Related papers: Some fundamental problems on real-analytic sets
In the first half of twentieth century the theory of complex analytic functions and of their zerosets was fully developed. The definition of holomorphic function has a local nature. Germs of holomorphic functions form a distinguished…
Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a…
The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…
Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made…
The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design…
We present here "the" cartesian closed theory for real analytic mappings. It is based on the concept of real analytic curves in locally convex vector spaces. A mapping is real analytic, if it maps smooth curves to smooth curves and real…
A critically challenging problem facing statisticians is the identification of a suitable framework which consolidates data of various types, from different sources, and across different time frames or scales (many of which can be missing),…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
The concept of category from mathematics happens to be useful to computer programmers in many ways. Unfortunately, all "good" explanations of categories so far have been designed by mathematicians, or at least theoreticians with a strong…
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
This article focuses on the study of cut groups, i.e., the groups which have only trivial central units in their integral group ring. We provide state of art for cut groups. The results are compiled in a systematic manner and have also been…
The central focus is on clarifying the distinction between sets and proper classes. To this end we identify several categories of concepts (surveyable, definite, indefinite), and we attribute the classical set theoretic paradoxes to a…
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
We show that a function is real analytic at the origin iff it is arc-analytic, has a subanalytic graph, and its restriction to every monomial curve is analytic. This complements recent results of Kucharz and Kurdyka.
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…
In mathematical applications, category theory remains a contentious issue, with enthusiastic fans and a skeptical majority. In a muted form this split applies to the authors of this note. When we learned that the only mathematically sound…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…