Higher topos theory applied to physics.
problem No specific problem stated.
method Exposition of higher topos theory.
result No specific key result mentioned.
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48 results for “higher topos”
The paper defines approximate fibrations in higher topos theory.
problem Defining approximate fibrations in a new mathematical framework.
method Introducing approximate fibrations for geometric morphisms of -topoi, providing characterizations and comparing to previous definitions.
result Generalization of shape-theoretic characterizations to a topos-theoretical proof.
Topo-MLP learns network representations without message passing.
problem Lack of efficient higher-order network modeling methods.
method Proposes Topo-MLP, a simplicial neural network algorithm using MLP and HONC loss.
result Demonstrates improved robustness and efficiency in representation learning.
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 -bundles is weak homotopy equivalent to -principal -bundles.
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.
Proper Orbifold Cohomologymath.AT
Unified theory of orbifolds and cohomology.
problem Formulating a general theory of orbifolds unifying differential and equivariant cohomology.
method Abstract axiomatization in higher topos theory and concrete models for various orbifolds.
result Fully faithful embedding of orbifolds into a cohesive infinity-topos with proper equivariant cohomology.
Cyclification of Orbifoldsmath.AT
Cyclification of orbifolds explained in cohesive higher topos theory.
problem Understanding cyclification of orbifolds in geometric and algebraic contexts.
method Cohesive higher topos theory and transgression of cohomological charges.
result Cyclification of orbifolds is a fundamental base-change construction.
We show that there is a fully faithful embedding of the category of manifolds with corners into the Cahiers topos, one of the premier models for Synthetic Differential Geometry. This embedding is shown to have a number of nice properties, such as preservation of open covers and transverse fibre products. We develop a t…
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive struct…
Proof confirms preservation of projective limits in synthetic differential geometry.
problem Prove preservation of projective limits in synthetic differential geometry.
method Detailed proof using synthetic differential geometry and Cahiers topos.
result Projective limits preserved in synthetic differential geometry.
It is well known that the category of Frolicher spaces and smooth mappings is Cartesian closed. The principal objective in this paper is to show that the full subcategory of Frolicher spaces that believe in fantasy that every Weil functor is really an exponentiation by the corresponding infinitesimal object is also Car…
We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman's ideas, we can endow it with a p…
Extends field theory foundations to infinitesimal spaces, simplifying complex concepts.
problem Develop rigorous foundations for field theory, especially for infinitesimal spaces.
method Formulates local Lagrangian field theory in a new category of thickened smooth sets.
result Establishes a firm foundation for field theory, including tangent bundles and perturbative considerations.
Let $\imath: M\to \RR^{p+2}$ be a smooth embedding from a connected, oriented, closed -dimesional smooth manifold to $\RR^{p+2}$, then there is a spin structure on canonically induced from the embedding. If an orientation-preserving diffeomorphism of extends over as an o…
In this paper, we study the problem of using representation learning to assist information diffusion prediction on graphs. In particular, we aim at estimating the probability of an inactive node to be activated next in a cascade. Despite the success of recent deep learning methods for diffusion, we find that they often…
This article constructs the moduli stack of torsionfree -jet-structures in homotopy type theory with one monadic modality. This yields a construction of this moduli stack for any -topos equipped with any stable factorization systems. In the intended applications of this theory, the factorization systems are …
Study moduli spaces of elliptic PDEs using derived -geometry.
problem Representability of moduli spaces of solutions of elliptic PDEs.
method Derived -geometry, stacks of relative jets, nonlinear Fredholm analysis.
result Moduli stack of solutions is relatively representable by quasi-smooth derived -schemes.
Unified classification of equivariant principal bundles using higher homotopy theory.
problem Unified classification of equivariant principal bundles.
method Smooth Oka principle, singular-cohesive homotopy theory, internally describing principal bundles.
result Unified classification results for equivariant principal bundles.
Frolicher spaces and smooth mappings form a cartesian closed category. It was shown in our previous paper [Far East Journal of Mathematical Sciences, 35 (2009), 211-233] that its full subcategory of Weil exponentiable Frolicher spaces is cartesian closed. By emancipating microlinearity from within a well-adapted model …
Homotopy theory of differentiable sheaves connects manifold properties to underlying homotopy types.
problem Understanding the homotopy type of manifolds using differentiable sheaves.
method Developed model structures and homotopical calculi on the -category to compute and compare shapes.
result The shape of any manifold coincides with various other notions of underlying homotopy types.
This paper uses sheaf theory to model virtual knots geometrically.
problem Defining and understanding virtual knots in a geometric framework.
method Sheaf theory applied to virtual knot diagrams to model them geometrically.
result A geometric model for virtual knots formalizes the intuitive notion of knots in a variable ambient space.
Introduces a new geometric framework for field theories.
problem Developing a rigorous mathematical framework for field theories.
method Introduces supergeometric homotopy theory to physics.
result Classical bosonic field theories fit naturally into smooth sets.
The paper gives a categorical approach to generalized manifolds such as orbit spaces and leaf spaces of foliations. It is suggested to consider these spaces as sets equipped with some additional structure which generalizes the notion of atlas. The approach is compared with the known ones that use the Grothendieck topos…
In category theory, logic and geometry cooperate with each other producing what is known under the name Synthetic Differential Geometry (SDG). The main difference between SDG and standard differential geometry is that the intuitionistic logic of SDG enforces the existence of infinitesimal objects which essentially modi…
In this article, we derive many properties of étale stacks in various contexts, and prove that étale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms. Moreover, we show that the bicategory of étale differentiable stacks and loca…
New algorithm optimizes DAGs by swapping node pairs to avoid cycles.
problem Optimizing DAGs with non-convex constraints to avoid cycles.
method Bi-level algorithm with iterative node swaps and off-the-shelf solvers.
result Guaranteed to find local minima or KKT points with lower scores.
The paper shows deep connections between exotic smoothings of small R^4, noncommutative algebras of foliations and quantization. At first, based on the close relation of foliations and noncommutative C*-algebras we show that cyclic cohomology invariants characterize some small exotic R^4. Certain exotic smooth R^4's de…
Paper constructs Chern character for higher twists and shows isomorphism between K-theory and cohomology.
problem Mapping higher twisted K-theory to higher twisted cohomology.
method Constructing Chern character for higher twists and showing isomorphism.
result Chern character gives isomorphism between higher twisted K-theory and higher twisted cohomology.
Higher Gauge Flow Models integrate higher geometry and symmetries into Generative Flow Models.
problem Improving generative models' performance.
method Integrates L-algebra into Generative Flow Models, leveraging higher geometry and symmetries.
result Substantial performance improvements on Gaussian Mixture Model datasets.
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.
Explains model structures for higher orbifolds and applies them to quantum cohomology.
problem Understanding quantum cohomology of higher orbifolds.
method Develops model structures on higher orbifolds and applies them to quantum cohomology.
result Model structures provide insights into quantum cohomology of higher orbifolds.
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…
Generalized Fáry's theorem to higher dimensions.
problem No specific problem stated; generalization of Fáry's theorem.
method Proof of a higher-dimensional version of Fáry's theorem.
result Proved a generalization of Fáry's theorem in higher dimensions.
Higher order higher spin operators are generalizations of -powers of the Dirac operator. In this paper, we study higher order higher spin operators defined on some conformally flat manifolds, namely cylinders and Hopf manifolds. We will also construct the kernels of these operators on these manifolds.
Holonomies match for higher local systems and principal 2-bundles.
problem Matching holonomies for higher local systems and principal 2-bundles.
method Higher Riemann-Hilbert correspondence and principal 2-bundles.
result Holonomies coincide for both formalisms.
This note discusses the higher K-energy functionals which were defined by Bando and Mabuchi, and integrate higher Futaki invariants. Two new formulas for the higher K-energy functionals are given, and the second K-energy is shown to be related to Donaldson's Lagrangian applied to metrics on the tangent bundle.
Many physical theories, including notably string theory, require non-abelian higher gauge fields defining higher holonomy. Previous approaches to such higher connections on categorified principal bundles require these to be fake flat. This condition, however, renders them locally gauge equivalent to connections on abel…
In this thesis, we employ simplicial methods to study actions, principal bundles, and bibundles of higher groupoids. Roughly, we use Kan fibrations to model actions of higher groupoids, we use pairs of a Kan fibration and a special acyclic fibration to model principal bundles of higher groupoids, we use inner Kan fibra…
-Bundlesmath.AT
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.
Transformed quadrics from 2D to higher dimensions.
problem Generalizing quadric transformations to higher dimensions.
method Bianchi's Hazzidakis transformation method.
result Generalization to higher dimensional quadrics.
Higher Gauge Theoryhep-th
Survey on advanced gauge theory concepts.
problem Understanding higher gauge theory structures.
method Introduction to higher structures and connections on higher principal bundles.
result Summarized applications and principles of higher gauge theories.
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
Higher gauge theory via differential nonabelian cohomology
problem Global infrared completion of higher gauge fields
method Maxwell-type higher gauge fields
result Electromagnetic flux quantization
The paper encourages Kleinian group thinking for higher rank Lie groups.
problem No specific problem stated; encouraging new thinking.
method Discussion of Kleinian group ideas applied to higher rank Lie groups.
result Encouragement to think about higher rank Lie groups using Kleinian group theory.
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…
Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
By developing a generalized cobordism theory, we explore the higher global symmetries and higher anomalies of quantum field theories and interacting fermionic/bosonic systems in condensed matter. Our essential math input is a generalization of Thom-Madsen-Tillmann spectra, Adams spectral sequence, and Freed-Hopkins's t…
For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann -invariants, which we call higher-order signatures. The higher-order genera o…