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 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.
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 -fold monoidal categories, arxiv/0907.3335