Monadic vs adjoint decomposition

It is known that the so-called monadic decomposition, applied to the adjunction connecting the category of bialgebras to the category of vector spaces via the tensor and the primitive functors, returns the usual adjunction between bialgebras and (restricted) Lie algebras. Moreover, in this framework, the notions of augmented monad and combinatorial rank play a central role. In order to set these results into a wider context, we are led to substitute the monadic decomposition by what we call the adjoint decomposition. This construction has the advantage of reducing the computational complexity when compared to the first one. We connect the two decompositions by means of an embedding and we investigate its properties by using a relative version of Grothendieck fibration. As an application, in this wider setting, by using the notion of augmented monad, we introduce a notion of combinatorial rank that, among other things, is expected to give some hints on the length of the monadic decomposition.