Related papers: Sheaves over complexes of groups and developabilit…
We characterise those objects in the derived category of a scheme which are a sheaf supported on a closed subscheme in terms of Koszul complexes. This is applied to generalize to arbitrary schemes the fully faithfullness criteria of an…
We propose a model for network community detection using topological data analysis, a branch of modern data science that leverages theory from algebraic topology to statistical analysis and machine learning. Specifically, we use cellular…
We observe that word reading is a crystal morphism. This leads us to prove that, in the case of the complex special linear group, the map from all galleries to MV cycles is a morphism of crystals.
Defining cellular sheaves beyond graph structures, such as on simplicial complexes containing higher-dimensional simplices, is an essential and intriguing topic in topological data analysis (TDA) and the development of sheaf neural…
We introduce a notion of compatibility for families $(\mathcal{F}_{\ell})_{\ell}$ of bounded constructible $\ell$-adic complexes of \'etale sheaves on schemes. For schemes of finite type over a field, this notion is preserved by the usual…
We introduce a sheaf-theoretic characterization of task solvability in general distributed computing models, unifying distinct approaches to message-passing models. We establish cellular sheaves as a natural mathematical framework for…
We describe the relationship between the notions of $M$-regular sheaf and $GV$-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic criterion characterizing it among the…
With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the…
Despite encouraging recent progresses in ensemble approaches, classification methods seem to have reached a plateau in development. Further advances depend on a better understanding of geometrical and topological characteristics of point…
We introduce a notion of complexity of a complex of ell-adic sheaves on a quasi-projective variety and prove that the six operations are "continuous", in the sense that the complexity of the output sheaves is bounded solely in terms of the…
If $X$ is an abelian variety over a field and $L$ is an invertible sheaf, we know that the degree of the 0-cycle $L^g$ is divisible by $g!$. As a 0-cycle, it is not, even over a field of cohomological dimension 1. But we show that over a…
Due to the complexity of the human body, most diseases present a high inter-personal variability in the way they manifest, i.e. in their phenotype, which has important clinical repercussions - as for instance the difficulty in defining…
This is the first of a series of papers on sheaf theory on smooth and topological stacks and its applications. The main result of the present paper is the characterization of the twisted (by a closed integral three-form) de Rham complex on…
Group cohomology of polynomial growth is defined for any finitely generated discrete group, using cochains that have polynomial growth with respect to the word length function. We give a geometric condition that guarantees that it agrees…
Let $\A$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and…
To analyze the evolutionary emergence of structural complexity in physical processes we introduce a general, but tractable, model of objects that interact to produce new objects. Since the objects--\emph{$epsilon$-machines}--have well…
A concept of "evolving categories" is suggested to build a simple, scalable, mathematically consistent framework for representing in uniform way both data and algorithms. A state machine for executing algorithms becomes clear, rich and…
Graph vertices are often organized into groups that seem to live fairly independently of the rest of the graph, with which they share but a few edges, whereas the relationships between group members are stronger, as shown by the large…
We classify the module categories over the double (possibly twisted) of a finite group.
We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results…