aa r X i v : . [ m a t h . C T ] O c t Metric-enriched categories and approximate Fra¨ıss´elimits
Wies law Kubi´s ∗ Institute of Mathematics, Jan Kochanowski University in Kielce, PolandInstitute of Mathematics, Academy of Sciences of the Czech Republic([email protected]) ¨K10
September 18, 2018
Abstract
We develop the theory of approximate
Fra¨ıss´e limits in the context ofcategories enriched over metric spaces. Among applications, we construct ageneric projection on the Gurari˘ı space and we present a simpler proof of arecent characterization of the pseudo-arc, due to Irwin and Solecki.
MSC (2010)
Primary: 18D20. Secondary: 18A30, 18B30, 03C15, 46B04,54B30, 54F15.
Keywords:
Metric-enriched category, almost homogeneity, Fra¨ıss´e limit, uni-versality.
Contents ∗ Research supported by the NCN grant DEC-2011/03/B/ST1/00419. Applications 17
Suppose that we are given a category K of some sort of embeddings; let us say thatthe objects of K are small . Now, assume that certain sequences in K (i.e. covariantfunctors from the set of natural numbers into K ) have co-limits in a bigger category,denoted by K ; the objects of K will be called big . We would like to have a “generic”sequence in K , whose co-limit (the K -generic object ) will accommodate all K -objects,and which will have the best possible homogeneity property. Specifically, a K -object U is K -homogeneous if given a K -object a , given K -arrows e , e : a → U , there existsan automorphism h : U → U such that h ◦ e = e .Unfortunately, it turns out that there exists a very natural category B fd , namely,all finite-dimensional Banach spaces with linear isometric embeddings, where thegeneric object is the Gurari˘ı space which is only almost homogeneous with respectto B fd . A general back-and-forth type argument says that the generic object isunique, therefore no generic separable Banach space can be B fd -homogeneous.Generic objects are in fact straight generalizations of Fra¨ıss´e limits , well knownin model theory. The example mentioned above would fit into the Fra¨ıss´e theory,however there are continuum many isometric types of finite-dimensional Banachspaces, therefore the existence of a generic sequence is not so obvious.Our aim is to find a general framework for these situations. It turns out thatdealing with a category enriched over metric spaces is a possible solution. On theother hand, referring to the example of Banach spaces, one quickly realizes that theGurari˘ı space cannot be fully explained in the category of isometric embeddings.The reason is precisely the fact that it lacks the homogeneity property: There existtwo 1-dimensional subspaces for which no bijective linear isometry maps the firstone onto the other. Thus, it is necessary to use isomorphic embeddings or, in otherwords, to work in the category of non-expansive linear embeddings between finite-dimensional spaces.The purpose of this work is developing the theory of approximate Fra¨ıss´e limits inthe context of categories enriched over metric spaces. One of the features is thatwe deal with such a category K together with a (usually, very natural) subcategory K ♥ with the same objects as K , such that all K ♥ -sequences have co-limits in K , and K ♥ satisfies some natural assumptions good enough for the existence of a genericsequence. This sequence leads to the Fra¨ıss´e limit U in K , which is almost K -homogeneous in the sense that given a K -object a , given K ♥ -arrows e , e : a → U ,given ε >
0, there exists an automorphism h : U → U in K ♥ such that h ◦ e is ε -close to e . As mentioned above, all these categories are enriched over metric spaces2hich means, roughly speaking, that a distance ̺ ( f, g ) is defined for each pair ofarrows f, g with common domain and common co-domain, allowing us to measurethe “commutativity” of diagrams.A prototype example is the category Ms of metric spaces with 1-Lipschitz mappings,where Ms ♥ consists of all isometric embeddings. Actually, this is the canonicalexample of a category enriched over itself. Another example is the category Comp of1-Lipschitz maps between metric compacta, where the arrows are formally reversed:an arrow from X to Y is a 1-Lipschitz map from Y into X . This is just because wewould like to consider inverse sequences whose limits, in the category of topologicalspaces, are again compact metrizable spaces. Here, Comp ♥ will be the subcategoryof all quotient maps in Comp .One has to admit that every category K is enriched over metric spaces by the zero-one metric. In this case, our main results are actually part of the category-theoreticFra¨ıss´e theory developed in [5] and [15].One has to mention that a parallel research in approximate Fra¨ıss´e theory has beenrecently done by Ben Yaacov [2] in continuous model theory (partially inspired bythe PhD thesis of Schoretsanitis [27]). Ben Yaacov’s main tool is the concept of bi-Katˇetov maps , smartly encoding almost isometric embeddings of metric structures.The concept of categories enriched over metric spaces goes back to Eilenberg &Steenrod [6], although one of the main inspirations for our study comes from a noteof Mioduszewski [21] on ε -commuting diagrams and inverse limits of compact metricspaces.The paper is organized as follows. Section 2 contains the main definitions, likemetric-enriched and norm category, sequences, almost amalgamation property, etc.Section 3 contains the main results, starting with the crucial notion of a Fra¨ıss´esequence, characterizing its existence and proving its main properties. Section 4contains selected applications: a simple description of the Gurari˘ı space, universalprojections, a new point of view on the Cantor set, and finally a discussion of thepseudo-arc, including its new characterization. Let Ms denote the category of metric spaces with non-expansive (i.e., 1-Lipschitz)mappings. A category K is enriched over Ms if for every K -objects a, b there is ametric ̺ on the set of K -arrows K ( a, b ) so that the composition operator is non-expansive on both sides. More precisely, we have(M) ̺ ( f ◦ g, f ◦ g ) ̺ ( f , f ) and ̺ ( h ◦ f , h ◦ f ) ̺ ( f , f )whenever the compositions make sense. This allows us to consider ε -commutativediagrams, with the obvious meaning. Formally, on each hom-set we have a differentmetric, but there is no ambiguity with using always the same letter ̺ . For the sakeof convenience, we allow + ∞ as a possible value of the metric.3ater on, we shall say that K is metric-enriched having in mind that K is enrichedover Ms . From now on, we fix a pair h K ♥ , K i , where K is a metric-enriched category and K ♥ is its subcategory with the same objects.Given f ∈ K , define µ ( f ) = inf { ̺ ( j ◦ f, i ) : i, j ∈ K ♥ } , where only compatible arrows i, j are taken into account. We allow the possibilitythat µ ( f ) = + ∞ , in particular when there are no i, j ∈ K ♥ with dom( j ) = cod( f ) =cod( i ) and dom( i ) = dom( f ). Obviously, K ♥ ⊆ { f ∈ K : µ ( f ) = 0 } , and in typical cases the equality holds. The meaning of µ ( f ) is “measure of distor-tion”. We call µ the norm induced by h K ♥ , K , ̺ i . An arrow f satisfying µ ( f ) = 0 willbe called a . It turns out that the inverse of a 0-isomorphism is a 0-arrow: Proposition 2.1.
Assume h : a → b is an isomorphism. Then µ ( h ) = µ ( h − ) .Proof. Given K ♥ -arrows i : a → c , j : b → c , we have ̺ ( j ◦ h, i ) = ̺ ( j ◦ h, i ◦ h − ◦ h ) ̺ ( j, i ◦ h − ) . By symmetry, ̺ ( j ◦ h, i ) = ̺ ( j, i ◦ h − ). Taking the infimum for all possible compatible K ♥ -arrows i, j we obtain µ ( h ) = µ ( h − ).From now on, the triple h K ♥ , K , ̺ i will be called a normed category . We shall usuallyomit the metric ̺ , just saying that h K ♥ , K i is a normed category. It is clear that thenorm (always denoted by µ ) is determined by the pair h K ♥ , K , ̺ i . Example 2.2.
Let B be the category of all finite-dimensional Banach spaces withlinear operators of norm B ♥ be the subcategory of all isometric embed-dings. It is clear that B is a metric-enriched category, where ̺ ( f, g ) = k f − g k for f, g : X → Y in B .Now assume f : X → Y is a linear operator satisfying( ∗ ) (1 − ε ) k x k k f ( x ) k k x k . We claim that µ ( f ) ε . In fact, consider Z = X ⊕ Y with the norm defined by thefollowing formula: kh x, y ik = inf {k u k X + k v k Y + ε k w k X : h x, y i = h u, v i + h w, − f ( w ) i , u, w ∈ X, v ∈ Y } . i : X → Z , j : Y → Z be the canonical injections. Note that k i − j ◦ f k ε ,just by definition. It remains to check that i , j are isometric embeddings, that is,they belong to the category B ♥ .It is clear that k i k k j k
1. On the other hand, if x = u + w and 0 = v − f ( w )then k u k X + k v k Y + ε k w k X > k u k X + (1 − ε ) k w k X + ε k w k X > k u + w k X = k x k X . This shows that k i ( x ) k = k x k X . A similar calculation shows that k j ( y ) k = k y k Y .Thus, if f satisfies ( ∗ ) then µ ( f ) ε . On the other hand, if k f ( x ) k = 1 − ε for some x with k x k = 1, then given isometric embeddings i , j , we have k i ( x ) − j ( f ( x )) k > |k i ( x ) k − k j ( f ( x )) k| = | − k f ( x ) k| = ε. Finally, we conclude that µ ( f ) = ε , where ε > ∗ ) holds.The example above is taken from [10], where it is proved that the embeddings i, j lead to a universal object in the appropriate category. A slightly more technicalargument is given in [16]. In order to speak about “big” objects, we shall introduce a natural category ofsequences. By this way, “big” objects will be identified with (equivalence classes of)sequences, having in mind their co-limits. For example, every separable completemetric space is the completion of the union of a chain of finite metric spaces, thereforeit can be described in terms of sequences in the category Ms . In general, we considersequences in which the bonding arrows are some sort of monics, therefore it isconvenient to work in a pair of categories. For our applications, it is sufficient todiscuss the case of normed categories.Let h K ♥ , K i be a fixed normed category. A sequence in K is formally a covariantfunctor from the set of natural numbers ω into K .Denote by σ ( K ♥ , K ) the category of all sequences in K ♥ with arrows in K . Thisis indeed a category with arrows being equivalence classes of semi-natural trans-formations. A semi-natural transformation from ~x to ~y is, by definition, a naturaltransformation from ~x to ~y ◦ ϕ for some increasing function ϕ : ω → ω . Slightlyabusing notation, we shall consider transformations (arrows) of sequences, having inmind their equivalence classes. Thus, an arrow from ~x to ~y is a sequence of arrows ~f = { f n } n ∈ ω together with an increasing map ϕ : ω → ω such that for each n < m the diagram y ϕ ( n ) y ϕ ( m ) ϕ ( n ) / / y ϕ ( m ) x nf n O O x mn / / x mf m O O
5s commutative.Since the category K is enriched over Ms , it is natural to allow more arrows in σ ( K ♥ , K ). An approximate arrow from a sequence ~x into a sequence ~y is ~f = { f n } n ∈ ω ⊆ K together with an increasing map ϕ : ω → ω satisfying the followingcondition:( (cid:9) ) For every ε > n such that all diagrams of the form y ϕ ( n ) y ϕ ( m ) ϕ ( n ) / / y ϕ ( m ) x nf n O O x mn / / x mf m O O are ε -commutative, i.e., ̺ ( f m ◦ x mn , y ϕ ( m ) ϕ ( n ) ◦ f n ) < ε whenever n n < m .It is obvious that the composition of approximate arrows is an approximate arrow,therefore σ ( K ♥ , K ) is indeed a category. It turns out that σ ( K ♥ , K ) is naturallymetric-enriched. Indeed, given approximate arrows ~f : ~x → ~y , ~g : ~x → ~y , define( ⇒ ) ̺ ( ~f , ~g ) := lim n →∞ lim m>n ̺ ( y mn ◦ f n , y mn ◦ g n ) , assuming that ~f and ~g are natural transformations. Replacing ~y by its cofinalsubsequence, we can make such assumption, without loss of generality. We need toshow that the limit above exists.Given n < m < ℓ , we have ̺ ( y ℓn ◦ f n , y ℓn ◦ g n ) = ̺ ( y ℓm ◦ y mn ◦ f n , y ℓm ◦ y mn ◦ g n ) ̺ ( y mn ◦ f n , y mn ◦ g n ) , therefore for each n ∈ ω the sequence { ̺ ( y mn ◦ f n , y mn ◦ g n ) } m>n is decreasing. Onthe other hand, given ε >
0, given n n < k < m such that ( (cid:9) ) holds for both { f n } n > n and { g n } n > n , we have that ̺ ( y mn ◦ f n , y mn ◦ g n ) ̺ ( y mk ◦ y kn ◦ f n , y mk ◦ f k ◦ x kn ) + ̺ ( y mk ◦ f k ◦ x kn , y mk ◦ g k ◦ x kn )+ ̺ ( y mk ◦ g k ◦ x kn , y mk ◦ y kn ◦ g n ) ̺ ( y mk ◦ f k , y mk ◦ g k )+ ̺ ( y kn ◦ f n , f k ◦ x kn ) + ̺ ( g k ◦ x kn , y kn ◦ g n ) < ̺ ( y mk ◦ f k , y mk ◦ g k ) + 2 ε. Passing to the limit as m → ∞ , we see that the sequence n lim m>n ̺ ( y mn ◦ f n , y mn ◦ g n ) o n ∈ ω is increasing. This shows that the double limit in ( ⇒ ) exists.6ne should mention that a much more natural definition for ̺ would be( ։ ) ̺ ( ~f , ~g ) = lim n →∞ ̺ ( f n , g n ) . The problem is that this limit may not exist in general. In practice however, weshall always have ̺ ( i ◦ f, i ◦ g ) = ̺ ( f, g )whenever i ∈ K ♥ . To be more precise, when K is a metric-enriched category, anarrow i ∈ K is called a monic (or a monomorphism ) if the above equation holdsfor arbitrary compatible arrows f, g ∈ K . This is an obvious generalization of thenotion of a monic in category theory. In fact, every category is metric-enriched overthe 0-1 metric: ̺ ( f, g ) = 0 iff f = g .In all natural examples of normed categories h K ♥ , K i the subcategory K ♥ consists ofarrows that are monics in K . Thus, we can use ( ։ ) instead of the less natural ( ⇒ )as the definition of ̺ ( ~f , ~g ). It is an easy exercise to check that ̺ is indeed a metricon each hom-set of σ ( K ♥ , K ) and that all composition operators are non-expansive.The important fact is that every normed category h K ♥ , K i naturally embeds into σ ( K ♥ , K ), identifying a K ♥ -object x with the sequence of identities x / / x / / x / / · · · and every K -arrow becomes a natural transformation between sequences of identities.Actually, it may happen that K is not a full subcategory of σ ( K ♥ , K ). Namely, everyCauchy sequence of K -arrows f n : x → y is an approximate arrow from x to y ,regarded as sequences. Now, if K ( x, y ) is not complete, there may be no f : x → y satisfying f = lim n →∞ f n . The problem disappears if all hom-sets of K are completewith respect to the metric ̺ . hhagg In practice, we can partially ignore the construction described above, because usuallythere is a canonical faithful functor from σ ( K ♥ , K ) into a natural category containing K and in which all countable sequences in K ♥ have co-limits. Two relevant examplesare described below. Example 2.3.
Let K be the category of all finite metric spaces with non-expansivemappings and let K ♥ be the subcategory of isometric embeddings. Let C be thecategory of all separable complete metric spaces. Given a K ♥ -sequence ~x , we canidentify it with a chain of finite metric spaces, therefore it is natural to considerlim ~x to be the completion of the union of this chain. This is in fact the co-limit of ~x in the category Ms . In particular, lim ~x = lim ~y , whenever ~x and ~y are equivalent.Furthermore, every approximate arrow ~f : ~x → ~y “converges” to a non-expansivemap F : lim ~x → lim ~y and again it is defined up to an equivalence of approximatearrows. By this way we have defined a canonical faithful functor lim : σ ( K ♥ , K ) → C .7nfortunately, since we are restricted to 1-Lipschitz mappings, the functor F is notonto. The simplest example is as follows.Let X = { , } ∪ {± /n : n ∈ N } with the metric induced from the real line. Let X n = {± /k : k < n } and let Y = { , } . Then X = lim n ∈ ω X n although thecanonical embedding e : Y → X is not the co-limit of any approximate arrow fromthe sequence { X n } n ∈ ω into Y (the space Y can be treated as the infinite constantsequence with identities).Note that if we change C to the category of all countable metric spaces then thecanonical co-limit is just the union of the sequence, however some approximatearrows of sequences would not have co-limits.Concerning applications, the situation where the canonical “co-limiting” functor isnot surjective does not cause any problems, since our main results say about theexistence of certain arrows (or isomorphisms) of sequences only. In the next examplewe have a better situation. Example 2.4.
Let B be, as in Example 2.2, the category of all finite-dimensionalBanach spaces with linear operators of norm B ♥ be the subcategory ofall isometric embeddings.Again, we have a canonical functor lim : σ ( B ♥ , B ) → C , where C is the categoryof all separable Banach spaces with non-expansive linear operators. It turns outthat this functor is surjective. Namely, fix two Banach spaces X = S n ∈ ω X n and Y = S n ∈ ω Y n , where { X n } n ∈ ω and { Y n } n ∈ ω are chains of finite-dimensional spaces.Fix a linear operator T : X → Y such that k T k T n = T ↾ X n . By aneasy induction, we define a sequence of linear operators T ′ n : X n → Y n so that T ′ n +1 extends T ′ n and k T n − T ′ n k < /n for each n ∈ ω . Note that k T ′ n k /n . Define T ′′ n = nn +1 T ′ n . Now ~t = { T ′′ n } n ∈ ω is a sequence of B -arrows and standard calculationshows that k T ′′ n − T n k < /n for every n ∈ ω . Thus ~t is an approximate arrow from { X n } n ∈ ω to { Y n } n ∈ ω with lim ~t = T .It is natural to extend the norm µ to the category of sequences. More precisely,given an approximate arrow ~f : ~x → ~y , we define µ ( ~f ) = lim n →∞ µ ( f n ) . This is indeed well defined, because given ε > n < m as in ( (cid:9) ), wehave that µ ( f n ) µ ( f m ) + ε . The function µ obviously extends the norm of h K ♥ , K i ,although it is formally not a norm, because it is defined in a different way, withoutreferring to any subcategory of σ ( K ♥ , K ). An approximate arrow ~f is a if µ ( ~f ) = 0. We shall be interested in 0-arrows only. Let K be a metric-enriched category. We say that K has almost amalgamationproperty if for every K -arrows f : c → a , g : c → b , for every ε >
0, there exist8 -arrows f ′ : a → w , g ′ : b → w such that the diagram b g ′ / / wc g O O f / / a f ′ O O is ε -commutative, i.e. ̺ ( f ′ ◦ f, g ′ ◦ g ) < ε . We say that K has the strict amalgamationproperty if for each f, g the diagram above is commutative (i.e. no ε is needed).It turns out that almost amalgamations can be moved to the bigger category K .Namely: Proposition 2.5.
Let h K ♥ , K i be with the almost amalgamation property. Then forevery ε, δ > , for every K -arrows f : c → a , g : c → b with µ ( f ) < ε , µ ( g ) < δ ,there exist K ♥ -arrows f ′ : a → w and g ′ : b → w such that the diagram b g ′ / / wc f / / g O O a f ′ O O is ( ε + δ ) -commutative.Proof. Fix η >
0. Find K ♥ -arrows i , j such that ̺ ( j ◦ f, i ) < µ ( f ) + η . Find K ♥ -arrows k , ℓ such that ̺ ( ℓ ◦ g, k ) < µ ( g ) + η . Using the almost amalgamationproperty, find K ♥ -arrows j ′ , ℓ ′ such that ̺ ( j ′ ◦ i, ℓ ′ ◦ k ) < η . Define f ′ := j ′ ◦ j , g ′ := ℓ ′ ◦ ℓ . Then ̺ ( f ′ ◦ f, g ′ ◦ g ) ̺ ( j ′ ◦ j ◦ f, j ′ ◦ i ) + ̺ ( j ′ ◦ i, ℓ ′ ◦ k ) + ̺ ( ℓ ′ ◦ k, ℓ ′ ◦ ℓ ◦ g ) < ̺ ( j ◦ f, i ) + η + ̺ ( k, ℓ ◦ g ) < µ ( f ) + µ ( g ) + 3 η. Thus, it is clear that if η is small enough then ̺ ( f ′ ◦ f, g ′ ◦ g ) < ε + δ .One more property needed later is that every two objects can be “mapped” into acommon one. Namely, we say that a category K is directed if for every a, b ∈ Ob ( K )there is c ∈ Ob ( K ) such that both hom-sets K ( a, c ) and K ( b, c ) are nonempty. Inmodel theory, this is usually called the joint embedding property .In the context of normed categories, we are interested in directedness of the smallercategory. In the presence of amalgamations, this turns out to be equivalent to thedirectedness of the bigger category.Let us say that a category K has the pseudo-amalgamation property if for every K -arrows f : c → a , g : c → b there exists d ∈ Ob ( K ) such that both hom-sets K ( a, d ), K ( b, d ) are nonempty. Proposition 2.6.
Let h K ♥ , K i be a normed category such that K ♥ has the pseudo-amalgamation property. The following statements are equivalent: K ♥ is directed. (b) Given a, b ∈ Ob (cid:0) K ♥ (cid:1) , there exist K -arrows f : a → d , g : b → d such that µ ( f ) < + ∞ and µ ( g ) < + ∞ .Proof. Evidently, (a) = ⇒ (b), because K ♥ and K have the same objects. Suppose(b) holds and fix a, b ∈ Ob (cid:0) K ♥ (cid:1) . Fix f , g as in (b). Since µ ( f ) < + ∞ , there exist K ♥ -arrows i : a → v , j : d → v such that ̺ ( i, j ◦ f ) < + ∞ . Similarly, there exist K ♥ -arrows k : b → w , ℓ : d → w such that ̺ ( k, ℓ ◦ g ) < + ∞ . Finally, using thepseudo-amalgamation property, find z ∈ Ob (cid:0) K ♥ (cid:1) and some K ♥ -arrows j ′ : v → z and ℓ ′ : w → z . The K ♥ -arrows j ′ ◦ i : a → z and ℓ ′ ◦ k : b → z witness the directednessof K ♥ . Proposition 2.7.
Assume h K ♥ , K i is a normed category with the almost amalga-mation property. Then for every compatible K -arrows f, g the following inequalitieshold: ( N ) µ ( f ◦ g ) µ ( f ) + µ ( g ) . ( N ) µ ( g ) µ ( f ) + µ ( f ◦ g ) .Proof. Fix ε > i, j, k, ℓ ∈ K ♥ such that ̺ ( j ◦ f, i ) < µ ( f ) + ε/ ̺ ( ℓ ◦ g, k ) < µ ( g ) + ε/ . Using the almost amalgamation property, we can find i ′ , ℓ ′ ∈ K ♥ such that ̺ ( i ′ ◦ i, ℓ ′ ◦ ℓ ) < ε/ . Combining these inequalities, we obtain that ̺ ( ℓ ′ ◦ k, i ′ ◦ j ) < µ ( f ) + µ ( g ) + ε , whichshows ( N ).A similar argument shows ( N ).It is well-known and easy to see that the category of finite metric spaces withisometric embeddings has the strict amalgamation property. The same result, whoseprecise formulation is given below, holds for Banach spaces. Most likely it belongsto the folklore, although we refer to [1] for a discussion of a more general statement. Proposition 2.8.
Let i : Z → X , j : Z → Y be linear isometric embeddings ofBanach spaces. Then there exist linear isometric embeddings i ′ : X → W , j ′ : Y → W such that Y j ′ / / WZ j O O i / / X i ′ O O is a pushout square in the category of Banach spaces with linear operators of norm . Furthermore, W = ( X ⊕ Y ) / ∆ , where ∆ = {h i ( z ) , − j ( z ) i : z ∈ Z } . Generic sequences
In the paper [15] we introduced and studied sequences leading to universal homo-geneous structures. We now adapt the theory to our setting. As usual, we assumethat h K ♥ , K i is a normed category with the almost amalgamation property.A sequence ~u : ω → K ♥ is Fra¨ıss´e in h K ♥ , K i if it satisfies the following two conditions:(U) For every K ♥ -object x , for every ε >
0, there exist n ∈ ω and a K -arrow f : x → u n such that µ ( f ) < ε .(A) Given ε >
0, given a K ♥ -arrow f : u n → y , there exist m > n and a K -arrow g : y → u m such that µ ( g ) < ε and ̺ ( u mn , g ◦ f ) < ε .Recall that we identify a sequence with all of its cofinal subsequences. It turns outthat the definition above is “correct” because of the almost amalgamation property: Proposition 3.1.
Assume h K ♥ , K i is a normed category with the almost amalgama-tion property. Let ~u be a sequence in K ♥ . The following conditions are equivalent. ( a ) ~u is Fra¨ıss´e in h K ♥ , K i . ( b ) ~u has a cofinal subsequence that is Fra¨ıss´e in h K ♥ , K i . ( c ) Every cofinal subsequence of ~u is Fra¨ıss´e in h K ♥ , K i .Proof. Implications (a) = ⇒ (c) and (c) = ⇒ (b) are obvious. In fact, the almostamalgamation property is used only for showing that (b) = ⇒ (a).Suppose M ⊆ ω is infinite and such that ~u ↾ M is Fra¨ıss´e in h K ♥ , K i . Fix n ∈ ω \ M and fix a K ♥ -arrow f : u n → y . Fix ε >
0. Using the almost amalgamation property,we can find K ♥ -arrows f ′ : u m → w and j : y → w such that m ∈ M , m > n and thediagram u n f ❆❆❆❆❆❆❆❆ u mn / / u m f ′ ! ! ❇❇❇❇❇❇❇❇ y j / / w is ε/ ~u ↾ M is Fra¨ıss´e, there is a K -arrow g : w → u ℓ with ℓ > m , µ ( g ) < ε , and such that the triangle u m f ′ ! ! ❈❈❈❈❈❈❈❈ u ℓm / / u ℓ w g > > ⑤⑤⑤⑤⑤⑤⑤ is ε/ µ ( g ◦ j ) µ ( g ) < ε and ̺ ( g ◦ j ◦ f, u ℓn ) < ε , whichshows that ~u satisfies (A). 11he following characterization of a Fra¨ıss´e sequence will be used later. Proposition 3.2.
Let h K ♥ , K i be a normed category and let ~u be a sequence in K ♥ satisfying (U). Then ~u is Fra¨ıss´e in h K ♥ , K i if and only if it satisfies the followingcondition: (B) Given ε > , given n ∈ ω , given a K -arrow f : u n → y with µ ( f ) < + ∞ , thereexist m > n and a K -arrow g : y → u m such that µ ( g ) < ε and ̺ ( g ◦ f, u mn ) < µ ( f ) + ε. Proof.
It is obvious that (B) implies (A). Suppose ~u is Fra¨ıss´e and choose K ♥ -arrows i : u n → w and j : y → w such that ̺ ( j ◦ f, i ) < µ ( f ) + ε/
2. Using (A), find m > n and a K -arrow k : y → u m such that µ ( k ) < ε and ̺ ( k ◦ i, u mn ) < ε/
2. Define g = k ◦ j .Then µ ( g ) µ ( k ) + µ ( j ) = µ ( k ) < ε and ̺ ( g ◦ f, u mn ) ̺ ( k ◦ j ◦ f, k ◦ i ) + ̺ ( k ◦ i, u mn ) < µ ( f ) + ε/ ε/ µ ( f ) + ε, which shows that ~u satisfies (A). We now turn to the question of existence of a Fra¨ıss´e sequence. Let h K ♥ , K i be anormed category.A subcategory F of K ♥ is dominating in h K ♥ , K i if( D ) Every K ♥ -object has K -arrows into F -objects of arbitrarily small norm. Moreprecisely, given x ∈ Ob (cid:0) K ♥ (cid:1) , given ε >
0, there exists f : x → y such that y ∈ Ob ( F ) and µ ( f ) < ε .( D ) Given ε >
0, a K ♥ -arrow f : a → y such that a ∈ Ob ( F ), there exist a K -arrow g : y → b and an F -arrow u : a → b such that µ ( g ) < ε and ̺ ( g ◦ f, u ) < ε .In some cases, it will be convenient to consider a condition stronger than ( D ),namely:( D ′ ) Given ε >
0, a K ♥ -arrow f : a → y such that a ∈ Ob ( F ), there exists a K -arrow g : y → b such that µ ( g ) < ε and g ◦ f ∈ F .We shall say that F is strongly dominating in h K ♥ , K i if it satisfies ( D ) and ( D ′ ).A normed category h K ♥ , K i is separable if there exists a countable F ⊆ K ♥ that isdominating in h K ♥ , K i .In the next proof, we shall use the simple folklore fact, known as the Rasiowa-Sikorski Lemma: given a directed partially ordered set P , given a countable family { D n } n ∈ ω of cofinal subsets of P , there exists an increasing sequence { p n } n ∈ ω ⊆ P such that p n ∈ D n for every n ∈ ω . 12 heorem 3.3. Let h K ♥ , K i be a directed normed category with the almost amalga-mation property. The following conditions are equivalent: ( a ) h K ♥ , K i is separable. ( b ) h K ♥ , K i has a Fra¨ıss´e sequence.Furthermore, if F is a countable directed dominating subcategory of h K ♥ , K i withthe almost amalgamation property, then there exists a sequence in F that is Fra¨ıss´ein h K ♥ , K i .Proof. Implication (b) = ⇒ (a) is obvious.Assume F ⊆ K ♥ is countable and dominating in h K ♥ , K i . Enlarging F if necessary,we may assume that it is directed and has the almost amalgamation property. Weare going to find a Fra¨ıss´e sequence in F , which by ( D ) and ( D ) must also beFra¨ıss´e in h K ♥ , K i . This will also show the “furthermore” statement.Define the following partially ordered set P : Elements of P are finite sequences in F (i.e. covariant functors from n < ω into F ). The order is end-extension, that is, ~x ~y if ~y ↾ n = ~x , where n = dom( ~x ).Fix n, k ∈ ω and fix an F -arrow f : a → b . We define D f,n,k ⊆ P to be the set of all ~x ∈ P such that dom( ~x ) > n and the following two conditions are satisfied:(1) There exists ℓ < dom( ~x ) such that F ( a, x ℓ ) = ∅ .(2) If a = x n then there exists an F -arrow g : b → x m such that n m < dom( ~x )and ̺ ( g ◦ f, x mn ) < /k .Since F is directed and has the almost amalgamation property, it is clear that allsets of the form D f,n,k are cofinal in P . It is important that there are only countablymany such sets. Thus, by the Rasiowa-Sikorski Lemma, there exists a sequence ~u < ~u < ~u < · · · such that for each suitable triple f, n, k there is r ∈ ω satisfying ~u r ∈ D f,n,k . It isnow rather clear that ~u := S n ∈ ω ~u n is a Fra¨ıss´e sequence in F which, by the remarksabove, is also a Fra¨ıss´e sequence in h K ♥ , K i .Let us note that the second part of Theorem 3.3 may give some additional informa-tion on the structure of the “approximate” Fra¨ıss´e limit associated to the Fra¨ıss´esequence. For example, this is the case with the Gurari˘ı space, where there is acountable dominating subcategory whose objects are precisely the ℓ n ∞ spaces, whichshows that the Gurari˘ı space is a so-called Lindenstrauss space (see [9] for moredetails). 13 .2 Approximate back-and-forth argument We now show that a Fra¨ıss´e sequence is “almost homogeneous” in the sense describedbelow.
Lemma 3.4.
Let h K ♥ , K i be a normed category and let ~u , ~v be Fra¨ıss´e sequences in h K ♥ , K i . Furthermore, let ε > and let h : u → v be a K -arrow with µ ( h ) < ε .Then there exists an approximate isomorphism H : ~u → ~v such that µ ( H ) = 0 andthe diagram ~u H / / ~vu u ∞ O O h / / v v ∞ O O is ε -commutative.Proof. Fix a decreasing sequence of positive reals { ε n } n ∈ ω such that µ ( h ) < ε < ε and 2 ∞ X n =1 ε n < ε − ε . We define inductively sequences of K -arrows f n : u ϕ ( n ) → v ψ ( n ) , g n : v ψ ( n ) → u ϕ ( n +1) such that(1) ϕ ( n ) ψ ( n ) < ϕ ( n + 1);(2) ̺ ( g n ◦ f n , u ϕ ( n +1) ϕ ( n ) ) < ε n ;(3) ̺ ( f n ◦ g n − , v ψ ( n ) ψ ( n − ) < ε n ;(4) µ ( f n ) < ε n and µ ( g n ) < ε n +1 ;We start by setting ϕ (0) = ψ (0) = 0 and f = h . We find g and ϕ (1) by usingcondition (B) of Proposition 3.2.We continue, using condition (B) for both sequences repeatedly. More precisely,having defined f n − and g n − , we first use property (B) of the sequence ~v is Fra¨ıss´e,constructing f n satisfying (3) and with µ ( f n ) < ε n ; next we use the fact that ~u satisfies (B) in order to find g n satisfying (2) and with µ ( g n ) < ε n +1 .We now check that ~f = { f n } n ∈ ω and ~g = { g n } n ∈ ω are approximate arrows. Fix n ∈ ω and observe that ̺ (cid:16) v ψ ( n +1) ψ ( n ) ◦ f n , f n +1 ◦ u ϕ ( n +1) ϕ ( n ) (cid:17) ̺ (cid:16) v ψ ( n +1) ψ ( n ) ◦ f n , f n +1 ◦ g n ◦ f n (cid:17) + ̺ (cid:16) f n +1 ◦ g n ◦ f n , f n +1 ◦ u ϕ ( n +1) ϕ ( n ) (cid:17) ̺ (cid:16) v ψ ( n +1) ψ ( n ) , f n +1 ◦ g n (cid:17) + ̺ (cid:16) g n ◦ f n , u ϕ ( n +1) ϕ ( n ) (cid:17) < ε n +1 + ε n . P n ∈ ω ε n is convergent, we conclude that { f n } n ∈ ω is an approximatearrow from ~u to ~v .By symmetry, we deduce that ~g is an approximate arrow from ~v to ~u . Conditions(2) and (3) tell us that the compositions ~f ◦ ~g and ~g ◦ ~f are equivalent to theidentities, which shows that H := ~f is an isomorphism. Condition (4) ensures usthat µ ( H ) = 0. Finally, recalling that h = f , we obtain ̺ ( v ∞ ◦ h, H ◦ u ∞ ) ∞ X n =0 ̺ (cid:16) v ψ ( n +1) ψ ( n ) ◦ f n , f n +1 ◦ u ϕ ( n +1) ϕ ( n ) (cid:17) < ∞ X n =0 ( ε n + ε n +1 ) = ε + 2 ∞ X n =1 ε n < ε. This completes the proof.The lemma above has two interesting corollaries. Recall that a is anisomorphism H with µ ( H ) = 0. In such a case also µ ( H − ) = 0. Theorem 3.5 (Uniqueness) . A normed category h K ♥ , K i may have at most oneFra¨ıss´e sequence, up to an approximate 0-isomorphism. Theorem 3.6 (Almost homogeneity) . Assume h K ♥ , K i is a normed category withthe almost amalgamation property and with a Fra¨ıss´e sequence ~u . Then for every K ♥ -objects a, b , for every approximate 0-arrows i : a → ~u , j : b → ~u , for every K -arrow f : a → b , for every ε > such that µ ( f ) < ε , there exists an approximate0-isomorphism H : ~u → ~u such that the diagram ~u H / / ~ua i O O f / / b j O O is ε -commutative. Note that the existence of a Fra¨ıss´e sequence automatically implies directedness.
Proof.
Recall that, by definition, i = { i n } n > n , where lim n > n ̺ ( u ∞ n ◦ i n , i ) = 0 andlim n > n µ ( i n ) = 0. The same applies to j . Choose δ > µ ( f ) < ε − δ .Choose k big enough so that(1) ̺ ( u ∞ k ◦ i k , i ) < δ and ̺ ( u ∞ k ◦ j k , j ) < δ holds and µ ( i n ) < δ , µ ( j n ) < δ whenever n > k . Let f = j k ◦ f . Then µ ( f ) µ ( f ) + µ ( j k ) < ε − δ . Using Proposition 2.5, we find K ♥ -arrows f : u k → w and g : u k → w such that(2) ̺ ( g ◦ f , f ◦ i k ) < ε − δ. ~u is Fra¨ıss´e, we find ℓ > k and g : w → u ℓ such that(3) µ ( g ) < δ and ̺ ( g ◦ g , u ℓk ) < δ. Define g = g ◦ f . Then µ ( g ) < δ and the sequences { u n } n > k , { u n } n > ℓ are Fra¨ıss´e,therefore by Lemma 3.4 there exists an approximate 0-isomorphism H : ~u → ~u satisfying(4) ̺ ( u ∞ ℓ ◦ g, H ◦ u ∞ k ) < δ. The situation is described in the following diagram a f (cid:15) (cid:15) f (cid:23) (cid:23) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ i k / / u k f ❆❆❆❆❆❆❆❆ / / · · · / / ~u H (cid:15) (cid:15) w g ❆❆❆❆❆❆❆❆ b j k / / u k g > > ⑥⑥⑥⑥⑥⑥⑥⑥ / / u ℓ / / · · · / / ~u where the first triangle is commutative, the second one is δ -commutative, and theinternal square is ( ε − δ )-commutative. Applying (1), (2), (3), the triangle inequalityand inequalities (M), we obtain ̺ ( j ◦ f , H ◦ i ) ̺ ( j ◦ f, u ∞ k ◦ j k ◦ f ) + ̺ ( u ∞ k ◦ f , u ∞ ℓ ◦ g ◦ g ◦ f )+ ̺ ( u ∞ ℓ ◦ g ◦ g ◦ f , u ∞ ℓ ◦ g ◦ f ◦ i k ) + ̺ ( u ∞ ℓ ◦ g ◦ i k , H ◦ u ∞ k ◦ i k )+ ̺ ( H ◦ u ∞ k ◦ i k , H ◦ i ) ̺ ( j, u ∞ k ◦ j k ) + ̺ ( u ℓk , g ◦ g ) + ̺ ( g ◦ f , f ◦ i k ) + ̺ ( u ∞ ℓ ◦ g, H ◦ u ∞ k )+ ̺ ( u ∞ k ◦ i k , i ) < δ + δ + ( ε − δ ) + δ + δ = ε. This completes the proof.
Below we show that all sequences “embed” into the Fra¨ıss´e sequence. In modeltheory, this is known as universality , although when using the language of cate-gory theory we should avoid confusion saying that the Fra¨ıss´e sequence ~u is weaklyterminal , that is, every other sequence has an approximate 0-arrow into the ~u . Theorem 3.7.
Assume h K ♥ , K i is a normed category with the almost amalgamationproperty and with a Fra¨ıss´e sequence ~u . Then for every sequence ~x in K ♥ there existsan approximate arrow ~f : ~x → ~u such that µ ( ~f ) = 0 . roof. We construct inductively a strictly increasing sequence of natural numbers { k n } n ∈ ω and a sequence of K -arrows f n : x n → u k n satisfying for each n ∈ ω thecondition( ∗ ) ̺ ( u k n +1 k n ◦ f n , f n +1 ◦ x n +1 n ) < · − n and µ ( f n ) < − n .We start by finding f using condition (U) of a Fra¨ıss´e sequence. Fix n ∈ ω andsuppose f n and k n have been defined already.Let ε = 2 − n . Since µ ( f n ) < − n , there exist K ♥ -arrows i : x n → v , j : u k n → v suchthat ̺ ( i, j ◦ f n ) < − n . Next, using the almost amalgamation property, we find K ♥ -arrows k : v → w and ℓ : x n +1 → w such that ̺ ( k ◦ i, ℓ ◦ x n +1 n ) < − n . Finally, using the fact that ~u is Fra¨ıss´e, find k n +1 > k n and a K -arrow g : w → u k n +1 such that µ ( g ) < − ( n +1) and ̺ ( g ◦ k ◦ j, u k n +1 k n ) < − n . The situation is described in the following diagram, where both internal squares andthe triangle are 2 − n -commutative: u k n / / j ! ! ❇❇❇❇❇❇❇ u k n +1 / / · · · v k / / w g O O x n / / f n O O i = = ④④④④④④④④ x n +1 ℓ O O / / · · · Define f n +1 := g ◦ ℓ . Then µ ( f n +1 ) µ ( g ) + µ ( ℓ ) = µ ( g ) < − ( n +1) . The diagramabove shows that condition ( ∗ ) is satisfied. This completes the inductive construc-tion.Finally, ~f = { f n } n ∈ ω is an approximate arrow from ~x to ~u satisfying µ ( ~f ) =lim n →∞ µ ( f n ) = 0. In this section we collect selected applications of Fra¨ıss´e sequences in normed cate-gories.First of all, summarizing the results of Section 3 we arrive at the following algorithmfor finding almost homogeneous structures:1. Choose a directed metric-enriched category K ♥ for which there is a chanceto have a sequence leading to an almost homogeneous object in some biggercategory. 17. Does K ♥ have the almost amalgamation property?3. If the answer to Question 2 is negative, STOP. Otherwise, find a natural biggercategory K with the same objects as K ♥ , such that h K ♥ , K i becomes a normedcategory.4. Is h K ♥ , K i separable?5. If the answer to Question 4 is negative, STOP. Otherwise, there is a uniqueFra¨ıss´e sequence ~u in h K ♥ , K i which is almost homogeneous with respect to K ♥ -objects.6. Interpret σ ( K ♥ , K ) in some natural category, that is, find a canonical “co-limiting” functor from σ ( K ♥ , K ) onto a category C containing K ♥ , in which allcountable K ♥ -sequences have co-limits.7. Finally, U = lim ~u is the desired C -object that is almost homogeneous withrespect to K ♥ . The object U is unique up to a 0-isomorphism and every other C -object has a 0-arrow into U .In the next subsections we demonstrate the use of this algorithm for obtaining sim-pler proofs of the existence and properties of some almost homogeneous structuresas well as for finding new ones.Of course, the prototype example is the category of finite metric spaces with isomet-ric embeddings and non-expansive mappings. When treated as a normed category, itis indeed directed, separable, and has the strict amalgamation property. Its Fra¨ıss´esequence leads to the well known Urysohn space, which is apparently homogeneouswith respect to finite metric spaces. Isometric uniqueness of the Urysohn space fol-lows easily from the homogeneity. We skip the details here, referring the readers tothe survey article of Melleray [20]. The Gurari˘ı space is the unique separable Banach space G satisfying the followingcondition:(G) Given ε >
0, given finite-dimensional Banach spaces X ⊆ Y , given an isometricembedding f : X → G , there exists an ε -isometric embedding g : Y → G suchthat g ↾ X = f .Recall that a linear operator T : E → F is an ε -isometric embedding if (1+ ε ) − k x k < k T x k < (1 + ε ) k x k holds for every x ∈ E .The Gurari˘ı space was constructed by Gurari˘ı [11] in 1966, where it was shown thatit is almost homogeneous in the sense that every linear isometry between finite-dimensional subspaces of G extends to a bijective ε -isometry of G . Furthermore, an18asy back-and-forth argument shows that the Gurari˘ı space is unique up to an ε -isometry for every ε >
0. The question of uniqueness of G up to a linear isometry wasopen for some time, solved by Lusky [19] in 1976, using rather advanced methods.The first completely elementary proof of the isometric uniqueness of G has beenfound very recently by Solecki and the author [16].It turns out that our framework explains both the existence of G , its isometricuniqueness and almost homogeneity with respect to isometries (already shown in[16]). Actually, a better understanding of the Gurari˘ı space was one of the mainmotivations for our study.Namely, as in Examples 2.2 and 2.4, let B and B ♥ be the category of finite-dimensional Banach spaces with linear operators of norm h B ♥ , B i is a normed category.By Proposition 2.8, B ♥ has strict amalgamations. Obviously, it is directed. Let F be the subcategory of B ♥ whose objects are Banach spaces of the form h R n , k · ki ,where the norm is given by the formula(Q) k x k = max i
0, there existfunctionals ϕ , . . . , ϕ m on Y such that k y k Y is ε/ k y k ′ = max i
1. Clearly, g is ε -close to h . Finally, g is 2 ε -close to h , therefore k g ( f ( x )) − x k = k g ( f ( x )) − h ( f ( x )) k ε k x k for x ∈ G n . This shows (A).Since B has the initial object { } , condition (U) follows from (A).It turns out that the converse to the lemma above is also true, because of theuniqueness of the Fra¨ıss´e sequence, up to a linear isometry of its co-limit: Lemma 4.3.
Let ~x , ~y be sequences in B ♥ and let ~f : ~x → ~y be an approximate 0-arrow in σ ( B ♥ , B ) . Let X and Y be the co-limits of ~x , ~y in the category of Banachspaces. Then lim ~f is an isometric embedding of X into Y .Proof. We may assume that ~x = { X n } n ∈ ω , ~y = { Y n } n ∈ ω are chains of (finite-dimensional) Banach spaces and that f n : X n → Y n for each n ∈ ω . Since µ ( f n ) → ε > n such that(1 − ε ) k x k k f n ( x ) k k x k holds for every n > n and for every x ∈ X n . Since { f n } n ∈ ω is an approximatearrow, for each x ∈ X k , the limit f ( x ) = lim n>k f n ( x )exists, because { f n ( x ) } n>k is a Cauchy sequence. Thus f ( x ) is defined on S n ∈ ω X n ,therefore it has a unique extension to a continuous linear operator from X to Y which is an ε -isometric embedding for every ε >
0. Thus, the extension of f is anisometric embedding.Thus, a Fra¨ıss´e sequence in h B ♥ , B i yields an isometrically unique separable Banachspace G , which must be the Gurari˘ı space by Lemma 4.2. As we have mentionedbefore, an elementary proof of its isometric uniqueness [16] (using Example 2.2)was one of the main inspirations for studying Fra¨ıss´e sequences in the context ofmetric-enriched categories. hhagg
20e shall now describe a natural category leading to a universal projection. Thiswill bring an improvement of a result due to Wojtaszczyk [29] and Lusky [18] oncomplemented subspaces of the Gurari˘ı space. In particular, we shall show that thereexists a projection P of the Gurari˘ı space G whose range and kernel are linearlyisometric to G and P is almost homogeneous with respect to finite-dimensionalsubspaces and “contains” all norm one operators between separable Banach spaces.The construction is inspired by a recent work of Pech & Pech [23] involving commacategories. Recall that a Banach space X is complemented in a space Y if X ⊆ Y and there is a bounded linear operator P : Y → X such that P ↾ X = id X ; suchan operator is called a projection . A space X is 1 -complemented in Y if there is aprojection of norm 1 witnessing that X is complemented in Y .From now on, we fix a separable Banach space S . We shall define a category K ( S )as follows. The objects will be linear operators of the form T : E → S satisfying k T k E is a finite-dimensional space. An arrow from T : E → S to T ′ : E ′ → S will be a linear operator f : E → E ′ such that k f k
1. We shall saythat f is a K ( S ) ♥ -arrow if it is an isometric embedding and T ′ ◦ f = T . There isan obvious metric on K ( S ), namely ̺ ( f, g ) = k f − g k , whenever f and g are K ( S )-arrows from T to T ′ . Note that K ( S ) ♥ is a comma category based on S , restrictedto finite-dimensional spaces.It is important to “decode” the norm given by the pair h K ( S ) ♥ , K ( S ) i . This is givenbelow. Lemma 4.4.
Let E , F be finite-dimensional Banach spaces, let T : E → Y , R : F → Y be linear operators of norm and let f : E → F be an injective linear operatorsuch that k f k , k f − k > − ε and k R ◦ f − T k ε , where ε > is fixed.Then there exist isometric embeddings i : E → Z , j : F → Z and a linear operator U : Z → Y , where Z is finite-dimensional, U ◦ i = T , U ◦ j = R , and k j ◦ f − i k ε. In other words, µ ( f ) ε , with respect to h K ( S ) ♥ , K ( S ) i .Proof. Looking at Example 2.2 above, it is not hard to show the following property:(*) Given linear operators p : E → V , q : F → V of norm k p − q ◦ f k ε , the unique linear operator S from Z = E ⊕ F with the normdefined in Example 2.2 into V , satisfying S ◦ i = p and S ◦ j = q , has norm Lemma 4.5.
The pair h K ( S ) ♥ , K ( S ) i is a separable directed normed category suchthat K ( S ) ♥ has the strict amalgamation property.Proof. It is clear that K ( S ) ♥ is directed and has the strict amalgamation property,the latter follows from the fact that the category of Banach spaces admits pushouts(see Proposition 2.8 above). It remains to show that it is separable.21ix a countable dense Q -linear subspace S of S and define F to be the family of all K ( S ) ♥ -arrows f from T : E → S to T ′ : E ′ → S , where E and E ′ are rational Banachspaces (i.e. E = R n , F = R m and the norms are induced by finitely many rationalfunctionals, see formula (Q) above), the operators T , T ′ map rational vectors into S , and f maps rational vectors to rational vectors (a vector in R k is rational if itscoordinates are rational). Obviously, F is countable and it is easy to check (usingLemma 4.4) that it is dominating in h K ( S ) ♥ , K ( S ) i .One has to stress out the importance of Lemma 4.4. Without it, we would onlyknow that h K ( S ) ♥ , K ( S ) i is a normed category, without having any description of itsnorm. In the extreme case, a normed category h K ♥ , K i can have the property that µ ( f ) = + ∞ whenever f ∈ K \ K ♥ .By Lemma 4.5, the normed category h K ( S ) ♥ , K ( S ) i has a Fra¨ıss´e sequence { U n : u n → S } n ∈ ω . Without loss of generality, we may assume that u n ⊆ u n +1 for each n ∈ ω .Denote by G ( S ) the completion of the chain { u n } n ∈ ω and let U S : G ( S ) → S be theunique linear operator satisfying U S ↾ u n = U n for every n ∈ ω .The next result, showing the properties of U S , is a straightforward application ofTheorems 3.6 and 3.7. Theorem 4.6.
The operator U S : G ( S ) → S has the following properties.(1) For every linear operator T : X → S of norm , with X separable, thereexists an isometric embedding j : X → G ( S ) such that U S ◦ j = T .(2) Given ε > , given finite-dimensional spaces E, E ′ ⊆ G ( S ) , given an ε -isometric embedding such that k U S ◦ f − U S ↾ E k < ε , there exists a bijectivelinear isometry h : G ( S ) → G ( S ) such that U S ◦ h = U S and k h ↾ E − f k < ε .(3) U S is right-invertible and its kernel is linearly isometric to the Gurari˘ı space.(4) Conditions (1) and (2) determine U S uniquely, up to a linear isometry.Proof. Property (1) follows from Theorem 3.7, knowing that every operator is the co-limit of a sequence in K ( S ) ♥ . Property (2) follows from Theorem 3.6, having in mindLemma 4.4. The fact that U S is left-invertible follows from (1) applied to the identityid S : S → S . Let G = ker U S . Notice that every operator h : G ( S ) → G ( S ) satisfying U S ◦ h = U S preserves G . Thus, property (2) applied to subspaces of G shows that G is almost homogeneous with respect to its finite-dimensional subspaces. Property(1) applied to zero operators 0 X : X → S shows that G contains isometric copies ofall separable spaces, therefore it is linearly isometric to G . Finally, uniqueness of U S follows from the fact that any of its decomposition into a chain of operators onfinite-dimensional spaces leads to a Fra¨ıss´e sequence in h K ( S ) ♥ , K ( S ) i . Remark . The work [10] contains a construction of a universal linear operatoron the Gurari˘ı space. Again, it is possible to describe it in the language of normedcategories. Namely, the objects of the category K are linear operators T : X → X ,22here X , X are finite-dimensional Banach spaces and k T k
1. An arrow from T to S is a pair h f , f i of linear operators of norm S ◦ f = f ◦ T .Obviously, K ♥ should be the subcategory of all pairs of isometric embeddings. Themain lemma in [10] says that h K ♥ , K i has the strict amalgamation property. Theremaining issues are easily solved and the Fra¨ıss´e sequence in h K ♥ , K i leads to theuniversal (almost homogeneous) linear operator whose domain and range turn outto be isometric to the Gurari˘ı space. The details can be found in [10], actuallywithout referring to normed categories. We are going to revisit some folklore facts about the Cantor set, the simplest reversedFra¨ıss´e limit in the sense of [14]. Some of our ideas are already contained in [4].Namely, let K be the opposite category of all non-expansive maps between nonemptycompact metric spaces and let K ♥ ⊆ K be the opposite category of all quotientmaps. Formally, a K -arrow f from X to Y is a non-expansive map f : Y → X .In particular, a sequence in K ♥ is an inverse sequence of compact metric spaces inwhich all bonding maps are non-expansive.It is clear that K is metric-enriched, by setting ̺ ( f, g ) = max t ∈ K d ( f ( t ) , g ( t )) , where f, g : K → L and d is the metric of L .The following simple fact will be needed later. Lemma 4.8.
Let h X, d i be a metric space and let { f i : X → Y i } i
Suppose µ ( f ) < ε and choose 1-Lipschitz quotient maps p : Z → Y , q : Z → X such that ̺ ( p, f ◦ q ) < ε . Fix y ∈ Y and choose z ∈ Z such that y = p ( z ). Then ε > ̺ ( p, f ◦ q ) > d Y ( p ( z ) , f ( q ( z ))). Since f ( q ( z )) ∈ f ◦ q [ Z ] = f [ X ], we concludethat dist( y, f [ X ]) < ε .Now suppose that f [ X ] is ε -dense in Y and define Z = {h x, y i ∈ X × Y : d Y ( y, f ( x )) ε } . Then Z is a closed subspace of X × Y . Endow it with the maximum metric. Let p : Z → Y , q : Z → X be the projections restricted to Z . It is obvious that q is onto.Given y ∈ Y , there is x ∈ X such that d Y ( y, f ( x )) < ε , therefore h x, y i ∈ Z and y = p ( x, y ). This shows that p is onto. Finally, d Y ( f ( q ( x, y )) , p ( x, y )) = d Y ( f ( x ) , y ) ε ,therefore ̺ ( f ◦ q, p ) ε , showing that µ ( f ) ε .Let F be the subcategory of K ♥ consisting of 1-Lipschitz quotient maps betweenfinite rational metric spaces. Recall that a metric space h X, d i is rational if d [ X × X ] ⊆ Q . It is clear that F is countable, directed and has the strict amalgamationproperty (the metric given by Lemma 4.8 is rational if all the involved metrics arerational). We are going to show that F is dominating in h K ♥ , K i . Below is thecrucial claim. Lemma 4.10.
Let X be a finite metric space and let ε > . Then there exista finite rational metric space Y and a -Lipschitz bijection h : Y → X such that Lip ( h − ) < ε .Proof. Fix η > r = min { d ( s, t ) : s, t ∈ X, s = t } . Let δ = η · r/
2. Wemay assume that X is a subspace of the Urysohn space U . Recall that the rationalUrysohn space U Q is dense in U , therefore for each s ∈ X we can find y s ∈ U Q such that d ( s, y s ) < δ . Let Y = { y s : s ∈ X } and let h : Y → X be the obviousbijection, i.e., h ( y s ) = s for s ∈ X . The space Y is rational, although h may not be1-Lipschitz. We shall later enlarge the metric of Y so that h will become 1-Lipschitz.First, note that given s, t ∈ X we have d ( h ( y s ) , h ( y t )) = d ( s, t ) < d ( y s , y t ) + 2 δ = d ( y s , y t ) + 2 η · r (1 + η ) d ( y s , y t ) . Similarly, d ( h ( y s ) , h ( y t )) > (1 − η ) d ( y s , y t ). It follows that Lip ( h ) < η andLip ( h − ) < (1 − η ) − .Now, suppose that η is rational and satisfies (1 + η )(1 − η ) − ε . Consider Y with the metric d ′ = (1+ η ) d . This is still a rational metric space and h is 1-Lipschitzwith respect to d ′ . Finally, Lip ( h − ) < (1 + η )(1 − η ) − ε with respect to themetric d ′ . 24 roposition 4.11. F is strongly dominating in h K ♥ , K i .Proof. Condition ( D ) follows directly from Lemma 4.10. Fix ε > X and fix a non-expansive quotient map f : K → X , where K is a compact metric space. Choose a finite ε -dense subset of K such that f [ S ] = X .Using Lemma 4.10, find a rational metric space Y and a non-expansive bijection h : Y → S such that Lip ( h − ) < ε . Now consider h as a map from Y to K . ByProposition 4.9, µ ( h ) ε . Finally, f ◦ h is a quotient map of finite rational metricspaces, therefore f ◦ h ∈ F . Corollary 4.12. h K ♥ , K i is a separable directed normed category with the strictamalgamation property. By Theorem 3.3 there exists a sequence ~u in F that is Fra¨ıss´e in h K ♥ , K i .We shall now get rid of the metrics, moving to the category of compact metricspaces. More precisely, define the “co-limiting” functor lim : σ ( K ♥ , K ) → Comp inthe obvious way: lim ~x should be the inverse limit of the sequence ~x in the categoryof topological spaces. In particular, the functor lim forgets the metric structuresof the sequence. Notice that, given any inverse sequence ~X of nonempty compactmetrizable spaces with quotient maps, by Lemma 4.8 and induction, there existcompatible metrics on each X n such that all bonding maps become 1-Lipschitz. Infact, Lemma 4.8 implies more: Given inverse sequences ~K , ~L of nonempty compactmetrizable spaces (with quotient bonding maps), given a natural transformation ~f : ~K → ~L , there exist compatible metrics such that all mappings in the diagram K f (cid:15) (cid:15) K o o f (cid:15) (cid:15) · · · o o K nf n (cid:15) (cid:15) o o K n +1 o o f n +1 (cid:15) (cid:15) · · · o o L L o o · · · o o L n o o L n +1 o o · · · o o become 1-Lipschitz.What is more important, every approximate arrow of sequences “converges” to acontinuous map of their inverse limits. Moreover, an approximate 0-arrow “con-verges” to a quotient map of the limits. This can be checked easily (see also [21]).Let C be the inverse limit of ~u in the category of topological spaces. As one canexpect, C is the Cantor set. Indeed, C is 0-dimensional, being the inverse limit offinite sets. Furthermore, C has the following property:(C) Given a quotient map of nonempty finite sets f : T → S , given a quotient map p : C → S , there exists a quotient map q : C → T such that f ◦ q = p .Indeed, assuming p is given, we find n ∈ ω such that p = p ′ ◦ u ∞ n , where u ∞ n : C → u n is the canonical projection. Since u n is a finite metric space, we find r > d ( s, t ) > r whenever s, t ∈ u n are distinct. Define a metric d on S by setting d ( x, y ) = r iff x = y . Define the same metric on T . By this way, both p ′ and f become 1-Lipschitz and we find q by using condition (A) of the Fra¨ıss´e sequence.25inally, it is well known and easy to check that a compact space satisfying (C) isdense-it-itself, therefore C is homeomorphic to the Cantor set.As an application of Theorem 3.7, we obtain the folklore fact that every compactmetric space is a quotient of the Cantor set. Below is the translation of almosthomogeneity: Theorem 4.13.
Let K be a compact metric space and let p : 2 ω → K , q : 2 ω → K be quotient maps. Then for every ε > there exists a homeomorphism h : 2 ω → ω such that ̺ ( q ◦ h, p ) < ε. It is natural to ask whether ε is needed in the statement above. The answer isnegative, as the following example shows. Namely, let p : 2 ω → I be a quotientmap such that p − (0) is a singleton and p − (1) contains more than one point. Let q = ϕ ◦ p , where ϕ ( t ) = 1 − t . Then there is no homeomorphism h satisfying q ◦ h = p .Finally, let us note that condition (C) characterizes the Cantor set among 0-dimen-sional compact metric spaces only. Indeed, take C = 2 ω × I and observe that forevery quotient map f : C → S with S finite, there exists a unique map g : 2 ω → S such that f = g ◦ p , where p : C → ω is the canonical projection. It follows that C satisfies condition (C), yet C ω . It turns out however that the condition inTheorem 4.13 characterizes the Cantor set: Theorem 4.14.
Assume C is a compact metrizable space that maps onto all poly-hedra and satisfies the assertion of Theorem 4.13 for every polyhedron K . Then C is homeomorphic to the Cantor set.Proof. By Freudenthal’s theorem [7], C is the inverse limit of a sequence of polyhedra { ∆ n } n ∈ ω with quotient maps δ mn : ∆ m → ∆ n ( n < m < ω ). By Theorem 3.5, itsuffices to show that { ∆ n } n ∈ ω is a Fra¨ıss´e sequence in h K ♥ , K i .Fix ε > n ∈ ω and fix a quotient map f : K → ∆ n . By assumption, there existsa quotient map g : C → K . Using the condition of Theorem 4.13 for the maps δ ∞ n and f ◦ q , we find a homeomorphism h : C → C such that ̺ ( f ◦ q ◦ h, δ ∞ n ) < ε/ . The approximation lemma of Eilenberg & Steenrod [6] (see also [21]) says that thereexist m > n and a map g : ∆ m → K such that ̺ ( g ◦ δ ∞ m , q ◦ h ) < ε/ . In particular, g [∆ m ] is ε/ K and hence µ ( g ) < ε . Using the inequalitiesabove, we get ̺ ( δ mn , f ◦ g ) = ̺ ( δ ∞ n , f ◦ g ◦ δ ∞ m ) ̺ ( δ ∞ n , f ◦ q ◦ h ) + ̺ ( f ◦ q ◦ h, f ◦ g ◦ δ ∞ m ) < ε. This shows that { ∆ n } n ∈ ω is Fra¨ıss´e and completes the proof.26 .3 The pseudo-arc We now describe the universal chainable continuum, known under the name pseudo-arc , as the limit of a Fra¨ıss´e sequence in a suitable metric category. Actually, weshall work in a normed category of the form h K , K i , so in particular µ = 0 and onlythe metric ̺ is relevant.Recall that a continuum is a compact connected metrizable space. A continuumis chainable (also called snake-like ) if it is homeomorphic to the limit of an inversesequence of quotient maps of the unit interval. In particular, a chainable continuummaps onto the unit interval and therefore cannot be degenerate.It turns out that we can restrict attention to piece-wise linear maps: Proposition 4.15.
Every inverse sequence of quotient maps of the unit intervalis equivalent to an inverse sequence of piece-wise linear quotient maps of the unitinterval.Proof.
Let ~f = { f mn } n Let ~q = { q mn } n 1. As a corollary, we get: Proposition 4.18. The category I has the strict amalgamation property. Let us note that the Mountain Climbing Theorem fails for arbitrary quotient maps ofthe unit interval; an example was first found by Minagawa (quoted in Homma [12])and independently by Sikorski & Zarankiewicz [28].As one can easily guess, I is separable. A natural countable dominating subcategoryis described below.We say that a quotient map f : I → I is rational if f (0) = 0, f (1) = 1, and there isa decomposition 0 = t < t < · · · < t n = 1 such that { t i } i The category F is countable and dominating in I . It is clear that the category I has a canonical “limiting” functor, which assigns theinverse limit to a sequence. From now on, let ~u be a Fra¨ıss´e sequence in σ I and let P = lim ←− ~u 28n the category of compact spaces. It turns out that P is the pseudo-arc. We explainthe details below.First of all, recall that formally the pseudo-arc is defined to be a hereditarily in-decomposable chainable continuum, which by a result of Bing [3] is known to beunique. Recall that K is indecomposable if it cannot be written as A ∪ B where A, B are proper subcontinua. A continuum K is hereditarily indecomposable if everysubcontinuum of K is indecomposable. Lemma 4.20. Let ~v = { v mn } n First, by Proposition 4.15, we may assume that all maps v mn are piece-wiselinear. Next, by an easy induction, we can “convert” ~v to a sequence in I ♥ . It is clearthat ( ¶ ) is preserved under the equivalence of sequences, therefore the “corrected”sequence still satisfies ( ¶ ). Now it is obvious that ( ¶ ) is equivalent to condition (A)of the Fra¨ıss´e sequence, since the map f can be approximated by a piece-wise linearquotient map which in turn can be made 1-Lipschitz by multiplying the metric of I by a large enough constant.Thus, if ~v satisfies ( ¶ ) then it is equivalent to a Fra¨ıss´e sequence which is uniquelydetermined, showing that lim ←− ~v ≈ P . Finally, if lim ←− ~v ≈ P , then ~v is equivalent to ~u , therefore it satisfies ( ¶ ).Lemma 4.20 allows us to work in the monoidal category of quotient maps of the unitinterval, endowed with the standard metric. This is formally not a metric-enrichedcategory, because the composition operator is not 1-Lipschitz, but in practice thisdoes not cause any trouble. Lemma 4.21. Every non-degenerate subcontinuum of P is homeomorphic to P .Proof. Let K be a subcontinuum of P and let K n = u n [ K ], where u n : P → I is thecanonical n -th projection. From some point on, K n is a non-degenerate interval.Without loss of generality, we may assume that this is the case for all n ∈ ω . Given n < m , let v mn : K m → K n be the restriction of u mn . Then ~v = { v mn } n 1. Composing f with a suitable quotient map,we may assume that f (0) = a and f (1) = b . Extend f to a piece-wise linear map29 ′ : [ − , → I in such a way that f ′ [[ − , , a ] and f ′ [[1 , b, − , 2] as the unit interval with multiplied metric. Thus, using the fact that ~u is Fra¨ıss´e, we find m > n and a piece-wise linear quotient map g : I → [ − , 2] suchthat ̺ ( f ′ ◦ g, u mn ) < ε . Note that g [ F m ] is ε -close to [ a, b ], therefore we can “correct” g so that g [ F m ] = [ a, b ], replacing ε by 3 ε . Finally, g ↾ F m witnesses that ~v satisfiescondition (A) of the definition of a Fra¨ıss´e sequence. Lemma 4.22. P is hereditarily indecomposable.Proof. In view of Lemma 4.21, it suffices to show that P is indecomposable. For thisaim, suppose P = A ∪ B , where A, B are proper subcontinua of P . Let A n = u n [ A ], B n = u n [ B ], where as before, u n is the canonical n th projection. Fix n ∈ ω such thatboth A n and B n are non-degenerate proper intervals. Without loss of generality, wemay assume that A n = [0 , a ], B n = [ b, < b a < 1. Let f : I → I be the tent map, that is, f ( t ) = 2 t for t ∈ [0 , ] and f ( t ) = − t for t ∈ [ , f − [ A n ] = [0 , s ] ∪ [ t, s = a and t = 1 − a > s . Furthermore, f is2-Lipschitz with respect to the standard metric, therefore multiplying the metric inthe domain of f by 2 we obtain a 1-Lipschitz piece-wise linear quotient map. Fix asmall enough ε > 0. Using property (A) of the Fra¨ıss´e sequence, we find m > n anda quotient map g : I → I such that ̺ ( f ◦ g, u mn ) < ε .Notice that J = g [ A m ] is an interval, therefore if ε is small enough then either J ∩ [0 , s ] = ∅ or J ∩ [ t, 1] = ∅ . This means that either [0 , s ] ⊆ g [ B m ] or [ t, ⊆ g [ B m ]. In particular, there is r ∈ B m such that g ( r ) ∈ { , } . On the other hand, d ( f ( g ( r )) , u mn ( r )) = d (0 , u mn ( r )) < ε , where d is the metric in the n th interval of thesequence ~u . Notice that u mn ( r ) ∈ u mn [ B m ] = B n . Thus, if ε < d (0 , b ) then we get acontradiction.The two lemmas above together with Bing’s uniqueness result [3] give Corollary 4.23. P is the pseudo-arc. Applying Theorem 3.7, we obtain another proof of the result of Mioduszewski [22]: Theorem 4.24. Every chainable continuum is a continuous image of the pseudo-arc. Almost homogeneity can be easily strengthened, obtaining the result of Irwin &Solecki [14] which in turn improves a result of Lewis (sketched after Thm. 4.2in [17]): Theorem 4.25. Let K be a chainable continuum with some fixed metric and let p, q : P → K be quotient maps. Then for each ε > there exists a homeomorphism h : P → P such that ̺ ( q ◦ h, p ) < ε . roof. Using the fact that K is the inverse limit of unit intervals, there is a quotientmap f : K → I such that all f -fibers have diameter < ε . A standard compactnessargument shows that f satisfies the following condition:( ⋆ ) ( ∀ s, t ∈ K ) | f ( s ) − f ( t ) | < δ = ⇒ d ( s, t ) < ε. Now let p ′ = f ◦ p and q ′ = f ◦ q . By Lemma 4.16, both p ′ , q ′ come from approximatearrows, therefore by Theorem 3.6, there is a homeomorphism h : P → P such that ̺ ( q ′ ◦ h, p ′ ) < δ . We have the following diagram, in which the upper triangle is δ -commutative and the side-triangles are commutative. P h / / p (cid:15) (cid:15) p ′ (cid:24) (cid:24) ✶✶✶✶✶✶✶✶✶✶✶✶✶ P q (cid:15) (cid:15) q ′ (cid:6) (cid:6) ✌✌✌✌✌✌✌✌✌✌✌✌✌ K f / / I K f o o Finally, condition ( ⋆ ) gives ̺ ( q ◦ h, p ) < ε .As one can guess, the property in Theorem 4.25 characterizes the pseudo-arc amongchainable continua. In fact, this has already been proved by Irwin & Solecki [14]. Theorem 4.26. A chainable continuum K is homeomorphic to P if and only if itsatisfies the following condition: (P) Given ε > , given quotient maps q : K → I , f : I → I , there exists a quotientmap g : K → I such that ̺ ( f ◦ g, q ) < ε .Proof. By Theorem 4.25, P satisfies condition (P). Now suppose that K satisfies(P) and choose a sequence ~v in I whose inverse limit is K . We shall check that ~v isa Fra¨ıss´e sequence. In fact, only condition (A) requires a proof.Fix n ∈ ω , ε > f : I → I . Let, as usual, v n : K → I be the n th canonical projection. Using (P), we find a quotient map p : K → I such that ̺ ( f ◦ p, v n ) < ε/ 2. By Lemma 4.16, there exist m > n and apiece-wise linear quotient map g : I → I such that ̺ ( g ◦ v m , p ) < ε/ 2. Thus we get ̺ ( f ◦ g ◦ v m , v mn ◦ v m ) ̺ ( f ◦ g ◦ v m , f ◦ p ) + ̺ ( f ◦ p, v n ) < ε and so ̺ ( f ◦ g, v mn ) < ε , because v m is a quotient map. This shows (A) and completesthe proof.As noticed at the beginning of the proof above, condition (P) easily follows fromTheorem 4.25. A direct proof of the converse implication would require the approx-imate back-and-forth argument, which is hidden in the proof of Lemma 3.4 above. Remark . As indicated in [14], one can try to prove the existence, properties anduniqueness of the so-called pseudo-circle , thus extending the work of Rogers [26]. Itseems that this can be done using piece-wise linear non-zero degree self-maps of thecircle, for which the uniformization theorem was proved by Rogers [26].31 eferences [1] A. Avil´es, F. Cabello, J. Castillo, M. Gonz´alez, Y. Moreno , Banachspaces of universal disposition , J. Funct. Anal. (2011) 2347–2361 2.3[2] I. Ben Yaacov , Fra¨ıss´e limits of metric structures , preprint, arXiv: 1203.44591[3] R.H. Bing , Concerning hereditarily indecomposable continua , Pacific J. Math. (1951) 43–51 4.3, 4.3[4] F. Cabello S´anchez , On continuous surjections from Cantor set , ExtractaMath., (2004) 35–37 4.2[5] Droste, M.; G¨obel, R. , A categorical theorem on universal objects andits application in abelian group theory and computer science , Proceedings ofthe International Conference on Algebra, Part 3 (Novosibirsk, 1989), 49–74,Contemp. Math., 131, Part 3, Amer. Math. Soc., Providence, RI, 1992. 1[6] S. Eilenberg, N. Steenrod , Foundations of Algebraic Topology , Princeton1952 1, 4.2[7] H. Freudenthal , Entwicklungen von R¨aumen und ihren Gruppen , Composi-tio Math. (1937) 145–234 4.2[8] J. Garbuli´nska , Isometric uniqueness of a complementably universal Banachspace for Schauder decompositions , preprint 4.1[9] J. Garbuli´nska, W. Kubi´s , Remarks on Gurari˘ı spaces , Extracta Math.26(2) (2011/2012) 235–269 3.1[10] J. Garbuli´nska, W. Kubi´s , A universal operator on the Gurari˘ı space ,preprint 2.1, 4.7[11] V.I. Gurari˘ı , Spaces of universal placement, isotropic spaces and a problemof Mazur on rotations of Banach spaces (in Russian), Sibirsk. Mat. ˇZ. (1966)1002–1013 4.1[12] T. Homma , A theorem on continuous functions , K¯odai Math. Semin. Rep. (1952) 13–16 4.3, 4.3[13] J.P. Huneke , Mountain climbing , Trans. Amer. Math. Soc. (1969) 383–391 4.3[14] T. Irwin, S. Solecki , Projective Fra¨ıss´e limits and the pseudo-arc , Trans.Amer. Math. Soc. , no. 7 (2006) 3077–3096 4.2, 4.3, 4.3, 4.273215] W. Kubi´s , Fra¨ıss´e sequences: category-theoretic approach to universal homo-geneous structures , preprint, arXiv: 0711.1683 1, 3[16] W. Kubi´s, S. Solecki , A proof of uniqueness of the Gurari˘ı space , to appearin Israel J. Math. 2.1, 4.1, 4.1[17] W. Lewis , Most maps of the pseudo-arc are homeomorphisms , Proc. Amer.Math. Soc. (1984) 147–154 4.3[18] W. Lusky , On separable Lindenstrauss spaces , J. Functional Analysis 26(1977), no. 2, 103120 4.1[19] W. Lusky , The Gurarij spaces are unique , Arch. Math. (Basel) (1976)627–635 4.1[20] J. Melleray , Some geometric and dynamical properties of the Urysohn space ,Topology Appl. (2008) 1531–1560 4[21] J. Mioduszewski , Mappings of inverse limits , Colloq. Math. (1963) 39–441, 4.2, 4.2[22] J. Mioduszewski , A functional conception of snake-like continua , Fund.Math. (1962/1963) 179–189 4.3[23] C. Pech, M. Pech , Universal homomorphisms, universal structures, and thepolymorphism clones of homogeneous structures , preprint, arXiv: 1302.5692 4.1[24] A. Pe lczy´nski , Universal bases , Studia Math. (1969) 247–268[25] A. Pe lczy´nski , Any separable Banach space with the bounded approximationproperty is a complemented subspace of a Banach space with a basis , StudiaMath. (1971) 239–243[26] J.T. Rogers, Jr. , Pseudo-circles and universal circularly chainable continua ,Illinois J. Math. (1970) 222–237 4.27[27] K. Schoretsanitis , Fra¨ıss´e theory for metric structures . Thesis (Ph.D.)–University of Illinois at Urbana-Champaign, 2007, 82 pp. ISBN: 978-0549-46476-1 1[28] R. Sikorski, K. Zarankiewicz , On uniformization of functions (I) , Fund.Math. (1955) 339–344 4.3, 4.3[29] P. Wojtaszczyk , Some remarks on the Gurarij space , Studia Math.41