Related papers: A Very Elementary Introduction to Sheaves
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…
This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory…
Essentials of sheaves are briefly presented, followed by related comments on presheaves, bundles, manifolds and singularities, aiming to point to their differences not only in their different formal mathematical structures, but also in the…
This document develops general concepts useful for extracting knowledge embedded in large graphs or datasets that have pair-wise relationships, such as cause-effect-type relations. Almost no underlying assumptions are made, other than that…
As data grows in size and complexity, finding frameworks which aid in interpretation and analysis has become critical. This is particularly true when data comes from complex systems where extensive structure is available, but must be drawn…
In this article, the theory of sheaves is studied from a categorical point of view. This perspective vastly generalizes the usual theory of sheaves of sets to a more abstract setting which allows us to investigate the theory of sheaves with…
There is an interplay between models, specified by variables and equations, and their connections to one another. This dichotomy should be reflected in the abstract as well. Without referring to the models directly -- only that a model…
Many complicated network problems can be easily understood on small networks. Difficulties arise when small networks are combined into larger ones. Fortunately, the mathematical theory of sheaves was constructed to address just this kind of…
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…
We aim to use the concept of sheaf to establish a link between certain aspects of the set of positive integers numbers, a topic corresponding to the elementary mathematics, and some fundamental ideas of contemporary mathematics. We hope…
We firstly introduce some key concepts in category theory, such as quotient category, completion of limits, $\mathrm{Mor}$ category, and so on; then give the concept of topology algebras and sheaves, and discuss how to restore the structue…
This is an introduction to the notion of local subgroupoid introduced by the author and R. Brown. It can also serve as an introduction to an application of sheaf theory, and so could be useful to beginners in that theory. The main results…
An introduction is given to the logic of sheaves of structures and to set theoretic forcing constructions based on this logic. Using these tools, it is presented an alternative proof of the independence of the Continuum Hypothesis; which…
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…
We define an equivalence relation among coherent sheaves on a projective variety called biliaison. We prove the existence of sheaves that are minimal in a biliaison class in a suitable sense, and show that all sheaves in the same class can…
Informal lecture notes with examples on sheaf theory and the derived category of sheaves; sheaves and Morse theory; perverse sheaves, and some applications to representation theory. Added Oct 2021: cellular perverse sheaves. Proofs are…
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…
A sensor integration framework should be sufficiently general to accurately represent all information sources, and also be able to summarize information in a faithful way that emphasizes important, actionable information. Few approaches…
This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular…
We provide an introductory exposition to the sheaf topos theoretic description of classical field theory motivated by the rigorous description of both $\bf{(i)}$ the variational calculus of (infinite dimensional) field-theoretic spaces, and…