Rational Betti numbers of toric diagram homotopy colimits are described via sheaf cohomology over CW posets, with Cohen-Macaulayness of the T-action equivalent to acyclicity of a sheaf.
Dugger,A primer on homotopy colimits, preprint, available at https://people.math.rochester.edu/ faculty/doug/otherpapers/hocolim.pdf
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Rational cohomology of toric diagrams
Rational Betti numbers of toric diagram homotopy colimits are described via sheaf cohomology over CW posets, with Cohen-Macaulayness of the T-action equivalent to acyclicity of a sheaf.