Cosheaves (original) (raw)
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A pr 2 00 5 Cohomology of the Grothendieck construction
2008
We consider cohomology of small categories with coefficients in a natural system in the sense of Baues and Wirsching. For any functor L : K → CAT, we construct a spectral sequence abutting to the cohomology of the Grothendieck construction of L in terms of the cohomology of K and of L(k), for k ∈ ObK. 2000 Mathematics Subject Classification : 18G40
Cohomology of the Grothendieck construction
manuscripta mathematica, 2006
We consider cohomology of small categories with coefficients in a natural system in the sense of Baues and Wirsching. For any functor L : K → CAT, we construct a spectral sequence abutting to the cohomology of the Grothendieck construction of L in terms of the cohomology of K and of L(k), for k ∈ Ob K.
Sheaves on Grothendieck constructions
Cornell University - arXiv, 2022
In this paper we introduce a generalisation of a covariant Grothendieck construction to the setting of sites. We study the basic properties of defined site structures on Grothendieck constructions as well as we treat the cohomological aspects of corresponding toposes of sheaves. Despite the fact that the toposes of G-equivariant sheaves Sh G (X) have been introduced in literature, their cohomological aspects have not been treated properly in a desired fashion. So in the end of the paper we study some of the acyclic families, introduce new type of acyclic resolutions which we call the G-equivariant Godement resolutions, the degree of actions, and some other basic cohomological concepts arising in Sh G (X).
Godement resolutions and sheaf homotopy theory
Collectanea Mathematica, 2014
The Godement cosimplicial resolution is available for a wide range of categories of sheaves. In this paper we investigate under which conditions of the Grothendieck site and the category of coefficients it can be used to obtain fibrant models and hence to do sheaf homotopy theory. For instance, for which Grothendieck sites and coefficients we can define sheaf cohomology and derived functors through it.
2016
It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafification) from the category of precosheaves on X with values in a category K, to the full subcategory of cosheaves, provided either K or K^op is locally presentable. If K is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category Pro K) of pro-objects in K. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in Pro( K) is smooth, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.
Comodule theories in Grothendieck categories and relative Hopf objects
2022
We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category S. We describe when the resulting category of comodules is locally finitely generated, locally noetherian or may be recovered as a coreflective subcategory of the noncommutative base change of a module category. We also introduce the category A S H of relative (A, H)-Hopf modules in S, where H is a Hopf algebra and A is a right H-comodule algebra. We study the cohomological theory in A S H by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in A S H. Finally, we use the theory of associated primes and support in noncommutative base change of module categories to give direct sum decompositions of minimal injective resolutions in A S H .
An homotopical description of small presheaves
2021
This article describes the cocompletion of a category C with finite limits as the homotopy category of some equivalence 2-groupoids in coproducts of elements of C. This yields a simple link between several definitions of an infinitary pretopos.
R A ] 2 9 Ju l 2 01 9 Cohomology of modules over H-categories and coH-categories
2019
Let H be a Hopf algebra. We consider H-equivariant modules over a Hopf module category C as modules over the smash extension C#H . We construct Grothendieck spectral sequences for the cohomologies as well as the H-locally finite cohomologies of these objects. We also introduce relative (D,H)-Hopf modules over a Hopf comodule category D. These generalize relative (A,H)-Hopf modules over an H-comodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants. MSC(2010) Subject Classification: 16S40, 16T05, 18E05
Supports in abstract module categories, local cohomology objects and spectral sequences
2022
We work with a strongly locally noetherian Grothendieck category S and we consider the category SR of R-module objects in S introduced by Popescu, where R is a commutative and noetherian k-algebra. Then, SR may be seen as an abstract category of modules over a noncommutative base change of R. Using what we call R-elementary objects in SR and their injective hulls, we develop a theory of supports and associated primes in the abstract module category SR. We use these methods to study associated primes of local cohomology objects in SR. In fact, we use a more general framework, extending the local cohomology with respect to a pair of ideals I, J ⊆ R introduced by Takahashi, Yoshino and Yoshizawa. We give a finiteness condition for the set of associated primes of local cohomology objects. Thereafter, we apply our theory to study a general functorial setup that requires certain conditions on the injective hulls of R-elementary objects and gives us spectral sequences for derived functors associated to two variable local cohomology objects, as well as generalized local cohomology and also generalized Nagata ideal transforms on SR. We note that this framework will give new results, even in the case we take S to be the category of modules over a noncommutative algebra that is strongly locally noetherian.