Related papers: Categorical Nonstandard Analysis
We propose a series-based nonparametric specification test for a regression function when data are spatially dependent, the `space' being of a general economic or social nature. Dependence can be parametric, parametric with increasing…
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and…
We try to convince the reader that the categorical version of differential geometry, called Synthetic Differential Geometry (SDG), offers valuable tools which can be applied to work with some unsolved problems of general relativity. We do…
This paper proposes a formal cognitive framework for problem solving based on category theory. We introduce cognitive categories, which are categories with exactly one morphism between any two objects. Objects in these categories are…
The interdefinability of the universal concepts of category theory has been introduced by Lawvere. The perfect interdefinability between the objects and arrows of some category, defines the class of Perfectly Symmetric Categories (PSC)…
In this paper, we will establish a general method of studying finite-dimensional normed spaces, and apply this method to classifying $3$-dimensional and $4$-dimensional normed spaces over a non-spherically complete field. For this purpose,…
This note has two principal aims: to portray an essence of Non-Standard Analysis as a particular structure (which we call lim-rim), noting its interplay with the notion of ultrapower, and to present a construction of Non-Standard Analysis,…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic…
In this article we present a new modelling framework for structured concepts using a category-theoretic generalisation of conceptual spaces, and show how the conceptual representations can be learned automatically from data, using two very…
This paper investigates the relationship between categorical entropy and von Neumann entropy of quantum lattices. We begin by studying the von Neumann entropy, proving that the average von Neumann entropy per site converges to the logarithm…
In this paper, we introduce a mathematical structure called Euclidean Universe. This structure provides a basic framework for Non-Archimedean Mathematics and in particular for Nonstandard Analysis.
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…
In this paper, we consider certain topological properties along with certain types of mappings on these spaces defined by the notion of ideal convergence. In order to do that, we primarily follow in the footsteps of the earlier studies of…
With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions…
We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a…
Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a…
The primary goal of this paper is to abstract notions, results and constructions from the theory of categories to the broader setting of plots. Loosely speaking, a plot can be thought of as a non-associative non-unital category with a…
The goal of this paper is to show that fundamental concepts in higher-order Fourier analysis can be nauturally extended to the non-commutative setting. We generalize Gowers norms to arbitrary compact non-commutative groups. On the…
For each $n$, we construct a separable metric space $\mathbb{U}_n$ that is universal in the coarse category of separable metric spaces with asymptotic dimension ($\mathop{asdim}$) at most $n$ and universal in the uniform category of…