Related papers: Compositories and Gleaves

The purpose of this note is to record a connection between sheaves on complete Boolean algebras and conditional sets. This connection yields a transfer principle for conditional set theory. On the other hand we use conditional set theory to…

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…

A graph theoretic approach is proposed for object shape representation in a hierarchical compositional architecture called Compositional Hierarchy of Parts (CHOP). In the proposed approach, vocabulary learning is performed using a hybrid…

We introduce an axiomatization of Grothendieck sites with additional structure, and we describe sheaves that reconstruct groupoids which are internal to the site structure. This setting applies to various concrete situations, where a Nash…

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…

We give characterizations, for various fragments of geometric logic, of the class of theories classified by a locally connected (resp. connected and locally connected, atomic, compact, presheaf) topos, and exploit the existence of multiple…

Many structured systems admit locally consistent descriptions that nevertheless fail to globalize when constrained by an ambient reference or feasibility condition. Diagnosing such failures is naturally an evaluative problem: given a fixed…

Complex systems of systems (SoS) are characterized by multiple interconnected subsystems. Typically, each subsystem is designed and analyzed using methodologies and formalisms that are specific to the particular subsystem model of…

The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like…

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…

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…

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…

Computational modelling of materials using machine learning, ML, and historical data has become integral to materials research. The efficiency of computational modelling is strongly affected by the choice of the numerical representation for…

For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection we classify thick…

We introduce the notion of (co-)filtered sheaves on quotients of moment graphs by a group action. We then introduce a (co-)filtered version of the canonical sheaves of Braden and MacPherson and show that their global sections are the…

This is the first in a series of papers math.AG/0503029, math.AG/0410267, math.AG/0410268 on "configurations" in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration (\sigma,\iota,\pi) is a finite…

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…

A formulation for a non-trivial composition of two classical gauge structures is given: Two parent gauge structures of a common base space are synthesized so as to obtain a daughter structure which is fundamental by itself. The model is…

We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the…

This paper considers the problem of consistently defining subsystems in gravitational theories. It is argued that a subsystem is a spacetime subregion in which the observables form a closed Poisson algebra. In a generally covariant theory,…