Constructs a new pairing for manifolds and higher categories.
problem Defining a pairing between manifolds and higher categories.
method Introduces vari-framing and constructs labeling systems on disk-stratified vari-framed manifolds.
result Defines factorization homology for (∞,n)-categories. Develops 2-Hilbert spaces for bundle gerbes in geometric quantization.
problem Higher geometric quantization of bundle gerbes.
method Introduces 2-Hilbert spaces, rig-categories, and duals for bundle gerbes.
result Constructs a 2-Hilbert space of sections for bundle gerbes.
We construct the Fukaya category of a closed surface equipped with an area form using only elementary (essentially combinatorial) methods. We also compute the Grothendieck group of its derived category.
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.
We summarize our axioms for higher categories, and describe the blob complex. Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$ to any n-manifold W. The 0-th homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should …
Second part of a series on higher coverings of racks and quandles.
problem Characterizing higher-dimensional centrality conditions in racks and quandles.
method Applying higher categorical Galois theory to racks and quandles.
result Identification and characterization of higher coverings, trivial coverings, and normal coverings.
Spaces over BO are equivalent to thickened manifolds.
problem Understanding embeddings of manifolds in higher dimensions.
method Formal identification of manifolds with their thickened versions, using geometric constructions.
result The infinity-category of thickened smooth manifolds is equivalent to the infinity-category of finite spaces over BO.
Novel A∞-categories derived from gauge theories for manifold homologies.
problem Categorifying manifold homologies via gauge theories.
method 3d and 8d gauged Landau-Ginzburg models, higher A∞-categories. result Derived novel A∞-categories for various manifold homologies. Generalizes Morse theory to n-categories using critical points and moduli spaces.
problem Constructing n-categories from Morse theory.
method Extending Cohen & Jones & Segal's flow category to n-categories by Morse theory on critical points and moduli spaces.
result The resulting structure is an 'almost strict' n-category.
Constructs 2-Hilbert space for line bundle gerbes.
problem Higher prequantisation and line bundle gerbes.
method Constructs a 2-Hilbert space category from morphism categories of line bundle gerbes.
result Shows the constructed 2-Hilbert space is semisimple and abelian.
We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We also obtain some general results on the relations between the fundamenta…
We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient…
Study higher arity self-distributive operations and their cohomology.
problem Understanding cohomology groups of higher arity operations.
method Introduced mutually distributive n-ary operations and defined a cohomology theory.
result Geometric interpretation of cohomology in terms of framed links.
In this technical paper, we present a new formulation of higher parallel transport in strict higher gauge theory required for the rigorous construction of Wilson lines and surfaces. Our approach is based on an original notion of Lie crossed module cocycle and cocycle 1- and 2-gauge transformation with a non standard do…
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properti…
Describes higher gauge theory on string 2-group principal 2-bundles.
problem Formalizing higher gauge theory on string 2-group principal 2-bundles.
method Constructs categories internal to Bibun to define principal 2-bundles with string 2-group as structure 2-group, Lie-differentiates the string 2-group model, finds Maurer-Cartan forms, and generalizes the differentiation process.
result Describes kinematical data of higher gauge theory on string 2-group principal 2-bundles, including their gauge symmetries.
Develops parametrised Poincaré duality for equivariant fixed points.
problem Understanding equivariant fixed points in non-presentable settings.
method Introduces parametrised Poincaré duality in parametrised higher category theory, proving basechange results.
result Generalises Cnossen's twisted ambidexterity to non-presentable settings and applies to isotropy separation methods.
Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to…
We construct what we call a Kirby category, a monoidal category whose morphisms are smooth 4-manifolds, projecting down to another monoidal category whose morphisms are orientable 3-manifolds, the projection being induced by the boundary map on manifolds. We construct a higher categorical generalization of such concept…
Study of higher spin bundles on Klein surfaces, linking to singularity theory.
problem Understanding the structure of spaces of higher spin bundles on Klein surfaces.
method Analysis of topological invariants and quotient spaces of Euclidean spaces.
result Connected components of the space of higher spin bundles are homeomorphic to quotients of Euclidean spaces.
The paper discusses strictification and non-strictification of monoidal categories.
problem Transforming monoidal categories into strict or non-strict ones.
method Explicit models and 2-adjunctions for strictification and non-strictification.
result Strictification and non-strictification are part of a pair of free-forgetful 2-adjunctions.
This paper upgrades instanton TQFT to infinity-categories for better simplification.
problem Developing a more structured approach to instanton TQFT.
method Constructing an infinity-cobordism category and a derived category for instantons.
result A simplified construction of the hypercube of chain complexes for link spectral sequences.
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 homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
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 compute Steenrod squares on Khovanov homology.
problem Computing Steenrod squares on Khovanov homology.
method Stable cup-i products on cochain complexes of augmented semi-simplicial objects in the Burnside category.
result Explicit formulas for cohomology operations on Khovanov homology.
Introduces principal bundles in a new geometric category.
problem No specific problem stated; introduces a new geometric category.
method Introduces Z2n-manifolds and principal bundles within this category. result Fundamental properties of classical principal bundles can be generalized to Z2n-manifolds. 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.
We describe an A∞-quasi-equivalence of dg-categories between the first authors' PA ---the category of category of prefect A0-modules with flat Z-connection, corresponding to the de Rham dga A of a compact manifold M--- and the dg-category of \emph{infinity-local syst…
Generalizes Riemann-Hilbert correspondence for curved local systems.
problem Higher Riemann-Hilbert correspondence with scalar curvature.
method Equivalence of dg-categories of curved local systems, graded vector bundles, and representations.
result Equivalence of dg-enhancements of twisted sheaves categories.
New smooth models for string groups defined in ∞-categories.
problem Defining string group models in smooth spaces.
method Homotopy-theoretic definition using singular complex functor.
result New smooth models for the string group.
The paper constructs braiding structures for a specific subfactor.
problem The challenge is to understand the braiding structures of a Jones-Wassermann subfactor.
method The approach involves constructing braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra.
result The braiding structures induce a projective unitary representation of the balanced superelliptic mapping class group.
In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a f…
In this paper we construct an A∞-category associated to a Legendrian submanifold of jet spaces. Objects of the category are augmentations of the Chekanov algebra A(Λ) and the homology of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearised Le…
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 …
Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by means of brackets and anchors. Moreover, we provide a geometric description of mo…
The procedure "Lie group --> Lie algebra" has a generalization "simplicial manifold --> L_infinity algebra", or yet better, "presheaf on the category of surjective submersions --> L_infinity algebra". We describe this generalization, together with its higher-order extensions.
Constructs a new invariant for 4-manifolds and 3+1 TQFTs.
problem Developing a new invariant for 4-manifolds and 3+1 TQFTs.
method Using a G-crossed braided spherical fusion category to construct a state-sum type invariant.
result Introduces a new invariant for 4-manifolds and 3+1 TQFTs.
5D gauge theories are dual to 3D and 2D models via Floer homologies.
problem Exploring dualities in 5D gauge theories and their 3D and 2D counterparts.
method Using Landau-Ginzburg models and Floer homologies, the paper establishes dualities between different gauge theories and their associated homologies.
result Dual A∞-categories of Floer homologies are derived, proving mirror symmetry and Langlands duality. In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a fo…
This reports on the fundamental objects revealed by Ross Street, which he called `orientals'. Street's work was in part inspired by Robert's attempts to use N-category ideas to construct nets of C*-algebras in Minkowski space for applications to relativistic quantum field theory: Roberts' additional challenge was that …
Defines odd Khovanov homology via categorification of q-Schur algebra.
problem Constructing odd Khovanov homology via representation theory.
method Supercategorification of q-Schur algebra, odd foams, tensor product on chain complexes.
result Odd Khovanov homology defined via categorification.
The abstract introduces a new A∞ duality via LSFT algebra.
problem Legendrian knot duality and its A∞ extension. method Using Ng's LSFT algebra, the abstract upgrades duality to a quasi-isomorphism of A∞ bimodules over Aug+. result Explicit construction of homotopy inverse for the A∞ Sabloff map. Defines a new symplectic Khovanov homology for links in fibered 3-manifolds.
problem No specific problem stated, but deals with Khovanov homology for links in fibered 3-manifolds.
method Defines a symplectic Khovanov type homology for a transverse link in a fibered closed 3-manifold with an auxiliary loop.
result Conjectural combinatorial dgas for surface categories, higher-dimensional analogs of strands algebras.
Proves nearby Lagrangian cocores are homotopically rigid in certain dimensions.
problem Homotopy rigidity of nearby Lagrangian cocores in Weinstein sectors.
method Spectral wrapped Donaldson-Fukaya category with orthogonal group coefficients.
result Inclusion followed by retract and quotient is null-homotopic.
The elliptic associator of Enriquez can be used to define an invariant of tangles embedded in the thickened torus, which extends the Kontsevich integral. This construction by Humbert uses the formulation of categories with elliptic structures. In this work we show that an extension of the LMO functor also leads to an e…
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.