Some categorical aspects of coarse spaces and balleans Nicol` o - PowerPoint PPT Presentation

Large scale geometry of metric spaces Beyond metric spaces The categories Coarse e Coarse / ∼ Some categorical aspects of coarse spaces and balleans Nicol` o Zava joint work with Dikran Dikranjan Toposym 2016 Twelfth Symposium on General Topology and its Relations to Modern Analysis and Algebra July 26, 2016 Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Beyond metric spaces The categories Coarse e Coarse / ∼ Contents 1. Basics on the classical theory of metric spaces: quasi-isometries and coarse equivalences; finitely generated groups; Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Beyond metric spaces The categories Coarse e Coarse / ∼ Contents 1. Basics on the classical theory of metric spaces: quasi-isometries and coarse equivalences; finitely generated groups; 2. Beyond metric spaces: coarse spaces; balleans; relationship between coarse spaces and balleans. Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Beyond metric spaces The categories Coarse e Coarse / ∼ Contents 1. Basics on the classical theory of metric spaces: quasi-isometries and coarse equivalences; finitely generated groups; 2. Beyond metric spaces: coarse spaces; balleans; relationship between coarse spaces and balleans. 3. Coarse category: definition of Coarse and Coarse / ∼ ; epimorphisms and monomorphisms in Coarse ; products, coproducts and quotients in Coarse ; epimorphisms and monomorphisms in Coarse / ∼ . Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Beyond metric spaces The categories Coarse e Coarse / ∼ Metric spaces and finitely generated groups Definition (Coarse equivalence) Let ( X , d ) and ( Y , d ′ ) be two metric spaces. A map f : X → Y is a coarse equivalence if: 1) f ( X ) is a net in Y (i.e., there exists ε ≥ 0 such that B ( f ( X ) , ε ) = Y ); + ∞ 2) there exist ρ − , ρ + : R ≥ 0 → R ≥ 0 such that ρ − , ρ + − − → + ∞ and, for every x , y ∈ X , ρ − ( d ( x , y )) ≤ d ′ ( f ( x ) , f ( y )) ≤ ρ + ( d ( x , y )) . Two spaces are coarsely equivalent if there exists a coarse equivalence between them. A quasi-isometry is a coarse equivalence such that ρ − and ρ + are affine. Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Beyond metric spaces The categories Coarse e Coarse / ∼ Metric spaces and finitely generated groups Definition (Coarse equivalence) Let ( X , d ) and ( Y , d ′ ) be two metric spaces. A map f : X → Y is a coarse equivalence if: 1) f ( X ) is a net in Y (i.e., there exists ε ≥ 0 such that B ( f ( X ) , ε ) = Y ); + ∞ 2) there exist ρ − , ρ + : R ≥ 0 → R ≥ 0 such that ρ − , ρ + − − → + ∞ and, for every x , y ∈ X , ρ − ( d ( x , y )) ≤ d ′ ( f ( x ) , f ( y )) ≤ ρ + ( d ( x , y )) . Two spaces are coarsely equivalent if there exists a coarse equivalence between them. A quasi-isometry is a coarse equivalence such that ρ − and ρ + are affine. Coarse equivalence and quasi-isometry are equivalence relations. The inclusion of a net into a metric space is a quasi-isometry. n 2 �→ n 3 is a coarse equivalence between { n 2 | n ∈ N } and { n 3 | n ∈ N } , but it is not a quasi-isometry. f : Z → { 0 } is not a coarse equivalence. Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Beyond metric spaces The categories Coarse e Coarse / ∼ A group G is finitely generated if there exists a finite set Σ ⊆ G of generators of G . Let G be a finitely generated group and Σ = Σ − 1 be a finite subset of generators of G . Define the word metric relative to Σ between two points g , h ∈ G the value � min { n ∈ N | ∃ σ 1 , . . . , σ n ∈ Σ : g − 1 h = σ 1 · · · σ n } if g � = h , d Σ ( g , h ) = 0 otherwise. d Σ is invariant under left multiplication (i.e., d Σ ( kg , kh ) = d Σ ( g , h ), for every g , h , k ∈ G ). Theorem (Indipendence from the generator set) Let G be a finitely generated group and Σ and ∆ be two symmetric finite generators subsets. Then ( G , d Σ ) and ( G , d ∆ ) are quasi-isometric. Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Coarse spaces Beyond metric spaces Balleans The categories Coarse e Coarse / Coarse spaces vs balleans ∼ Beyond metric spaces: coarse spaces and balleans Definition (Roe, 2003) Let X be a set. A coarse structure E on X is a subset of P ( X × X ) s.t.: 1) ∆ X = { ( x , x ) | x ∈ X } ∈ E ; 2) E is closed under subsets; 3) E is closed under finite unions; 4) if E , F ∈ E , then E ◦ F = { ( x , z ) | ∃ y : ( x , y ) ∈ E , ( y , z ) ∈ F } ∈ E ; 5) if E ∈ E , then E − 1 = { ( y , x ) | ( x , y ) ∈ E } ∈ E . ( X , E ) is a coarse space. Properties (2) and (3) imply that E is an ideal of subsets of X × X . Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Coarse spaces Beyond metric spaces Balleans The categories Coarse e Coarse / Coarse spaces vs balleans ∼ Beyond metric spaces: coarse spaces and balleans Definition (Roe, 2003) Let X be a set. A coarse structure E on X is a subset of P ( X × X ) s.t.: 1) ∆ X = { ( x , x ) | x ∈ X } ∈ E ; 2) E is closed under subsets; 3) E is closed under finite unions; 4) if E , F ∈ E , then E ◦ F = { ( x , z ) | ∃ y : ( x , y ) ∈ E , ( y , z ) ∈ F } ∈ E ; 5) if E ∈ E , then E − 1 = { ( y , x ) | ( x , y ) ∈ E } ∈ E . ( X , E ) is a coarse space. Properties (2) and (3) imply that E is an ideal of subsets of X × X . The definition is quite similar to the one of uniformity. Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Coarse spaces Beyond metric spaces Balleans The categories Coarse e Coarse / Coarse spaces vs balleans ∼ Definition (Roe, 2003) Let X be a set. A coarse structure E on X is a subset of P ( X × X ) s.t.: 1) ∆ X = { ( x , x ) | x ∈ X } ∈ E ; 2) E is closed under subsets; 3) E is closed under finite unions; 4) if E , F ∈ E , then E ◦ F = { ( x , z ) | ∃ y : ( x , y ) ∈ E , ( y , z ) ∈ F } ∈ E ; 5) if E ∈ E , then E − 1 = { ( y , x ) | ( x , y ) ∈ E } ∈ E . ( X , E ) is a coarse space. T X = { E ⊆ ∆ X } is the trivial coarse X structure over X . E R M X = P ( X × X ) is the indiscrete coarse R structure over X . B ( x, R ) If ( X , d ) is a metric space, the family of all E ⊆ X × X such that x X E ⊆ E R = { ( x , y ) | d ( x , y ) ≤ R } , for some R ≥ 0, is the metric coarse structure. Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Coarse spaces Beyond metric spaces Balleans The categories Coarse e Coarse / Coarse spaces vs balleans ∼ Morphisms A subset L of a coarse space ( X , E ) is large in X if exists E ∈ E such that E [ L ] = { y | ( x , y ) ∈ E , x ∈ L } = X . Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Coarse spaces Beyond metric spaces Balleans The categories Coarse e Coarse / Coarse spaces vs balleans ∼ Morphisms A subset L of a coarse space ( X , E ) is large in X if exists E ∈ E such that E [ L ] = { y | ( x , y ) ∈ E , x ∈ L } = X . Two maps f , g : S → ( X , E ) from a non-empty set to a coarse space are close ( f ∼ g ) if { ( f ( x ) , g ( x )) | x ∈ S } ∈ E . Nicol` o Zava Some categorical aspects of coarse spaces and balleans

Large scale geometry of metric spaces Coarse spaces Beyond metric spaces Balleans The categories Coarse e Coarse / Coarse spaces vs balleans ∼ Morphisms A subset L of a coarse space ( X , E ) is large in X if exists E ∈ E such that E [ L ] = { y | ( x , y ) ∈ E , x ∈ L } = X . Two maps f , g : S → ( X , E ) from a non-empty set to a coarse space are close ( f ∼ g ) if { ( f ( x ) , g ( x )) | x ∈ S } ∈ E . A map f : ( X , E ) → ( Y , F ) between coarse spaces is: bornologous (coarsely uniform) if ( f × f )( E ) = { ( f ( x ) , f ( y )) | ( x , y ) ∈ E } ∈ E , for every E ∈ E ; effectively proper if ( f × f ) − 1 ( F ) = { ( x , y ) | ( f ( x ) , f ( y )) ∈ F } ∈ E , for every F ∈ F ; a coarse embedding if it is bornologous and effectively proper; an asymorphism if it is bijective and both f and f − 1 are bornologous; a coarse equivalence if it is a coarse embedding and f ( X ) is large in Y . Nicol` o Zava Some categorical aspects of coarse spaces and balleans