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$\mathbb{P}^n$-functorsMay 14 2019We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, ... More On uniqueness of P-twistsNov 17 2017Jun 23 2018We prove that for any $\mathbb{P}^n$-functor $F$, split or non-split, all the convolutions (double cones) of the three-term complex $FHR \rightarrow FR \rightarrow$ Id defining its $\mathbb{P}$-twist are isomorphic. Bar category of modules and homotopy adjunction for tensor functorsDec 30 2016Jul 31 2018Given a DG-category A we introduce the bar category of modules Modbar(A). It is a DG-enhancement of the derived category D(A) of A which is isomorphic to the category of DG A-modules with A-infinity morphisms between them. However, it is defined intrinsically ... More Spherical DG-functorsSep 19 2013Oct 19 2015For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A -> B. We construct its associated autoequivalences: the twist T of D(B) and the co-twist F of D(A). We give powerful sufficiency criteria for a quasi-functor to be ... More