  We present a method to compute integral cohomology of posets. This toolbox is
applicable as soon as the sub-posets under each object possess certain
structure. This is the case for simplicial complexes and simplex-like posets.
The method is based on homological algebra arguments in the category of
functors and on a spectral sequence built upon the poset. We show its relation
to discrete Morse theory. As application we give an alternative proof of Webb's
conjecture for saturated fusion systems and we compute the cohomology of
Coxeter complexes for finite and infinite Coxeter groups.
