Euclidean spaces to , a differential structure is given by maps from to , and a Frölicher structure is given by maps from to $X…
Ambrose-Singer theorem on diffeological bundles and complete integrability of the KP equationmath.DG
In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in \cite{Ma2}, and conclude with a differential geometric treatment of KP hierarchy. The example…
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.
New construction provides non-trivial representations for geometric quantisation.
problem Geometric quantisation of non-integral symplectic structures.
method Construction from Noncommutative Differential Geometry adapted to diffeology.
result The construction provides non-trivial representations.
It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note we consider a way to circumnavigate this issue, by introducing a notion of pseudo-metric, which, said informally, is the least-degenera…
We review the basic definitions and properties concerning smooth structures, convenient spaces, diffeological spaces and tangent structures. The relation betwen them is described. A tangent structure is constructed for each pre-convenient space. This one is proved to be convenient iff the space and the tangent fibres c…
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…
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 -bundles is weak homotopy equivalent to -principal -bundles.
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 …
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.
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.
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.
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.
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.
Abstract Lie algebroids generalize Lie algebroids to abstract categories.
problem Generalizing Lie algebroids to abstract categories.
method Generalized differentiation procedure to groupoid objects in categories with tangent structures.
result Abstract Lie algebroids are defined and examples include various groupoids.
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 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.