Related papers: Stacks associated to abelian tensor categories
We study aisles in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence,…
In this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infinity-categories of quasi-coherent complexes. It can be viewed as a derived analogue of Tannaka…
Bisets can be considered as categories. This note uses this point of view to give a simple proof of a Mackey-like formula expressing the tensor product of two induced bimodules.
A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore--Seiberg relations. A functor to N is constructed…
We explore the notion of representation of an affine extension of an abelian variety -- such an extension is a faithfully flat affine morphism of $\Bbbk$-group schemes $q:G\to A$, where $A$ is an abelian variety. We characterize the…
A graded tensor category over a group $G$ will be called a strongly $G$-graded tensor category if every homogeneous component has at least one multiplicativily invertible object. Our main result is a description of the module categories…
Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed…
Let X be a stratified space on which the Juteau-Mautner-Williamson theory of parity sheaves is available. We develop a "nearby cycles formalism" in the framework of the homotopy category of parity sheaves on X, also known as the mixed…
In this article, we investigate the category $\mathcal{A}^G$ of equivariant objects of an additive category $\mathcal{A}$ with respect to an action of a finite group $G$. We show that if $G$ is solvable then we can reconstruct $\mathcal{A}$…
An abelian stack is a stacky generalization of an abelian variety that was introduced by Brochard. Just as an abelian variety has a dual, an abelian stack $\mathcal{A}$ has a dual $\mathfrak{D}(\mathcal{A})$ which generalizes the classical…
In this article, we develop a general technique for gluing subcategories of $\infty$-categories. We obtain categorical equivalences between simplicial sets associated to certain multisimplicial sets. Such equivalences can be used to…
We study the non-abelian tensor square $G\otimes G$ for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G/G' so that $G\otimes G$ is isomorphic to the direct product of…
A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a…
We show that Lurie's results on Tannaka duality for geometric stacks hold without any tameness hypotheses. We deduce this as a consequence of an affineness theorem in the theory of sheaves of categories. This affineness result is also…
Given a positively graded commutative coherent ring A which is finitely generated as an A_0-algebra, a bijection between the tensor Serre subcategories of qgr A and the set of all subsets Y\subseteq Proj A of the form…
For an abelian category, a category equivalent to its derived category is constructed by means of specific projective (injective) multicomplexes, the so-called homological resolutions.
Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories to construct semi-orthogonal decompositions (or Bousfield localizations, or hereditary torsion pairs) in various derived categories of…
Alexander L. Rosenberg has constructed a spectrum for abelian categories which is able to reconstruct a quasi-separated scheme from its abelian category of quasi-coherent sheaves. In this note we present a detailed proof of this result…
We extend \cite{G} to the nonsemisimple case. We define and study exact factorizations $\B=\A\bullet \C$ of a finite tensor category $\B$ into a product of two tensor subcategories $\A,\C\subset \B$, and relate exact factorizations of…
We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary…