Monadic vs adjoint decomposition

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