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. We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. O…
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 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.
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.
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. 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. 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. 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.
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…
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…
Lie ∞-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie ∞-groupoids called `Lie ∞-groups' by integrating finite type Lie n-algebras. In order to study the compatibility between this…
We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbif…
Curved L∞ spaces form a category of fibrant objects.
problem Understanding the structure of curved L∞ spaces. method Proving L∞ spaces over dg manifolds form a category of fibrant objects. result Transitive L∞ algebroids over dg manifolds also form a category of fibrant objects. We show that conically smooth stratified spaces embed fully faithfully into ∞-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each ∞-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…
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.
The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends…
In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid L-and the natural generalization to dg Lie algebroids-provides an (essentially unique) L∞ space. More precisely, we construct a faithful functor from the category of Lie algebroids …
The paper develops a theory of C∞-superrings and their superschemes.
problem Developing a theory for C∞-superrings and superschemes. method Proving an equivalence between categories of fair affine C∞-superschemes and fair C∞-superrings. result A key equivalence between fair affine C∞-superschemes and fair C∞-superrings. Foundations of derived geometry in smooth settings.
problem Building tools for moduli spaces in differential geometry.
method Abstract structured spaces and universal properties in (∞,2)-categories. result Established derived flatness results for derived C∞-rings. A new category generates 1D tangle invariants.
problem Developing a new category for 1D tangles.
method Proving a new (∞,1)-category has universal mapping property. result The new category generates link invariants.
This paper introduces a new distance metric for filtered A-infinity categories, focusing on Lagrangian submanifolds.
problem Measuring the distance between filtered A-infinity categories associated with Lagrangian submanifolds.
method Developed a Gromov-Hausdorff distance to measure the difference between these categories.
result Established that the sequence of filtered A-infinity categories forms a Cauchy sequence in Gromov-Hausdorff distance.
Let X be a compact real analytic manifold, and let T∗X be its cotangent bundle. Let Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on X. In this paper, we develop a Fukaya A∞-category Fuk(T∗X) whose objects are exact, not necessarily compact Lagrangian branes in…
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the ∞-category of curved Lie algebras using homotopy theory of algebras over a complete operad. result Equivalence of ∞-categories between Lie algebroids and certain kinds of curved Lie algebras. 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.
It is proved that the category of simplicial complete bornological spaces over R carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particu…
New model for Calabi-Yau-X categories using decorated marked surfaces.
problem Constructing models for Calabi-Yau-X categories. method Using graded decorated marked surfaces and string models.
result Isomorphism between braid twist group and spherical twist group.
We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition l…
Extends Poincaré-Lefschetz duality to pairs of ∞-categories.
problem Generalizing Poincaré-Lefschetz duality to ∞-categories.
method Introduces Poincaré duality pairs of ∞-categories and uses them to study various diagrams of spaces.
result Unified treatment of Wall's Poincaré ads and iterated Poincaré cobordisms.
Constructs a cyclic, filtered, strictly unital curved A∞ category for Lagrangian submanifolds and develops Floer theory.
problem Proving that any Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by a given collection of Lagrangian submanifolds.
method Develops a cyclic, filtered, strictly unital curved A∞ category and uses it to prove the above statement. result Any Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by a given collection of Lagrangian submanifolds.
We construct a functor from the derived category of homotopy Gerstenhaber algebras with finite-dimensional cohomology to the purely geometric category of so-called F∞-manifolds. The latter contains Frobenius manifolds as a subcategory (so that a pointed Frobenius manifold is itself a homotopy Gerstenhaber alg…
This paper extends a category equivalence to an A-infinity quasi-equivalence for compact Lie groups.
problem Equivalence of DG categories for smooth singular chains on Lie groups.
method Construction of A-infinity quasi-isomorphisms and use of Van Est map, De Rham theorem.
result Extension of equivalence to A-infinity quasi-equivalence for compact Lie groups.
This article reviews ∞-bundles and their applications in geometry and physics.
problem Understanding higher bundles in geometry and physics.
method An ∞-categorical formulation of higher bundles. result Identification of higher bundles in various contexts.
This paper introduces ∞- and n-fold vector bundles as special functors from the ∞- and n-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of n-fold vector bundles and we prove that any n-fold vector bundle admits a non-canonical isomorphism to a decomposed …
Develops L∞ spaces over dg manifolds and establishes an equivalence with L∞ algebroids.
problem Defining and comparing L∞ spaces and algebroids over dg manifolds. method Establishes an equivalence between categories of L∞ algebroids and L∞ spaces, constructs a faithful functor. result Detects weak equivalences between L∞ algebroids and L∞ spaces. Categorifies Chern-Weil theory for infinite local systems.
problem Categorifying Chern-Weil theory for infinite local systems.
method Using DG functors and A∞-natural isomorphisms. result Provides a categorification of the Chern-Weil homomorphism.
Study of embedding calculus using infinite operads.
problem Spaces of embeddings and automorphisms of manifolds.
method Infinite operadic towers and Morita (∞,2)-categories. result Generalization of embedding calculus to bordism categories.
Introduces new connections in higher geometry.
problem Defining connections in higher geometry.
method Develops formal differentiation and integration of maps to derived stacks.
result Establishes new L∞-algebras of higher symmetries. We study the behavior of Pin(2)-monopole Floer homology under connected sums. After constructing a (partially defined) A∞-module structure on the Pin(2)-monopole Floer chain complex of a three manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to …
Research resolves sign conventions in Floer theory for Morse-Bott case.
problem Sign conventions in filtered A∞-operations for Lagrangian Floer theory. method Defined filtered A∞-operations and verified formulae using de Rham model. result Resolved sign issues in Bott-Morse setting.
The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We conjecture that these categories become A∞-equivalent after a twist by a kind o…
In this article we apply ideas from homotopy theory to the study of singular foliations. We verify that a technical lemma remains valid for left semi-model categories. When applied to the category of L∞-algebroids thanks to the work of Nuiten, this lemma enables to recover results very similar to those of Laure…
We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a "thickening…
The paper studies higher geometric structures and connections on manifolds, constructing moduli stacks and proving equivalence criteria.
problem Classifying and understanding higher geometric structures and connections on manifolds.
method Constructing smooth higher symmetry groups, moduli stacks, and higher gauge actions; proving equivalence criteria.
result Construction and classification of moduli stacks of higher geometric data and connections.
Vector fields on schemes have flows if rings are finitely generated.
problem Understanding vector fields and flows on schemes.
method Analyzing vector fields on affine C∞-schemes with finitely generated rings. result Vector fields on affine C∞-schemes with finitely generated rings have flows and are groupoid internal maps. We study an A∞ category associated to Legendrian links in R3 whose objects are n-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a …
The abstract defines G2-structures and connects them to octonion algebras.
problem Classifying G2-structures and understanding their geometric properties. method Established an isomorphism between G2-structures and octonion algebras over C∞(M). result The classification of G2-structures agrees with a parametrisation of octonion algebras with isometric norm.