  We prove that a connected commutator (or NC) complete associative algebra can
be recovered in the derived setting from its abelianization together with its
natural induced structure. Specifically, we prove an equivalence between
connected derived commutator complete associative algebras and connected
commutative algebras equipped with a coaction of the comonad arising from the
adjunction between associative and commutative algebras. This provides a Koszul
dual description of connected derived commutator (or NC) complete associative
algebras and furthermore may be interpreted as a theory of relative Koszul
duality for the associative operad relative to the commutative operad. We also
prove analogous results in the setting of En-algebras. That is, we develop a
theory of commutator complete En-algebras and a theory of relative Koszul
duality for the En operad relative to the commutative operad. We relate the
derived commutator filtration on associative and En-algebras to the filtration
on the associative and En operad whose associated graded is the Poisson and
shifted Poisson operad Pn. We argue that the derived commutator filtration on
associative algebras (and En-algebras) are a relative analogue of the
Goodwillie tower of the identity functor on the model category of associative
algebras (and En-algebras) relative to the model category of commutative
algebras.

Algebraic Topology;Category Theory;Quantum Algebra