  Motivated by generalizing Khovanov's categorification of the Jones
polynomial, we study functors $F$ from thin posets $P$ to abelian categories
$\mathcal{A}$. Such functors $F$ produce cohomology theories
$H^*(P,\mathcal{A},F)$. We find that CW posets, that is, face posets of regular
CW complexes, satisfy conditions making them particularly suitable for the
construction of such cohomology theories. We consider a category of tuples
$(P,\mathcal{A},F,c)$, where $c$ is a certain $\{1,-1\}$-coloring of the cover
relations in $P$, and show the cohomology arising from a tuple
$(P,\mathcal{A},F,c)$ 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.
