  Given a multisymplectic manifold $(M,\omega)$ and a Lie algebra $\frak{g}$
acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a
homotopy (co-)moment as an $L_{\infty}$-algebra-homomorphism from $\frak{g}$ to
the observable algebra $L(M,\omega)$ associated to $(M,\omega)$, in analogy
with and generalizing the notion of a co-moment map in symplectic geometry. We
give a cohomological characterization of existence and unicity for homotopy
co-moment maps and show its utility in multisymplectic geometry by applying it
to special cases as exact multisymplectic manifolds and simple Lie groups and
by deriving from it existence results concerning partial co-moment maps, as
e.g. covariant multimomentum maps and multi-moment maps.
