The existence of a (co)moment is an important prerequisite for many constructions in symplectic geometry. The recent discovery of a multisymplectic comoment, that is a morphism of $L_\infty$-algebras, opened the pathway to reduction and quantization procedures in the multisymplectic realm.
In this talk we will give a geometric characterization of the existence of multisymplectic comoments. We will look at the case of actions on n-dimensional spheres and discuss what can still be done, when no homotopy comoment exists.