Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

Cellular sheaves of lattices and the Tarski Laplacian

Pages: 325 – 345

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a16

Authors

Robert Ghrist (Department of Mathematics, Department of Electrical & Systems Engineering, David Rittenhouse Lab, University of Pennsylvania, Philadelphia, Pa., U.S.A.)

Hans Riess (Department of Electrical & Systems Engineering, University of Pennsylvania, Philadelphia, Pa., U.S.A.)

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.

Keywords

cellular sheaves, lattice theory, non-abelian homological algebra

2010 Mathematics Subject Classification

05C50, 18B35, 18F20, 55N30

Received 13 July 2020

Received revised 21 April 2021

Accepted 26 April 2021

Published 13 April 2022