We study the functor of points and the local functor of points (here called the Weil--Berezin functor) for smooth and holomorphic supermanifolds, providing characterization theorems and fully discussing the representability issues. In the end we examine applications to differential calculus including the transitivity t…
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48 results for “functor of point approach”
This paper studies supergrassmannians as homogeneous spaces for super Lie groups.
problem Defining and understanding supergrassmannians as homogeneous spaces.
method Using the functor of point approach, the paper constructs the action of super Lie group GL(m|n) on G_{k|l}(m|n).
result Explicit construction of the action of super Lie group GL(m|n) on supergrassmannian G_{k|l}(m|n).
Reformulates -manifolds using functor of points.
problem Understanding -graded manifolds in a categorical framework.
method Uses functor of points to embed -manifolds into contravariant functors of -points.
result Equivalence of -manifolds and locally trivial functors.
Gauge theory connects principal bundles and functors.
problem Understanding the relationship between principal bundles and functors.
method Characterized association functors and established an equivalence between principal bundles and functors.
result Established an equivalence between principal bundles and functors using association functors.
Generalizes Lie supergroups to Lie superheaps.
problem No specific problem stated; generalization of Lie supergroups.
method Uses functor of points to show isomorphism between categories of pointed Lie superheaps and Lie supergroups.
result There is an isomorphism between the categories of pointed Lie superheaps and Lie supergroups.
In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differen…
In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differen…
We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in se…
New map constructed from equivariant spectra for manifold study.
problem Understanding equivariant parametrized h-cobordism in non-manifold settings.
method Constructed a map from suspension G-spectrum to equivariant A-theory spectrum, compatible with tom Dieck splitting formulas.
result Fiber of constructed map is wedge of stable h-cobordism spectra.
A functorial extension of the Magnus representation to the category of three-dimensional cobordismsmath.GT
A new functor extends Magnus representation to 3D cobordisms.
problem Extending Magnus representation to 3D cobordisms.
method Constructing a monoidal functor from Cob_G to pLagr_R.
result Relates the Magnus functor to the Alexander functor.
We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the…
For an oriented knot , we construct a functor from the category of pointed quandles to the category of quandles in three different ways. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariant of knots, and study their properties.
Constructs Khovanov spectra for periodic links, proving rank inequalities.
problem Understanding Khovanov homology for periodic links.
method Equivariant Khovanov spectra using Burnside functor construction.
result Rank inequalities for Khovanov homologies and annular filtrations of prime-periodic links.
We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. We show that such a bundle has an underlying reflection space, and we investigate the corresponding forgetful functor both from t…
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.
Study embedding calculus and link invariants using functor calculus.
problem Detect Milnor invariants using embedding towers of string links.
method Use functor calculus and Goodwillie-Weiss embedding calculus.
result Embedding tower detects Milnor invariants.
This thesis bridges Lie theory and sketch theory using tangent categories.
problem Two diverging lines of research in Lie theory.
method Developing involution algebroids and using tangent categories to connect Lie algebroids and Weil algebras.
result The category of Lie algebroids is a functor category, and the Lie functor is a composition with a tangent categorical functor.
The paper constructs supergrassmannians using gluing and describes their actions.
problem Constructing and describing the actions of supergrassmannians.
method Gluing of superdomains and action of super Lie group in functor of points language.
result Concrete proof of the transitively of the action and gluing of local charts.
Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le-Murakami-Ohtsuki invariant of homology three-spheres to a functor from the category of Lagrangian cobordisms to a certain category of Jacobi diagrams. We prov…
Study of equivariant scalar curvature groups for proper group actions.
problem Understanding equivariant scalar curvature groups for discrete group actions.
method Definition of fundamental groupoid functor, construction of classifying spaces, geometric result.
result Stolz's equivariant R-group depends only on the fundamental groupoid functor of the space.
Unified Long-Moody and Katz methods for constructing local systems.
problem Constructing representations of braid groups and local systems.
method Katz-Long-Moody functor unifying Long-Moody and Katz methods.
result Extends Katz algorithm to various topological spaces.
Manifold calculus of functors, due to M. Weiss, studies contravariant functors from the poset of open subsets of a smooth manifold to topological spaces. We introduce "multivariable" manifold calculus of functors which is a generalization of this theory to functors whose domain is a product of categories of open sets. …
Study on polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
problem Understanding polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
method Analyzing polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
result Results generalize previous work by Katada and study polynomiality and outer nature of these functors.
A modular functor is constructed from non-semisimple 3d TFTs.
problem Constructing modular functors from non-semisimple 3d topological field theories.
method Using a 3d TFT defined in [arXiv:1912.02063], a symmetric monoidal 2-functor is constructed from a 2-category of bordisms to a 2-category of finite linear categories.
result A modular functor is explicitly described as a symmetric monoidal 2-functor.
Paper constructs infinitely many tangent functors on diffeological spaces.
problem Tangent spaces in diffeological spaces are not uniquely defined.
method Introduced and constructed infinitely many non-isomorphic tangent functors.
result The choice of tangent functor is not unique outside smooth manifolds.
The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study …
Study homology manifolds using spectral sheaves and spectral six functor formalism.
problem Characterize and understand homology manifolds through spectral sheaves.
method Adapt six functor formalism to spectral sheaves on locally compact Hausdorff spaces.
result Prove that compact ANR homology manifolds are Poincaré duality complexes.
Three functors link Lorentzian geometry concepts.
problem Finiteness results, singularity theorems, boundary constructions.
method Review of three functors from Lorentzian categories.
result Novel functor from ordered measure spaces to Lorentzian pre-length spaces.
The study explores how different Grothendieck topologies and functors between categories preserve locality.
problem Exploring relationships between different Grothendieck topologies and functors.
method Using Grothendieck topologies and functors to relate categories and geometric objects.
result Objects like sheaves, groupoids, and functors are invariant under equivalences of Grothendieck topologies and certain functors.
Constructs a 2-functor for tangles extending Burau and Alexander.
problem No specific problem stated; focuses on extending existing concepts.
method Constructs a weak 2-functor from tangles to Lagrangian cospans.
result Extends Burau and Alexander concepts to tangles.
New algebraic formalism for differential calculus in Diolic algebras.
problem Studying differential calculus in vector bundles.
method Introducing functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this to Diolic algebras.
result Recovery of well-known objects and notions from ordinary differential, symplectic, and Poisson geometry, with unique aspects.
Grothendieck reformulated Teichmüller's construction using category theory.
problem Defining Teichmüller space and its universal curve rigorously.
method Using category theory to generalize Teichmüller's construction.
result Reformulated Teichmüller's construction into a functorial perspective.
In terms of category theory, the Gromov homotopy principle for a set valued functor asserts that the functor can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor holds if the functor can be induced from a (co)homology functor. We examin…
Develops theory of Cartan geometries on skeletons and morphisms induced by extension functors.
problem Describing categories of Cartan geometries with additional morphisms.
method Using extension functors to define new categories of Cartan geometries and studying their properties.
result Shows functors between categories of Cartan geometries with morphisms induced by extension functors and categories of Cartan geometries modeled on skeletons.
We give an overview of how calculus of the embedding functor can be used for the study of long knots and summarize various results connecting the calculus approach to the rational homotopy type of spaces of long knots, collapse of the Vassiliev spectral sequence, Hochschild homology of the Poisson operad, finite type k…
Parity functorsmath.GT
Parity functors assign labels to knot diagrams based on crossing parity.
problem Assigning consistent labels to knot diagrams.
method Define parity functors for knot diagrams and surfaces.
result Universal oriented parity functors for free knots and fixed surface knots.
Mapping spaces of supermanifolds are usually thought as exclusively in functorial terms (i.e. trough the Grothendieck functor of points). In this work we provide a geometric description of such mapping spaces in terms of infinite-dimensional super-vector bundles.
This paper defines a functor for -modules and applies it to Khovanov homology.
problem Defining a functor for -modules and proving its properties.
method Abstract approach using morphisms of -algebras.
result Restriction of scalars defines a functor between -modules.
The theory of product preserving functors and Weil functors is partly extended to infinite dimensional manifolds, using the theory of -algebras.
The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects o…
We reformulate superalgebra and supergeometry in completely categorical terms by a consequent use of the functor of points. The increased abstraction of this approach is rewarded by a number of great advantages. First, we show that one can extend supergeometry completely naturally to infinite-dimensional contexts. Seco…
We construct the Weil functor corresponding to a general Weil algebra : this is a functor from the category of manifolds over a general topological base field or ring (of arbitrary characteristic) to the category of manifolds over . This result simultaneously generalizes results known for o…
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an ass…
Motivated by our attempt to recast Cartan's work on Lie pseudogroups in a more global and modern language, we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem. From this point of view, the novelty of this paper is that w…
On --modular functormath.QA
In this paper, we extend the notion of modular functor and fusion category to what we called equivariant modular functor and equivariant fusion category, where is a finite group, and establish a correspondence between between these notions.
Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
Cheptea, Habiro and Massuyeau constructed the LMO functor, which is defined on a certain category of cobordisms between two surfaces with at most one boundary component. In this paper, we extend the LMO functor to the case of any number of boundary components, and our functor reflects relations among the parts correspo…
New jet functors generalize classical notions in noncommutative geometry.
problem Defining and understanding jet functors in noncommutative settings.
method Constructing and proving properties of jet functors , , and .
result Holonomic jet functor satisfies jet exact sequence under specific conditions.