Related papers: A categorical approach to cyclic duality

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of…

REVISED VERSION: We have re-organized the paper, and included some new results. Most important, we prove that the (truncated) Weil complexes compute the cyclic cohomology of the Hopf algebra (see the new Theorem 7.3). We also include a…

We show that the category of partial modules over a Hopf algebra $H$ is a biactegory (a bimodule category) over the category of global $H$-modules. The corresponding enrichment of partial modules over global modules is described, and the…

We introduce the notion of a partial corepresentation of a given Hopf algebra $H$ over a coalgebra $C$ and the closely related concept of a partial $H$-comodule. We prove that there exists a universal coalgebra $H^{par}$, associated to the…

Given a scheme in characteristic p together with a lifting modulo p^2, we construct a functor from a category of suitably nilpotent modules with connection to the category of Higgs modules. We use this functor to generalize the…

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study…

Given a crossed module $\chi$, we introduce Hopf $\chi$-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of…

Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…

We construct algebraic cycles in Bloch's cubical cycle group which correspond to multiple polylogarithms with generic arguments. Moreover, we construct out of them a Hopf subalgebra in the Bloch-Kriz cycle Hopf algebra. In the process, we…

The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…

What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phung Ho Hai. The construction…

Bialgebroids, separable bialgebroids, and weak Hopf algebras are compared from a categorical point of view. Then properties of weak Hopf algebras and their applications to finite index and finite depth inclusions of von Neumann algebras are…

The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category…

The aim of section 1 is to define the homotopic functor to category of Abelian groups, connected with the special classes of bundles with fiber matrix algebra or projective space. The aim of section 2 is to define some generalization of the…

If A is a bialgebra over a field k and M, N are either left-right Yetter-Drinfel'd modules or left-right Hopf modules over A, we construct deformation cohomologies H^*(M,N) as total cohomologies of certain double complexes Y(M,N) and…

We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a…

We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly…

We replace a ring with a small $\mathbb C$-linear category $\mathcal{C}$, seen as a ring with several objects in the sense of Mitchell. We introduce Fredholm modules over this category and construct a Chern character taking values in the…

Grothendieck-Verdier duality is a powerful and ubiquitous structure on monoidal categories, which generalises the notion of rigidity. Hopf algebroids are a generalisation of Hopf algebras, to a non-commutative base ring. Just as the…

Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the…