相关论文: Hegel and Modern Topology

In a previous publication, we introduced an abstract logic via an abstract notion of quantifier. Drawing upon concepts from categorical logic, this abstract logic interprets formulas from context as subobjects in a specific category, e.g.,…

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…

A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes a relatively accessible example of such a thing, suggested by…

In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…

We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to…

Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…

Questions such as whether we live in a spatially finite universe, and what its shape and size may be, are among the fundamental open problems that high precision modern cosmology needs to resolve. These questions go beyond the scope of…

We firstly introduce some key concepts in category theory, such as quotient category, completion of limits, $\mathrm{Mor}$ category, and so on; then give the concept of topology algebras and sheaves, and discuss how to restore the structue…

This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…

We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove…

This overview article makes the case for how topological concepts can enrich research in machine learning. Using the Euler Characteristic Transform (ECT), a geometrical-topological invariant, as a running example, I present different use…

In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as…

The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately…

Positive logic is a generalisation of full first-order logic that does not have negation built in. Still, many model-theoretic ideas, tools and techniques work perfectly fine in positive logic. Importantly, there is a compactness theorem.…

The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are…

Network theory has proven to be a powerful tool in describing and analyzing systems by modelling the relations between their constituent objects. In recent years great progress has been made by augmenting `traditional' network theory.…

From the perspective of the physics of complex systems (1) we deal with the current state of modern physics including the crisis in physics demonstrated through its epistemological, psychological, economical as well as the social context;…

The main purpose of this paper is to introduce and study minimal and maximal ideals defined on ideal topological spaces. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of $2^X$, where $2^X$…

The study of complex systems through the lens of category theory consistently proves to be a powerful approach. We propose that cognition deserves the same category-theoretic treatment. We show that by considering a highly-compact cognitive…

This work concludes a series of four papers on the foundational theory of orbifolds and stacks. We apply the abstract theory, developed in its predecessors, to orbifolds derived from manifolds. Specifically, we show how the very concrete…