Appl Categor Struct (2017) 25:569-602 DOI 10.1007/s10485-016-9461-z

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Katetov Functors

Wiestaw Kubis1'2 • Dragan MasuloviC3

Received: 20 July 2015 / Accepted: 1 August 2016 / Published online: 24 August 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We develop a theory of Katetov functors which provide a uniform way of constructing Frai'sse limits. Among applications, we present short proofs and improvements of several recent results on the structure of the group of automorphisms and the semigroup of endomorphisms of some Frai'sse limits.

Keywords Katetov functor • Amalgamation • Frai'sse limit

Mathematics Subject Classification (2010) 03C50 • 18A22 • 03C30

1 Introduction

The theory of Frai'sse limits has a long history, inspired by Cantor's theorem saying that the set of rational numbers is the unique, up to isomorphisms, countable linearly ordered set without end-points, such that between any two points there is another one. In the fifties of the last century, Roland Frai'sse [12] realized that the ideas behind Cantor's theorem are much more general, thus developing his theory of limits. Namely, given a class A of finitely generated first-order structures with certain natural properties, there exists a unique countably generated structure L (called the Frai'sse limit of A) containing isomorphic copies of all structures from A and having a very strong homogeneity property, namely, every isomorphism between finitely generated substructures of L extends to an automorphism of L. Frai'sse theory can now be called classical and is part of almost every textbook in model theory. Independently of Frai'sse, around thirty years earlier, Urysohn [28] constructed a

H Wieslaw Kubis

kubisw@gmail.com

1 Cardinal Stefan Wyszyiiski University, College of Science, Warsaw, Poland

2 Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

3 Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia

universal separable complete metric space U which has the same homogeneity property as Frai'sse limits: every finite isometry extends to a bijective isometry of U. Around the eighties of the last century, merely sixty years after Urysohn's work, Katetov [19] found a uniform way of extending metric spaces, leading to a new and simple construction of the Urysohn space U. It turns out that Katetov's construction is functorial, namely, it can be extended to all nonexpansive mappings between metric spaces.

One has to point out that the crucial ideas in Frai'sse theory (including its existing generalizations) are of purely category-theoretic nature. This has been recently demonstrated by the first author [23], subsequently applied by Caramello [7] in topos theory. It seems that the first work presenting category-theoretic approach to Frai'sse limits is by Droste and Gobel [11] (1989), with applications in algebra and theoretical computer science, simply by considering classes of models with certain restrictions on embeddings. Another work, independent of the above, is by Irwin and Solecki [18] (2006), where the authors simply reverse the arrows in the classical Frai'sse theory, replacing embeddings by quotient epimorphisms. They apply their theory (which is in fact an instance of the general category-theoretic Frai'sse theory from [11] or [23]) for proving new properties of the so-called pseudo-arc, a special compact connected 1-dimensional planar set which has been in the scope of interest of geometric topologists throughout the last decades.

We address the question when a functorial way of constructing a Frai'sse limit exists. Namely, we define the concept of a Katetov functor capturing simple extensions of finitely generated structures, whose infinite power gives the Frai'sse limit. The existence of a Katetov functor implies directly that the automorphism group of the Frai'sse limit is universal for the class of all automorphism groups of countably generated structures from the given Frai'sse class. Papers [10] and [4] discuss some of the issues addressed in this note, without realizing that what one deals with are actually functorial constructions.

The authors would like to thank the Editor and the Referee for drawing our attention to the fact that our Katetov construction (presented in Section 3 below) is a special case of Reit-erman's functorial construction, published with Koubek [21] in 1979 (see also [1] for more, explanations and details) and for pointing out references [1] and [21]. This paper, therefore, can be thought of as another display of the versatility of Reiterman's important construction.

As we have mentioned above, our principal motivation comes from Katetov's construction of the Urysohn space [19], which we briefly recall here in case of the rational Urysohn space. Let X be a metric space with rational distances. A Katetov function over X is every function a : X — Q such that

\a(x) - a(y)\ < d(x, y) < a(x) + a(y)

for all x, y € X. Let K(X) be the set of all Katetov functions over X. The sup metric turns K(X) into a metric space. There is a natural isometric embedding X — K(X) which takes a € X to d(a, •) € K(X). Hence we get a chain of embeddings

X — K(X) — K 2(X) — K 3(X) — • • •

whose colimit is easily seen to be the rational Urysohn space.

It was observed by several authors (see, e.g., [3, 29]) that the construction K is actually functorial with respect to embeddings. Our principal observation is that more is true: if A is the category of all finite metric spaces with rational distances and nonexpansive mappings, and C is the category of all countable metric spaces with rational distances and nonexpansive mappings, then K can be turned into a functor from A to C. We present the details in the appendix to this note.

Our aim is to introduce and study the concept of a Katetov functor for a Frai'sse class. We have decided to work within the classical framework of model-theoretic Frai'sse theory, although most of our results can be adapted to a more general setting of Frai'sse categories (i.e., classes of objects with abstractly defined embeddings, see [23]).

The paper is organized as follows. Section 2 contains the main concept of a Katetov functor, its basic properties, examples, and a discussion of sufficient conditions for its existence. We prove, in particular, that a Kaetov functor exists if embeddings have pushouts in the category of all homomorphisms. In Section 3 we show how iterations of a Kaetov functor lead to Frai'sse limits. It turns out that the Frai'sse limit can be viewed as a fixed point of the countable infinite power of a Katetov functor and all orbits of this functor "tend" to the Frai'sse limit, resembling the Banach contraction principle. Section 4 deals with the semigroup Bergman property. We prove that in the presence of a Katetov functor, under some mild additional assumptions the endomorphism monoid End(L) of the Frai'sse limit L is strongly distorted and its Sierprnski rank is at most five. Applying a result from [25], we conclude that if End(L) is not finitely generated, then it has the Bergman property. This extends a recent result of Dolinka [8]. The last Section 5 is an appendix containing description of the original Katetov functor on metric spaces with nonexpansive mappings. We finish with a short discussion of possible generalizations of our results in category-theoretic setting.

1.1 The Setup

Let A = R U F U C be a first-order language, where R is a set of relational symbols, F a set of functional symbols, and C a set of constant symbols. We say that A is a purely relational language if F = C = 0. For a A-structure A and X C A, by (X)a we denote the substructure of A generated by X. We say that A is finitely generated if A = (X)a for some finite X C A. The fact that A is a substructure of B will be denoted by A ^ B.

Let C be a category of A-structures. A chain in C is a sequence of objects and embeddings of the form Q — C2 — C3 — • • •. Note that although there may be other kinds of morphisms in C, a chain always consists of objects and embeddings. We shall say that L is a standard colimit of the chain Q — C2 — • • • if it is a colimit of this chain in the usual sense and moreover, after forgetting the structure L is still a colimit in the category of sets. In other words, if the embeddings are inclusions, that is, Q < C2 < • • • then a standard colimit is L = n Cn with an appropriate A-structure making it a colimit in C. We shall say that C has standard colimits of chains if every chain in C has a standard colimit in C. Given C e Ob(C), let Aut(C) denote the permutation group consisting of all automorphisms of C, and let End(C) denote the transformation monoid consisting of all C-morphisms C — C. It may be the case that End(C) consists of all embeddings of C into C (if C consists of embeddings only). We shall sometimes write Endc (C) instead of End(C) in order to emphasize that we consider C-morphisms only. We say that C has the joint-embedding property (briefly: (JEP)) if every two structures in C embed into a common structure in C.

Standing Assumption Throughout the paper we assume the following. Let A be a first-order language, let C be a category of countably generated A-structures and some appropriately chosen class of morphisms that includes all embeddings (and hence all isomorphisms). Let A be the full subcategory of C spanned by all finitely generated structures in C. In particular, A is hereditary in the sense that given A e Ob(A), every finitely generated substructure1 of A is an object of A.

1 Recall that substructures of finitely generated structures may not be finitely generated. For example, the free group with 2 generators has a subgroup isomorphic to the free group with infinitely many generators.

We assume that the following holds:

• C has standard colimits of chains;

• every C e Ob (C) is a colimit of some chain Ai — A 2 — ■ ■ ■ in A;

• A has only countably many isomorphism types; and

• A has the joint embedding property (JEP).

For an object C e Ob(C) let age(C) denote the class of all finitely generated objects that embed into C. Note that age(C) C Ob (A) for every C e Ob(C).

We say that C e Ob(C) is a one-point extension of B e Ob(C) if there is an embedding j : B c—y C and an x e C \ j(B) such that C = {j(B) U {x}>c. In that case we write j : B — C or simply B — C.

The following lemmas are immediate consequences of the fact that C is a category of A-structures and that A is spanned by finitely generated objects in C.

Lemma 1.1 (Reachability) (a) For all A, B e Ob (A) and an embedding A — B which is not an isomorphism, there exist an n e N and Ai,...,An e Ob(A) such that A — Ai — A2 <——■■■ — An = B.

(b) For all C, D e Ob(C) and an embedding f : C — D which is not an isomorphism, there exist Ci,C2 ... e Ob(C) such that

is a colimit diagram in C.

Lemma 1.2 Let C, D G Ob(C) be structures such that f : C D and let Ai A2 ^ ... be a chain in A whose colimit is C. Then there exists a chain Bi — B2 — ... in A whose colimit is D and the following diagram commutes

where the curvy arrows are canonical embeddings into the colimits.

Proof Without loss of generality we can assume that C < D, and that Ai < A2 < ... < C, so that C = !€n Ai. Since D is a one-point extension of C, there exists an x e D \ C such that D = {C U {x}>D. Put Bi = {Ai U {x}>D. □

The next lemma is rather obvious, as we assume that colimits are standard.

Lemma 1.3 (Factoring through the colimit of a chain) Let

be a chain in C and let L be its colimit with the canonical embeddings ik : Ck L. Then for every A e Ob and every morphism f : A ^ L there is an n e N and a morphism g : A ^ Cn such that f o g = in. Moreover, if f is an embedding, then so is g.

Lemma 1.4 For every C e Ob(C) we have that age(C) C Ob(^).

Proof Take any C e Ob(C), and let Ai A2 • • • be a chain in A whose colimit is C. Take any B e age(C). Then B C, so by Lemma 1.3 there are n e N and an embedding g : B An such that

Bc—c

Therefore, B An e Ob(A), so the assumption that A is hereditary yields B e Ob(A).

2 Katetov Functors

Definition 2.1 A functor K0 : A ^ C is a Katetov functor if:

• K0 preserves embeddings, that is, if f : A ^ B is an embedding in A, then K0(f) : K0(A) ^ K0(B) is an embedding in C; and

• there is a natural transformation n0 : ID ^ K0 such that for every one-point extension A B where A, B e Ob(A), there is an embedding g : B K0(A) satisfying

A t^L K°(A)

Theorem 2.2 If there exists a Katetov functor K0 : A ^ C then there is a functor K : C ^ C such that:

• K is an extension of K0 (that is, K and K0 coincide on A);

• there is a natural transformation n : ID ^ K which is an extension of n0 (that is, nA = nA whenever A e Ob (A));

• K preserves embeddings.

Proof The obvious candidate for K is the left Kan extension of K0 along the inclusion functor E : A — C (which acts identically on both objects and morphisms of A). To show

that such an extension exists it suffices to show that the diagram (E 4 C) —> A —> C has a colimit in C for every C e Ob(C), where n is the projection functor from the comma category (E 4 C) to A which takes an object (A, h : A — C) of the comma category to its first coordinate A, and acts on morphisms accordingly [24].

Takeany C e Ob(C) andletAC — a^ — ■ ■ ■ be a chain in A whose colimit is C .Let ir : AC — C be the canonical embeddings. Recall that for every B e Ob(A) and every morphism f : B — C there is an n and a morphism fn : B — A^ such that i^ o fn = f (Lemma 1.3):

■"n + l

The diagram (E 4 C) —> A —> C then takes the form

K°(fu)

K°(B)

Let D be the colimit of the chain K0(A^) ^ K0(ACC) ■ ■ ■ with the canonical embeddings D : K 0(AC) ^ D. For each f : B ^ C in A let f' = D ◦ K0(fn) : K 0(B) ^ D:

K°{Acn) ^ K\Acn+1)

K°(fu)

K°(B)

Then it is easy to show that D is the colimit of the diagram (E 4 C) —> A —> C in C. Therefore, K0 has the left Kan extension K along E.

Let us show that K preserves embeddings. Take any embedding f : C — D in C. Let AC — AC — ■ ■ ■ be a chain in A whose colimit is C and let AD — ADD — ■ ■ ■ be a chain in A whose colimit is D. Moreover, let i^ : A^ — C and i D : AD — D be the corresponding canonical embeddings. By Lemma 1.3, for every k there is an nk and a morphism fk : A'C — AD (which is necessarily an embedding) such that

A'jH <—^ C

Without loss of generality nk 's can be chosen in such a way that ni < n2 < .... In the extension we then have

K°(fk) K°(AP)

"fc' k(d) Lnk

whence follows that K(f) is also an embedding.

Analogous argument provides a construction of the natural transformation n ■ ID — K which extends n0. Consider the diagram:

Since C is the colimit of the chain AC

■ ■ ■, there is a unique morphism nc ■ C ^ K(C) to the tip of the competing compatible cone. The morphism nc is clearly an embedding and it is easy to check that all the morphisms nc constitute a natural transformation n ■ ID ^ K. □

We also say that K is a Katetov functor and from now on we denote both K and K0 by K, and both n and n0 by n. An obvious yet important property of K is that all its powers remain Katetov. Specifically, for n e N define nn ■ ID ^ Kn as nC = rlKn-i(c) ◦. ..◦ nK(C) ◦ nc ■ C ^ Kn(C). Then nn is a natural transformation witnessing that Kn is a Katetov functor. We shall elaborate this in Section 3. For now, we state the following important property of finite iterations of K.

Lemma 2.3 Let K ■ A ^ C be a Katetov functor. Then for every embedding g ■ A B, where A, B e Ob (A), there is an n e N and an embedding h ■ B — Kn(A) satisfying h o g = nnA.

Proof If g is an isomorphism, take n = 1 and h = nA o g-1. Assume, therefore, that g is not an isomorphism. Then by Lemma 1.1(a) there exist n e N and Ai,...,An e Ob (A) such that

A — A1 —- A2 <-—■■■<-— An = B.

Fig. 1 The proof of Lemma 2.3

It is easy to see that the diagram in Fig. 1 commutes: the triangles commute by the definition of a Katetov functor, while the parallelograms commute because n is a natural transformation. So, take h = Kn-1(f1) o Kn-2(f2) o ... o K(fn-1 ) o fn. □

2.1 Examples

Below we collect some examples of Katetov functors. The second one shows in particular that Katetov functors (as well as their powers) do not necessarily have the extension property for embeddings between objects of C. In other words, Lemma 2.3 does not hold for embeddings between C-objects.

Example 2.4 A Katetov functor on the category of finite metric spaces with rational distances and nonexpansive mappings

This is a small modification of the original Katetov functor, in order to fit into Frai'sse theory. The details are explained in Section 5 below.

Let P2(X) = {Y C X : |Y | = 2}, andlet Vfm(X) denote the set of all finite subsets of X.

Example 2.5 A Katetov functor on the category of graphs and graph homomorphisms. Let {V, E) be a graph, where E C P2(V). Put K({V, E}) = {V*E*) where

V* = V U Pfin(V),

E* = E U {{v, A}: A e Vfm(V), v e A}.

For a graph homomorphism f : {Vi, Ei) ^ {V2, E2) let f * = K(f) be a mapping from V* to V2* defined by f *(v) = f(v) for v e V1 and f *(A) = f(A) for A e Pfm(V1). Then it is easy to show that f * is a graph homomorphism from {Vi*, E*) to {V2*, E**). Moreover, if f is an embedding, then so is f*.

Now let G be an infinite graph. Let H = G U {v}, where v is connected to all the vertices of G. Note that each vertex of K(G) \ G has a finite degree in K(G). Thus, there is no embedding of H extending nG : G ^ K(G). This shows that K does not have the extension property for embeddings in the bigger category consisting of all countable graphs. The same

holds for Kn for every n > 1 (and even for its «-power), because all "new" vertices in Kn(G) have finite degrees in G.

Example 2.6 A Katetov functor on the category of Kn -free graphs and graph embeddings. Fix an integer n > 3. Let (V, E) be a Kn-free graph, where E is the set of some 2-element subsets of V. Put K((V, E)) = (V*, E*) where

V * = V U V',

V' = {A e Pfin(V) ■ (A,E sec P*(A)) is Kn-1-free}, E* = E U{{v, A} ■A e V ',v e A}.

For a graph embedding f ■ (V1, E1) — (V2, E2) let f * = K(f) be a mapping from V* to V2* defined by f *(v) = f(v) for v e V1 and f *(A) = f(A) for A e V1/. It is easy to show that f * is a graph embedding from (V* E*) to (V**, E*).

Example 2.7 A Katetov functor on the category of digraphs and digraph homomorphisms. Let (V, E) be a digraph, where E C V2 is an irreflexive relation satisfying (x, y) e E ^ (y, x) e E. Put K((V, E)) = (V*, E*) where

V * = V U V',

V' = {(AB) ■ A, B e Pfin(V) such thatA sec B = 0}, E* = E U {(v(AB)) ■ v e V, (AB) e V',v e A} U {((AB, v) ■v e V, (AB) e V',v e B}.

For a digraph homomorphism f ■ (V1, E1) — (V2, E2) let f * = K(f) be a mapping from V! to V2* defined by f *{v) = f(v) for v e V1 and f *((AB)) = (f(A)f(B)) for (AB) e V1. It is easy to show that f * is a digraph homomorphism from (V*, E*) to (V2*, E**). Moreover, if f is an embedding, then so is f *.

Example 2.8 A Katetov functor on the category of all finite linear orders and monotone mappings. For a linear order (A, <) put K((A, <)) = (A*, <*) where

A* = A U A',

A = {(U, V) ■ {U, V} is a partition of A and Vu e U Vv e V (u < v)}, <* = < U {(a, (U, V)) ■ a e U}U {((U, V), a) ■ a e V} UMU1V1), (U*V*))■ V1 sec U* = 0} U{((U, V), (U, V) ■ (U, V) e A'}.

Then it is easy to see that <* is a linear order on A*. For a monotone map f ■ (A1, <1) — (A2, <*) let f * = K(f) be the mapping from A* to A* defined by f *(a) = f(a) for a e A1 and for (U, V) e A1 we put f *((U, V)) = (A* \ W, W) where W = f(V). It is easy to show that f * is a monotone map from (A*, <*) to (A*, <*). Moreover, if f is an embedding, then so is f*.

The description of K((A, <)) in case (A, <) is a countable linear order is more involved. As an illustration, let us just say that K((Q, <)) is of the form Q1 U Q*, where both Q1 and Q* are dense in K((Q, <)) and Q1 serves as a copy of (Q, <) while Q* serves as the set of all one-point extensions of finite subsets of Q1, ordered in a suitable way.

Example 2.9 A Katetov functor on the category of partially ordered sets and monotone mappings. For a partially ordered set {A, <) put K({A, <)) = {A*, <*) where

A* = A U A',

A' = {{U, V) : U, V e Pfin(A) and Vu e U Vv e V (u < v)}, <* = < U {{a, {U, V)) : 3u e U (a < u)} U{{{U, V),a) :3v e V (v < a)} U {{{Ui Vi), {U2V2)) : 3v e Vi 3u e U2 (v < u)} U{{{U, V), {U, V)) : {U, V) e A'}.

It is easy to see that <* is a partial order on A*. For a monotone map f : {Ai, <i) ^ {A2, <2) let f * = K(f) be amapping from A* to A* defined by f *(a) = f(a) for a e Ai and for {U, V) e A we put f *({U, V)) = {f(U)f(V)). Then it is easy to show that f * is a monotone map from {A*, <*) to {A*, <2). Moreover, if f is an embedding, then so is f *.

Example 2.10 A Katetov functor on the category of tournaments and embeddings. Recall that a tournament is a digraph {V,E) such that for every x, y e V exactly one of the possibilities holds: either x = y or (x, y) e E or (y, x) e E.

For a finite set A and a positive integer n let A<n be the set of all sequences {ai,...,ak) of elements of A where k e {0, i,...,n}. In case of k = 0 we actually have the empty sequence {), as we will be careful to distinguish the i-element sequence {a) from a e A. For a sequence s e A<n let |s| denote the length of s. For a tournament T = {V, E), where E C V2, let n = | V | and let T<n be the tournament whose set of vertices is V<n and whose set of edges is defined lexicographically as follows:

• if s and t are sequences such that |s| < |f |, put s ^ t in T<n;

• if s = {si, ...,sk) and t = {ti ,...,tk) are distinct sequences of the same length, find the smallest i such that si = ti and then put s ^ t in T <n if and only if si ^ ti in T.

For a tournament T = {V, E) define K(T) = {V*, E*), where

V* = V U V<n,

E* = E U E(T<n) U {{v, s) : v e V, s e V<n, v appears as an entry ins} U {{s, v) : v e V, s e V<n, v does not appear as an entry ins}.

It is easy to see that {V*, E*) is a tournament realizing all one-point extensions of {V, E). For an embedding f : {Vi, Ei) ^ {V2, E2) let f* = K(f) be the mapping from Vi* to V2* defined by f *(v) = f(v) for v e Vi and for {si,. ..,sk) e Vi<n we put f *({si, ..., sk)) = {f(si), ..., f(sk)). Then it is easy to show that f * is an embedding from {Vi*, E*) to {V2*, E*). Finally, K is a Katetov functor which is witnessed by the obvious natural transformation mapping T = {V, E) to its copy in K(T).

Example 2.11 A Katetov functor on the category of all Boolean algebras. For a finite set A let B(A) denote the finite Boolean algebra whose set of atoms is A. For a finite Boolean algebra B(A) put K(B(A)) = B({0, i} x A) and let nB(A) : B(A) B({0, i} x A) be the unique homomorphism which takes a e A to {0,a)v{i,a) e B({0, i}x A). Clearly, nB(A) is an embedding. Let us define K on homomorphisms between finite Boolean algebras as follows. Let f : B(A) ^ B(A') be a homomorphism and assume that for a e A we have f(a) = \J S(a) for some S(a) C A', with the convention that \J 0 = 0. Then for i e {0, i}

let K(f)((i, a)) = \J({i} x S(a)). This turns K into a functor from the category of finite Boolean algebras into itself which preserves embeddings and such that n ■ ID — K is a natural transformation.

Let us show that K is indeed a Katetov functor. Let j ■ B(A) — B(A'). Then B(A') = (B(A) U {x}) since B(A') is a one-point extension of B(A), so A' = A{x A j(a),x A j(a)}) \ {0}. Let g ■ B(A') — K(B(A)) be the embedding defined on the atoms of B(A') as follows:

• if x A j(a) = j(a) (and consequently x A j(a) = 0) or x A j(a) = 0 (and consequently x A j(a) = j(a)) let g take j(a) to (0,a) V (1,a),

• if x A a = 0 and x A a = 0 let g take x A a to (0, a) and x A a to (1, a),

and which extends to the rest of B(A') in the obvious way. Then it is easy to see that g ◦ j = nB(A).

2.2 Sufficient Conditions for the Existence of Katetov Functors

Let A be a purely relational language, let A be a A-structure, and let B1, B* be A-structures such that A is a substructure of both of them and A = B1 n B*. The free amalgam of the B1, B* over A is the A-structure C with universe B1 U B* such that both B1, B* are substructures of C and for every R e A we have that RC = RB1 U RB* (in other words, no tuple which meets B1 \ A and B* \ A satisfies any relation symbol in A). Following [4], we say that A has the free amalgamation property if every triple A, B1, B* as above has the free amalgam in A. The next result is implicit in [4] (see Definition 3.7 in [4] and the comment that follows).

Theorem 2.12 (implicit in [4]) If A has free amalgamations then a Katetov functor K ■ A — C exists.

The following theorem is a strengthening of this as well as of the main result of [10]. We say that A has one-point extension pushouts [resp. mixed pushouts] in C if for every morphism f ■ A0 — A1 in A and a one-point extension [resp. embedding] g ■ A0 — A* in A there exists a B e Ob (A), an embedding p ■ A1 — B and a morphism q ■ A* — B such that p o f = q o g and this commuting square is a pushout square in the category Chom of all homomorphisms between C-objects.

Note that free amalgamations are particular examples of pushouts. Note also that typical categories of models with embeddings rarely have pushouts. Namely, recall that a pair of morphisms (p, q) provides the pushout of (f, g) if p o f = q o g and for every other pair (p', q') satisfying p' o f = q' o g there exists a unique morphism h such that h o p = p' and h o q = q'. Now, if f = g and (p', q') consists of identities then clearly h cannot be an embedding. That is why, in the definition above, we have to consider pushouts in the category of all homomorphisms.

Lemma 2.13 Suppose

ai b 1 p

is a pushout square in the category Chom of all homomorphisms. If g is a one-point extension then so is p.

Proof Let Bi be the substructure of B generated by p[Ai] U {q(s)}, where s e A2 is such that g[A0] U {s} generates A2. Notice that q[A2] C Bi. In other words, the square

is commutative, where pi and qi denote the same mappings as p and q, respectively. By the universality of a pushout, there is a unique homomorphism h : B ^ Bi such that h o p = pi and h o q = qi. Let hi the composition of h with the inclusion Bi C B. Again by the universality of a pushout, hi : B ^ B is the unique homomorphism satisfying hi o p = p and hi o q = q .It follows that hi = idB and hence Bi = B. This completes the proof. □

It turns out that both variants of the definition above are equivalent. In practice however, it is usually easier to verify the existence of pushouts for one-point extensions.

Proposition 2.14 The following properties are equivalent:

(a) A has the one-point extension pushouts in C.

(b) A has mixed pushouts in C.

Proof Only implication (a) (b) requires a proof. Fix f : Ao ^ Ai and g : Ao ^ A2 as above and assume that g = gn o ■ ■ ■ o gi is the composition of n one-point extensions gi : E( Ei+i, where Ei = Ao, En+i = A2. By Lemma 2.13 we have the following sequence of pushout squares in Chom.

91 ß2C

Qn—l

pi * p2

Clearly, the composition of all these squares is a pushout in

Theorem 2.15 If A has has one-point extension pushouts in C then a Katetov functor K : A^ C exists.

Proof Let us first show that every countable source (A — Bn)nGjq has a pushout in C, where A, Bi, B2,... G Ob (A). Let en : A — Bn be the embeddings in this source. Let P2 G Ob (A) together with the embeddings f2 : B1 — P2 and g2 : B2 — P2 be the pushout of ei and e2. Next, let P3 G Ob (A) together with the embeddings /3 : P2 — P3 and g3 : B3 — P3 be the pushout of /2 o ei and e3. Then, let P4 G Ob (A) together with the embeddings /4 : P3 — P4 and g4 : B4 — P4 be the pushout of /3 o /2 o ei and e4, and so on:

Let P G Ob(C) be the colimit of the chain B1 — P2 — P3 — P4 — • • •. It is easy to show that P is the pushout of the source (A—Bn)nGN.

Let us now construct the Katetov functor as the pushout of all the one-point extensions of an object in A. More precisely, for every A G Ob (A) let us fix embeddings en : A — Bn, where Bi, B2, ... is the list of all the one-point extensions of A, where every isomorphism type is taken exactly once to keep the list countable. Define K(A) to be the pushout of the source (en : A — Bn)n. This is how K acts on objects.

Let us show how K acts on morphisms. Take any morphism h : A — A' in A. Let (ei : A — Bi)iGi be the source consisting of all the one-point extensions of A (where every isomorphism type is taken exactly once), and let let (ej : A' — Bj )jgj be the source consisting of all the one-point extensions of A' (where again, every isomorphism type is taken exactly once). By the assumption, for every i G I there exists an m(i) G J and a morphism hi : Bi — B'm^ such that the following is a pushout square in C:

Now, K(A') is a pushout of the source (ej : A'

Bj)j€j so let us denote the canonical

embeddings Bj

K(A') by ij, j G J. Therefore, (t'm(i) o h

K(A'))i€i is a

compatible cone over the source (ei : A — Bi)iGi, so there is a unique mediating morphism hi: K(A) — K(A'). Then we put K(h) = h. □

Note that the category of graphs and homomorphisms has pushouts, while the category of Kn-free graphs has pushouts of embeddings only. On the other hand, categories like tournaments or linear orderings do not have pushouts, even when considering all homomorphisms.

3 Katetov Construction

Definition 3.1 Let K : C — C be a Katetov functor. A Katetov construction is a chain of the form:

C ^ K(C) K\C) & K\C) • • •

where C e Ob(C). We denote the colimit of this chain by KA(C). An object L e Ob(C) can be obtained by the Katetov construction starting from C if L = KA(C). We say that L can be obtained by the Katetov construction if L = KA(C) for some C e Ob(C).

Note that KA is actually a functor from C into C. Namely, for a morphism f : A — B let KA(f) be the unique morphism KA(A) ^ Km(B) from the colimit of the Katetov construction starting from A to the competitive compatible cone with the tip at Km(B) and morphisms o Kn (f))neN:

K»(f)

K»(B)

It is clear that Km preserves embeddings (the colimit of embeddings is an embedding). Moreover, the canonical embeddings nA : A — Km(A) constitute a natural transformation nw : ID — KA. Thus, we have:

Theorem 3.2 KA : C — C is a Katetov functor.

Recall that a countable structure L is ultrahomogeneous if every isomorphism between two finitely generated substructures of L extends to an automorphism of L. More precisely, L is ultrahomogeneous if for all A, B e age(L), embeddings ja : A — L and jb : B — L, and for every isomorphism f : A — B there is an automorphism f * of L such that

JB o f = f *o ja.

One of the crucial points of the classical Frai'sse theory is the fact that every ultrahomogeneous structure is the Fraisse limit of its age, and every Fraisse limit is ultrahomogeneous.

Analogously, we say that a countable structure L is C-morphism-homogeneous, if every C-morphism between two finitely generated substructures of L extends to a C-endomorphism of L. More precisely, L is C-morphism-homogeneous if for all A, B e age(L), embeddings ja : A — L and jb : B — L, and for every C-morphism f : A — B there is a C-endomorphism f * of L such that jb o f = f * o ja. In particular, if C is the category of all countable A-structures with all homomorphisms between them, instead of saying that L is C-morphism-homogeneous, we say that L is homomorphism-homogeneous. The study of homomorphism-homogeneity was initiated by Cameron &Nesetfil [6].

The first part of the next result can be viewed as an analogy to Banach's contraction principle: iterating a Katetov functor, starting from an arbitrary object, one always "tends" to the Fraisse limit, which can be regarded as a "fixed point" of the Katetov functor.

Theorem 3.3 If there exists a Katetov functor K : A — C, then A is an amalgamation class, it has a Fraisse limit L in C, and L can be obtained by the Katetov construction starting from an arbitrary C e Ob(C). Moreover, L is C-morphism-homogeneous.

Proof Take any C G Ob(C), let

« K\C) & K3(C)

be the Katetov construction starting from C, and let L e Ob (C) be the colimit of this chain. Let in : Kn(C) — L be the canonical embeddings of the colimit diagram.

Let us first show that age(L) = Ob(A). Lemma 1.4 yields age(L) C Ob(A), so let us show that Ob (A) C age(L). Take any B e Ob (A) and let A1 — A2 — • • • be a chain whose colimit is C. Since A has (JEP) there is a D e Ob (A) such that A1 — D ^ B. Lemma 2.3 then ensures that there is an n e N such that D — Kn(Ai). On the other hand, A1 — C implies Kn(A1) — Kn(C). Therefore, B — D— Kn(A1) — Kn(C) — L, so B e age(L). This completes the proof that age(L) = Ob(A).

Next, let us show that L realizes all one-point extensions, that is, let us show that for all A, B e Ob (A) such that A — B and every embedding f : A — L there is an embedding g : B — L such that:

Take any A, B e Ob (A) such that A — B and let f : A — L be an arbitrary embedding. By Lemma 1.3 there is an n e N and an embedding h : A — Kn(C) such that f o h = in. Note that the following diagram commutes:

A Kn{C)

K(A) Kn+1(C)

(the triangle on the left commutes due to the definition of the Katetov functor, the parallelogram in the middle commutes because n is a natural transformation, while the triangle on

the right commutes as part of the colimit diagram for the chain (2)). Let g = in+i o K{h)o j. Having in mind that f = in o h, from the last commuting diagram we immediately get that the diagram (3) commutes for this particular choice of g.

Therefore, L realizes all one-point extensions, so L is an ultrahomogeneous countable structure whose age is Ob (A). Consequently, L is the Frai'sse limit of Ob (A), whence we easily conclude that A is an amalgamation class. Moreover, the Frai'sse limit of A can be obtained by the Katetov construction starting from an arbitrary C e Ob(C).

Finally, let us show that L is C-morphism-homogeneous. Take any A, B e age(L), fix embeddings jA : A ^ L and jB : B L, and let f : A ^ B be an arbitrary morphism. Then

Ku(f) v ' (4)

Having in mind that Km(A) and Km(B) are colimits of Katetov constructions starting from A and B, respectively, we conclude that both Km(A) and Km(B) are isomorphic to L. Since L is ultrahomogeneous, there exist isomorphisms s : Km(A) ^ L and t : Km(B) ^ L such that

Putting diagrams (4) and (5) together we obtain

whence follows that f * = t o Km(f) o s 1 is a C-endomorphism of L which extends f. So, L is C-morphism-homogeneous. □

Consequently, if the Katetov functor is defined on a category of countable A-structures and all homomorphisms between A-structures, the Frai'sse limit of A is both ultrahomoge-neous and homomorphism-homogeneous.

Example 3.4 Let n > 3 be an integer, let Cn be the category of all countable Kn-free graphs together with all graph homomorphisms, and let An be the full subcategory of Cn spanned by all finite Kn-free graphs. Then there does not exist a Katetov functor K : An ^ Cn, for if there were one, the Henson graph Hn - the Frai'sse limit of An - would be homomorphism-homogeneous, and we know this is not the case.

(Proof. Since Hn is universal for all finite Kn-free graphs, it embeds both Kn-i and the star Sn, which is the graph where one vertex is adjacent to n — 1 independent vertices. Let f

be a partial homomorphism of Hn which maps the n — 1 independent vertices of the star Sn onto the vertices of Kn—1. If Hn were homomorphism-homogeneous, f would extend to an endomorphism f * of Hn, so f * applied to the center of the star Sn would produce a vertex adjacent to each of the vertices of Kn—1 inducing thus a Kn in Hn, which is not possible).

Note however that there exists a Katetov functor from the category A'n of all finite Kn-free graphs together with all graph embeddings to the category Cn of all countable Kn-free graphs together with all graph embeddings (see Example 2.6).

3.1 Characterizations of the existence of a Katetov functor

The following theorem gives a necessary and sufficient condition for a Katestov functor to exist. It depends on a condition that resembles the Herwig-Lascar-Solecki property (see [17, 27]).

Definition 3.5 A partial morphism of C e Ob(C) is a triple (A, f, B) where A, B < C are finitely generated and f : A — B is a C-morphism. We say that C e Ob (C) has the morphism extension property in C if for any choice f1,f2,... of partial morphisms of C there exist D e Ob(C) and m1, m.2,... e End(D) such that C is a substructure of D, mi is an extension of fi for all i, and the following coherence conditions are satisfied for all i, j and k:

• if fi = (A, idA, A) then mi = idD,

• if fi is an embedding, then so is mi, and

• if fi o fj = fk then mi o mj = mk.

We say that C has the morphism extension property if every C e Ob(C) has the morphism extension property in C.

Theorem 3.6 The following are equivalent:

(1) there exists a Katetov functor K : A — C;

(2) A has (AP) and C has the morphism extension property;

(3) A has (AP) and the Fraisse limit of A has the morphism extension property in C.

Proof (1) ^ (2): From Theorem 3.3 we know that A is an amalgamation class, it has a Fraisse limit L in C, and L can be obtained by the Katetov construction starting from an arbitrary C e Ob(C). Now, take any C e Ob(C) and let us show that C has the morphism extension property in C. Since L is universal for Ob(C), without loss of generality we can assume that C < L. For every finitely generated A < C fix an isomorphism ja : Km(A) — L such that

a^l ku(a)

cc—^ l

(such an isomorphism exists because L is ultrahomogeneous). Now, for any family (Ai, fi, Bi), i e I, of partial morphisms of C it is easy to see that L together with its endo-

morphisms mi = jßi o K°\fi) o jAl, i e I, is an extension of C and its partial morphisms

fi, i e I: i

tfA. jA. ai k"(ai) l

K"(fi)

bi k»{bi) l

The coherence requirements are satisfied since Km is a functor which preserves embeddings.

(2) ^ (3): Trivial.

(3) ^ (1): Let L be the Frai'sse limit of A. For every A e Ob(A) fix an embedding j A ■ A — L. Then every A-morphism f ■ A — B induces a partial morphism p(f) : ]a(A) — jB(B) of L by p(f) = jB ◦ f ◦ j—1. Since L is a countable structure, there are

only countably many partial morphisms p(f), say, pi, p2,____By the assumption of (3)

there exist D e Ob(C) and mi,m2,... e End(D) such that L is a substructure of D, mi is an extension of pi for all i, and the coherence conditions are satisfied. Let e ■ L < D be the inclusion of L into D.

Define a functor K ■ A — C on objects by K(A) = D and on morphisms by K(f) = mi, where p(f) = pi. Let us show that K is indeed a functor. First, note that K(idA) = idD = idK(A): let p(idA) = pi; since pi = p(idA) = idjA(A) coherence requirements force that mi = idD. Then, let us show that K(g o f) = K(g) o K(f), where f ■ A — B and g ■ B — C. Let k and l be positive integers such that p(f) = pk = jB o f o j-1 and p(g) = pi = jc o g o j-1. Let s be an integer such that ps = jc o g o f o j-1. Then pi o pk = ps, so the coherence requirements imply that mi o mk = ms. Finally, K(g o f) = ms = mi o mk = K(g) o K(f). The coherence requirements also ensure that K preserves embeddings.

Let us now show that the set of arrows nA = e o jA constitutes a natural transformation n ■ ID — K. Take any A-morphism f ■ A — B. Then p(f) = pi = jB o f o j-1 is a partial morphism of L whose extension is mi. This is why the following diagram commutes (where the dashed arrow indicates a partial morphism):

Finally, let us show that K(A) embeds all one-point extensions of A. Let A — B. Then there is an h ■ B — L such that

since L is the Frai'sse limit of A. Therefore,

which concludes the proof.

Note that the Henson graph Hn, n > 3, does not have the morphism extension property with respect to all graph homomorphisms (for otherwise there would be a Katetov functor defined on the category of all finite Kn-free graphs and all graph homomorphisms, and we know that such a functor cannot exist).

The following theorem shows that the existence of a Katetov functor for varieties of algebras understood as categories whose objects are the algebras of the variety and morphisms are embeddings is equivalent to the amalgamation property for the category of finitely generated algebras of the variety.

Theorem 3.7 Let A be an algebraic language and let V be a variety of A-algebras understood as a category whose objects are A-algebras and morphisms are embeddings. Let A be the full subcategory of V spanned by all finitely generated algebras in V and let C be the full subcategory of V spanned by all countably generated algebras in V. Assume additionally that there are only countably many isomorphism types in A Then there exists a Katetov functor K : A ^ C if and only if A is an amalgamation class.

Proof Immediately from Theorem 3.3.

The category of all A-algebras has pushouts [13, §28] and a pushout of finitely generated algebras is finitely generated. Thus if A is an amalgamation class then it has mixed pushouts in C. The statement now follows from Proposition 2.14 and Theorem 2.15. □

Corollary 3.8 A Katetov functor exists for the category of all finite semilattices, the category of all finite lattices and for the category of all finite Boolean algebras.

Proof The proof follows immediately from the fact that all the three classes of algebras are well-known examples of amalgamation classes. □

3.2 Automorphism Groups and Endomorphism Monoids

The existence of a Katetov functor enables us to quickly conclude that the automorphism group of the corresponding Frai'sse limit is universal, as is the monoid of C-endomorphisms. As an immediate consequence of Theorem 3.3 we have:

Corollary 3.9 Let K : A ^ C be a Katetov functor and let L be the Frai'sse limit of A (which exists by Theorem 3.3). Then for every C e Ob(C):

• Aut(C) ^ Aut(L);

• Endc (C) ^ Endc (L).

Proof Since Km is a functor, we immediately get that Aut(C) Aut(K®(C)) via f ^ Km(f) and that Endc(C) ^ Endc (Km(C)) via f ^ Kw(f). But, Km(C) = L due to Theorem 3.3. □

Recall that Endc (X) may be just the set of all embeddings of X into itself, in case other homomorphisms are not in C. This is the case, for example, in the class of Kn-free graphs, where there is no Katetov functor acting on all homomorphisms.

Corollary 3.10 For the following Frai'sse limits L we have that Aut(L) embeds all permutation groups on a countable set:

• the random graph (proved originally in [16]),

• Henson graphs (proved originally in [16]),

• the random digraph,

• the rational Urysohn space (follows also from[29]),

• the random poset,

• the countable atomless Boolean algebra,

• the random semilattice,

• the random lattice,

For the following Fraisse limits L we have that End(L) embeds all transformation monoids on a countable set:

• the random graph (proved originally in [5]),

• the random digraph,

• the rational Urysohn space,

• the random poset (proved originally in [9]),

• the countable atomless Boolean algebra.

Proof Having in mind Corollary 3.9, in each case it suffices to show that the corresponding category C contains a countable structure whose automorphism group embeds Sym(N) and whose endomorphism monoid embeds Nn considered as a transformation monoid. For example, in case of the rational Urysohn space it suffices to consider the metric space (N, d) where d(m, n) = 1 for all m, n e N, while in the case of the random Boolean algebra it suffices to consider the free Boolean algebra on ^o generators. □

For some applications it is important to know whether the embeddings mentioned in Corollary 3.9 above are topological embeddings, when Aut(X) and End(X) are endowed

with the pointwise topology, that is, the topology inherited from the power XX, where X carries the discrete topology. This natural topology makes the composition operation (and the inverse, in case of Aut(X)) continuous. Note that Aut(X) C End(X) are closed in XX (not being a homomorphism is witnessed by a finite set). In case where X is countable, XX is the well-known Baire space, a canonical Polish space, and Aut(X) is isomorphic to a closed subgroup of the countable infinite symmetric group Sx. The importance of such groups is demonstrated in the pioneering work [20] connecting Frai'sse theory with general Ramsey theory and topological dynamics. As we shall see in a moment, every Katetov functor embeds hom-sets preserving their pointwise topology.

Given C-objects X, Y, denote by C(X, Y) the set of all C-morphisms from X to Y, endowed with the pointwise topology, that is, the topology inherited from the product XY with X discrete. Note that a sequence fn e C(X, Y) converges to f e C(X, Y) if and only if for every finite set S C X there is no such that fn \ S = f \ S for every n > no.

Proposition 3.11 Let K ■ C — C be a Katetov functor. Then for every C-objects X, Y, the mapping

C(X, Y) 9 f — K(f ) e C(K(X), K(Y)) is a topological embedding.

Proof From the definition of a Katetov functor, we know that the mapping above (which we also denote by K) is one-to-one, as K(f) can be viewed as an extension of f (the natural transformation n consists of embeddings). Let fn be a sequence in C(X, Y). If K(fn) — K(f ) pointwise, then fn — f pointwise, due to the remark above. Now suppose fn — f pointwise and fix a e K(X). Choose a finite S C X such that the structure A = (S) generated by S has the property that a e K(A), after identifying K(A) with a suitable substructure of K(X) (recall that K(X) is the standard colimit of K(F), where F runs over all finitely generated substructures of X). There is no such that fn \ S = f \ S whenever n > no. Then also fn \ A = f \ A for every n > no. Hence K(fn) \ K(A) = K(f) \ K(A) whenever n > no, showing that fn(a) — f(a) in the discrete topology. Finally, note that the topology on C (X, Y) is always metrizable (and therefore determined by sequences), because X is countably generated and C(X, Y) is homeomorphic (via the restriction operator) to a subspace of YG, where G is a countable set generating X. □

Corollary 3.12 The embeddings appearing in Corollary 3.9 are topological with respect to the pointwise topology.

4 Semigroup Bergman property

Semigroup-theoretic investigations of endomorphism monoids of Frai'sse limits have recently gained a significant momentum. In this section we generalize one of those results (see [8]) to demonstrate the power of the Katetov construction. We prove that in the presence of a Katetov functor, additional mild assumptions ensure that the endomorphism monoid End(L) of the Frai'sse limit L is strongly distorted and its Sierprnski rank is at most five. Applying a result from [25], we conclude that if End(L) is not finitely generated, then it has the Bergman property. The constructions we present here are rather technical, and the reader might find it helpful to get acquainted with the constructions from [8] before going on.

Following [25], we say that a semigroup S is semigroup Cayley bounded with respect to a generating set U if S = U U U2 U ... U Un for some n e N. We say that a semigroup S has the semigroup Bergman property if it is semigroup Cayley bounded with respect to every generating set.

A semigroup S has Sierpinski rank n if n is the least positive integer such that for any countable T C S there exist sj ,...,sn e S such that T C (sj ,...,sn). If no such n exists, the Sierpinski rank of S is said to be infinite. A semigroup S is strongly distorted if there exists a sequence of natural numbers lj ,l2,l-,... and an N e N such that for any sequence aj, a2, a-},... e S there exist si,...,sn e S and a sequence of words wj, w2, w-,... over the alphabet {xj, x2,..., xn} such that |wn| < ln and an = wn(sj,..., sn) for all n.

Lemma 4.1 ([25]) If S is a strongly distorted semigroup which is not finitely generated, then S has the semigroup Bergman property.

It was shown in [26] that End(^), the endomorphism monoid of the random graph, is strongly distorted and hence has the semigroup Bergman property since it is not finitely generated. The idea from [26] was later directly generalized to classes of structures with coproducts in [8]. Here, we present a general treatment in the context of classes for which a Katetov functor exists, and where the (JEP) can be carried out constructively in the sense of the following definition.

Definition 4.2 A category C has natural (JEP) if there exists a covariant functor F : C x C ^ C such that

• for all C, D e Ob(C) there exist embeddings kC : C ^ F(C, D) and pD : D ^ F(C, D), and

• for every pair of morphisms f : C ^ C' and g : D ^ D' the diagram below commutes:

F(C,D) D

F(C', D')

We also say that F is a natural (JEP) functor for C.

A category C has retractive natural (JEP) if C has natural (JEP) and the functor F has the following additional property: for every C e Ob (C) there exist morphisms pC : F(C, C) ^ C such that p*C o pC = idC = ^C ◦ XC.

Remark 4.3 Note that since F is a covariant functor, the following also holds:

• F(idC, idD) = idF(C,D) for all C,D e Ob(C),

• for all fj : Bj ^ Cj, f2 : B2 ^ C2, gj : Cj ^ Dj, g2 : C2 ^ D2 we have F(gj o fj, g2 o f2) = F(gj, g2) o F(fj, f2), and

C and P

Q implies F(A,P)F-^ F(C,Q)

F(h,9i) F(B,R)

F(h,9i)

iX/2,32)

F(D,S)

Example 4.4 Any category with coproducts (such as the category of graphs, posets, digraphs) has retractive natural (JEP): just take F(C, D) to be the coproduct of C and D.

Example 4.5 The category of all countable metric spaces with distances in [o, 1]q = Q sec[o, 1] and nonexpansive mappings has retractive natural (JEP): take F(C, D) to be the disjoint union of C and D where the distance between any point in C and any point in D is 1.

On the other hand, it is easy to show that the category of all countable metric spaces with distances in Q and nonexpansive mappings does not have natural (JEP). Suppose, to the contrary, that there exists a functor F which realizes the natural (JEP) in this category, let U be the rational Urysohn space and let W = F(U, U). Let ao, bo e U be arbitrary but fixed, and let 5 = dw(Xu(ao), PU(bo)). Take any a,b e U, let ca ■ U — U ■ x — a and cb ■ U — U ■ x — b be the constant maps and put $ = F(ca,cb). Then dw(Xu(a), Pu(b)) = dw(^u(ca(ao)), Pu(cb(bo))) = dw($(Xu(ao)),$(pu(bo))) < dw(Xu(ao), Pu(bo)) = 5, because $ is nonexpansive. Now, for ai,a2 e U we have du (ai, a2) = dw(^u(ai),Xu(a2)) < dw(Xu(ai), Pu(b)) + dw(Xu(a2), Pu(b)) < 25. Hence, diam(U) < 25. Contradiction.

Example 4.6 Let A be the language consisting of function symbols and constant symbols only so that A-structures are actually A-algebras, and assume that A contains a constant symbol 1. Then the category of A-algebras has retractive natural (JEP): take F(C, D) to be C x D where XC ■ c — (c, 1d), pD ■ d — (1c, d), XC = n and pD = n2.

Our aim in this section is to prove the following theorem:

Theorem 4.7 Assume that there exists a Katetov functor K ■ A — C and assume that C has retractive natural (JEP). Let L be the Fraisse limit of A (which exists by Theorem 3.3). Assume additionally that there is a retraction r ■ K(L) — L such that r o nL = idL. Then Endc (L) is strongly distorted and its Sierpinski rank is at most 5. Consequently, if Endc (L) is not finitely generated then it has the Bergman property.

The proof of the theorem requires some technical prerequisites. Let us denote the functor which realizes (JEP) in C by (•, •) so that (C, D) denotes its action on objects, and (f, g) its action on morphisms. For objects Ci,C2,C3,...,Cn and morphisms f, g, fi, f2, f3,...,fn of C let

[Ci, C2, C3,..., Cn] = ((((Ci, C2), C3),...), Cn), [fi, f2, f3,..., fn] = ((((fi, f2), f3), . . .), fn),

[f, g]n = [f, g._^g], with [f, g]o = f.

Moreover, let

L1 = L,

Ln = (Ln-i,L) = [L,L,...,L], forn > 2.

Let C denote the colimit of the following chain in C with the canonical embeddings denoted by in:

Let L be the Frai'sse limit of A, which exists by Theorem 3.3. We know that Km(C) = l, so let us fix an isomorphism

a ■ Km(C) L. The following diagram commutes because (•, •) is a natural (JEP) functor:

Li Lo <~

so the following diagram also commutes:

a ¿2 Ai3

-- ¿3C

[Al.idili

[**lml]2

Therefore, there is a compatible cone with the tip at L and the morphisms idL, xL, XL o [xL, idL]i, X*Ll o X*Ll o [xL, idL]2 ... over the chain li — l2 — l3 <—•••. Since C is a colimit of the chain, there is a unique f ■ C — L such that

In particular,

ß o ¿1 = idL.

As the next step in the construction, note that the following diagram commutes (again due to the fact that (•, •) is a natural (JEP) functor):

A £,„

\PL№L\I

[Pi,,idz,]2

a £,.

[Pi,.idz,]2

az,4 ai,5

[/>l>idz,]3

L4 (——>■

Therefore, there is a compatible cone with the tip at C and the morphisms ¿2 о pi, 13 о [PL, idi] 1,14 o[pi, idi]2 ...

• • •. Since C is a colimit of the chain, there is a

over the chain Li — L2 unique a : C — C such that

11 Ьг

or, explicitly,

a о in = i„+i о [pl, idL]„-i, for alln > 1. An easy induction on n then suffices to show that

an о ii = in+i о [pi, idi]n-i о ... о [pi, idi]i о pi, for alln > 1. Also, there is a compatible cone with the tip at C and the morphisms ii о pL, 12 о [pi, idi]i,

13 о [pi, idi]2 such that

over the chain L2 — L3 — L4

so there is a unique т : C — C

or, explicitly,

т о in+i = in о [pi, idi]n-i, for alln > i.

Another easy induction on n suffices to show that

Tn o in+j = ij o p*L o [pL, idL]j o ... o [pL, idiln-j, for alln > L

Let f = (fj, f2,...) be a sequence of C-endomorphisms of L. As the final step, we shall now construct an endomorphism $(f) : C ^ C which encodes the sequence f. Using once more the fact that (•, •) is a natural (JEP) functor, we immediately get that the following diagram commutes:

A L„

[Л.Л]

— l2c

[/b/2,/3]

-- L3C

[/l ,/2 ,/3 ,/4]

so there is a unique ф(f) : C — C such that

or, explicitly,

0(f) ◦ in = in o [fi, fi,..., fn], for all n > 1.

Lemma 4.8 (a) 0(f) o ii = ii o fi;

(b) 0(f) o i2 o Pl = ii o Pl o fi;

(c) 0(f)o in o[pl, idL]n-2 o .. . o [pl, idL]i o PL = in o[pl, idL]n-2 o . . .o[pL, idL]i o PL o fn, for all n > 3.

Proof (a) This is immediate from the construction of 0(f).

(b) It suffices to note that the diagram below commutes. The square on the left commutes because (-, •) is natural, while the square on the right commutes by the construction of 0(f).

u l2 c

l, lo c

(c) This follows by induction on n. Just to illustrate the main ideas (which are straightforward, anyhow) we show the case n = 4. The following diagram commutes:

\plML\T.

[/21/3 if4]

[/1 ,/2 ,/3 ,/4]

[PlA&L\2

The leftmost square commutes because (-, •) is natural, while the rightmost square commutes by the construction of 0(f). To see that the second square in this row commutes, just apply the functor (-, •) to the following two commutative squares (see Remark 4.3):

[/2J3]

The same argument suffices to show that the third square in the row commutes too. □

Lemma 4.9 (a) ß o 0(f) o ii = fi;

(b) ß o Tn o 0(f) o an o ii = fn+i.

Proof In order to make it easier to follow the calculations we underline the expression that is to be reduced in the following step.

(a) f o 0(f ) o 11 = f o 11 o fi = fi, by Lemma 4.8 and (6).

(b)ß o Tn o 4(f) o (an o ¿1) =

5[by (7)] = ß o Tn o 4(f) o ¿n+i o [pL, id^ln-i o ... o [pL, id^li o pL [Lemma 4.8] = ß o Tn o ¿n+i o[pL, idLln-1 o ...o [pL, idLli o PL o fn+i [by (8)] = ß o ¿1 o pL o [pL, idLli o ... o [p*L, idLln-i o

o [pL, idLln-1 o ...o [pL, idLli o PL o fn+i [by (6)] = p*Lo[p*L, idLli o ...o[pL, idLln-1 o[pL, idLln-1 o o [pl, idLln-2 o ...o [pl, idLli o pl o fn+i

= ... = fn+1,

since [pL, idLly o [pl, idLly = idL, for all j. We are now ready to prove Theorem 4.7.

Proof (of Theorem 4.7) We are going to show that End(KC(C)), which is isomorphic to End(L) because L = KC(C), is strongly distorted and that the Sierpihski rank of End(KC(C)) is at most 5. Take any countable sequence fi,f2,... e End(KC(C)), and let us construct a, f, 5, t, (f> e End(KC(C)) as follows, with the notation introduced above.

Let a = nc o ii o a ■ Km(C) — Km(C). We shall construct f ■ Km(C) — Km(C) such that f o nC = a-i o f. Since n is natural, the diagram on the left below commutes, so by taking f i = r o K(f ) we have that the diagram on the right also commutes:

Analogously, the following diagrams also commute where f2 = r o K(fi): Pi T Pi

And so on. We get a sequence of morphisms ßn : Kn (C) ^ L such that

k"(c) l

vkn(c)

kn+1(c)

Since KC(C) is the colimit of the chain

c ^^ k{c)

rvk(c)

k\c) k3(c)

there is a unique mediating morphism fC : KC (C) ^ L such that

In particular, fC o nC = f. Now put f = a 1 o fC.

Finally, let o = KC(o) and t = Kc(t), let = a o fn o a-1, and let 0 = KC(0(g)) where g = (f^, f% ,■■■). Then

fo 0o a = fo Kw(0(g)) o nC o i1 o a [nw is natural] = fo nC o 0(g) o 11 o a [definition of ff] = a-1 o f o 0(g) o 11 o a [Lemma 4.9] = a-1 o f{* o a = f1,

fo t n o 0 o on o a = fo KC(Tn o 0 o on) o n°C o i1 o a [nC is natural] = f o nC o Tn o 0 o on o 11 o a [definition of f ] = a-1 o f o Tn o 0 o on o 11 o a [Lemma 4.9] = a-1 o f^+1 o a = fn+1■

This shows that every fn belongs to the semigroup generated by a, f, o, t and 0, and we uniformly have that the length of the word representing fn is 2n +1. Therefore, End(KC (C)) is strongly distorted and the Sierpihski rank of End(KC(C)) is at most 5. Lemma 4.1 now yields that End(L) has the Bergman property if it is not finitely generated. □

Corollary 4.10 For the following Frai'sse limits L we have that End(L) has the Bergman property:

• the random graph,

• the random digraph,

• the random poset,

• the rational Urysohn sphere (the Fraisse limit of the category of all finite metric spaces with rational distances bounded by 1),

• the countable atomless Boolean algebra.

Proof It is easy to see that each of the categories involved has retractive natural (JEP). In the first four cases the existence of a retraction r : K(L) ^ L such that r o nL = idL where L is the corresponding Fraisse limit follows from the explicit construction of the Katetov functor (Section 2.1).

Let (U, q) denote the Urysohn sphere. Note that each p e K(U) is determined by a finite set F C U in the sense that

Q(p, u) = min^Q(p, x) + q(x, u)) (9)

(see Section 5 for more details, in particular, formula (10)). Note also that enlarging the set F, the equation above remains true, because of the triangle inequality.

Suppose that U C Xo C K(U) is such that Xo \ U is finite and a nonexpansive retraction r : Xo — U has already been defined. Fix p e K(U)\ Xo. Choose a finite set F containing r [X0\U], such that (9) holds. Let A = (X0\U)UF .Then r \ A : A — U is a nonexpansive mapping (which is identity on F) therefore by [22, Thm. 3.18] it extends to a nonexpansive mapping on r : A U {p} — U. Let q = r(p) and let r' = r U idy. We claim that r': Xo U {p} — U is nonexpansive.

Fix u e U \ F. Using (9), we have q(p, u) = q(p, s) + q(s, u) for some s e F. Thus

Q(r'(p), r'(u)) = Q(r(p), u) < Q(r(p), s) + q(s, u) < Q(p, s) + q(s, u) = Q(p, u).

The last inequality follows from the fact that r is nonexpansive on A and r(s) = s. This shows that r' is a nonexpansive extension of r. Easy induction shows the existence of a nonexpansive retraction of K(U) onto U.

Let us finally show that a retraction r : K(L) ^ L also exists in case of the category of finite Boolean algebras. Recall from Example 2.11 that for a finite Boolean algebra B(A) whose set of atoms is A we have the Katetov functor K(B(A)) = B({o, 1}x A) where nB(A) : B(A) B({o, 1} x A) is the unique homomorphism which takes a e A to (o, a) V (1, a) e B({o, 1} x A). It is now easy to see that for each finite Boolean algebra B(A) there is a retraction rB(A) : K(B(A)) — B(A) which takes (i, a) to a (i e {o, 1}) and extends to the rest of K(B(A)) in an obvious way. Clearly, rB(A) ◦ VB(A) = idB(A). Let L be the countable atomless Boolean algebra and let Bx — B2 — ... be a chain of finite Boolean algebras whose colimit is L. Then the colimit of the chain K(B1) — K(B2) — ... is K(L) and the following diagram commutes:

Since K(L) is the colimit of the bottom chain, there is a unique mapping rL : K(L) — L such that the diagram commutes. In particular, rL o nL = idL. D

5 Appendix: The Original Katetov Construction

For the sake of completeness we present the details of Katetov's construction in the case of finite spaces. Actually, we were unable to find any source were Katetov's extensions of metric spaces are explicitly treated as a functor acting on nonexpansive mappings. We remark that all the considerations below are valid in the case of rational metric spaces; the main difference is that the values of the Katetov functor are countable rational metric spaces, instead of complete separable metric spaces.

Given a metric space X we shall denote its metric either by q or by qx . Fix a finite metric space X and denote by K(X) the set of all functions ^X ^ [0, satisfying

\v(xo) - V(x1) \ < Q,(x0,X1) < p(xo) + vfa)

for every xo, X1 e X. Elements of K(X) are called Katetov functions on X. Given a e X, the function~a(x) = q(x, a) is Katetov therefore it is natural to define nx : X ^ K(X) by nx(x) = x. Endow K(X) with the metric

q(0, ft) = max \0(x) - ft(x)\■ xeX

It is easy to see that nx is an isometric embedding. Note that K(X) is a Polish space, being a closed subspace of RX.

We now fix a nonexpansive map f : X ^ Y between finite metric spaces. Given 0 e K(X), define

0f(y) = xmn(QY(y, f(x)) + 0(x))■ (10)

Lemma 5.1 0f e K(Y) for every 0 e K(X). Furthermore, given x e X, we have that 0f (f(x)) < 0(x) and 0f (f(x)) = 0(x) whenever f is an isometric embedding.

Proof Fix yo, yi e Y and assume (yt) = Qy(yt, f(xt)) + 0(xt) for i = 0, 1. Then

0f(yo) < QY(yo,f(xi)) + 0(xi)

< Qj(yo, f(xi)) - $Y(yi, f(xi)) + Qy(yi, f(xi)) + 0(xi)

< QY(yo, yi) + 0f(yi)-

Similarly, 0f (yi) < QY(yo, yi) + &f (yo)- Furthermore, using the fact that f is nonexpansive and 0 is Katetov, we get

QY(yo, yi) < QY(yo, f(xo)) + QY(f(xo), f(xi)) + QY(yi, f(xi))

< Qy (yo, f(xo)) + Qx(xo ,xi) + qy (yi, f(xi))

< QY(yo, f(xo)) + 0(xo) + 0(xi) + qy (yi, f(xi)) = 0f(yo) + 0f(yi).

This shows that 0f is a Katetov function. Inequality 0f (f(x)) < 0(x) is trivial. Finally, suppose f is an isometric embedding and fix x e X. Choose xi e X so that 0f (f(x)) = QY(f(x), f(xi)) + 0(xi). Then

0f(f(x)) = Qx(x,xi) + 0(xi) > 0(x),

because 0 is Katetov. This completes the proof. □

Lemma 5.2 Q(0f, ftf) < q(0, ft) for every 0, ft e K(X). Equality holds whenever f is an isometric embedding.

Proof Fix y e Y. Find x0 e X such that <f (y) = Q(y, f(xo)) + <(x0). Then ff (y) < Q(y, f(xo)) + f(xo), therefore

ff(y) - <f(y) < f(xo) - <(xo) < \f(xo) - <(xo)\ < q(<, f).

By symmetry, <f (y) - ff(y) < q(<, f). Thus \<f(y) - ff(y)\ < Q(<,f) and hence Q(<f, ff) < q(<, f). Finally, if f is an isometric embedding and q(<, f) = \<(xo) -f(xo)\ then, using Lemma 5.1, we get

Q(<f, ff) > \<f(x) - ff(x)\ = \<(x) - f(x)\

for every x e X, which implies that Q(<f, ff) > q(<, f). d

Lemma 5.3 Given nonexpansive mappings f : X ^ Y, g : Y ^ Z between finite metric spaces, it holds that <gof = (<f)g for every < e K(X).

Proof Fix z e Z. We have

(<f)g(z) = min^Qz(z, g(y)) + <f

,(qz(z, g(y)) + QY(y, f(x)) ■

= min \Qz(z,g(y)) + QY(y,f(x)) +

yeY, xeX

> minex(Qz(z' g(y)) + Qz(g(y), gf(x)) + <(x))

> mn(Qz(z, gf(x)) + <(x) = <gof(z). On the other hand, using Lemma 5.1, we get

(<f)g(z) < min(Q(z,gf(x)) + <f(f(x))J

< min(Q(z, gf(x)) + <(x)j = <gof(z).

It is obvious that 0ldx = 0, therefore defining

K (f)(0) =

we obtain a covariant functor K from the category of finite metric spaces into the category of Polish metric spaces, both considered with nonexpansive mappings. Furthermore, K preserves isometric embeddings (by the second part of Lemma 52).

Lemma 5.4 Given a nonexpansive mapping of finite metric spaces f : X ^ Y, the following diagram is commutative.

f K(J)

Proof Fix x e X. We have (K(f) o nx)(x) = K(f)(x) = (x)f and (nY o f)(x) = nY(f(x)) = f(x). It remains to show that (x)f = f(x). We have

cx)f(y) = mn{Q(y, f(t))+q(x, )

> min(Q(y, f(t)) + Q(f(x), f(t) > Q(y, f(x)) = Jlx)Cy)-On the other hand,

x)f(y) < Q(y, f(x)) + Q(x, x) = Q (y, f(x)) = f^Cyy Hence (x)f = f(x). □

The lemma above says that n is a natural transformation from the identity functor into K. The last fact just says that K is a Katetov functor.

Proposition 5.5 Let e : X ^ Y be an isometric embedding such that X is finite and \Y \ X\ = 1. Then there exists an isometric embedding g : Y ^ K(X) such that g o e = nX.

Proof We may assume that Y = X Ujs} and e is the inclusion. Let 0(x) = Qy(x, s). Then 0 is a Katetov function on X and hence, setting g(s) = 0 and g(x) = x for x e X, we obtained the required embedding. □

As we have already mentioned, exactly the same arguments show the existence of a Katetov functor for finite metric spaces with rational distances, leading to the rational Urysohn space. Finally, one can restrict the set of distances to the unit interval [0, 1] obtaining a Katetov functor leading to the Urysohn sphere (or its rational variant). On the other hand, knowing that the category of finite metric spaces has one-point extension pushouts, Theorem 2.15 provides another Katetov functor on the category of finite rational metric spaces. The original Katetov functor is better in the sense that, when working in the category of all finite metric spaces, its values are complete separable metric spaces, which can be viewed as "minimal" spaces realizing all one-point extensions.

6 Conclusion

As we have seen above, the original Katetov construction deals with complete metric spaces, therefore it is formally out of the scope of our model-theoretic approach. The same applies to the recent Ben Yaacov's construction [3] of a Katetov functor on separable Banach spaces, leading to the so-called Gurarii space, the unique universal separable Banach space that is almost homogeneous, namely, isometries between finite-dimensional subspaces can be approximated by bijective isometries of the entire space. Both examples can be presented in the framework of continuous model theory [2]. In the definition of a Katetov functor one would need to relax the extension property, as the Gurarii space satisfies only its approximate variant.

With some effort, one can adapt most of our arguments to categories of continuous models, obtaining in particular the universality result of Uspenskij [29] as well as its counterpart concerning monoids of nonexpansive mappings. We have decided to present the theory of Katetov functors in discrete model-theoretic setting in order to make it more clear and accessible.

It is possible to provide a purely category-theoretic framework for Katetov functors. Actually, the only problem is to formulate the extension property properly, as it involves one-point extensions. Taking into account that every embedding of finitely generated structures is a finite composition of one-point extensions, in general one could consider a fixed family of morphisms generating the given category in this sense. Furthermore, if one wants to generalize homomorphisms of models, then the necessary structure should be a pair consisting of a fixed category K and its subcategory Ko with the same objects as K, such that the arrows of Ko correspond to embeddings. Furthermore, one has to fix a family S generating Ko. Within this setup, a functor K defined on K is declared to be Katetov if there is a natural transformation n from the identity to K such that, among others, the following condition is satisfied: For every K-object a, for every f e S with f : a ^ b, there is g e Ko such that g o f = na, where na : a ^ K(a). As one can see, there is a need to consider a bigger category a K containing K and such that countable chains in Ko have colimits in a K, serving as the range of the functor K and allowing iterations of K. Summarizing, purely category-theoretic generalization of the theory of Katetov functors is possible, yet seems to be much more technical than the theory presented in this work. On the other hand, we believe that such a generalization, (in particular, capturing continuous model theory) definitely deserves developing. Finally, another direction of further research is studying uncountable iterations of Katetov functors, obtaining models of arbitrarily high cardinality that are homogeneous with respect to their finitely generated substructures. This will be done elsewhere.

The results of Section 2 motivate the following natural

Conjecture 6.1 Every Frai'sse limit has the morphism extension property with respect to embeddings.

It turns out that this conjecture is false. Namely, recently Grebi k [14] has found a relational Frai'sse class (with countable infinite relational language) without a Katetov functor for embeddings. It remains open whether there exists a Frai'sse class with a finite language, which is a counterexample to the above conjecture. A particular example for which we do not know the answer is the class of finite groups, where the amalgamation property is not trivial (its Frai'sse limit is Hall's group [15]).

Acknowledgments The authors would like to thank Igor Dolinka and Christian Pech for their helpful comments on the early version of this paper. The authors are grateful to the Referee and the Editor handling the paper for bringing out information on functorial injective replacements, in particular, Reiterman's construction. Research of the first author supported by NCN grant 2oi1/o3/B/ST1/oo419 (Poland). Research of the second author supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.o International License (http://creativecommons.org/licenses/by/4.o/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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