Homology of coloured posets: a generalisation of Khovanov's cube construction

TL;DR

This paper introduces a homology theory for coloured posets with representations, generalizing Khovanov's cube complex homology, and shows it coincides with Khovanov's in the case of Boolean lattices.

Contribution

It generalizes Khovanov's cube homology to a broader class of coloured posets with representations, establishing an isomorphism in the Boolean lattice case.

Findings

Homology theory for coloured posets with representations is defined.

The theory generalizes Khovanov's cube complex homology.

In Boolean lattices, the new homology matches Khovanov's homology.

Abstract

We define a homology theory for a certain class of posets equipped with a representation. We show that when restricted to Boolean lattices this homology is isomorphic to the homology of the "cube" complex defined by Khovanov.