New classifying spaces constructed from Ore categories with Garside families.
problem Constructing classifying spaces for fundamental groups.
method Using Ore categories with Garside families, introducing Zappa–Szép product.
result New categories and groups constructed, including Braided T of type F∞. New geometric structures in tangent categories.
problem Defining affine structures in tangent categories.
method Structural differential geometry, axiomatization of tangent categories, flat torsion-free connections.
result Affine objects form a tangent category and provide new characterizations of flat torsion-free connections.
Finite spaces can be or not coproducts of subspaces.
problem When finite-dimensional diffeological vector spaces are coproducts of their subspaces.
method Reviewing the question in diffeological vector spaces and comparing with other categories.
result Finite-dimensional spaces can be coproducts, but not always.
Direct proof given for a category of free groups.
problem Understanding the category of finitely generated free groups.
method Direct, combinatorial proof without Lawvere theories.
result Direct proof of the category being a PROP for commutative Hopf algebras.
We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We prese…
Adversarial MoE learns category-specific models for product search.
problem Variations in product features and importance across categories.
method Mixture of Experts with adversarial regularization and soft gating constraints.
result Improved clustering of gate output vectors and shared experts among similar categories.
The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the "modern" defi…
We consider the problem of learning soft assignments of N items to K categories given two sources of information: an item-category similarity matrix, which encourages items to be assigned to categories they are similar to (and to not be assigned to categories they are dissimilar to), and an item-item similarity mat…
Characterizes quasi-isometric embeddings in coarsely Lipschitz category.
problem Understanding quasi-isometric embeddings in geometric terms.
method Formalizes quasi-isometric embeddings as regular monomorphisms in coarsely Lipschitz category.
result Quasi-isometric embeddings are equivalently characterised as effective, strong, or extremal monomorphisms.
Generalizes string-net modular functors to non-spherical categories.
problem Extending string-net models to non-spherical categories.
method Using non-semisimple string-nets and Drinfeld centers.
result Equivalence between string-net and Lyubashenko modular functors.
New method detects novel node categories in graphs with distribution shifts.
problem Detecting novel node categories in graphs with distribution shifts.
method Recall-Constrained Optimization with Selective Link Prediction (RECO-SLIP).
result RECO-SLIP outperforms existing methods in detecting novel node categories.
New method transplants specific neural networks into generic ones.
problem Learning massive tasks and categories requires collecting samples for all at once.
method Designing a functionally interpretable generic network and transplanting specific modules.
result Generic network can learn new categories without sample annotations.
We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…
In principle, zero-shot learning makes it possible to train a recognition model simply by specifying the category's attributes. For example, with classifiers for generic attributes like \emph{striped} and \emph{four-legged}, one can construct a classifier for the zebra category by enumerating which properties it posses…
Let Ag,d be the (topological) cobordism category of orientable surfaces whose connected components are homeomorphic to either S1×I with one incoming and one outgoing boundary component or the surface Σg,d of genus g and d boundary components that are all incoming. In this paper, we stu…
New method transplants specific neural networks to generic ones without training samples.
problem Learning massive tasks and categories requires collecting samples for all at once.
method Designs a functionally interpretable generic network and uses back-distillation for transplanting.
result Method without training samples outperforms with 100 samples.
Study categorifies link invariants using Soergel bimodules.
problem Categorification of link invariants.
method Explicit computation of derived traces of Soergel bimodules.
result Derived annular Khovanov-Rozansky link invariant.
Research improves open-set learning by leveraging unlabelled data.
problem Learning between observed and unobserved novel categories.
method Unified policy of positive and unlabelled learning, semi-supervised learning, and open-set recognition.
result Achieves state-of-the-art results in open-set learning.
Study braid group actions on exceptional sequences using branched coverings.
problem Transitivity of braid group action on full exceptional sequences.
method Relate exceptional sequences to branched coverings, apply Birman--Hilden theory.
result Counterexamples to Bondal--Polishchuk conjecture on braid group transitivity.
The paper improves consumer preference modeling by considering multiple product categories.
problem Estimating consumer preferences across multiple product categories with varying attributes and price sensitivity.
method Extends matrix factorization techniques to account for time-varying product attributes and out-of-stock products, pooling information across categories to estimate heterogeneity in preferences.
result The model improves over traditional approaches, accurately estimating consumer preferences and price sensitivity.
This paper introduces tangent display maps to simplify tangent category theory.
problem The category of smooth manifolds does not admit all pullbacks, complicating tangent category theory.
method Develops tangent display maps as a special class of maps well-behaved with respect to pullbacks.
result Tangent display maps simplify previous work in tangent categories and provide a new way to define open subobjects.
A TQFT is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2+1-cobordism category is built from manifolds which are equipped with an extra structure such as a …
Modified invariants from quantum sl(2|1) for 3-manifolds.
problem Quantum sl(2|1) modules have vanishing quantum dimensions, complicating category construction.
method Specialize q to a root of unity, quotient by morphisms with zero modified quantum dimension, show resulting category is finite and semi-simple.
result Obtained relative G-spherical categories from modified quantum dimensions.
Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is play…
Frölicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as C^\infty(M,G) or \Diff(M), but also projective limits of Lie groups are in a natural way objects of that category, and group operations are morphisms in the category…
We show that the geometry of a Riemannian manifold (M,g) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted cat_{LS}(M). Here we introduce a Riemannian analogue of cat_{LS}(M), called the systolic category of M. It is denoted cat_{sys}(M), and d…
Paper develops an algorithm with PAC guarantees for detecting alien categories.
problem Detecting alien categories not seen in training data reliably.
method Develops an algorithm with PAC-style guarantees for alien detection under known upper bounds on alien fraction.
result Empirical results show the algorithm's effectiveness in detecting aliens.
New probabilistic invariants bound classical topological complexity and category.
problem Bounding classical topological complexity and category.
method Developed probabilistic variants of one-category and diagonal topological complexity.
result Identified new invariants with distributional category and complexity on Eilenberg-Mac Lane spaces.
Mathematical study supports connection between 3D manifolds and modular tensor categories.
problem Connecting geometric topology and quantum topology using Chern-Simons invariants and Reidemeister torsions.
method Developed an algorithm to generate modular T-matrices and quantum dimensions from Seifert fibered spaces and torus bundles over the circle. result Mathematically constructed premodular categories from Seifert fibered spaces and torus bundles over the circle, conjecturing their modularity under specific conditions.
Develops graphical calculus for monoidal categories with twisted pivotal structures.
problem Constructing modules for surfaces with Morse functions or foliations.
method Graphical calculus and string nets for monoidal categories with twisted pivotal structures.
result Twisted string net modules assemble in an oriented categorified 2-TQFT.
We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an étale topological or differentiable stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of 2-categori…
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.
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …
We improve stochastic gradient estimators for large categories using Rao-Blackwellization.
problem Computing gradients over large or infinite categories.
method Rao-Blackwellization to reduce variance of stochastic gradient estimators.
result Improves performance on semi-supervised classification and pixel attention tasks.
The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hop…
For each integer k≥4 we describe diagrammatically a positively graded Koszul algebra Dk such that the category of finite dimensional Dk-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type Dk or Bk−1, constructible with respect…
SEVEN tackles verification challenges with semi-supervised deep learning.
problem Verification with scarce labeled examples across many categories.
method SEVEN combines generative and discriminative components for end-to-end training.
result SEVEN significantly outperforms other techniques in verification tasks with limited labeled data.
Geometrically generates Fukaya categories of Weinstein manifolds.
problem Generating Fukaya categories of Weinstein manifolds.
method Geometric approach using surgery formula and Floer cohomology.
result Wrapped Fukaya category generated by index n critical points.
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.
Similarity encoding improves learning from messy categorical data.
problem Learning from categorical variables with high cardinality and redundancy.
method Similarity encoding, a generalization of one-hot encoding that uses similarities between categories.
result Similarity encoding significantly outperforms traditional encoding methods in prediction accuracy.
Framework orders categories and measures differences from real data.
problem Inferring orders and differences between categories.
method Nonparametric Estimation statistics framework.
result Scalable framework for large datasets.
Unified clustering model handles both pairwise and cardinality constraints for better performance.
problem Clustering with specific constraints (pairwise and cardinality) to improve clustering quality.
method Unified integer programming formulation, binary and quadratic constraints, reformulated as continuous constraints, solved using ADMM.
result Unified model outperforms single category constraints and achieves better clustering performance.
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…
Studies amenable category's monotonicity and its relation to topological complexity.
problem Monotonicity of amenable category for degree-one maps.
method Uses amenable covers and compares with topological complexity.
result Establishes a relation between amenable category and topological complexity.
To each oriented surface S, we associate a differential graded category Ko(S). The homotopy category Ho(Ko(S)) is a triangulated category which satisfies properties akin to those of the contact categories studied by K. Honda. These categories are also related to the algebraic contact categories of Y. Tian and to the bo…
Study integrates attentional and spacing factors to improve category learning models.
problem Understanding the impact of training sequences on category learning.
method Introduced a novel integration of attentional factors and spacing into logistic knowledge tracing models.
result Enhanced model predicts students' learning outcomes better than existing models.
Localizes smooth spaces to study their homotopy properties.
problem Understanding the homotopy theory of smooth spaces.
method Model category localization, Quillen equivalences, fibrant replacement.
result Localisation of smooth spaces agrees with motivic-style R-localisation. Innovative 2-categories create 4-manifold invariants.
problem Constructing invariants for 4-manifolds.
method Semisimple 2-categories, fusion 2-categories, and state-sum construction.
result Construct a state-sum invariant for 4-manifolds.