Local-to-global spectral sequences for the cohomology of diagrams

We construct local-to-global spectral sequences for the cohomology of a diagram, which compute the cohomology of the full diagram in terms of smaller pieces. These are motivated by the obstruction theory of D. Blanc et al. [D. Blanc, M.W. Johnson, J.M. Turner, On realizing diagrams of ΠΠ-algebras, Algebraic Geom. Topol. 6 (2006) 763–807] for realizing a diagram of ΠΠ-algebras, but are valid in quite general algebraic settings.