Sheaves on Graphs and their Differential Calculi

TL;DR

This paper investigates the relationship between sheaves on graphs and noncommutative geometry, demonstrating how sheaf-theoretic methods can generalize and improve concepts like Laplacians and connections in discrete noncommutative geometry.

Contribution

It introduces a sheaf-theoretic framework for noncommutative geometric concepts on graphs, extending existing theories with new methods and insights.

Findings

Generalization of Laplacians using sheaves

Enhanced understanding of connections in discrete geometry

Application of (semi)simplicial sets to noncommutative concepts

Abstract

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 include notions such as Laplacians and connections, important in the theory of discrete noncommutative geometry, that are here explored with sheaf theoretic methods and using the language of (semi)simplicial sets.