# Cellular Sheaves of Lattices and the Tarski Laplacian

Published in arXiv, 2020

Recommended citation: Ghrist, R. & H. Riess. (2020). Cellular Sheaves of Lattices and the Tarski Laplacian. arXiv preprint. Submitted. http://hans-riess.github.io/files/tarski-laplacian.pdf

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.