Related papers: Enlargements of Categories
In order to apply nonstandard methods to questions of algebraic geometry we continue our investigation from "Enlargements of categories" (Theory Appl. Categ. 14 (2005), No. 16, 357--398) and show how important homotopical constructions…
In this article we use our constructions from "Enlargements of Categories" (Theory and Applications of Categories, 14:357-398) to lay down some foundations for the application of A. Robinson's nonstandard methods to modern Algebraic…
We study varieties defined over nonstandard fields using techniques of nonstandard mathematics.
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
Categories of models of algebraic theories have good categorical properties except for gluing. Building upon insights and examples from Synthetic Differential Geometry, we introduce a generalisation of models of algebraic theories to…
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
We will present the benefits of using methods of non-standard analysis in dynamic projective geometry. One major application will be the desingulariazation of geometric constructions.
We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly…
We set up some foundations of generalised scheme theory related to new incompressible symmetric tensor categories. This is analogous to the relation between super schemes and the category of super vector spaces.
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
We study objects in triangulated categories which have a two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general…
We show how one can do algebraic geometry with respect to the category of simplicial objects in an exact category. As a biproduct, we get a theory of derived analytic geometry.
In this article, we will introduce methods of non-standard analysis into projective geometry. Especially, we will analyze the properties of a projective space over a non-Archimedean field. Non-Archimedean fields contain numbers that are…
In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while…
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 develop some nonstandard techniques for bornological and coarse spaces. We first generalise the notion of bornology to prebornology, which better fits to coarse spaces. We then give nonstandard characterisations of some basic large-scale…
We construct a category, $\Omega$, of which the objects are pointed categories and the arrows are pointed correspondences. The notion of a "spec datum" is introduced, as a certain relation between categories, of which one has been given a…
We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the "standard part" of certain algebraic nonstandard schemes. We construct a category of…
This is the seventh part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part VII), we give sufficient…
New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> makes an important such contribution.