Related papers: Descent and forms of tensor categories
We develop a homotopical framework for small categories that extends classical invarints of algebraic topology to the categorical setting. Our approach is based on the construction of genuine path category, obtained trough a localization…
We introduce new techniques for working with presentations for a large class of (strict) tensor categories. We then apply the general theory to obtain presentations for partition, Brauer and Temperley-Lieb categories, as well as several…
Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.
We introduce some deformations of the biset category and prove a semisimplicity property. We also consider another group category, called the subgroup category, whose morphisms are subgroups of direct products, the composition being star…
We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…
Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum of 2-dimensional torsion forms. This is obtained from a more…
In this expository paper we explain in detail how to construct bicategorical colimits of several kinds of tensor categories, for example essentially small finitely cocomplete K-linear tensor categories. The constructions are direct and…
This is an introduction to Grothendieck's descent theory, with some stress on the general machinery of fibered categories and stacks.
Category theoretic aspects of non-rational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain…
Restriction categories were established to handle maps that are partially defined with respect to composition. Tensor topology realises that monoidal categories have an intrinsic notion of space, and deals with objects and maps that are…
An extension of order theory is presented that serves as a formalism for the study of dendroidal sets analogously to way the formalism of order theory is used in the study of simplicial sets.
The tensor ideal localising subcategories of the stable module category of all, including infinite dimensional, representations of a finite group scheme over a field of positive characteristic are classified. Various applications concerning…
The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the…
We present an overview of the notions of exact sequences of Hopf algebras and tensor categories and their connections. We also present some examples illustrating their main features; these include simple fusion categories and a natural…
We examine the use of classes to formulate several categorical notions. This leads to two proposals: an explicit structure for working with subobjects, and a hierarchy of $k$-classes. We apply the latter to both ordinary and higher…
We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular,…
In this paper a method of constructing a semiorthogonal decomposition of the derived category of $G$-equivariant sheaves on a variety $X$ is described, provided that the derived category of sheaves on $X$ admits a semiorthogonal…
We will introduce formal frames of manifolds, which are a generalization of ordinary frames. Their fundamental properties are discussed. In particular, canonical forms are introduced, and torsions are defined in terms of them as a…
We study the derived categories of small categories over commutative noetherian rings. Our main result is a parametrization of the localizing subcategories in terms of the spectrum of the ring and the localizing subcategories over residue…
This is the second part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part II), we develop logarithmic formal…