Rectifying stable infinity-categories and relative koszul duality for operads



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This thesis is divided into two main portions. The first portion of this thesis describes a comparison between pretriangulated differential graded categories and certain stable infinity-categories. Specifically, we use a model category structure on small differential graded categories over k (a commutative ring with unit) where the weak equivalences are the Morita equivalences, and where the fibrant objects are in particular pretriangulated differential graded categories. We show the underlying infinity-category of this model category is equivalent to the infinity-category of small idempotent-complete k-linear stable infinity-categories. The second portion of this thesis proves 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) is 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.