group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Twisted Bredon cohomology, or equivariant ordinary cohomology with local coefficients is the twisted generalization of Bredon cohomology, and the equivariant-generalization of twisted ordinary cohomology.
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Amiya Mukherjee, Goutam Mukherjee, Bredon-Illman cohomology with local coefficients, The Quarterly Journal of Mathematics, Volume 47, Issue 2, June 1996, Pages 199–219 (doi:10.1093/qmath/47.2.199)
Marek Golasiński, Equivariant cohomology with local coefficients, Mathematica Slovaca (1997) Volume: 47, Issue: 5, page 575-586 (dml:34467)
Goutam Mukherjee, Neeta Pandey, Equivariant cohomology with local coefficients, Proc. Amer. Math. Soc. 130 (2002), 227-232 (doi:10.1090/S0002-9939-01-06377-8)
Goutam Mukherjee, Debasis Sen, Equivariant simplicial cohomology with local coefficients and its classification, Topology and its Applications, Volume 157, Issue 6, 15 April 2010, Pages 1015-1032 (doi:10.1016/j.topol.2010.01.004)
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