A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Extends topological groupoids and studies their properties.
Studies geometric structures on Lie groupoids and differentiable stacks.
As groupoids generalize groups, motivated by group extensions we consider a kind of fibrations of Lie groupoids, called locally topological product Lie groupoid fibrations with fiber , i.e., \[ 1\rightarrow {\sf A} \rightarrow {\sf G} \rightarrow {\sf K}\rightarrow 1 \] where and are Lie gr…
New Euler characteristics for groupoids generalize orbifold Euler characteristics.
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
We consider principal bundles as generalized morphisms between topological groupoids. In the category of these generalized morphisms two topological groupoids are isomorphic if and only if they are Morita equivalent. We show that the fibers of a generalized morphism from H to G induce a singular foliation of the topolo…
A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to pr…
Analytic linearization and holomorphic extensions for proper groupoids.
For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks fa…
Homflypt skein theory and string topology linked via 2-groupoids.
The paper defines a stratification for Lie groupoids in a tame topology context.
Introduces fat Lie theory for Lie groupoids and algebroids.
In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.
Groupoids are mathematical structures able to describe symmetry properties more general than those described by groups. They were introduced (and named) by H. Brandt in 1926. Around 1950, Charles Ehresmann used groupoids with additional structures (topological and differentiable) as essential tools in topology and diff…
In this paper we investigate some connections between Topological Dynamics, the theory of G-Principal Bundles, and the theory of Locally Trivial Groupoids.
Lie groupoid equivariant neural networks are a new type of neural network.
We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topol…
We observe that any regular Lie groupoid G over an manifold M fits into an extension of a foliation groupoid E by a bundle of connected Lie groups K. If $\FF$ is the foliation on M given by the orbits of E and T is a complete transversal to $\FF$, this extension restricts to T, as an extension $K_{T}\to…
Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent…
We discuss two generalizations of Lie groupoids. One consists of Lie -groupoids defined as simplicial manifolds with trivial . The other consists of stacky Lie groupoids $\cG\rra M$ with $\cG$ a differentiable stack. We build a 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to …
In the following paper we investigate the question: when is a transitive topological groupoid continuously isomorphic to a Lie groupoid? We present many results on the matter which may be considered generalizations of the Hilbert's fifth problem to this context. Most notably we present a "solution" to the problem for p…
In the following paper we investigate the question: when is a transitive topological groupoid continuously isomorphic to a Lie groupoid? We present many results on the matter which may be considered generalizations of the Hilbert's fifth problem to this context. Most notably we present a "solution" to the problem for p…
We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid -extensions, which we call "connections on gerbes", and study the induced connections on various associated bundles. We also prove analogues of the Bianchi id…
Affine deformations of cotangent groupoids
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admi…
In this paper, we introduce a notion of a central -extension of a double Lie groupoid and show that it defines a cocycle in the certain triple complex.
The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it i…
We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds wit…
We define multiplicative Poisson-Nijenhuis structures on a Lie groupoid which extends the notion of symplectic-Nijenhuis groupoid introduced by Stié23non and Xu. We also introduce a special class of Lie bialgebroid structure on a Lie algebroid , called P-N Lie bialgebroid, which defines a hierarchy of compatible Lie…
The purpose of this thesis is to use the language of orbifold groupoids to describe the geometry and topology of orbifolds, highlighting advantages and disadvantages of this language as they arise.
New Virasoro-like structures for circle diffeomorphisms with breaks.
Extends Cheeger's method to Lie groupoid actions on manifolds.
Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for…
Let S be a compact connected oriented surface with one boundary component. We extend each of Johnson's and Morita's homomorphisms to the Ptolemy groupoid of S. Our extensions are canonical and take values into finitely generated free abelian groups. The constructions are based on the 3-dimensional interpretation of the…
The paper studies deformations of symplectic groupoids using cohomology and spectral sequences.
The paper introduces Morse theory for Lie groupoids and proves inequalities.
This paper extends symplectic reduction to cosymplectic groupoids and explores their properties.
Extends connections on Lie groupoids, proving completeness conditions.
Extends particle classification to curved space-times using groupoids.
Every singular foliation has an associated topological groupoid, called holonomy groupoid (see arXiv:math/0612370). In this note we exhibit some functorial properties of this assignment: if a foliated manifold is the quotient of a foliated manifold along a surjective submersion w…
Reconstruct Lie structures from functional-analytic data on groupoids.
We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation on a smooth vector bundle, i.e., one whose kernel consists only of ineffective arrows. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as a…
We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numero…
We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion of Lie subalgebroid by dropping the constant rank requirement. We lay the bases of a Lie theory for singular subalgebroids: we construct the associated holonomy groupoids, adapting the procedure of Androulidakis-Skandalis for singu…
Study of symplectic groupoids from tt*-Toda equations.
In this paper, we study geometric properties of quotient spaces of proper Lie groupoids. First, we construct a natural stratification on such spaces using an extension of the slice theorem for proper Lie groupoids of Weinstein and Zung. Next, we show the existence of an appropriate metric on the groupoid which gives th…
Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.
Lie groupoids generalize Lie groups with multiplication defined for certain pairs.