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.