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This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give sufficient conditions for a collection of spherical functors to yield a weak representation of the category of tangles, and prove a structure theorem for such representations under certain restrictions.
Given 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 intr
For 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
We characterize the category of co-semi-analytic functors and describe an action of semi-analytic functors on co-semi-analytic functors.
Each Gr-functor of the type $(varphi,f)$ of a Gr-category of the type $(Pi,C)$ has the obstruction be an element $overline{k}in H^3(Pi,C).$ When this obstruction vanishes, there exists a bijection between congruence classes of Gr-functors of the type
We describe an abstract 2-categorical setting to study various notions of polynomial and analytic functors and monads.