Related papers: Constructive higher sheaf models with applications…

We extend Stone duality to a fully faithful embedding of condensed sets into fpqc sheaves over an arbitrary field, which preserves colimits and finite limits. We study how familiar notions from condensed mathematics/topology and algebraic…

In this note we relate three topics for arithmetic schemes: a general duality for \'etale constructible torsion sheaves, an \'etale homology theory, and a Gersten-Bloch-Ogus-Kato complex. The results in this paper have been used in other…

From descent theory to higher geometry, the idea of gluing has been embedded in many elegant and powerful techniques, proving instrumental for the solution of many problems. In this paper, we introduce a framework that allows to link…

We regard a geometric theory classified by a topos as a syntactic presentation for the topos and develop tools for finding such presentations. Extensions of geometric theories, which can add axioms, symbols and sorts, are treated as objects…

A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…

This book provides an inviting tour through sheaf theory, from the perspective of applied category theory and pitched at a less specialized audience than is typical with introductions to sheaves. The book makes it as easy as possible for…

A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…

In recent years, several models have improved the capacity to generate synthetic tabular datasets. However, such models focus on synthesizing simple columnar tables and are not useable on real-life data with complex structures. This paper…

While there is substantial need for dependence models in higher dimensions, most existing models quickly become rather restrictive and barely balance parsimony and flexibility. Hierarchical constructions may improve on that by grouping…

We explore a new connection between synthetic domain theory and Grothendieck topoi related to the distributive lattice classifier. In particular, all the axioms of synthetic domain theory (including the inductive fixed point object and the…

We develop a denotational semantics for general reference types in an impredicative version of guarded homotopy type theory, an adaptation of synthetic guarded domain theory to Voevodsky's univalent foundations. We observe for the first…

This is a foundation for algebraic geometry, developed internal to the Zariski topos, building on the work of Kock and Blechschmidt. The Zariski topos consists of sheaves on the site opposite to the category of finitely presented algebras…

In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model…

We introduce and develop the theory of metric sheaves. A metric sheaf $\A$ is defined on a topological space $X$ such that each fiber is a metric model. We describe the construction of the generic model as the quotient space of the sheaf…

In this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with the…

In this paper we explore the link between the theory of sheaves on graphs and noncommutative geometry showing that many concepts and constructions in the latter can be generalized and enhanced using methods coming from the former. They…

A brief survey of how classical field theory emerges synthetically in cohesive homotopy type theory. Extended Conference Abstract submitted to the proceedings of the Conference on Type Theory, Homotopy Theory and Univalent Foundations in…

This paper has several purposes. We present through a critical review the results from already published papers on the constructive semigroup theory, and contribute to its further development by giving solutions to open problems. We also…

We develop foundations for abstract homotopy theory based on Grothendieck's idea of a "derivator". The theory is model-independent, and does not depend on model categories, nor on simplicial sets. It is designed to accomodate all the usual…

This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways…