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. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications.”

In section 2.5 of this paper, they describe a cohomology theory for functors on face posets of cell complexes (by this I believe they mean regular CW complexes). This theory coincides exactly with my construction in “Thin Posets, CW Posets, and Categorification”. When I wrote my paper, I had thought I was the first to attempt to develop such a theory, which to me was the natural generalization of Khovanov’s cube construction for Khovanov homology. At the time, I was using the words “functors on posets”. If at the time I had realized that these were the same thing as “presheafs on posets”, I would have found that I was not the first to consider such a construction. To me, it seems that Turner and Everitt gave the first and most complete version of the theory I was hoping to build. Anyway, I was surprised not to see Turner and Everitt’s paper Cellular Cohomology of Posets with Local Coefficients referenced in this new paper by Ghrist and Riess, but they seem to go a quite different direction than Turner and Everitt and end up with some really interesting results.

As I finish my read through of the papers mentioned here, I will come back to this post and comment on comparisons between these papers and mine. I expect that many of the open questions from my paper here are answered somewhere among them. Actually I am starting with a read-through of the wonderful PhD thesis of Justin Curry, which provides a really nice hands-on exposition of some of the background material needed to understand this stuff, in particular categorical limits and colimits. I would recommend anybody interested in this kind of thing to start here.