This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the wo…
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48 results for “differential graded categories”
The paper integrates DGLA to DGLG using HCPs and Hopf algebras.
problem Integrating DGLA to DGLG.
method Definition of DGLG and HCPs, use of graded Hopf algebras.
result Construction of DGLG from DGLA and vice versa.
Formal Contact Categoriesmath.GT
New categories for surfaces link to contact geometry.
problem Understanding contact structures on surfaces.
method Associate differential graded categories to surfaces.
result Homotopy category of these categories is triangulated.
We construct families of differential graded algebras R and R \boxtimes R and give an algebraic formulation of the contact category of a disk through the differential graded category DGP(R) generated by some distinguished projective differential graded R-modules. The homology category H^0(DGP(R)) is a triangulated cate…
This paper proves equivalence between derived manifolds and differential graded manifolds.
problem Characterizing derived manifolds and their relationship to differential graded manifolds.
method Proving equivalence between the infinity categories of derived manifolds and differential graded manifolds.
result The infinity category of differential graded manifolds is equivalent to that of derived manifolds.
The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.
problem Splitting supermanifolds and understanding their structure.
method Using -fold vector bundles and graded manifolds, the abstract generalizes a construction for splitting supermanifolds.
result The images of these embeddings into the category of graded manifolds satisfy universal properties of graded coverings or semicoverings for Lie supergroups and Lie superalgebras.
Develops theory of differential graded schemes for derived stacks.
problem Creating a theory for derived stacks using dg schemes.
method Formulates dg schemes as homotopy sites, equates to stacks on dg algebras.
result Infinity category of stacks represented by dg schemes is derived schemes.
Categorifies Jones polynomial for odd primes.
problem Categorification of Jones polynomial for odd primes.
method Using -differential graded link homologies and homotopy categories of finite-dimensional -complexes.
result Categorification of Jones polynomial evaluated at odd prime roots of unity.
A formula connects two algebraic structures derived from a category.
problem Connecting two algebraic structures derived from a category.
method Using differential graded modular functors and Calabi-Yau structures.
result The action of a specific mapping class group element transforms one algebraic structure into another.
Examples of SL(2, Z) actions on differential graded categories are defined and explored.
This paper explores A-infinity structures in contact categories and strand algebras.
problem Understanding A-infinity structures in contact categories and strand algebras.
method Explicit constructions and properties of A-infinity operations are established.
result Conditions for the vanishing and nonvanishing of A-infinity operations are derived.
Study graded coverings for supermanifolds, proving their universal properties.
problem Constructing obstructions for splitting supermanifolds.
method Introduce and prove properties of infinite prolongations of differential operators.
result Infinite prolongations form a covering of supermanifolds in graded manifolds.
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…
VB-algebroids control deformations of Lie algebroids structures.
problem Deformation of Lie algebroid structures.
method Attach differential graded Lie algebra to VB-algebroids to control deformations.
result Controlled deformations of VB-algebroids through DG Lie algebra.
Introduces holomorphic string algebroids and classifies them.
problem Classifying holomorphic string algebroids.
method Using Courant extensions and inner morphisms of holomorphic Courant algebroids.
result Classification of string algebroids via Cech cohomology.
New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.
problem Generalizing Brauer graph algebras to new dg-algebras.
method Derived categories, mixed-angulations of surfaces, stability conditions, and quadratic differentials.
result Spaces of stability conditions on derived categories of these algebras are described in terms of spaces of quadratic differentials.
Study multiplicity-free covering of graded manifolds, proving equivalence of categories.
problem Equivalence of categories of graded manifolds and symmetric vector bundles.
method Defined and computed multiplicity-free covering, showed deck transformation group isomorphic to .
result Equivalence of categories of graded manifolds and symmetric -fold vector bundles.
Categorifies colored Jones polynomial at roots of unity.
problem Categorification of colored Jones polynomial.
method Differential on triply-graded homology, compatible with p-differential structure.
result Categorification of the colored Jones polynomial at a root of unity.
3D HQFTs constructed using graded monoidal categories.
problem Constructing 3D HQFTs with specific targets.
method Using spherical χ-fusion categories and the state sum method.
result 3D HQFTs constructed with target Bχ.
Introduces principal bundles in a new geometric category.
problem No specific problem stated; introduces a new geometric category.
method Introduces -manifolds and principal bundles within this category.
result Fundamental properties of classical principal bundles can be generalized to -manifolds.
Study Legendrian links using representations and sheaves.
problem Understanding Legendrian links through algebraic and geometric representations.
method Investigate an category of -dimensional representations and conjecture equivalence to sheaves.
result Established cohomological equivalence for Legendrian torus links.
Three definitions of graded vector bundles are shown to be equivalent.
problem Defining graded vector bundles in three different ways.
method Equivalence of categories among sheaves, graded modules, and locally trivial graded manifolds.
result All three approaches to graded vector bundles are equivalent.
Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham coh…
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as cons…
The paper studies graded manifolds and their functorial relationship.
problem Understanding the functor between two categories of graded manifolds.
method Examines polynomial filtrations and homogeneity structures, applying the Batchelor-Gawedzki theorem and Borel-Whitney theorem.
result The functor is full and surjective on objects between the categories of graded vector bundles and manifolds.
The paper defines and studies the category of Z-graded manifolds, including their intrinsic structure and formal properties.
problem Understanding the categorical properties and intrinsic structure of Z-graded manifolds.
method Describing local models, explaining formality, and formulating analogues of theorems.
result Proper definitions of objects and morphisms in the category of Z-graded manifolds, and formulation of Batchelor's theorem.
The paper proves a category of dg manifolds with finite positive amplitude.
problem Understanding the structure of dg manifolds with finite positive amplitude.
method Using path spaces and homotopy transfer theorem for curved -algebras.
result Proves that dg manifolds of finite positive amplitude form a category of fibrant objects.
We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddin…
This paper studies graded manifolds of type Δ and their equivalence with n-fold vector bundles.
problem Understanding the relationship between graded manifolds and vector bundles.
method Geometrization process for -graded manifolds of type Δ.
result Established an equivalence between a subcategory of n-fold vector bundles and graded manifolds of type Δ.
Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a R[n]-bundle over T[1]M is measured by the G equivariant…
Develops non-semisimple ETQFTs for 3-manifolds.
problem Creating ETQFTs for non-semisimple categories.
method Introducing relative modular categories and using 2-categorical universal construction.
result Extends ETQFTs to non-semisimple cases.
This thesis generalizes structures on -manifolds and Lie -algebroids.
problem Representation theory and linear structures of -manifolds and Lie -algebroids.
method Introduces differential graded modules and representations up to homotopy, defines Weil algebra, and studies VB-Lie -algebroids.
result Establishes an equivalence between VB-Lie -algebroids and -term representations up to homotopy of Lie -algebroids.
New rings relate to Soergel categories, categorifying a representation.
problem Categorifying a representation of the braid group.
method Construction of graded rings and categorical braid group action.
result Categorification of the Burau representation.
We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformat…
We give a diagrammatic presentation of the category of -tilting modules for being a root of unity and introduce a grading on . This grading is a "root of unity phenomenon" and might lead to new insights about link and -manifold invariants deduced from $…
A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain sub…
New model for Calabi-Yau- categories using decorated marked surfaces.
problem Constructing models for Calabi-Yau- categories.
method Using graded decorated marked surfaces and string models.
result Isomorphism between braid twist group and spherical twist group.
Inspired by a work of Kapranov, we define the notion of Dolbeault complex of the formal neighborhood of a closed embedding of complex manifolds. This construction allows us to study coherent sheaves over the formal neighborhood via complex analytic approach, as in the case of usual complex manifolds and their Dolbeault…
Extends supersymmetry to include exotic -graded spinors.
problem Developing a new mathematical framework for supersymmetry.
method Using -graded (Majorana) spinor coordinates and the category of -manifolds.
result A new mathematical formalism that resembles -extended superspace but with unique properties.
Inverse function theorem and homotopy description for L-infinity bundles.
problem Inverse function theorem and homotopy description for L-infinity bundles.
method Local sections composed of elementary morphisms.
result Simple description of homotopy category of L-infinity bundles.
Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various …
Diagrammatically describes algebra equivalent to perverse sheaves on isotropic Grassmannians.
problem Equivalence between algebra modules and perverse sheaves on isotropic Grassmannians.
method Uses a Koszul algebra and a folding procedure from a Khovanov arc algebra to establish equivalence.
result Category of -modules is equivalent to perverse sheaves on isotropic Grassmannians.
We show that the category of Lie triple systems is equivalent to the category of Lie algebras graded by Z/(2Z) such that the odd component generates the algbera and the second graded cohomology group coefficients in any trivial module is zero. As a corollary we obtain an analogous result for symmetric spaces and Lie gr…
Defines smooth actions of a group on manifolds and vector spaces.
problem Representing the general linear group and its actions.
method Restricted functor of points and category theory.
result Smooth actions on -graded vector spaces and manifolds.
New equivalences found between graded supermanifolds and vector bundles.
problem Understanding equivalences between graded supermanifolds and vector bundles.
method Explicit geometric constructions using supergeometry tools.
result Desuperization equivalence functor as a composition of canonical equivalences.
Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differentia…
Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.
problem Understanding transverse link invariants in the annular setting.
method Constructs a stable homotopy type for annular links and defines a map to the Khovanov skein spectrum.
result At extreme gradings, the map from the Khovanov spectrum to the Khovanov skein spectrum recovers the cohomotopy transverse invariant.
Geometrically computes sheaves linking HOMFLY-PT homology to Hilbert schemes.
problem Linking HOMFLY-PT homology to geometric structures on Hilbert schemes.
method Geometric sheaf theory, Hochschild homology formality, Hilbert schemes of points.
result Established formalism connecting HOMFLY-PT homology to coherent sheaves on Hilbert schemes.