Is there a homology theory of posets which computes topological invariants (e.g., number of $k$-faces, etc.) of the associated Hasse diagrams (viewed as simplicial/cellular/singular complexes) as perhaps ranks of homology groups or values of Hilbert polynomials of nice rings (e.g., Stanley-Risner, Cohen-Macaulay, etc.)?
For example, the Hasse diagram of the Boolean lattice $B_{n}$ has $2^{n-k} \binom{n}{k}$ $k$-faces and therefore euler characteristic $\sum_k (-1)^k f_k = 1$. Is there a homology theory which relates the Boolean lattice to a contractible space? (Note the solid $n$-cube is homotopy equivalent to the $n$-ball.)