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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.
In this paper, by Jacob White, the focus is on the study of Cohen-Macaulay Hopf monoids in the category of species. Some insights from the abstract: “The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras.
I found this paper on the arXiv a couple of months ago and from time to time I go back to it to try and understand this new diagrammatic language. I think the abstract is quite informative (more so than anything I can say at the moment) so I will show that below:
In this paper by Alexander McCleary and Amit Patel, they build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite lattice, and the output is a persistence diagram defined as the Mobius inversion of a certain monotone integral function.