Cellular Sheaves of Lattices and the Tarski Laplacian

TL;DR

This paper develops a discrete Hodge theory for cellular sheaves valued in lattices, introducing the Tarski Laplacian, which connects fixed points of an endomorphism to cohomology and has applications in network consensus and optimization.

Contribution

It introduces the Tarski Laplacian for cellular sheaves in lattice categories, linking fixed points to cohomology and enabling new applications in networks.

Findings

Tarski Laplacian fixed points correspond to cohomology in degree zero.

Application to consensus and distributed optimization problems.

Framework extends discrete Hodge theory to lattice-valued sheaves.

Abstract

This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points yield a cohomology that agrees with the global section functor in degree zero. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications.