Generic objects in topology and functional analysis

Transcription

1 Generic objects in topology and functional analysis Wiesław Kubiś Institute of Mathematics, Czech Academy of Sciences and Cardinal Stefan Wyszyński University in Warsaw, Poland 46th Winter School in Abstract Analysis Svratka, January 2018 W.Kubiś ( Generic objects January / 36

2 1 Part 1 2 Part 2 3 Part 3 W.Kubiś ( Generic objects January / 36

3 Part 1 W.Kubiś ( Generic objects January / 36

4 Categories A category K consists of a class of objects Obj(K), W.Kubiś ( Generic objects January / 36

5 Categories A category K consists of a class of objects Obj(K), a class of arrows a,b Obj(K) K(a, b), where f K(a, b) means a is the domain of f and b is the codomain of f, W.Kubiś ( Generic objects January / 36

6 Categories A category K consists of a class of objects Obj(K), a class of arrows a,b Obj(K) K(a, b), where f K(a, b) means a is the domain of f and b is the codomain of f, a partial associative composition operation defined on arrows, where f g is defined the domain of g coincides with the domain of f. W.Kubiś ( Generic objects January / 36

7 Categories A category K consists of a class of objects Obj(K), a class of arrows a,b Obj(K) K(a, b), where f K(a, b) means a is the domain of f and b is the codomain of f, a partial associative composition operation defined on arrows, where f g is defined the domain of g coincides with the domain of f. Furthermore, for each a Obj(K) there is an identity id a K(a, a) satisfying id a g = g and f id a = f for f K(a, x), g K(y, a), x, y Obj(K). W.Kubiś ( Generic objects January / 36

8 Definition A sequence in K is a functor x from N into K. W.Kubiś ( Generic objects January / 36

9 Definition A sequence in K is a functor x from N into K. x0 1 x1 2 x2 3 x 0 x 1 x 2 W.Kubiś ( Generic objects January / 36

10 Definition A sequence in K is a functor x from N into K. x0 1 x1 2 x2 3 x 0 x 1 x 2 Definition Let x be a sequence in K. The colimit of x is a pair X, {xn } n N with : x n X satisfying: x n 1 x n = x m x m n for every n < m. 2 If Y, {y n } n N with y n : x n Y satisfies y n = y m y m n for every n < m then there is a unique arrow f : X Y satisfying f x n = y n for every n N. W.Kubiś ( Generic objects January / 36

11 The Banach-Mazur game Definition The Banach-Mazur game BM (K) played on K is described as follows. W.Kubiś ( Generic objects January / 36

12 The Banach-Mazur game Definition The Banach-Mazur game BM (K) played on K is described as follows. There are two players: Eve and Odd. Eve starts by choosing a 0 Obj(K). W.Kubiś ( Generic objects January / 36

13 The Banach-Mazur game Definition The Banach-Mazur game BM (K) played on K is described as follows. There are two players: Eve and Odd. Eve starts by choosing a 0 Obj(K). Then Odd chooses a 1 Obj(K) together with a K-arrow a 1 0 : a 0 a 1. W.Kubiś ( Generic objects January / 36

14 The Banach-Mazur game Definition The Banach-Mazur game BM (K) played on K is described as follows. There are two players: Eve and Odd. Eve starts by choosing a 0 Obj(K). Then Odd chooses a 1 Obj(K) together with a K-arrow a 1 0 : a 0 a 1. More generally, after Odd s move finishing with an object a 2k 1, Eve chooses a 2k Obj(K) together with a K-arrow a 2k 2k 1 : a 2k 1 a 2k. W.Kubiś ( Generic objects January / 36

15 The Banach-Mazur game Definition The Banach-Mazur game BM (K) played on K is described as follows. There are two players: Eve and Odd. Eve starts by choosing a 0 Obj(K). Then Odd chooses a 1 Obj(K) together with a K-arrow a0 1 : a 0 a 1. More generally, after Odd s move finishing with an object a 2k 1, Eve chooses a 2k Obj(K) together with a K-arrow a2k 1 : a 2k 1 a 2k. Next, Odd chooses a 2k+1 Obj(K) together with a K-arrow : a 2k a 2k+1. And so on... a 2k+1 2k W.Kubiś ( Generic objects January / 36

16 The Banach-Mazur game Definition The Banach-Mazur game BM (K) played on K is described as follows. There are two players: Eve and Odd. Eve starts by choosing a 0 Obj(K). Then Odd chooses a 1 Obj(K) together with a K-arrow a0 1 : a 0 a 1. More generally, after Odd s move finishing with an object a 2k 1, Eve chooses a 2k Obj(K) together with a K-arrow a2k 1 : a 2k 1 a 2k. Next, Odd chooses a 2k+1 Obj(K) together with a K-arrow a 2k+1 2k : a 2k a 2k+1. And so on... The result of a play is a sequence a: a 1 0 a2k 1 a 0 a 1 a 2k 1 a 2k W.Kubiś ( Generic objects January / 36

17 Generic objects General assumption: K L. W.Kubiś ( Generic objects January / 36

18 Generic objects General assumption: K L. Definition We say that U Obj(L) is K-generic if Odd has a strategy in the Banach-Mazur game BM (K) such that the colimit of the resulting sequence a is always isomorphic to U, no matter how Eve plays. W.Kubiś ( Generic objects January / 36

19 Generic objects General assumption: K L. Definition We say that U Obj(L) is K-generic if Odd has a strategy in the Banach-Mazur game BM (K) such that the colimit of the resulting sequence a is always isomorphic to U, no matter how Eve plays. Proposition A K-generic object, if exists, is unique up to isomorphism. W.Kubiś ( Generic objects January / 36

20 Generic objects General assumption: K L. Definition We say that U Obj(L) is K-generic if Odd has a strategy in the Banach-Mazur game BM (K) such that the colimit of the resulting sequence a is always isomorphic to U, no matter how Eve plays. Proposition A K-generic object, if exists, is unique up to isomorphism. Proof. The rules for Eve and Odd are the same. W.Kubiś ( Generic objects January / 36

21 Part 2 W.Kubiś ( Generic objects January / 36

22 Example 1 Let K be the category of all finite linearly ordered sets. Then Q, < is K-generic. W.Kubiś ( Generic objects January / 36

23 Example 1 Let K be the category of all finite linearly ordered sets. Then Q, < is K-generic. Example 2 Let M fin be the category of finite metric spaces with isometric embeddings. Then the Urysohn space U is M fin -generic. W.Kubiś ( Generic objects January / 36

24 The Gurarii space W.Kubiś ( Generic objects January / 36

25 Theorem (Gurarii 1966) There exists a separable Banach space G with the following property. (G) For every ε > 0, for every finite-dimensional normed spaces E F, for every linear isometric embedding e : E G there exists a linear ε-isometric embedding f : F G such that f E = e. W.Kubiś ( Generic objects January / 36

26 Theorem (Gurarii 1966) There exists a separable Banach space G with the following property. (G) For every ε > 0, for every finite-dimensional normed spaces E F, for every linear isometric embedding e : E G there exists a linear ε-isometric embedding f : F G such that f E = e. Theorem (Lusky 1976) Among separable spaces, property (G) determines the space G uniquely up to linear isometries. W.Kubiś ( Generic objects January / 36

27 Theorem (Gurarii 1966) There exists a separable Banach space G with the following property. (G) For every ε > 0, for every finite-dimensional normed spaces E F, for every linear isometric embedding e : E G there exists a linear ε-isometric embedding f : F G such that f E = e. Theorem (Lusky 1976) Among separable spaces, property (G) determines the space G uniquely up to linear isometries. Elementary proof: Solecki & K W.Kubiś ( Generic objects January / 36

28 Theorem (K. 2018) The Gurarii space G is generic over the category B fd of finite-dimensional normed spaces with linear isometric embeddings. W.Kubiś ( Generic objects January / 36

29 Theorem (K. 2018) The Gurarii space G is generic over the category B fd of finite-dimensional normed spaces with linear isometric embeddings. Key Lemma (Solecki & K.) Let X, Y be finite-dimensional normed spaces, let f : X Y be an ε-isometry with 0 < ε < 1. Then there exist a finite-dimensional normed space Z and isometric embeddings i : X Z, j : Y Z such that i j f ε. W.Kubiś ( Generic objects January / 36

30 W.Kubiś ( Generic objects January / 36

31 The amalgamation property Definition We say that K has amalgamations at z Obj(K) if for every K-arrows f : z x, g : z y there exist K-arrows f : x w, g : y w such that f f = g g. y g w g f z f x W.Kubiś ( Generic objects January / 36

32 The amalgamation property Definition We say that K has amalgamations at z Obj(K) if for every K-arrows f : z x, g : z y there exist K-arrows f : x w, g : y w such that f f = g g. y z g w g f We say that K has the amalgamation property (AP) if it has amalgamations at every z Obj(K). f x W.Kubiś ( Generic objects January / 36

33 Theorem (Universality) Assume K has the AP and X = lim x, where x is a sequence in K. Assume U is K-generic. W.Kubiś ( Generic objects January / 36

34 Theorem (Universality) Assume K has the AP and X = lim x, where x is a sequence in K. Assume U is K-generic. Then there exists an arrow e : X U. W.Kubiś ( Generic objects January / 36

35 W.Kubiś ( Generic objects January / 36

36 References V.I. Gurarii, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces (in Russian), Sibirsk. Mat. Ž. 7 (1966) W. Lusky, The Gurarij spaces are unique, Archiv der Mathematik 27 (1976) W. Kubiś, S. Solecki, A proof of uniqueness of the Gurarii space, Israel Journal of Mathematics 195 (2013) F. Cabello Sánchez, J. Garbulińska-Wegrzyn, W. Kubiś, Quasi-Banach spaces of almost universal disposition, Journal of Functional Analysis 267 (2014) W. Kubiś, Game-theoretic characterization of the Gurarii space, Archiv der Mathematik 110 (2018) W.Kubiś ( Generic objects January / 36

37 Part 3 W.Kubiś ( Generic objects January / 36

38 The generic linear operator W.Kubiś ( Generic objects January / 36

39 The generic linear operator Theorem (Cabello Sánchez, Garbulińska-Wegrzyn, K. 2014) There exists a norm-one linear operator Ω: G G satisfying the following condition. (E) For every ε > 0, for every finite-dimensional Banach spaces E F, for every non-expansive linear operator T : F G, for every linear isometric embedding e : E G with Ω e = T E, there exists an ε-isometric embedding f : F G such that f E = e and Ω f = T. W.Kubiś ( Generic objects January / 36

40 The generic linear operator Theorem (Cabello Sánchez, Garbulińska-Wegrzyn, K. 2014) There exists a norm-one linear operator Ω: G G satisfying the following condition. (E) For every ε > 0, for every finite-dimensional Banach spaces E F, for every non-expansive linear operator T : F G, for every linear isometric embedding e : E G with Ω e = T E, there exists an ε-isometric embedding f : F G such that f E = e and Ω f = T. Theorem For every non-expansive linear operator S : X G with X separable, there exists a linear isometric embedding e : X G such that Ω e = S. W.Kubiś ( Generic objects January / 36

41 Theorem The operator Ω is generic. W.Kubiś ( Generic objects January / 36

42 Theorem The operator Ω is generic. Theorem (Bargetz, Kakol, K. 2017) There exists a unique graded separable Fréchet space G satisfying: (E) For every ε > 0, for every finite-dimensional graded Fréchet spaces E F, for every linear isometric embedding e : E G there exists an ε-isometric embedding f : F G such that f E = e. W.Kubiś ( Generic objects January / 36

43 The Cantor set W.Kubiś ( Generic objects January / 36

44 The Cantor set Fix a compact 0-dimensional space K. Define the category K K as follows. W.Kubiś ( Generic objects January / 36

45 The Cantor set Fix a compact 0-dimensional space K. Define the category K K as follows. The objects are continuous mappings f : K S with S finite. W.Kubiś ( Generic objects January / 36

46 The Cantor set Fix a compact 0-dimensional space K. Define the category K K as follows. The objects are continuous mappings f : K S with S finite. An arrow from f : K S to g : K T is a surjection p : T S satisfying p g = f. W.Kubiś ( Generic objects January / 36

47 The Cantor set Fix a compact 0-dimensional space K. Define the category K K as follows. The objects are continuous mappings f : K S with S finite. An arrow from f : K S to g : K T is a surjection p : T S satisfying p g = f. g T K p f S W.Kubiś ( Generic objects January / 36

48 Let L K be the category whose objects are continuous mappings f : K X with X metrizable compact 0-dimensional. W.Kubiś ( Generic objects January / 36

49 Let L K be the category whose objects are continuous mappings f : K X with X metrizable compact 0-dimensional. An L K -arrow from f : K X to g : K Y is a continuous surjection p : Y X satisfying p g = f. W.Kubiś ( Generic objects January / 36

50 Let L K be the category whose objects are continuous mappings f : K X with X metrizable compact 0-dimensional. An L K -arrow from f : K X to g : K Y is a continuous surjection p : Y X satisfying p g = f. g Y K p f X W.Kubiś ( Generic objects January / 36

51 Theorem (Bielas, Walczyńska, K.) Let 2 ω denote the Cantor set. A continuous mapping η : K 2 ω is K K -generic η is a topological embedding and η[k ] is nowhere dense in 2 ω. W.Kubiś ( Generic objects January / 36

52 Theorem (Bielas, Walczyńska, K.) Let 2 ω denote the Cantor set. A continuous mapping η : K 2 ω is K K -generic η is a topological embedding and η[k ] is nowhere dense in 2 ω. Corollary (Knaster & Reichbach 1953) Let h : A B be a homeomorphism between closed nowhere dense subsets of 2 ω. Then there exists a homeomorphism H : 2 ω 2 ω such that H A = h. W.Kubiś ( Generic objects January / 36

53 The pseudo-arc Let I be the category of all continuous surjections from the unit interval [0, 1] onto itself. W.Kubiś ( Generic objects January / 36

54 The pseudo-arc Let I be the category of all continuous surjections from the unit interval [0, 1] onto itself. Let C be the category of all chainable continua. W.Kubiś ( Generic objects January / 36

55 The pseudo-arc Let I be the category of all continuous surjections from the unit interval [0, 1] onto itself. Let C be the category of all chainable continua. Theorem The pseudo-arc is I-generic. W.Kubiś ( Generic objects January / 36

56 Amalgamations Definition We say that K has amalgamations at z Obj(K) if for every K-arrows f : z x, g : z y there exist K-arrows f : x w, g : y w such that f f = g g. y g w g f z f x W.Kubiś ( Generic objects January / 36

57 Amalgamations Definition We say that K has amalgamations at z Obj(K) if for every K-arrows f : z x, g : z y there exist K-arrows f : x w, g : y w such that f f = g g. y z g w g f We say that K has the amalgamation property (AP) if it has amalgamations at every z Obj(K). f x W.Kubiś ( Generic objects January / 36

58 Definition A category K is directed if for every x, y Obj(K) there is z Obj(K) such that K(x, z) and K(y, z). W.Kubiś ( Generic objects January / 36

59 Definition A category K is directed if for every x, y Obj(K) there is z Obj(K) such that K(x, z) and K(y, z). x z y W.Kubiś ( Generic objects January / 36

60 Fraïssé theory Theorem Assume K is a countable directed category of finitely generated models with embeddings. If K has the AP then there exists a K-generic (countably generated) model, called the Fraïssé limit of K. W.Kubiś ( Generic objects January / 36

61 Fraïssé theory Theorem Assume K is a countable directed category of finitely generated models with embeddings. If K has the AP then there exists a K-generic (countably generated) model, called the Fraïssé limit of K. Theorem (Fraïssé 1954) Let K be as above, let U = n N u n with u n Obj(K) for every n N. The following conditions are equivalent. (a) U is the Fraïssé limit of K. (b) Every K-object embeds into U and for every embeddings e : a b, f : a U with a, b Obj(K) there exists an embedding g : b U such that f = g e. W.Kubiś ( Generic objects January / 36

62 a f U e b g W.Kubiś ( Generic objects January / 36

63 Fact Finite graphs of vertex degree 2 fail the amalgamation property. W.Kubiś ( Generic objects January / 36

64 Weakenings of amalgamation Definition We say that K has the cofinal amalgamation property (CAP) if for every z Obj(K) there is a K-arrow e : z z such that K has amalgamations at z. W.Kubiś ( Generic objects January / 36

65 Weakenings of amalgamation Definition We say that K has the cofinal amalgamation property (CAP) if for every z Obj(K) there is a K-arrow e : z z such that K has amalgamations at z. Definition (Ivanov, 1999) We say that K has the weak amalgamation property (WAP) if for every z Obj(K) there is a K-arrow e : z z such that for every K-arrows f : z X, g : z y there exist K-arrows f : x w, g : y w such that f f e = g g e. W.Kubiś ( Generic objects January / 36

66 CAP and WAP y g w g f e z f x z W.Kubiś ( Generic objects January / 36

67 CAP and WAP y g w g f e z f x z Proposition Finite graphs of vertex degree 2 have the CAP. W.Kubiś ( Generic objects January / 36

68 Theorem (Krawczyk & K. 2016) Let K be a countable directed category of finitely generated models with embeddings. W.Kubiś ( Generic objects January / 36

69 Theorem (Krawczyk & K. 2016) Let K be a countable directed category of finitely generated models with embeddings. The following conditions are equivalent: (a) There exists a K-generic model. W.Kubiś ( Generic objects January / 36

70 Theorem (Krawczyk & K. 2016) Let K be a countable directed category of finitely generated models with embeddings. The following conditions are equivalent: (a) There exists a K-generic model. (b) K has the WAP. W.Kubiś ( Generic objects January / 36

71 Theorem (Krawczyk & K. 2016) Let K be a countable directed category of finitely generated models with embeddings. The following conditions are equivalent: (a) There exists a K-generic model. (b) K has the WAP. Theorem (Krawczyk & K. 2016) Let K be as above and let U be a countably generated model. The following properties are equivalent: (a) U is K-generic. (b) Eve does not have a winning strategy in BM (K, U). W.Kubiś ( Generic objects January / 36

72 A more concrete setup We assume that K is a full subcategory of L and the following conditions are satisfied. (L0) All L-arrows are monic. (L1) Every L-object is the co-limit of a sequence in K. (L2) Every sequence in K has a co-limit in L. (L3) Every K-object is ω-small in L. W.Kubiś ( Generic objects January / 36

73 Weak injectivity Definition An object V Obj(L) is weakly K-injective if every K-object has an L-arrow into V, and for every L-arrow e : a V there exists a K-arrow i : a b such that for every K-arrow f : b y there is an L-arrow g : y V satisfying g f i = e. W.Kubiś ( Generic objects January / 36

74 Weak injectivity Definition An object V Obj(L) is weakly K-injective if every K-object has an L-arrow into V, and for every L-arrow e : a V there exists a K-arrow i : a b such that for every K-arrow f : b y there is an L-arrow g : y V satisfying g f i = e. i a b y f e V g W.Kubiś ( Generic objects January / 36

75 Theorem (K. 2017) Assume K L satisfy (L0) (L3) and K is locally countable. Given V Obj(L), the following conditions are equivalent. W.Kubiś ( Generic objects January / 36

76 Theorem (K. 2017) Assume K L satisfy (L0) (L3) and K is locally countable. Given V Obj(L), the following conditions are equivalent. (a) V is weakly K-injective. (b) V is K-generic. (c) Eve does not have a winning strategy in BM (K, V ). W.Kubiś ( Generic objects January / 36

77 Theorem (K. 2017) Assume K L satisfy (L0) (L3) and K is locally countable. Given V Obj(L), the following conditions are equivalent. (a) V is weakly K-injective. (b) V is K-generic. (c) Eve does not have a winning strategy in BM (K, V ). Remark If there exists a weakly K-injective object then K is directed and has the WAP. W.Kubiś ( Generic objects January / 36

78 C. Bargetz, J. Kakol, W. Kubiś, A separable Fréchet space of almost universal disposition, Journal of Functional Analysis 272 (2017) A. Krawczyk, W. Kubiś, Games on finitely generated structures, preprint, arxiv: W. Kubiś, Weak Fraïssé categories, preprint, arxiv: W. Kubiś, Metric-enriched categories and approximate Fraïssé limits, preprint, arxiv: THE END W.Kubiś ( Generic objects January / 36