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Sheaves on Graphs and their Differential Calculi

Rita Fioresi, Angelica Simonetti, Ferdinando Zanchetta

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.

Sheaves on Graphs and their Differential Calculi

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.
Paper Structure (25 sections, 25 theorems, 80 equations, 6 figures)

This paper contains 25 sections, 25 theorems, 80 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Graphs as semisimplicial sets
  3. The categories of (semi)simplicial sets and digraphs
  4. Undirected Graphs
  5. Grothendieck Topologies and Sheaves
  6. Grothendieck topologies, gross coverings, bisheaves
  7. Locally constant sheaves
  8. Laplacian pairs
  9. Sheaves on graphs
  10. Grothendieck sites on graphs
  11. Sheaves on digraphs
  12. Graphs, finite topological spaces and preordered sets
  13. Sheaves on graphs as preordered sets
  14. Homology and cohomology on directed graphs
  15. Sheaf Laplacians on directed graphs
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $G\in \mathrm{diGraphs}_{\leq 1}$ be a bidirected graph, $\mathcal{F}$ a vector bundle of rank $n$ on $G$, $M$ the free $A_G$-bimodule associated to it. Assume to have isomorphisms $M \cong M^*$ and $\Gamma^1\cong (\Gamma^1)^*$ as in Def. def:bochnerlapl and consider $(,)$ the generalized quantu

Figures (6)

  • Figure 1: Étale non standard cover of $G$ by the disconnected digraph $H$.
  • Figure 2: The undirected graph $G$ and its image via $\Theta$
  • Figure 3: Irreducible open sets for the topology in Ex. \ref{['graph-alex']}.
  • Figure 4: The undirected graph $G$ and its image $H=\Theta(G)$ via $\Theta$
  • Figure 5: The triangular cliques $\omega_{x\to y \to z}$ (right) and $\omega_{x\to y \to x}$ (left).
  • ...and 1 more figures

Theorems & Definitions (96)

  • Theorem 1.1: \ref{['theo:sheaf_lap']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 86 more