In operad theory, a minimal model of a dg-operad $\mathcal{P}$ (or dga A) is given by a quasi-free resolution $(\mathcal{T}(E),\partial) \overset{f}{\longrightarrow} P$ where $f$ is a quasi-isomorphism and the differential $\partial$ is decomposable, that is, $\partial(e) \in \mathcal{T}(E)^{(\geq 2)}$ for generators $e$, which means the trees appearing in $\partial(e)$ have at least two vertexes.
I am wondering what the significance is of the decomposability of the differential? What are its implications? Which advantages does it provide that a cofibrant resolution does not?
The main implication that I expect, is that it is used to prove unicity of the minimal model (see operads in Algebra, Topology and Physics). However, this proof also requires the operads involved to have no $0$-ary operation and only the identity as $1$-ary operation. This uniqueness also does not extend to props (see a resolution (minimal model) of the prop for bialgebras by Markl), yet the decomposability is also imposed for its minimal model.
I would be very interested in seeing where this condition plays a role. Especially with respect to operadic cohomology of $\mathcal{P}$-algebras and modules.