group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
Gerstenhaber and Schack introduced a cohomology related to deformation theory of bialgebras. Teillefer has proven that this cohomology computes in fact the Ext-groups in certain abelian category of “tetramodules” over the bialgebra. A somewhat more systematic writeup of the proof is in the appendix Shoikhet 09.
Please distinguish the Gerstenhaber-Schack bialgebra cohomology from Shahn Majid‘s bialgebra cohomology? which is (in full generality) nonabelian and cohomologies associated to other categories of “modules” in bialgebra theory (Hopf modules, Yetter-Drinfeld modules etc.). In the $n$Lab we choose to use the term bialgebra cocycle unadorned for the Majid’s cocycles and always use Gerstenhaber-Schack cocycle/cohomology for the latter, as it is often referred nowdays.
Murray Gerstenhaber, Samuel D. Schack, Bialgebra cohomology, deformations, and quantum groups, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), no. 1, 478–481, MR90j:16062, article
Rachel Taillefer, Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product, J. Algebra 276 (2004), no. 1, 259–279, MR2005f:16067, doi
Boris Shoikhet, Hopf algebras, tetramodules, and $n$-fold monoidal categories, arxiv/0907.3335