Related papers: Cohomology of the Grothendieck construction

We construct spectral sequences in the framework of Baues-Wirsching cohomology and homology for functors between small categories and analyze particular cases including Grothendieck fibrations. We also give applications to more classical…

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…

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on…

We introduce cohomology and homology theories for small categories with general coefficient systems from simplex categories first studied by Thomason. These theories generalize at once Baues-Wirsching cohomology and homology and other more…

Let $C_\bullet$ be a simplicial object in the category $Cat$ of small categories. For a field $k$, taking the Grothendieck groups of isomorphism classes of $kC_n$-modules gives rise to a cochain complex, whose cohomology, which we refer to…

Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this paper we define and study the section functor on $\mathcal{A}$ with respect to an open subset of ASpec$\mathcal{A}$. Next we define and study local cohomology theory…

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphism and…

We define the partial group cohomology as the right derived functor of the functor of partial invariants, we relate this cohomology with partial derivations and with the partial augmentation ideal and we show that there exists a…

The Grothendieck construction is a process to form a single category from a diagram of small categories. In this paper, we extend the definition of the Grothendieck construction to diagrams of small categories enriched over a symmetric…

The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As…

Cohomology of a topological space with coefficients in stacks of abelian 2-groups is considered. A 2-categorical analog of the theorem of Grothendieck is proved, relating cohomology of the space with coefficients in a 2-stage spectrum and…

Let $D$ be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from $D$ to simplicial sets. As an application we construct homotopy localization functors on the…

We introduce a new method to construct a Grothendieck category from a given colored quiver. This is a variant of the construction used to prove that every partially ordered set arises as the atom spectrum of a Grothendieck category. Using…

We consider the category of comodules over a smash coproduct coalgebra $C\smashco H$. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from $C\smashco H$-colinear,…

For a compact Lie group G we show that if the representing spectrum for Borel cohomology generates its category of modules if G is connected. For a closed subgroup H of G we consider the map C^*(BG)--->C^*(BH) and establish the sense in…

We define a cohomology for an arbitrary $K$-linear semistrict semigroupal 2-category $(\mathfrak{C},\otimes)$ (called in the paper a Gray semigroup) and show that its first order (unitary) deformations, up to the suitable notion of…

The Baues--Wirsching cohomology is one of the cohomologies of a small category. Our aim is to describe the first Baues--Wirsching cohomology of the small category generated by a finite quiver freely. We consider the case where the…

The Grothendieck construction is a fundamental link between indexed categories and opfibrations. This work is a detailed study of the Grothendieck construction over a small tight bipermutative category in the context of Cat-enriched…

A notion of support for objects in any Grothendieck category is introduced. This is based on the spectral category of a Grothendieck category and uses its Boolean lattice of localising subcategories. The support provides a classification of…

Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories.…