Related papers: Barr-Exact Categories and Soft Sheaf Representatio…
It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a…
We construct a quasi-coherent sheaf of associative algebras which controls a category of $AV$-modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in \'etale charts these associative algebras…
Schwartz functions, or measures, are defined on any smooth semi-algebraic ("Nash") manifold, and are known to form a cosheaf for the semi-algebraic restricted topology. We extend this definition to smooth semi-algebraic stacks, which are…
Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can…
In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra…
Given a general finite group $G$, we consider several categories built on it, their Grothendieck topologies and resulting sheaf categories. For a certain class of transporter categories and their quotients, equipped with atomic topology, we…
Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of local monoidal categories on a topological space. Every small stiff monoidal category…
We show that a projective space P^\infty(Z/2) endowed with the Alexandrov topology is a classifying space for finite closed coverings of compact quantum spaces in the sense that any such a covering is functorially equivalent to a sheaf over…
The purpose of this paper is to develop an efficient computational model for Abelian categories of coherent sheaves over certain classes of varieties. These categories are naturally described as Serre quotient categories. Hence, our…
In the framework of Abstract Differential Geometry, we show that to a given principal sheaf and a representation of its stuctural sheaf in $A^n$, where A is a sheaf of associative, commutative, unital algebras (over R or C), we associate a…
We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational…
In this paper, we investigate the properties of $A$-coherent and $A$-quasi-coherent sheaves within the framework of algebraic geometry over non-algebraically closed fields. We define an $\mathcal{O}_X$-module to be $A$-coherent (resp.…
The main goal of this paper is to establish close relations among sheaves of modules on atomic sites, representations of categories, and discrete representations of topological groups. We characterize sheaves of modules on atomic sites as…
Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an equivariant $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of…
Sheaves are objects of a local nature: a global section is determined by how it looks locally. Hence, a sheaf cannot describe mathematical structures which contain global or nonlocal geometric information. To fill this gap, we introduce the…
Let $X$ be a smooth manifold and $\mathbf{k}$ be a commutative (or at least $\mathbb{E}_2$) ring spectrum. Given a smooth exact Lagrangian $L\hookrightarrow T^*X$, the microlocal sheaf theory (following Kashiwara--Schapira) naturally…
Quivers (directed graphs) and species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their…
Using the results of J. Arthur on the representation theory of classical groups with additional work by Colette Moeglin and its relation with representations of affine Hecke algebras established by the author, we show that the category of…
In this note I introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group $G$ as a sheaf on some geometric object (a stack). This point of…
In this note we realize the sheaf of Cherednik algebras $H_{1, c, X, G}$ on a general good complex orbifold $X/G$, originally introduced by Etingof for smooth complex varieties with an action by a finite group, by gluing sheaves of flat…