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.

scholarly journals Extremal balleans

2019 ◽  
Vol 20 (1) ◽  
pp. 297
Author(s):  
Igor Protasov
Keyword(s):  
Coarse Structure ◽  
Unbounded Subset ◽  
Coarse Space ◽  
Extremal Properties

<p>A ballean (or coarse space) is a set endowed with a coarse structure. A ballean X is called normal if any two asymptotically disjoint subsets of X are asymptotically separated.  We say that a ballean X is ultra-normal (extremely normal) if any two unbounded subsets of X are not asymptotically disjoint (every unbounded subset of X is large).   Every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold.   A normal ballean is ultranormal if and only if the Higson′s corona of X is a singleton.   A discrete ballean X is ultranormal if and only if X is maximal.  We construct a series of concrete balleans with extremal properties.</p>

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.

Free coarse groups

2019 ◽  
Vol 22 (4) ◽  
pp. 775-782
Author(s):  
Igor Protasov ◽  
Ksenia Protasova
Keyword(s):  
Topological Groups ◽  
Coarse Structure ◽  
Variety Of Groups ◽  
And Inversion ◽  
Coarse Space

AbstractA coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. Let {(X,\mathcal{E})} be a coarse space, and let {\mathfrak{M}} be a variety of groups different from the variety of singletons. We prove that there is a coarse group {F_{\mathfrak{M}}(X,\mathcal{E})\in\mathfrak{M}} such that {(X,\mathcal{E})} is a subspace of {F_{\mathfrak{M}}(X,\mathcal{E})}, X generates {F_{\mathfrak{M}}(X,\mathcal{E})} and every coarse mapping {(X,\mathcal{E})\to(G,\mathcal{E}^{\prime})}, where {G\in\mathfrak{M}}, {(G,\mathcal{E}^{\prime})} is a coarse group, can be extended to coarse homomorphism {F_{\mathfrak{M}}(X,\mathcal{E})\to(G,\mathcal{E}^{\prime})}. If {\mathfrak{M}} is the variety of all groups, the groups {F_{\mathfrak{M}}(X,\mathcal{E})} are asymptotic counterparts of Markov free topological groups over Tikhonov spaces.

scholarly journals Coarse structures on groups defined by conjugations

2021 ◽  
Vol 32 (1) ◽  
pp. 65-75
Author(s):  
I. Protasov ◽  
◽  
K. Protasova ◽  
Keyword(s):  
Asymptotic Properties ◽  
Infinite Group ◽  
Locally Finite ◽  
Coarse Structure ◽  
Dedekind Group ◽  
Algebraic Properties ◽  
Coarse Structures ◽  
Coarse Space

For a group G, we denote by G↔ the coarse space on G endowed with the coarse structure with the base {{(x,y)∈G×G:y∈xF}:F∈[G]<ω}, xF={z−1xz:z∈F}. Our goal is to explore interplays between algebraic properties of G and asymptotic properties of G↔. In particular, we show that asdim G↔=0 if and only if G/ZG is locally finite, ZG is the center of G. For an infinite group G, the coarse space of subgroups of G is discrete if and only if G is a Dedekind group.

scholarly journals Selectors and orderings of coarse spaces

2021 ◽  
Vol 18 (1) ◽  
pp. 71-79
Author(s):  
Igor Protasov
Keyword(s):  
Linear Orders ◽  
Coarse Structure ◽  
Coarse Space

Given a coarse space $(X, \mathcal{E})$, we consider linear orders on $X$ compatible with the coarse structure $\mathcal E$ and explore interplays between these orders and macro-uniform selectors of $(X, \mathcal{E})$.

Lattices of theories

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Duality Theorem ◽  
Explicit Description ◽  
Local Operator ◽  
Algebra Structure ◽  
Topos Theory ◽  
Grothendieck Topology ◽  
Geometric Morphism ◽  
Geometric Logic ◽  
Grothendieck Topologies

In this chapter, by using the duality theorem established in Chapter 3, many ideas and concepts of elementary topos theory are transferred into the context of geometric logic; these notions notably include the coHeyting algebra structure on the lattice of subtoposes of a given topos, open, closed, quasi-closed subtoposes, the dense-closed factorization of a geometric inclusion, coherent subtoposes, subtoposes with enough points, the surjection-inclusion factorization of a geometric morphism, skeletal inclusions, atoms in the lattice of subtoposes of a given topos, the Booleanization and DeMorganization of a topos. An explicit description of the Heyting operation between Grothendieck topologies on a given category and of the Grothendieck topology generated by a given collection of sieves is also obtained, as well as a number of results about the problem of ‘relativizing’ a local operator with respect to a given subtopos.

scholarly journals Closeness and linkness in balleans

2020 ◽  
Vol 53 (1) ◽  
pp. 100-108
Author(s):  
I.V. Protasov ◽  
K. Protasova
Keyword(s):  
General Question ◽  
Coarse Structure ◽  
Coarse Space

A set $X$ endowed with a coarse structure is called ballean or coarse space. For a ballean $(X, \mathcal{E})$, we say that two subsets $A$, $B$ of $X$ are close (linked) if there exists an entourage $E\in \mathcal{E}$ such that $A\subseteq E [B]$, $B\subseteq E[A]$ (either $A, B$ are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness.

scholarly journals Fibrations and Grothendieck topologies

1976 ◽  
Vol 14 (1) ◽  
pp. 111-128
Author(s):  
Howard Lyn Hiller
Keyword(s):  
Base Change ◽  
Grothendieck Topology ◽  
Definition Of ◽  
A Site ◽  
Grothendieck Topologies

Given a site T, that is, a category equipped with a fixed Grothendieck topology, we provide a definition of fibration for morphisms of the presheaves on T. We verify that the notion is well-behaved with respect to composition, base change, and exponentiation, and is trivial on the topos of sheaves. We compare our definition to that of Kan fibration in the semi-simplicial setting. Also we show how we can obtain a notion of fibration on our ground site T and investigate the resulting notion in certain ring-theoretic situations.

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