相关论文: On Generalization of Homotopy of Words and Its App…

This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book "New Spaces for Mathematics and Physics" (ed. Gabriel Catren and…

Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to…

Motivic homotopy theory is meant to play the role of algebraic topology, in particular homotopy theory, in the context of algebraic geometry. As proved by Oliver Rondigs and Paul Arne Ostvaer, this theory is closely connected to Voevodsky's…

This paper surveys work on generalized Johnson homomorphisms and tools for studying them. The goal is to unite several related threads in the literature and to clarify existing results and relationships among them using Hodge theory. We…

In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new…

Word embeddings represent language vocabularies as clouds of $d$-dimensional points. We investigate how information is conveyed by the general shape of these clouds, instead of representing the semantic meaning of each token. Specifically,…

This paper defines homology in homotopy type theory, in the process stable homotopy groups are also defined. Previous research in synthetic homotopy theory is relied on, in particular the definition of cohomology. This work lays the…

This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and…

The average uncertainty associated with words is an information-theoretic concept at the heart of quantitative and computational linguistics. The entropy has been established as a measure of this average uncertainty - also called average…

Partial generalizations of virtual polyhedra theory (sometimes under different names) appeared recently in the theory of torus manifolds. These generalizations look very different from the original virtual polyhedra theory. They are based…

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by…

The necessity of a theory of General Topology and, most of all, of Algebraic Topology on locally finite metric spaces comes from many areas of research in both Applied and Pure Mathematics: Molecular Biology, Mathematical Chemistry,…

The aim of this paper is to introduce the concepts of homotopical smallness and closeness. These are the properties of homotopical classes of maps that are related to recent developments in homotopy theory and to the construction of…

This paper aims to examine the version of the topological group structure in proximity and especially descriptive proximity spaces, that is, the concepts of proximal group and descriptive proximal group are introduced. In addition, the…

This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice…

This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often…

We discuss some recent results by a number of authors regarding word maps on algebraic groups and finite simple groups, their mixing properties and the geometry of their fibers, emphasizing the role played by equidistribution results in…

A virtual $n$-string is a chord diagram with $n$ core circles and a collection of arrows between core circles. We consider virtual $n$-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given…

General probabilistic theories are designed to provide operationally the most general probabilistic models including both classical and quantum theories. In this letter, we introduce a systematic method to construct a series of entropies,…

Understanding the features of and mechanisms behind language learning can provide insights into the general process of knowledge acquisition. Recent methods from network science applied to language learning have advanced the field,…