Thin posets, CW posets, and Categorification

Motivated by generalizing Khovanov's categorification of the Jones polynomial, we study functors from thin posets to abelian categories. Such functors produce cohomology theories, and we find that CW posets--face posets of regular CW complexes--satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category whose objects are tuples (P,A,F,c) where P is a CW poset, A is an abelian category, F is a functor from P to A, and c is a certain coloring of the Hasse diagram of P making intervals of length 2 anticommute. We show the cohomology arising from a tuple (P,A,F,c) is functorial, and independent of the coloring c up to natural isomorphism. Such a construction provides a framework for the categorification of a variety of familiar topological/combinatorial invariants, anything expressible as a rank alternating sum over a thin poset.