Retracts of universal homogeneous structures

Transcription

1 Retracts of universal homogeneous structures Wiesław Kubiś Czech Academy of Sciences (CZECH REPUBLIC) and Jan Kochanowski University, Kielce (POLAND) wkubis/ TOPOSYM, Prague 2011

2 Dedicated to the memory of my friend Paweł Waszkiewicz

3 Outline 1 The goal 2 Motivation 3 Injectivity 4 Pushouts 5 Main result 6 HH objects 7 The end

4 Main goal Let X, d be a separable complete metric space. TFAE: (a) X, d is a non-expansive retract of the Urysohn space U. (b) X, d is finitely hyperconvex, that is, given a finite family of closed balls F = {B(x 0, r 0 ),..., B(x n 1, r n 1 )} with F =, there exist i < j < n such that d(x i, x j ) > r i + r j. Remark Implication (a) = (b) is easy.

5 Main goal Let X, d be a separable complete metric space. TFAE: (a) X, d is a non-expansive retract of the Urysohn space U. (b) X, d is finitely hyperconvex, that is, given a finite family of closed balls F = {B(x 0, r 0 ),..., B(x n 1, r n 1 )} with F =, there exist i < j < n such that d(x i, x j ) > r i + r j. Remark Implication (a) = (b) is easy.

6 Main goal Let X, d be a separable complete metric space. TFAE: (a) X, d is a non-expansive retract of the Urysohn space U. (b) X, d is finitely hyperconvex, that is, given a finite family of closed balls F = {B(x 0, r 0 ),..., B(x n 1, r n 1 )} with F =, there exist i < j < n such that d(x i, x j ) > r i + r j. Remark Implication (a) = (b) is easy.

7 Let K be a compact space of weight ℵ 1. TFAE: (a) K is a topological retract of ω. (b) K is a 0-dimensional F-space, that is, disjoint open F σ sets have disjoint closures. Remark Implication (a) = (b) is trivial.

8 Let K be a compact space of weight ℵ 1. TFAE: (a) K is a topological retract of ω. (b) K is a 0-dimensional F-space, that is, disjoint open F σ sets have disjoint closures. Remark Implication (a) = (b) is trivial.

9 Let K be a compact space of weight ℵ 1. TFAE: (a) K is a topological retract of ω. (b) K is a 0-dimensional F-space, that is, disjoint open F σ sets have disjoint closures. Remark Implication (a) = (b) is trivial.

10 Motivation (Dolinka 2011) Let M be a nice Fraïssé class of finite models and let U be its Fraïssé limit. Given a countable model X, TFAE: (a) X is a retract of U. (b) X is algebraically closed.

11 Motivation (Dolinka 2011) Let M be a nice Fraïssé class of finite models and let U be its Fraïssé limit. Given a countable model X, TFAE: (a) X is a retract of U. (b) X is algebraically closed.

12 Injectivity Definition Let K C be a pair of categories. We say that X Ob(C) is K-injective in C if for every K-arrow i : a b and for every C-arrow f : a X, there exists a C-arrow g : b X such that g i = f. a i b f g X

13 The pushout of f, g y g w g f z f x

14 The pushout of f, g g v g y g w f f z f x

15 The pushout of f, g g y g g h w f f v z f x

16 The mixed pushout property Mixed pushout: f, g K and f, g L y g w g f z f x

17 Let K L be two categories with the same objects and satisfying the following conditions: (h1) K has a weakly initial object. (h2) K, L has the mixed pushout property. (h3) K has a Fraïssé sequence U. Let X be a sequence in K. The following properties are equivalent. (a) X is K-injective. (b) X is a retract of U.

18 Let K L be two categories with the same objects and satisfying the following conditions: (h1) K has a weakly initial object. (h2) K, L has the mixed pushout property. (h3) K has a Fraïssé sequence U. Let X be a sequence in K. The following properties are equivalent. (a) X is K-injective. (b) X is a retract of U.

19 Let K L be two categories with the same objects and satisfying the following conditions: (h1) K has a weakly initial object. (h2) K, L has the mixed pushout property. (h3) K has a Fraïssé sequence U. Let X be a sequence in K. The following properties are equivalent. (a) X is K-injective. (b) X is a retract of U.

20 Let K L be two categories with the same objects and satisfying the following conditions: (h1) K has a weakly initial object. (h2) K, L has the mixed pushout property. (h3) K has a Fraïssé sequence U. Let X be a sequence in K. The following properties are equivalent. (a) X is K-injective. (b) X is a retract of U.

21 Let K L be two categories with the same objects and satisfying the following conditions: (h1) K has a weakly initial object. (h2) K, L has the mixed pushout property. (h3) K has a Fraïssé sequence U. Let X be a sequence in K. The following properties are equivalent. (a) X is K-injective. (b) X is a retract of U.

22 About the proof x 0 u l0 x 1 w 1,1 u l1 x 2 w 2,1 w 2,2 u l2 x 3 w 3,1 w 3,2 w 3,3 u l

23 Definition (Cameron & Nešetřil 2006) A countable relational structure X is homomorphism homogeneous (HH) if every homomorphism between its finite substructures extends to an endomorphism of X. Let K L be a pair of categories with the same objects, satisfying conditions (h1) (h3) above. Let X be a sequence in K. The following properties are equivalent: (a) X is HH. (b) X is a retract of a Fraïssé sequence of some subcategory of K satisfying (h1) (h3).

24 Definition (Cameron & Nešetřil 2006) A countable relational structure X is homomorphism homogeneous (HH) if every homomorphism between its finite substructures extends to an endomorphism of X. Let K L be a pair of categories with the same objects, satisfying conditions (h1) (h3) above. Let X be a sequence in K. The following properties are equivalent: (a) X is HH. (b) X is a retract of a Fraïssé sequence of some subcategory of K satisfying (h1) (h3).

25 Definition (Cameron & Nešetřil 2006) A countable relational structure X is homomorphism homogeneous (HH) if every homomorphism between its finite substructures extends to an endomorphism of X. Let K L be a pair of categories with the same objects, satisfying conditions (h1) (h3) above. Let X be a sequence in K. The following properties are equivalent: (a) X is HH. (b) X is a retract of a Fraïssé sequence of some subcategory of K satisfying (h1) (h3).

26 THE END

27 Selected bibliography FRAÏSSÉ, R., Sur quelques classifications des systèmes de relations, Publ. Sci. Univ. Alger. Sér. A. 1 (1954) DOLINKA, I., A characterization of retracts in certain Fraïssé limits, to appear in MLQ. Mathematical Logic Quarterly (2011) CAMERON, P.; NEŠETŘIL, J., Homomorphism-homogeneous relational structures, Combin. Probab. Comput. 15 (2006) KUBIŚ, W., Fraïssé sequences: category-theoretic approach to universal homogeneous structures, preprint