A remark on the general nature of the Kat etovs construction - PowerPoint PPT Presentation

A remark on the general nature of the Katˇ etov’s construction Dragan Maˇ sulovi´ c Department of Mathematics and Informatics University of Novi Sad, Serbia joint work with Wiesłav Kubi´ s SE � � � � � � OP 2014, Novi Sad, 18 Aug 2014

The Urysohn space P. U RYSOHN : Sur un espace m´ etrique universel. Bull. Math. Sci. 51 (1927), 43–64, 74–90 U — complete separable metric space which is homogeneous and embeds all separable metric spaces. U = U Q

Katˇ etov’s construction of the Urysohn space M. K AT ˇ ETOV : On universal metric spaces. General topology and its relations to modern analysis and algebra. VI (Prague, 1986), Res. Exp. Math. vol. 16, Heldermann, Berlin, 1988, 323–330 A Katˇ etov function over a finite rational metric space X is every function α : X → Q such that | α ( x ) − α ( y ) | � d ( x , y ) � α ( x ) + α ( y ) K ( X ) = all Katˇ etov functions over X , which is a rational metric space under sup metric → K 2 ( X ) ֒ → K 3 ( X ) ֒ colim ( X ֒ → K ( X ) ֒ → · · · ) = U Q

Katˇ etov’s construction of the Urysohn space M. K AT ˇ ETOV : On universal metric spaces. General topology and its relations to modern analysis and algebra. VI (Prague, 1986), Res. Exp. Math. vol. 16, Heldermann, Berlin, 1988, 323–330 Observation 1. U Q is countable and homogeneous. Observation 2. K ( X ) contains all 1-point extensions of X . Observation 3. K is functorial.

Homogeneity automorphism A isomorphism

Fra¨ ıss´ e theory age ( A ) — the class of all finitely generated struct’s which embed into A amalgamation class — a class K of fin. generated struct’s s.t. ◮ there are countably many pairwise noniso struct’s in K ; ◮ K has (HP); ◮ K has (JEP); and ◮ K has (AP): for all A , B , C ∈ K and embeddings v f : A ֒ → B and g : A ֒ → C , there exist D ∈ K C → D ֒ and embeddings u : B ֒ → D and v : C ֒ → D → → g u ֒ ֒ such that u ◦ f = v ◦ g . A ֒ → B f

Fra¨ ıss´ e theory Theorem. [Fraisse, 1953] 1 If A is a countable homogeneous structure, then age ( A ) is an amalgamation class. 2 If K is an amalgamation class, then there is a unique (up to isomorphism) countable homogeneous structure A such that age ( A ) = K . 3 If B is a countable structure younger than A (that is, age ( B ) ⊆ age ( A ) ), then B ֒ → A . Definition. If K is an amalgamation class and A is the countable homogeneous structure such that age ( A ) = K , we say that A is the Fra¨ ıss´ e limit of K and write A = Flim ( K ) .

Some prominent Fra¨ ıss´ e limits ıss´ Q — the Fra¨ e limit of the class of all linear orders U Q — the Fra¨ ıss´ e limit of the class of finite metric spaces with rational distances (the rational Urysohn space) R — the Fra¨ ıss´ e limit of the class of all finite graphs (the Rado graph) H n — the Fra¨ ıss´ e limit of the class of all finite K n -free graphs, n � 3 (Henson graphs) P — the Fra¨ ıss´ e limit of the class of all finite posets (the random poset)

Recall: M. K AT ˇ ETOV : On universal metric spaces. General topology and its relations to modern analysis and algebra. VI (Prague, 1986), Res. Exp. Math. vol. 16, Heldermann, Berlin, 1988, 323–330 Katˇ etov’s construction → K 2 ( X ) ֒ → K 3 ( X ) ֒ colim ( X ֒ → K ( X ) ֒ → · · · ) = U Q Observation 1. U Q is countable and homogeneous. Observation 2. K ( X ) contains all 1-point extensions of X . Observation 3. K is functorial.

� � �� � Katˇ etov functors A — a category of fin generated L -struct’s with (HP) and (JEP) C — the category of all colimits of ω -chains in A Definition. A functor K : A → C is a Katˇ etov functor if η A A � � K ( A ) 1 K preserves embeddings, and � � 2 there exists a natural transformation · f g η : ID → K such that for every emb- B edding f : A ֒ → B in A where B is a 1-point extension of A there is an → K ( A ) satisfying ր embedding g : B ֒

Katˇ etov functors K ( A ) A K ( A ) is “a functorial amalgam” of all 1-point extensions of A .

Why is it hard to construct a Katˇ etov functor by hand? Example. Tournaments.

Why is it hard to construct a Katˇ etov functor by hand? Example. Tournaments.

Why is it hard to construct a Katˇ etov functor by hand? Example. Tournaments. How to add edges in a “functorial” way?

Why is it hard to construct a Katˇ etov functor by hand? Example. Tournaments. T = ( V , E ) — a tournament with n vertices T � n — the tournament with vertices V � n and edges defined by: ◮ if s and t are seq’s such that | s | < | t | , put s → t in T � n ; ◮ if s = � s 1 , . . . , s k � and t = � t 1 , . . . , t k � are distinct sequences of the same length, find the smallest i such that s i � = t i and then put s → t in T � n if and only if s i → t i in T . Put K ( T ) = ( V ∗ , E ∗ ) where V ∗ = V ∪ V � n , E ∗ = E ∪ E ( T � n ) ∪ { v → s : v ∈ V , s ∈ V � n , v appears in s } ∪ { s → v : v ∈ V , s ∈ V � n , v does not appear in s } .

Why is it hard to construct a Katˇ etov functor by hand? Example. Tournaments. approx. 2 n n new vertices

Katˇ etov functors A — a category of fin generated L -struct’s with (HP) and (JEP) C — the category of all colimits of ω -chains in A Theorem. If there exists a Katˇ etov functor K : A → C , then 1 A is an amalgamation class, 2 its Fra¨ ıss´ e limit F can be obtained by the “Katˇ etov construction” starting from an arbitrary A ∈ A : → K 2 ( A ) ֒ → K 3 ( A ) ֒ F = colim ( A ֒ → K ( A ) ֒ → · · · ) , 3 F is C -morphism-homogeneous .

C -morphism-homogeneity C -endomorphism F C -morphism Definition. A structure F is C -morphism-homogeneous if every C -morphism between finitely induced substructures of F extends to a C -endomorphism of F .

C -morphism-homogeneity C -endomorphism F C -morphism P. J. C AMERON , J. N E ˇ SET ˇ RIL : Homomorphism-homogeneous relational structures. Combin. Probab. Comput, 15 (2006), 91–103

Katˇ etov functors: Examples A Katˇ etov functor exists for the following categories A : ◮ finite linear orders with order-preserving maps, ◮ finite graphs with graph homomorphisms, ◮ finite K n -free graphs with embeddings, ◮ finite digraphs with digraph homomorphisms, ◮ finite tournaments with homomorphisms = embeddings. ◮ finite rational metric spaces with nonexpansive maps, ◮ finite posets with order-preserving maps, ◮ finite boolean algebras with homomorphisms, ◮ finite semilattices/lattices/distributive lattices with embeddings. A Katˇ etov functor does not exist for the category of finite K n -free graphs and graph homomorphisms.

Existence of Katˇ etov functors A — a category of fin generated L -struct’s with (HP) and (JEP) C — the category of all colimits of ω -chains in A Theorem. There exists a Katˇ etov functor K : A → C if and only if A is an amalgamation class with the morphism extension property.

Morphism extension property C — a category Definition. C ∈ C has the morphism extension property in C if for any choice f 1 , f 2 , . . . of partial C -morphisms of C there exist D ∈ C and m 1 , m 2 , . . . ∈ End C ( D ) such that C is a substructure of D , m i is an extension of f i for all i , and the following coherence conditions are satisfied for all i , j and k : ◮ if f i = id A , A � C , then m i = id D , ◮ if f i is an embedding, then so is m i , and ◮ if f i ◦ f j = f k then m i ◦ m j = m k . We say that C has the morphism extension property if every C ∈ C has the morphism extension property in C .

Existence of Katˇ etov functors for algebras L — algebraic language V — a variety of L -algebras understood as a category of L -algebras with embeddings A — the full subcategory of V spanned by all finitely generated algebras in V C — the full subcategory of V spanned by all countably generated algebras in V Theorem. There exists a Katˇ etov functor K : A → C if and only if A is an amalgamation class.

� � � � The Importance of Being ✘✘✘✘✘ ✘ Earnest Functor Theorem. Let K : A → C be a Katˇ etov functor and let F be the Fra¨ ıss´ e limit of A . Then for every object C in C : ◮ Aut ( C ) ֒ → Aut ( F ) ; ◮ End C ( C ) ֒ → End C ( F ) . Proof (Idea). Take any f : C → C . Then: η � K ( C ) � � η � � � � η � K 2 ( C ) � · · · C � F K 2 ( f ) K ( f ) f f ∗ C � � η � K ( C ) � � � K 2 ( C ) � � � · · · � F � η η

The Importance of Being ✘✘✘✘✘ ✘ Earnest Functor Theorem. Let K : A → C be a Katˇ etov functor and let F be the Fra¨ ıss´ e limit of A . Then for every object C in C : ◮ Aut ( C ) ֒ → Aut ( F ) ; ◮ End C ( C ) ֒ → End C ( F ) . Moreover, if K is locally finite (that is, K ( A ) is finite whenever A is finite), then the above embeddings are countinuous w.r.t. the topology of pointwise convergence.

The Importance of Being ✘✘✘✘✘ ✘ Earnest Functor ıss´ Corollary. For the following Fra¨ e limits F we have that Aut ( F ) embeds all permutation groups on a countable set: ◮ Q , ◮ the random graph [Henson 1971], ◮ Henson graphs [Henson 1971], ◮ the random digraph, ◮ the rational Urysohn space [Uspenskij 1990], ◮ the random poset, ◮ the countable atomless boolean algebra, ◮ the random semilattice, ◮ the random lattice, ◮ the random distributive lattice.