Timothy Logvinenko
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Families of $G$-constellations over Resolutions of Quotient Singularities Open
Let G be a finite subgroup of GLn(C). A study is made of the ways in which resolutions of the quotient space Cn/G can parametrise G-constellations, that is, G-regular finite length sheaves. These generalise G-clusters, which are used in th…
Unbounded twisted complexes Open
We define unbounded twisted complexes and bicomplexes generalising the notion of a (bounded) twisted complex over a DG category [BK90]. These need to be considered relative to another DG category B admitting countable direct sums and shift…
$A_{\infty}$-structures in monoidal DG categories and strong homotopy unitality Open
We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of stron…
Unbounded twisted complexes Open
We define unbounded twisted complexes and bicomplexes generalising the notion of a (bounded) twisted complex over a DG category [BK90]. These need to be considered relative to another DG category $B$ admitting countable direct sums and shi…
The Heisenberg category of a category Open
Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical a…
$\mathbb{P}^n$-functors Open
We 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 adjoi…
On uniqueness of P-twists Open
We prove that for any $\mathbb{P}^n$-functor all the convolutions (double cones) of the three-term complex $FHR \xrightarrowψ FR \xrightarrow{tr} Id$ defining its $\mathbb{P}$-twist are isomorphic. We also introduce a new notion of a non-s…
Spherical DG-functors Open
For two DG-categories A and B we define the notion of a spherical Morita quasi-functor \mathcal A \to \mathcal B . We construct its associated autoequivalences: the twist T \in \mathrm {Aut} \mathcal D(\mathcal B) and the cotwist F \in \ma…
Bar category of modules and homotopy adjunction for tensor functors Open
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 d…