  We categorify the inclusion-exclusion principle for partially ordered
topological spaces and schemes to a filtration on the derived category of
sheaves. As a consequence, we obtain functorial spectral sequences that
generalize the two spectral sequences of a stratified space and certain
Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs
in the Grothendieck ring of varieties. As an application, we give an
algebro-geometric proof of Vakil and Wood's homological stability conjecture
for the space of smooth hypersurface sections of a smooth projective variety.
In characteristic zero this conjecture was previously established by Aumonier
via topological methods.
