# thin poset homology

## Cellular Sheaves of Lattices

In this paper, by Robert Ghrist and Hans Riess, the authors initiate a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. Their abstract goes on to explain “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.

## A Broken Circuit Model for Chromatic Homology Theories

Using the tools of algebraic Morse theory and the thin poset approach to constructing homology theories, we give a categorification of Whitney's broken circuit theorem for the chromatic polynomial and Stanley's chromatic symmetric function. As an application, we obtain bounds on the homological span for both homology theories.