The thesis uses simplicial methods to study actions, bundles, and bibundles of higher groupoids.
problem Understanding actions, bundles, and bibundles of higher groupoids.
method Employing simplicial methods to model actions, principal bundles, and bibundles of higher groupoids.
result The simplicial definitions agree with categorification approaches and prove a theorem on differentiation of higher Lie groupoids.
Lie Calculus connects differential and Lie theory using groupoids.
problem Understanding the relationship between differential and Lie theories.
method Using groupoids to link differential and Lie theories.
result Higher order theory involves higher algebra (n-fold groupoids).
Extends adjoint representation concept to higher Lie groupoids.
problem Defining adjoint representation for higher Lie groupoids.
method Generalizes standard construction to higher Lie groupoids using simplicial vector bundles.
result Adjoint representation up to homotopy is well-defined and unique.
We study higher-degree generalizations of symplectic groupoids, referred to as {\em multisymplectic groupoids}. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe "higher'' versions of Poisson structures by identifying the infinitesimal counterparts of mul…
Develops higher gauge theory for categorified spaces, solving tensor field equations.
problem Finding non-Abelian self-dual tensor field equations in six dimensions.
method Uses higher groupoids and connections on higher groupoid bundles.
result Obtains six-dimensional superconformal field theories via higher gauge structure.
This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored.
Proof that m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.
problem Consistent definition of symplectic structures on higher Lie groupoids under Morita equivalence.
method Rigorous proof of m-shifted symplectic forms preservation.
result m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.
Study derived Lie ∞-groupoids and algebroids in higher differential geometry.
problem Addressing problems in higher differential geometry using derived Lie ∞-groupoids and algebroids.
method Construct CFO structures, study L∞-algebroids, homotopical algebras, and homotopy-coherent representations.
result Construct Atiyah classes for L∞-algebroids pairs and study singular foliations and their holonomies.
Geometric models for representations up to homotopy using simplicial vector bundles.
problem Geometric models for representations up to homotopy of Lie groupoids.
method Application of higher analogs of cleavages in simplicial fibrations to geometric models.
result An equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage.
Geometric structures are lifted to higher tangent bundles preserving statistical properties.
problem Lifting statistical structures to higher tangent bundles while maintaining their properties.
method Natural lifts of geometric objects and potentials to higher tangent bundles, preserving statistical manifold structures.
result Lifted statistical structures on higher tangent bundles maintain pseudo-Riemannian metrics and are again statistical manifolds.
We discuss two sorts of generalization of Lie groupoids. One is Lie n-groupoids defined as simplicial manifolds with trivial πk≥n+1. The other is the stacky Lie groupoid $\cG\rra M$ with $\cG$ a differentiable stack. We build 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain…
This paper proves cohomology invariants for differentiable stacks.
problem Understanding cohomology of differentiable stacks.
method Simplicial approach to representations up to homotopy.
result Cohomology with coefficients in a representation up to homotopy is a Morita invariant of the underlying stack.
We discuss two generalizations of Lie groupoids. One consists of Lie n-groupoids defined as simplicial manifolds with trivial πk≥n+1. 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 …
The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.
New models for symplectic structures on classifying stacks.
problem Building models for symplectic structures on classifying stacks.
method Introducing m-shifted symplectic Lie n-groupoids and constructing explicit symplectic Morita equivalences. result Explicit symplectic Morita equivalences between models of the 2-shifted symplectic structure on classifying stacks.
Defines duals of higher vector bundles for Lie 2-groupoids.
problem Constructing duals for higher vector bundles over Lie 2-groupoids.
method Develops theory of n-duals for simplicial vector spaces, defines n-duals for Lie 2-groupoids, and studies their properties.
result Proposes a new construction for VB 2-duals of VB 2-groupoids, showing they are VB 2-groupoids themselves and have nondegenerate canonical dual pairings up to homotopy.
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
This work explores symplectic structures on graded manifolds and higher Lie groupoids.
problem Understanding symplectic structures on graded manifolds and their global counterparts.
method Introduction and study of graded manifolds, symplectic Q-manifolds, higher Lie groupoids, and their symplectic structures.
result Developed a graded analogue of Weinstein's tubular neighborhood theorem and explored its applications.
New approach to principal groupoid bundles with connections using dg-Lie groupoids.
problem Developing a new perspective on principal bundles with connections.
method Using dg-Lie groupoids and additional adjustment data for Lie groupoids.
result Adjusted connections provide a global formulation of curved Yang-Mills-Higgs theories.
Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale L…
Introduces derived Lie n-groupoids with shifted symplectic structures.
problem Defines structures for higher groupoids and their symplectic properties.
method Introduced derived Lie n-groupoids and their shifted symplectic structures, defining shifted lagrangian structures and proving composition well-defined.
result Shows that the framework includes various reduction procedures.
Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples th…
The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an A∞ de Rham theorem and constructing an integration functor. result An equivalence between ∞-representations of L∞-algebroids and ∞-representations of Lie ∞-groupoids for foliations. Affine structures on Lie groupoids are studied, showing rich algebraic properties.
problem Understanding affine structures on Lie groupoids.
method Analyzing affine k-vector fields, k-forms, and (p,q)-tensors, and showing their algebraic properties. result The space of affine structures forms a 2-vector space over multiplicative structures, and affine multivector fields have a Lie 2-algebra structure.
In this paper we introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (i.e., Poisson group(oid)s) and multiplicative closed 2-forms (e.g., symplectic groupoids). We prove that for every source simply connected Lie groupoid G with Lie …
Solves differentiation for Lie ∞-groups using formal groupoids.
problem Differentiation of Lie ∞-groups.
method Develops homotopy theory of formal ∞-groupoids and analyzes Dold-Kan adjunction for cosimplicial algebras.
result Differentiation functor from finite-dimensional Lie ∞-groups to finite-type Lie ∞-algebras is homotopically well-behaved.
New representations of loop braid group from higher gauge theory effects.
problem Understanding representations of loop braid group in higher gauge theory.
method Introducing W-bikoids and constructing representations from groupoid algebras.
result Candidate for higher quantum group and flux metamorphosis.
New duality concept for vector spaces and groupoids.
problem Defining and studying duals for vector spaces and groupoids.
method Introducing and analyzing n-duals for simplicial vector spaces and n-groupoid objects. result Non-degenerate canonical pairing up to homotopy for homotopy n-types. Constructs a Lie groupoid integrating singular foliations.
problem Integrating singular foliations into higher Lie groupoids.
method Recursive use of bi-submersions and geometric resolutions.
result Finite-dimensional Lie groupoid integrating singular foliations.
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infin…
New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.
problem Provides a Morita invariant definition of Lie and Courant algebroids over Lie groupoids.
method Views vector fields as Maurer-Cartan elements in a differential graded Lie algebra and as functors and natural transformations.
result Obtains a unifying conceptual framework for studying various algebraic structures.
Develops connections on principal 2-bundles using Lie 2-algebra-valued forms.
problem Defines connections on principal 2-bundles for strict Lie 2-groups.
method Introduces Lie 2-algebra-valued differential forms and connections on Lie groupoids.
result Provides a consistent global perspective to higher gauge theory.
Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.
problem Understanding sheaves of Lie-Rinehart algebras and their morphisms.
method Introduced morphisms and comorphisms, proved factorization theorems, and defined higher homotopy groups and groupoids.
result Sheaves of Lie-Rinehart algebras over smooth manifolds induce partitions into orbits of the fundamental groupoid.
Simplified 3D Dijkgraaf-Witten theory with defects explained geometrically.
problem Constructing 3D Dijkgraaf-Witten theory with defects.
method Symmetric monoidal functor from defect cobordism category to vector spaces, using geometric and homotopy theoretic methods.
result Explicit construction of 3D untwisted Dijkgraaf-Witten theory with defects.
We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical lifts of the higher Johnson homomorphisms τm, for m≥1, to the Torelli groupoid, and we provide a recursive combinatorial …
Abstract: New geometric incarnation of isomonodromy functors.
problem Classical isomonodromic deformations.
method Functorial upgrade of isomonodromic deformations using Lie groupoids.
result Geometric incarnation of isomonodromy functors as Morita equivalences.
This thesis explores symplectic foliations and local Lie groupoids, with applications in current theory.
problem Understanding calibratable symplectic foliations and local Lie groupoids.
method Applying de Rham's and Sullivan's theories to symplectic foliations and generalizing Mal'cev and Olver's theorems.
result Generalizations of theorems by Mal'cev and Olver for local Lie groupoids and algebroids.
Calculates the index of a geometric Dirac operator on manifolds with corners using glueing and Lie groupoid.
problem Calculating the Fredholm index of a geometric Dirac operator with mixed boundary conditions.
method Introduces a glueing construction and Lie groupoid to describe the Dirac operator. Uses a heat kernel method with rescaling to derive an index formula.
result Derives a general index formula of the Atiyah-Singer type.
What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and …
In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations fro…
The study explores weightings on submanifolds and their geometric properties.
problem Understanding weightings on submanifolds and their geometric implications.
method Detailed exploration of weighted normal bundles, weighted deformation spaces, and weighted blow-ups.
result A description of weightings in terms of subbundles of higher tangent bundles, leading to new concepts for Lie algebroids and groupoids.
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.
We describe various equivalent ways of associating to an orbifold, or more generally a higher étale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising from a Lie groupoid G, the weak homo…
We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obta…
Homotopy theory for Lie ∞-groupoids aids integrating L-infinity algebras.
problem Integrating L-infinity algebras and compatibility with homotopy theory.
method Developed a homotopy theory for Lie ∞-groupoids, showing they form an incomplete category of fibrant objects.
result Henriques' integration functor is exact with respect to quasi-split fibrations.
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a diff…
Formally equates two quantization methods and constructs non-commutative algebras.
problem Equivalence of deformation and geometric quantization methods.
method Symplectic reduction and Lie 2-groupoid quantization.
result Recovery of strict deformation quantizations and non-associative products.