  We show that if an operad is at the same time a cosimplicial object such that
the respective structure maps are compatible with the operadic composition in a
natural way, then one obtains a Gerstenhaber algebra structure on cohomology,
and if the operad is cyclic, even that of a BV algebra. In particular, if a
cyclic opposite module over an operad with multiplication is itself a cyclic
operad that meets the cosimplicial compatibility conditions, the cohomology of
its cyclic dual turns into a BV algebra. This amounts to conditions for when
the cyclic dual of a BV module is endowed with a BV algebra structure, a result
we exemplify by looking at classical and less classical (co)homology groups in
Hopf algebra theory.
