scholarly journals TOPOLOGIES ON SCHEMES AND MODULUS PAIRS

2020 ◽  
pp. 1-31
Author(s):  
BRUNO KAHN ◽  
HIROYASU MIYAZAKI
Keyword(s):  
Grothendieck Topologies

We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs, which were introduced in previous papers. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs.

scholarly journals Coarse cohomology with twisted coefficients

2020 ◽  
Vol 70 (6) ◽  
pp. 1413-1444
Author(s):  
Elisa Hartmann
Keyword(s):  
Sheaf Cohomology ◽  
Grothendieck Topology ◽  
Coarse Structure ◽  
Relative Cohomology ◽  
Number Of Ends ◽  
Constant Coefficients ◽  
Coarse Space ◽  
Grothendieck Topologies

AbstractTo 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 coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

Grothendieck Topologies

Categories ◽  
1972 ◽  
pp. 291-319
Author(s):  
Horst Schubert
Keyword(s):  
Grothendieck Topologies

Quotients of a theory of presheaf type

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Semantic Representation ◽  
Sufficient Conditions ◽  
Grothendieck Topology ◽  
Rigidity Property ◽  
The Given ◽  
Grothendieck Topologies

In this chapter the quotients of a given theory of presheaf type are investigated by means of Grothendieck topologies that can be naturally attached to them, establishing a ‘semantic’ representation for the classifying topos of such a quotient as a subtopos of the classifying topos of the given theory of presheaf type. It is also shown that the models of such a quotient can be characterized among the models of the theory of presheaf type as those which satisfy a key property of homogeneity with respect to a Grothendieck topology associated with the quotient. A number of sufficient conditions for the quotient of a theory of presheaf type to be again of presheaf type are also identified: these include a finality property of the category of models of the quotient with respect to the category of models of the theory and a rigidity property of the Grothendieck topology associated with the quotient.

scholarly journals What is a noncommutative topos?

2019 ◽  
Vol 18 (06) ◽  
pp. 1950107 ◽  
Author(s):  
Karin Cvetko-Vah ◽  
Jens Hemelaer ◽  
Lieven Le Bruyn
Keyword(s):  
Topological Space ◽  
Grothendieck Topologies

In [Cvetko-Vah, Non-commutative frames, J. Algebra Appl. (2018), https://doi.org/10.1142/S0219498819500117 ] noncommutative frames were introduced, generalizing the usual notion of frames of open sets of a topological space. In this paper, we extend this notion to noncommutative versions of Grothendieck topologies and their associated noncommutative toposes of sheaves of sets.

Grothendieck topologies on Chu spaces

2009 ◽  
Vol 19 (3) ◽  
pp. 192-210 ◽  
Author(s):  
E. E. Skurikhin ◽  
A. G. Sukhonos
Keyword(s):  
Chu Spaces ◽  
Grothendieck Topologies
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