Injective objects and retracts of Fra"isse limits

Injectiv e ob jects and retracts of F ra ¨ ıss ´ e limits Wies la w Kubi ´ s ∗ Mathematica l Institute, A cadem y of Sciences of the Czec h Repub lic and Institute of Mathematics, Jan Ko c hanowski Univ ersit y in K ielce , Pola nd kubis@ma th.cas.cz , wkubis@p u.kielce.pl Ma y 2 8, 2022 De dic ate d to the memo ry of my frie nd Pawe l Waszkiewicz Abstract W e presen t a p urely category-theo retic charac terization of retracts of F ra ¨ ıss ´ e limits. F or this aim, w e consider a natural v ersion of injectivit y with resp ect to a p air of categories (a category and its sub category). It turns out that retracts of F ra ¨ ıss´ e limits are pr ecisely the ob jects that are injectiv e relativ ely to suc h a pair. One of the applicatio n s is a c haracterizatio n of non-expansiv e r etracts of Urysohn’s univ ersal metric space. MSC (2010 ): Pr im ary : 18A30, 18B3 5; S econdary: 03C13, 03C50, 08A35. Keywor ds and phr ases: F ra ¨ ıss´ e limit, retract, injectiv e ob ject, amalgamation, pushout. Con te n ts 1 In tr o duction 2 1.1 Categories of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 F ra ¨ ıs s ´ e sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Main result 5 2.1 Remarks on absolute retracts . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Extensions of the main result . . . . . . . . . . . . . . . . . . . . . . . 12 ∗ Research supported in part b y the GA ˇ CR grant No. P 201/12 / 0290. 1 3 Applications 13 3.1 F ra ¨ ıs s ´ e classes and algebraically closed mo dels . . . . . . . . . . . . . . 16 3.2 A note on ho momorphism-homogeneous structures . . . . . . . . . . . . 19 3.3 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Banac h spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5 Linear orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 In tro duction Throughout nearly a ll areas of mathematics one can find certain canonical o b jects that are uniquely determined by their homogeneit y-like prop erties. Historically , the first de- tected example of this sort was t he set of ratio nal num b ers Q , c haracterized b y Can to r as the unique dense coun table linearly ordered set with no end-p oin ts. Another exam- ple is Urysohn’s univ ersal metric space, the unique separable complete metric space U con taining isometric copies of all separable metric spaces and with the property that ev ery isometry b et w een finite subsets of U extends to a bijectiv e isometry o f U . In 1954, Roland F ra ¨ ıss ´ e dev elop ed a general theory in the language of first-ordered structures, curren tly kno wn a s F r a ¨ ıss´ e the ory . After his w ork, sev eral univers al homo geneous struc- tures (called F r a ¨ ıss´ e limits ) had b een iden tified and studied, being imp ortan t ob jects in v arious areas of mathematics, computer science and ev en mathematical phys ics [ 11 ]. One needs to admit that the Urysohn space had b een almost forgott en fo r many y ears, and not link ed to F r a ¨ ıss ´ e theory un til a relativ ely recen t line of researc h dealing with top ological dynamics of automorphism g roups. A notable w ork in this area is [ 19 ]. F or more informa tion on curren t status of F ra ¨ ıs s ´ e theory w e refer to a recen t surv ey a rticle of Macpherson’s [ 26 ]. *** Recall that a F r a ¨ ıss´ e class is a countable class F o f finitely generated mo dels of a fixed first-order language, satisfying the following conditions: (i) G iv en a, b ∈ F t here exists d ∈ F suc h that b ot h a and b em b ed into d (Joint Em b edding Prop erty). (ii) G iv en a, b ∈ F and em b eddings i : c → a , j : c → b , there exist w ∈ F and em b eddings k : a → w and ℓ : b → w suc h that k ◦ i = ℓ ◦ j (Amalga mation Prop ert y). (iii) G iv en a ∈ F , ev ery substructure of a is isomorphic to an elemen t of F . 2 The F r a ¨ ıss´ e lim it of F is a countable mo del U suc h that, up to isomorphism, F = { a ⊆ U : a is a finitely generated substructure of U } and for ev ery isomorphism h : a → b b et w een finitely g enerated substructures of U there exis t s an automorphism H : U → U suc h that H ⊇ h . The latter property is called ultr ah omo geneity . It is a classical theorem of F ra ¨ ıss ´ e [ 13 ] that t he F ra ¨ ıss ´ e limit exists and is unique, up to isomorphism. Unc oun table v ersions of F ra ¨ ıss ´ e limits were studied b y J´ onsson [ 17 , 18 ], supplemen ted by Morley and V a ugh t [ 31 ]. A recen t res ult of D olink a [ 8 ] c haracterizes, under certain assumptions, countable mo dels that are em b eddable as retracts into the F ra ¨ ıss ´ e limit. Namely , he pr o v es that, under certain c onditions on t he F ra ¨ ıss ´ e class, retracts of the F ra ¨ ıss ´ e limit are precisely t he (coun table) algebraically closed mo dels. F ur ther study , in the con text transforma tion semigroups and p erm utat ion group theory has b een done in a r ecen t PhD thes is of McPhee [ 29 ]. The aim of this note is to extend D olink a’s c haracterization to the case of category- theoretic F ra ¨ ıss ´ e limits, at the same time weak ening the assumption on the class of ob jects. In particular, Dolink a’s result assumes that mo dels are finite and for eac h nat- ural n umber n there exist only finitely man y isomorphic types of mo dels generated b y a set of cardinality n . W e do not make an y of these assumptions. O ur result relates retracts of F ra ¨ ıss ´ e limits t o a natural v arian t of injectivit y . Among new applications, w e characterize non-expansiv e retracts of the univ ersal metric space of Urysohn. This metric space is formally not a F ra ¨ ıss ´ e limit, b ecause the category of finite metric spaces is uncoun table. Ho wev er, it can b e “approximated” b y F ra ¨ ıss ´ e limits of countable sub- categories (e.g. b y considering ra tional distances only). Category-theoretic approach to F ra ¨ ıss ´ e limits comes from the author’s pap er [ 20 ], mo- tiv ated b y a m uch ear lier w ork o f Droste a nd G ¨ obel [ 12 ] and by a recen t w ork of Irwin and Soleck i [ 16 ] on pr o jectiv e F ra ¨ ıss´ e limits. In [ 20 ] the ke y notion is a F r a ¨ ı ss ´ e se quenc e rather than a F ra ¨ ıss ´ e limit. This turns out to b e conv enien t, allo wing to work in a single category (corresp onding to finitely g enerated mo dels), forgetting ab out the existence o r non-existence of colimits. In o rder to sp eak ab out retractions, we need to w ork with a pair o f categories, b o th with the same o b jects; the first one allows “embeddings” o nly , while the second one a llo ws all p ossible homomorphisms. 1.1 Categories of sequences Fix a category K . W e shall treat K as the class of arrows , the class of ob jects will b e denoted b y O b ( K ) and the se t of K - arro ws with domain x and co domain y will b e denoted by K ( x, y ). A se quenc e in K is simply a co v ariant functor from ω into K . One can think that the ob jects of K are “small” structures (e.g. finitely generated mo dels of a fixed language). Sequences in K f orm a bigger category of “large” structures. F or category-theoretic notions w e refer to [ 25 ]. 3 W e shall use the fo llo wing con v ention: Sequences in K will b e denoted b y capital letters X , Y , Z , . . . and the o b jects of K will be denoted b y small letters x, y , z , . . . . Fix a sequence X : ω → K . Recall that formally X assigns to eac h natural num b er n an ob ject X ( n ) of K and X assigns a K -ar ro w X ( n, m ) : X ( n ) → X ( m ) fo r eac h pair h n, m i of natural n um b ers suc h that n 6 m . W e shall alwa ys write x n instead of X ( n ) and x m n instead of X ( n, m ). Note t hat b eing a functor imp oses the conditio ns x n n = id x n and x m k = x m ℓ ◦ x ℓ k for k 6 ℓ 6 m . An arrow from a sequenc e X to a sequence Y is, by definition, a natural tra nsformation from the functor X in to the functor Y ◦ ψ , where ψ : ω → ω is increasing (i.e. ψ is a cov aria n t functor from ω to ω ). W e iden tify arro ws that “ p oten tially con v erge” to t he same limit. More precisely , giv en natura l transformations τ 0 and τ 1 from t he sequence X to Y ◦ ψ 0 and Y ◦ ψ 1 , resp ectiv ely , w e say that τ 0 is e quivalent to τ 1 , if the diagram consisting of b oth sequences X , Y t ogether with all arrow s induced b y τ 0 and τ 1 is comm utative. This is indeed an equiv alence relation and it comm utes with the comp osition, t herefore σ K b ecomes a category . In order to illustrate this idea, observ e that ev ery sequence is isomorphic t o its cofinal subsequenc e. Indeed, if X is a sequence and k = { k n } n ∈ ω is a strictly increasing sequence of natural n um b ers, then the σ K -arrow I : X ◦ k → X defined b y I = { i n } n ∈ ω , where i n = id x k n , is an isomorphism. Its in v erse is J = { j n } n ∈ ω , whe re j n = x k m n and m = min { s : k s > n } . The comp osition I ◦ J is formally { j n } n ∈ ω regarded as an arrow from X to X . Clearly , I ◦ J is equiv a len t to the iden tity { id x n } n ∈ ω . Similarly , J ◦ I is equiv alen t to the iden tity of X ◦ k . The original category K may b e regarded as a sub category of σ K , iden tifying an ob ject x with a sequenc e x id x / / x id x / / x id x / / . . . Th us, w e shall alw ays assume that K ⊆ σ K . G iv en a sequence X and n ∈ ω , w e s hall denote by x ∞ n the arrow from x n to X induced b y t he n th ob j ect of X . F ormally , x ∞ n is the equiv a lence class of { x m n } m > n . 1.2 F ra ¨ ıss ´ e sequences F ra ¨ ıss ´ e classes and limits can b e describ ed using categories. Let K b e a fixed category . A F r a ¨ ıss´ e se quenc e in K is a sequen ce U satisfying the following tw o conditions. (F1) F o r ev ery ob ject x in K there exist n ∈ ω and a K -arro w x → u n . (F2) F o r ev ery n ∈ ω and fo r ev ery K -arrow f : u n → y there ex ist m > n and a K - arro w g : y → u m suc h that g ◦ f = u m n . Recall that K has the amal gamation pr op erty if for eve ry K -arro ws f : c → a , g : c → b there exist K -arrows f ′ : a → w , g ′ : b → w satisfying f ′ ◦ f = g ′ ◦ g . A F ra ¨ ıss ´ e sequenc e exists whenev er K has the amalgamation property , the joint em b edding prop ert y and 4 has coun tably man y isomorphic ty p es of ar ro ws. A F ra ¨ ıss ´ e sequence is unique up t o isomorphism. W e refer to [ 20 ] for t he details. A standard induction sho ws that the amalgamatio n prop ert y partially extends to the category of sequenc es. Namely: Prop osition 1.1. Assume K has the amalgamation pr op erty. Then for every σ K -arr ows f : c → A , g : c → B w ith c ∈ Ob ( K ) , ther e exist σ K -arr ows f ′ : A → W , g ′ : B → W satisfying f ′ ◦ f = g ′ ◦ g . Ho we v er, it is shown in [ 20 ] that in general the amalgamation prop erty of K do es not imply the same prop ert y of σ K . No w, let K ⊆ L b e a pair of categories suc h that K has the same ob jects as L . F or in- stance, K is a category of finitely generated mo dels of a fixed language with em b eddings and L allows a ll homomorphisms. Note that σ K a s a subcategory of σ L . W e s ha ll need to deal with the catego ry R = σ ( K , L ) whose ob jects are ω -sequence s in K and t he ar ro ws come f rom L , i.e., Ob ( R ) = Ob ( σ K ) and R ( X , Y ) = σ L ( X, Y ) f or X , Y ∈ Ob ( R ). F or example, if K , L are as ab o v e, σ ( K , L ) is the category of countable mo dels with all p ossible homomorphisms, while σ K is the category of coun table mo dels with em b ed- dings. 2 Main resu lt Let K ⊆ L b e tw o fixed cat egories with the same o b jects. W e sa y that h K , L i has the mixe d a malgamation pr op erty if for ev ery ar ro ws f : c → a and g : c → b suc h that f ∈ K and g ∈ L , there exist arrows f ′ : a → w , g ′ : b → w satisfying f ′ ◦ f = g ′ ◦ g and suc h that g ′ ∈ K and f ′ ∈ L . The mixed amalgamation is described in the following diagram, where / / / / denotes an arrow in K . a f ′ / / w c O O f O O g / / b O O g ′ O O W e sa y that h K , L i has the amalgamate d ex tension pr op erty if for ev ery comm utativ e L -diagram a f / / x c O O i O O / / j / / b g O O with i, j ∈ K , there exist K -arro ws e : x → y , k : a → w , ℓ : b → w and an L -arrow h : w → y suc h tha t e ◦ f = h ◦ k , e ◦ g = h ◦ ℓ and k ◦ i = ℓ ◦ j . That is, t he following 5 diagram is comm utat iv e. y x ? ? e ? ? ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ a f 6 6 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ / / k / / w h 5 5 c O O i O O / / j / / b g G G ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ ✍ O O ℓ O O W e no w define the follow ing axioms for a pa ir o f categories h K , L i , needed for o ur main result. ( H 0 ) K ⊆ L and Ob ( K ) = Ob ( L ). ( H 1 ) K has b oth the amalg amation prop erty and the joint em b edding pro perty . ( H 2 ) h K , L i has the mixed amalgamation prop erty . ( H 3 ) h K , L i has the amalgamated extension prop erty . Definition 2.1. A pair of categories h K , L i has pr op erty ( H ) if it satisfi es conditions ( H 0 ) – ( H 3 ). It is necessary to make some commen ts on the prop erties described ab ov e. Namely , the condition Ob ( K ) = Ob ( L ) can b e remov ed from ( H 0 ), it app ears there for the sak e of con v enience only . The role of L is o ffering more a rro ws than K , some of them will b e needed for construc ting retractions. One can think of the K - arro ws as “ em b eddings”. In most cases, t hese will b e indeed monics. Condition ( H 1 ) is needed mainly f or t he ex- istence and g o o d prop erties of a F ra ¨ ıss ´ e sequence in K . Recall t hat the joint em b edding prop ert y follows from amalgamations, whe nev er K has an initial ob ject (or at least a w eakly initial ob ject). Condition ( H 2 ) will b e crucial for proving that the F ra ¨ ıss ´ e se- quence and its retra cts are K - injectiv e (see the definition b elo w). Finally , the somewhat tec hnical condition ( H 3 ) will b e neede d for the argumen t in the main lemma relating K -injectiv e ob jects with the F ra ¨ ıss ´ e sequence. If L has a terminal ob ject then ( H 3 ) im- plies that K has the amalgamation prop erty . Summarizing, if K has a w eakly initial ob ject a nd L has a t erminal o b ject, then we ma y ignore condition ( H 1 ). Condition ( H 3 ) b ecomes tr ivial if K ha s pushouts in L . W e sa y that K has pushouts in L if for ev ery pair of K -arro ws i : c → a , j : c → b , there exist K -arro ws k : a → w , ℓ : b → w suc h that a / / k / / w c O O i O O / / j / / b O O ℓ O O 6 is a pushout square in L . It is obvious from the definition of a pushout tha t h K , L i ha s the amalg amated extension property (with y = x and e = id x ) whenev er K has pushouts in L . Let us remark that for a ll examples with prop ert y ( H ) app earing in this note, the amalgamated extension prop erty holds with x = y and e = id x (see the definition and diagram ab o ve). Belo w is the crucial notion, whose v ariations app ear of ten in the literature (see, e.g., [ 1 ], where a definition similar t o ours can b e found). Definition 2.2. Let K ⊆ L b e tw o categories with the same ob jects. W e sa y that A ∈ Ob ( σ K ) is K -inje ctive in σ ( K , L ) if for ev ery K -a rro w i : a → b , for eve ry σ ( K , L )- arro w f : a → A , there exists a σ ( K , L )-arr o w f : b → A suc h that f ◦ i = f . a   i   f / / A b f 8 8 q q q q q q q This defin ition obv iously generalizes to an arbitra ry pair of categories K ⊆ R . W e restrict atten tion to the sp ecial case R = σ ( K , L ), since more general v ersions will not b e needed. F ollow ing is a useful criterion for injectivity . Prop osition 2.3. Assume h K , L i has the mixe d amalgamation pr op erty and X ∈ Ob ( σ K ) . T hen X i s K -inje ctive in σ ( K , L ) if a nd o nly if for every n ∈ ω , for ev ery K -arr ow f : x n → y , ther e exis t m > n and an L -arr ow g : y → x m satisfying g ◦ f = x m n . Pr o of. Supp ose X is K -injectiv e and fix a K -a rro w f : x n → y . Applying K -injectivit y for x ∞ n : x n → X , we find G : y → X suc h that G ◦ f = x ∞ n . The arro w G f actors through some L - arro w g : y → x m for some m > n , that is, G = x ∞ m ◦ g . Finally , g ◦ f = x m n . Supp ose no w that X satisfies the condition ab ov e and fix a K -arrow j : a → b and a σ ( K , L )-arrow F : a → X . Then F = x ∞ n ◦ f for some L -arr o w f , where n ∈ ω . Applying the mixed amalgamation prop ert y , find a K - arro w h : x n → y and an L -arrow g : b → y suc h that g ◦ j = h ◦ f . By assumption, there exist m > n and an L -arro w k : y → x m suc h that the following diagra m comm utes. a   j   f / / x n   h   ! ! x m n ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ b g / / y k / / x m Finally , taking G = x ∞ m ◦ k ◦ g , w e get G ◦ j = F . 7 Our in t erest in K - injectivit y comes from the following fact, whic h is an immediate consequenc e of the criterion ab o ve. Prop osition 2.4. Assume h K , L i has the mixe d amalga mation pr op erty and U is a F r a ¨ ıss´ e se quenc e in K . Then U is K -inje ctive i n σ ( K , L ) . W e shall need the follo wing “injectiv e” v ersion of amalgamated extension prop erty . Lemma 2.5. Assume h K , L i satisfies ( H ) and X ∈ Ob ( σ K ) is K -inje ctive in σ ( K , L ) . Then for every K - arr ows i : c → a , j : c → b and for every σ ( K , L ) -arr ows F : a → X , G : b → X such that F ◦ i = G ◦ j , ther e ex ist K -arr ows k : a → w , ℓ : b → w and a σ ( K , L ) -arr ow H : w → X s uch that the diagr am a   k   ❅ ❅ ❅ ❅ ❅ ❅ ❅ F * * ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ c ? ? i ? ?          j   ❂ ❂ ❂ ❂ ❂ ❂ ❂ w H / / X b ? ? ℓ ? ?        G 5 5 ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ c ommutes. Pr o of. Find n suc h that F = x ∞ n ◦ f and G = x ∞ n ◦ g for some L -arrows f , g , where x ∞ n : x n → X is the canonical arro w induced by t he n th ob ject of the se quence X . Using prop erty ( H 3 ), w e find K -arrows k : a → w , ℓ : b → w , e : x n → y and an L -arro w h : w → y suc h tha t h ◦ k = e ◦ f and h ◦ ℓ = e ◦ g . Using the K - injectivit y of X w e can find a σ ( K , L )-arrow P : y → X such that P ◦ e = x ∞ n . Let H = P ◦ h . Then H ◦ k = P ◦ h ◦ k = P ◦ e ◦ f = x ∞ n ◦ f = F . Similarly , H ◦ ℓ = G . The follow ing lemma is crucial. Lemma 2.6. Assume h K , L i ha s p r op erty ( H ) and A is a K -inje ctive obje ct in σ ( K , L ) . F urthermor e, assume U is a F r a ¨ ıs s ´ e se quenc e i n K and F : X → A is an arbitr ary σ ( K , L ) -arr ow. Then ther e exist a σ K -arr ow J : X → U an d a σ ( K , L ) -arr ow G : U → A such that G ◦ J = F . X   J   F / / A U G 8 8 q q q q q q q 8 Pr o of. Recall that w e use the usual conv en tion fo r ob jects x n = X ( n ), u n = U ( n ), a nd for a rro ws x m n = X ( n, m ), u m n = U ( n, m ). W e shall construct inductive ly the follo wing “triangular matrix” in K , together with commuting σ ( K , L )-arrow s F i,j : w i,j → A f or j 6 i + 1, where w e agree that w i, 0 = x i and w i,i +1 = u ℓ i . x 0     / / / / u ℓ 0     # # # # ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ x 1     / / / / w 1 , 1     / / / / u ℓ 1     # # # # ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ x 2     / / / / w 2 , 1     / / / / w 2 , 2     / / / / u ℓ 2     " " " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ x 3     / / / / w 3 , 1     / / / / w 3 , 2     / / / / w 3 , 3     / / / / u ℓ 3     ❆ ❆ ❆ ❆ ❆ ❆ . . . . . . . . . . . . . . . . . . The first column in the diagram ab o ve is the sequence X , while the diagonal is a cofinal subseque nce of U . Our initial assumption o n F i,j is that { F n, 0 } n ∈ ω = F . It is clear ho w to start the construction: Using the F ra ¨ ıss ´ e prop erty of U , w e find ℓ 0 and a K - arro w e 0 : x 0 → u ℓ 0 . Next, using the K - injectivit y of A , we find F 0 , 1 : u ℓ 0 → A satisfying F 0 , 1 ◦ e 0 = F 0 , 0 . Supp ose the n th ro w has already b een constructed, together with arrows F i,j for i 6 n , j 6 n + 1 . Starting fro m K -a rro ws x n +1 n : x n → x n +1 and x n → w n, 1 , using Lemma 2.5 , we find w n +1 , 1 ∈ Ob ( K ) and K - arro ws w n, 1 → w n +1 , 1 , x n +1 → w n +1 , 1 , and a σ ( K , L )-arrow F n +1 , 1 : w n +1 , 1 → A suc h that the diagram w n, 1 $ $ $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ F n, 1 ( ( x n < < < < ③ ③ ③ ③ ③ ③ ③ ③ " " x n +1 n " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ w n +1 , 1 F n +1 , 1 / / A x n +1 : : : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ F n +1 , 0 6 6 comm utes. Con tinuing this w ay , using Lemma 2.5 , we obtain the ( n + 1 )st row and σ ( K , L )-arrow s F n +1 ,i for i 6 n + 1 whic h comm ute together with the follo wing diagram. x n     / / / / w n, 1     / / / / w n, 2     / / / / . . . / / / / w n,n − 1     / / / / u ℓ n     x n +1 / / / / w n +1 , 1 / / / / w n +1 , 2 / / / / . . . / / / / w n +1 ,n − 1 / / / / w n +1 ,n 9 No w, using the F ra ¨ ıss ´ e prop ert y of U we find ℓ n +1 > ℓ n and a K -arrow w n +1 ,n → u ℓ n +1 making the triangle u ℓ n     % % % % ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ w n +1 ,n / / / / u ℓ n +1 comm utative . Using Lemma 2.5 again, w e get an arro w F n +1 ,n +2 : u ℓ n +1 → A commuting with F n +1 ,n , F n,n +1 and the triangle ab o ve. Finally , the comp ositions of the horizon tal arro ws in the tr iangular “matrix” constructed ab o ve induce a n a rro w of sequences J : X → U in σ K . The inductiv e construction also giv es a sequence of arrow s { F n,n +1 } n ∈ ω that turns in to a σ ( K , L )-arrow G : U → A satisfying G ◦ J = F . This completes the pro of. Theorem 2.7. L et h K , L i b e a p air of c ate gories with pr op erty ( H ) . Assume K has a F r a ¨ ıss´ e se quenc e U and let X b e an arbitr ary se quenc e in K . The fol lowing pr op erties ar e e quivale nt. (a) X is K -inje ctive in σ ( K , L ) . (b) Ther e exi st a σ K -arr ow J : X → U and a σ ( K , L ) arr ow R : U → X such that R ◦ J = id X . (c) X is a r etr act of U in σ ( K , L ) . Note that condition (c) is formally w eake r than ( b), since it is not required in (c) that the righ t inv erse of a retraction R : U → X is a σ K -arro w. Pr o of. (a) = ⇒ (b) Applying L emma 2.6 t o the iden tity id X : X → X , w e get a σ K -arr o w J : X → U a nd a σ ( K , L )-arrow R : U → X suc h tha t R ◦ J = id X . (b) = ⇒ (c) This is obv io us. (c) = ⇒ (a) Let J : X → U and R : U → X b e σ ( K , L )-arrows suc h that R ◦ J = id X . Fix a K -arrow i : a → b and an L -arro w F : a → X . By Proposition 2.4 , U is K - injectiv e in σ ( K , L ), so there exists G : b → U suc h that G ◦ i = J ◦ F . Finally , w e hav e R ◦ G ◦ i = R ◦ J ◦ F = F . 2.1 Remarks on absolute retracts One can hav e a false impression after reading our result characterizing retracts of a F ra ¨ ıss ´ e s equence, namely that every embedding (i.e. a σ K -a rro w) of a K -injective σ K - ob ject in to the F ra ¨ ıss ´ e sequence admits a left inv erse in σ ( K , L ). This is not tr ue in general. W e make a brief discussion of this problem. T o be more concrete, w e assume h K , L i has prop ert y ( H ) and let U be a F ra ¨ ıss ´ e sequence in K . The problem stated ab ov e is strictly related to the followin g w ell know n concept: 10 Definition 2.8. Let K ⊆ L b e as ab ov e. W e sa y that W ∈ Ob ( σ K ) is an abs olute r etr act in σ ( K , L ) if for ev ery σ K -arrow J : W → Y there exists a σ ( K , L )-arr o w R : Y → W suc h that R ◦ J = id W . This notion is well kno wn esp ecially in top ology . In par ticular, there is a ric h theory of absolute retracts in geometric top olog y (see [ 5 ]). One of the main asp ects is the existence of some “canonical” ob jects whic h can b e used for c hec king whether a giv en ob ject is an absolute retract or not. F or instance, in the category of compact top ological spaces, an absolute retract is simply a retract of a Tikhonov cub e. In the category of metric spaces with contin uous maps, absolute retracts are retracts of con ve x sets in normed linear spaces. In the categor y of metric spaces with non-expansiv e maps, absolute retracts are h yp ercon vex metric spaces [ 2 ]. Notice that our defi nition is relativ e to a fixed category of “sm all” ob jects, e.g. spaces of w eigh t less than a fixed cardinal num b er. In the case of compact top ological (or metric) spaces, the “canonical” ob jects (e.g. Tikhonov cub es) turn out t o b e absolute retracts without restrictions on the w eigh t of the spaces and therefore b eing an absolute retra ct in a catego ry of o b jects of restricted “size” is equiv alen t to b eing an absolute retra ct in the big categor y , with no “ size” restrictions. The fo llo wing fact is rather standard. Prop osition 2.9. Assume K ⊆ L is a p air of c ate gories such that h σ K , σ ( K , L ) i has the mix e d amalgamation pr op erty. Given W ∈ Ob ( σ K ) the fol lowing two pr op erties ar e e quivalent. (a) W is an absolute r etr act in σ ( K , L ) . (b) W is σ K -inje ctive in σ ( K , L ) . Pr o of. Only (a) = ⇒ (b) requires an argumen t. Fix a σ K -arrow I : X → Y and a σ ( K , L )-arrow F : X → W . Using the mixed amalgamation pro p ert y we find a σ K - arro w J : W → V and a σ ( K , L )-arrow G : Y → W for which the diagra m W / / J / / V X F O O / / I / / Y G O O is comm utativ e. Let R : V → W b e suc h that R ◦ J = id W . Then R ◦ G ◦ I = F . No w the problem arises whether K - injectivit y implies σ K -injectivit y . The next r esult c haracterizes this prop ert y using a F ra ¨ ıss ´ e sequence. Theorem 2.10. L et K ⊆ L b e a p air of c ate go ries with pr o p erty ( H ) , and let U ∈ Ob ( σ K ) b e a F r a ¨ ıs s ´ e s e quenc e in K . Assume further that h σ K , σ ( K , L ) i has the mixe d amalgamation p r op erty . Then the fol lowing statements ar e e quivalent. 11 (a) K -inje ctivity implies σ K -inje ctivity in σ ( K , L ) . (b) U is σ K -in je ctive in σ ( K , L ) . (c) F or every σ K -arr ow J : U → U ther e exists a σ ( K , L ) -arr ow R : U → U s uch that R ◦ J = id U . Pr o of. Implication (a) = ⇒ (b) follows from the fact t hat U is K -injectiv e (Prop osi- tion 2.4 ). Implication (b) = ⇒ (c) is trivial. Implication (b) = ⇒ (a) follows directly from Theorem 2.7 , b ecause a retra ct of a σ K -injectiv e ob ject is ob viously σ K -injectiv e. It remains to sho w that (c) = ⇒ (b). Supp ose U is not σ K - injectiv e. By Prop osition 2.9 , there exists a σ K -arrow I : U → Y whic h is not left-in vertible in σ ( K , L ). Here w e ha ve used the mixed amalgamation prop ert y fo r h σ K , σ ( K , L ) i . Since U is F ra ¨ ıss ´ e, there exists a σ K -arrow J : Y → U . Using (c), w e find a σ ( K , L )-ar ro w R : U → U suc h that R ◦ J ◦ I = id U . But no w R ◦ J is a left inv erse to I , a contradiction. An in teresting consequence of the res ult ab ov e is that whenev er K -injectivit y is differen t from σ K -injectivity , it is witnessed by some σ K - arro w J : U → U with no left in vers e in σ ( K , L ). In other w ords, U carries all the information ab out σ K -inj ectivit y . 2.2 Extensions of the main r esult Theorem 2.7 has a natural generalization to uncoun table F ra ¨ ıss ´ e sequenc es. More pre- cisely , let κ be an uncoun ta ble regular cardinal and assume that all sequences in K of length < κ hav e colimits in L , where the colimiting co cones are K -arrows. In this case w e sa y that K is κ -c ontinuous in L . Under this assumption, a v ersion of Lemma 2.6 for κ -sequences is tr ue, with almost the same pro of—usual induction is r eplaced b y trans- finite induction. Prop osition 2.4 is v alid for arbitrary F ra ¨ ıss ´ e sequences, the coun table length of the sequen ce w as nev er used in the pro of. Let Seq 6 κ ( K ) denote the catego ry of all sequences in K of length 6 κ , w ith arrows induced b y natura l transformations (lik e in the countable case). Let Seq 6 κ ( K , L ) denote the category with the same ob jects a s Seq 6 κ ( K ), and with arro ws tak en from Seq 6 κ ( L ). W e can no w formulate an “ uncoun table” v ersion of our main result. Theorem 2.11. L et κ b e an unc ountable r e gular c ar dinal and let h K , L i b e a p air of c ate gories with pr op erty ( H ) , such that K is κ -c ontinuous in L . Assume K has a F r a ¨ ıss´ e se quenc e U of length κ . Given a se quenc e X in K of length 6 κ , the fol lowing pr op erties ar e e quivale nt. (a) X is K -inje ctive in Seq 6 κ ( K , L ) . (b) Ther e exist a Seq 6 κ ( K ) -arr ow J : X → U and a Seq 6 κ ( K , L ) arr ow R : U → X such that R ◦ J = id X . 12 (c) X is a r etr a ct of U in Seq 6 κ ( K , L ) . Let us no w come bac k to the coun table case. As sume h K , L i has prop erty ( H ) and moreo ve r K has pushouts in L . Let us lo ok at the pro of of Lemma 2.6 . W e can assume that all squares in the infinite “ triangular mat rix” constructed there a re pushouts in L . Using the notation from the pro of of Lemma 2.6 , let W n denote the sequence coming from the n th column. O bserv e that the arrow from W n to W n +1 is determine d b y the “horizon t al” K -arrow w n +1 ,n → w n +1 ,n +1 . In other w o rds, all other K -arrows come as a result of the corresp onding pushout square. An arrow of sequence s F : V → W determined b y pushouts fr om a single K -arro w will b e called pushout gener ate d from K . D enote b y σ PO K the category whose ob jects are ω -sequenc es in K , while arrow s are pushout generated from K . A deep er analysis of the pro of of Lemma 2.6 giv es the follo wing observ ation, which may b e of indep enden t in terest. Prop osition 2.12. Assume h K , L i is a p air of c ate gories with pr op erty ( H ) an d K has pushouts i n L . L et X ∈ Ob ( σ K ) b e K -inje ctive in σ ( K , L ) . Then: (1) X is σ PO K -inje ctive in σ ( K , L ) . (2) L et U b e a F r a ¨ ıss´ e se quenc e in K . Ther e ex ists a se quenc e X 0 → X 1 → X 2 → . . . in σ PO K such that X 0 = X an d U is the c olimit of this se quenc e in σ K . Clearly , (1) and (2 ) imply immediately that X is a retract of U . 3 Applicatio n s W e start with some more commen ts on prop erty ( H ). In many cases (esp ecially in mo del- theoretic categories), it is m uch easier to prov e the (mixed) a malgamation prop erty f or sp ecial “primitiv e” arro ws rather tha n for arbitrary arro ws. In order t o formalize this idea, fix a pair of categories h K , L i satisfying condition ( H 0 ) and fix a collection F ⊆ K (actually F migh t b e a prop er class). W e sa y that K is gen er ate d by F if for ev ery f ∈ K there exist n ∈ ω and g 0 , . . . , g n − 1 ∈ F suc h that f = g n − 1 ◦ . . . ◦ g 0 . F or example, if K is the category of em b eddings of finite mo dels of a fixed first-order la nguage, F ma y b e the class of em b eddings f : S → T suc h that T is generated by f [ S ] ∪ { b } for some b ∈ T . W e define the amalg amation prop ert y for F and the mixed amalgamation prop ert y for hF , L i , as b efore. Prop osition 3.1. L et K ⊆ L b e two c ate gories with the sam e obje cts, w her e K ha s the joint em b e dding pr op erty. Assume further that K is gener ate d by a family F such that F has the amalgamation pr op erty and hF , L i has b oth the mixe d amalgamation pr op e rty and the amalga mate d ex tension pr op erty. T hen h K , L i has pr op erty ( H ) . 13 Pr o of. Give n an arro w f ∈ K , w e sa y that f h as length 6 n if f = g n − 1 ◦ . . . ◦ g 0 , where g 0 , . . . , g n − 1 ∈ F . In particular, all arrow s in F hav e length 1. Easy induction sho ws that if i : c → a , j : c → b are K -arrows such that the length of i is 6 m and the length of j is 6 n , then there exist K -arrows k : a → w , ℓ : b → w suc h that k ◦ i = ℓ ◦ j and k has length 6 n , while ℓ has length 6 m . Since ev ery K - arro w has a finite length, t his sho ws that K has the amalga mation prop erty . A similar induction on the length of K - arro ws sho ws that h K , L i has t he amalgamated extension prop erty . Finally , using the fact that hF , L i has the mix ed amalgamatio n prop ert y , w e prov e b y induction tha t for ev ery K -arrow i : c → a o f length 6 n , and for ev ery L -arro w f : c → b , there exist an L -arrow g : a → w and a K -arrow ℓ : b → w of length 6 n suc h tha t g ◦ i = ℓ ◦ f . This show s that h K , L i ha s the mixed amalga mation prop ert y . Another simplification for pro ving prop erty ( H ) is the concept of mixed pushouts. Let K ⊆ L b e tw o catego ries with the same o b jects. W e say that h K , L i ha s the mi xe d pushout pr op erty if fo r eve ry arrows f : c → a and g : c → b suc h that f ∈ K and g ∈ L , there exist arro ws f ′ : a → w and g ′ : b → w suc h that f ′ ∈ L , g ′ ∈ K and a f ′ / / w c O O f O O g / / b O O g ′ O O is a pushout square in L . Note that if b oth f , g a re K -arrows in the definition abov e, then so are f ′ , g ′ , b y uniqueness of the pushout. The definition ab o ve mak es sense (and is applicable) in case where K is an a rbitrary family of arrow s, not necessarily a sub category . This is presen ted in the next statemen t. Prop osition 3.2. L et K ⊆ L b e two c ate gorie s with the sa me obje cts. Assume that K has the joint emb e dding pr op erty and F ⊆ K is such that hF , L i has the mixe d pushout pr op erty and F gen er ates K . Then h K , L i has pr op erty ( H ) . Pr o of. Supp ose first that F = K . The amalgamation prop ert y (condition ( H 1 )) follow s from the remark ab ov e, namely that the pushout of t w o K -arr o ws consists of K -a rro ws. Mixed amalgamation prop ert y (condition ( H 2 )) is just a w eak er v ersion of the mixed pushout property . Finally , the amalgama ted extension property (condition ( H 3 )) follo ws immediately from the definition of a pushout. Supp ose now that F 6 = K . It suffices to prov e t hat h K , L i has the mixed pushout prop ert y . Lik e in the pro of of Prop osition 3.1 , w e use induction o n the length of K - arro ws, b earing in mind that the obv ious comp osition of tw o pushout squares is a pushout sq uare. More precisely , the indu ctiv e h yp othesis sa ys: Giv en a K - arro w i : c → a 14 of length < n , and an L -arro w f : c → b , there exist an L -arro w g : a → w and a K -arro w ℓ : b → w of length < n suc h that b / / ℓ / / w c f O O / / i / / a g O O is a pushout square in L . Man y natural pairs of categories, in pa rticular coming from mo del theory , ha v e the mixed pushout prop ert y . Concrete w ell kno wn examples are finite graphs, partially ordered sets, semilattices. Eac h of these classes is considered as a pair of t wo categories, the first one with em b eddings and the s econd one with all ho momorphisms. These examples a re men tioned in [ 8 ]. A t ypical example of a pair h K , L i with prop erty ( H ), failing the mixed pushout property is the category L of a ll finite linear orders with increasing (i.e. order preserving) functions and K the categor y of all finite linear orders with em b eddings. In con trast to the ab ov e results, it is w orth men tioning a F ra ¨ ıss ´ e class that do es not fit into our fra mew ork. Namely , the F ra ¨ ıss ´ e class of finite K n -free g raphs (where K n denotes the complete graph with n v ertices and n > 2) has the pushout prop erty (formally t he class of em b eddings has pushouts in the class of all homomorphisms), y et the corresp onding pair of cat egories fails to ha ve mixed amalgama tions. Sp ecifically , a graph is meant to b e a structure with one sy mmetric irreflexiv e binary relation, so a homomorphism o f g raphs cannot identify v ertices connected b y edges. In o ther words, ev ery gr aph homomorphism restricted to a complete subgraph b ecomes an em b edding. It ha s b een pro v ed b y Mudrinski [ 32 ] that for n > 2, the F ra ¨ ıss ´ e limit of K n -free gr aphs (called the Henson gr aph H n ) is retract rigid, i.e. iden tit y is the only retraction of H n . On the other hand, w e ha v e the followin g easy fa ct (stated in a differen t form in [ 9 , Example 3.3]). Prop osition 3.3. No K n -fr e e gr aph with n > 2 is inje ctive for fi nite K n -fr e e gr aphs. Pr o of. Supp ose X is suc h a g raph. Using injectivit y for S = ∅ and T = K n − 1 , w e see that X con tains an isomorphic cop y K o f K n − 1 . No w let S b e a graph with n − 1 v ertices and no edges and let f : S → X b e a bijection on to K . Let T = S ∪ { v } , where v is connected to all t he v ertices of S . By injectivit y , there exists a homomorphism g : T → X extending f . But now K ∪ { g ( v ) } ⊆ X is a cop y of K n , a con tr adiction. Before discussing concrete examples of pairs with prop ert y ( H ), we make one more remark on injectivit y . Recall that an ar ro w j : x → y is le ft-invertible in L if there exists f ∈ L suc h that f ◦ j = id x . The f ollo wing is an easy consequence of our main result. 15 Corollary 3.4. L et h K , L i b e a p air of c ate gories such that every K -arr ow is left- invertible in L . Assume that h K , L i has p r op erty ( H ) and U is a F r a ¨ ıss ´ e se quenc e in K . Then for every se quenc e X ∈ O b ( σ K ) ther e exist a σ K -arr ow J : X → U an d a σ ( K , L ) -arr ow R : U → X such that R ◦ J = id X . Pr o of. In view of Theorem 2.7 , it suffices to sho w that eve ry sequence is K -injectiv e in σ ( K , L ). Fix X ∈ Ob ( σ K ), a K -ar ro w j : a → b , and a σ ( K , L )-arrow f : a → X . Cho ose an L -arr o w r : b → a suc h that r ◦ j = id a . The n g = f ◦ r ha s the prop ert y that g ◦ j = f . This shows that X is K -injectiv e in σ ( K , L ). This corolla ry applies to finite Bo olean algebras (also noted in [ 8 ]) and, as w e shall see later, to finite linear orderings. 3.1 F ra ¨ ıss ´ e classes and algebraically closed mo dels Let M b e a class of finitely g enerated mo dels of a fixed first-order language L . It is natural to consider the category hom M whose ob jects are all elemen ts of M a nd arro ws are all homomorphisms (i.e. maps that preserv e all relatio ns, functions and constan ts). It is also natura l to consider the category em b M whose ob j ects ar e ag ain all elemen ts of M , while arrow s are em b eddings only . In man y cases, h em b M , hom M i has prop ert y ( H ). Simplifying the notation, w e shall say that M has the pushout pr op erty or mixe d amal- gamation pr op e rty if h em b M , hom M i has suc h a prop ert y . D enote b y M the class of all (coun table) mo dels that are unio ns of ω -c hains of mo dels from M . It is clear that σ em b M is equiv alen t to M with em b eddings and σ ( em b M , hom M ) is equiv alen t to M with all homomorphisms. Recall that a mo del X ∈ M is algebr aic al ly close d if for ev ery formula ϕ ( x 0 , . . . , x k − 1 , y 0 , . . . , y ℓ − 1 ) that is a finite conjunction of atomic formulae, for ev ery a 0 , . . . , a k − 1 ∈ X , if there exists an extension X ′ ⊇ X in M satisfying X ′ | = ( ∃ y 0 , . . . , y ℓ − 1 ) ϕ ( a 0 , . . . , a k − 1 , y 0 , . . . , y ℓ − 1 ) then there exist b 0 , . . . , b ℓ − 1 ∈ X such that X | = ϕ ( a 0 , . . . , a k − 1 , b 0 , . . . , b ℓ − 1 ). Prop osition 3.5. L et M b e a class of finitely gener ate d mo dels of a fixe d firs t-or der language. Every M -i nje ctive mo del in M is algeb r aic al ly clo se d. Pr o of. Fix a n M -injectiv e mo del X ∈ M . Fix X ′ ⊇ X a nd assume X ′ | = ( ∃ ~ y ) ϕ ( ~ a, ~ y ) for some k -tuple ~ a of elemen ts of X , where ϕ ( ~ x, ~ y ) is a finite conjunction o f atomic form ula e and ~ x, ~ y are shortcuts for ( x 0 , . . . , x k − 1 ) and ( y 0 , . . . , y ℓ − 1 ), resp ectiv ely . Let S ∈ M b e a submo del of X that contains ~ a . Let T ∈ M b e a submo del of X ′ con taining S and a fixed tuple ~ b suc h that X ′ | = ϕ ( ~ a, ~ b ). Then also T | = ϕ ( ~ a, ~ b ), b ecause 16 this prop erty is a bsolute for ϕ . Using the M - injectivit y of X , find a homomorphism f : T → X satisfying f ↾ S = id S . Finally , let ~ c = ( f ( b 0 ) , . . . , f ( b ℓ − 1 )), where ~ b = ( b 0 , . . . , b ℓ − 1 ). Since f is a homomorphism and ϕ is a conjunction of atomic formulae, w e hav e that X | = ϕ ( ~ a, ~ c ). W e shall say that a structure M is n -gener ate d if there exists S ⊆ M suc h that | S | 6 n and S generates M , that is, no prop er submo del of M con tains S . Recall that a first- order language is fi nite if it con tains finitely man y predicates (constan t, relation and function sym b ols). Prop osition 3.6. L et M b e a class of finite m o dels of a fixe d first-or der language L . As- sume that either L is fi nite or for every n ∈ ω ther e exist fi nitely many isomorphic typ es of n -gener ate d mo dels in M . Assume furthermor e that M has the mixe d amalga mation pr op erty. Then every alg ebr aic al ly c lose d L -mo del X ∈ M is M -inje ctive. Pr o of. Fix S, T ∈ M suc h that S is a submo del of T . Fix a homomorphism f : S → X . Using the mixed amalgamation, w e can find an extension X ′ ∈ M of X and a homomorphism f ′ : T → X ′ suc h that f ′ ↾ S = f . Let G b e the set of all f unctions g : T → X satisfying g ↾ S = f . W e need to show that some g ∈ G is a homomo rphism. Supp ose first that there exist only finitely many | T | -generated structures in M and let N ⊆ M b e a finite set that con tains isomorphic types of all of them. Giv en g ∈ G , denote by g ′ a fixed isomorphism from the submo del generated b y g [ T ] on to a fixed mo del from the collection N . Note that g is a homomor phism if and only if g ′ ◦ g is a homomorphism. Now observ e that the set H = { g ′ ◦ g : g ∈ G } is finite. Let S = { s i } i k so that f k ◦ i = f and f n +1 ◦ y n +1 n = f n for n > k . This giv es rise to an arro w of sequence s F = { f n } n > k satisfying F ◦ j = f . Letting X = Y in t he lemma ab ov e, w e obtain: Corollary 3.10. L et K ⊆ L b e a p air of c ate gories. Every K -inj e ctive obje ct is L - homo gene ous in σ ( K , L ) . The equiv alence (b) ⇐ ⇒ (c) in the next statemen t, in the contex t of mo del theory , has b een noticed b y Dolink a [ 9 , Prop. 3 .8]. Prop osition 3.11. L et K ⊆ L b e a p air of c ate gories and let K have a F r a ¨ ıss´ e s e quenc e U ∈ Ob ( σ K ) . The fol lowing pr op erties ar e e quivalent: (a) U is K -inje ctive in σ ( K , L ) . (b) U is L -homo gene ous in σ ( K , L ) . (c) h K , L i has the mixe d amalgamation pr op erty. Pr o of. Implication (c) = ⇒ (a ) has b een prov ed in Prop osition 2.4 . Implication (a) = ⇒ (b) is a consequence of Coro llary 3 .10 . It remains to sho w that (b) = ⇒ (c). Supp ose U is L -homogeneous in σ ( K , L ) and fix a K -arro w j : c → a and a n L -ar ro w f : c → b . Using the prop ert y of b eing a F r a ¨ ıss ´ e sequence, find K -arrow s i : a → u k and e : b → u ℓ with some k , ℓ < ω . Since U is L -homogeneous, there exists a σ ( K , L )- arro w F : U → U satisfying F ◦ u ∞ k ◦ i ◦ j = u ∞ ℓ ◦ e ◦ f . Finally , find a n L -a rro w g : u k → u m with m > ℓ , suc h that u ∞ m ◦ g = F ◦ u ∞ k . The situation is describ ed in the follo wing diagram. c / / j / / f   a / / i / / u k g ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ / / / / U F   b / / e / / u ℓ / / u m ℓ / / u m / / / / U Th us, j and f are amalg amated b y a K -arrow u m ℓ ◦ e and an L -arrow g ◦ i . Under certain nat ural assumptions, w e are able to c haracterize homomorphism-homo- geneous ob jects. In the next statemen t we deal with coun ta ble categories, but what w e really hav e in mind is the existence of coun t ably man y isomorphic types of arrows. F or example, the category of finite sets is a prop er class, y et it is o b viously equiv alen t to a coun table category . 20 Theorem 3.12. L et K ⊆ L b e a p air of c ate gories such that h K , L i has the mixe d pushout pr op erty, L is c o untable, and K ha s the initial obje ct 0 . F or a se quenc e X ∈ Ob ( σ K ) , the fol low ing pr op erties ar e e quivalent. (a) X is L -homo gene ous in σ ( K , L ) . (b) T her e exists a sub c ate gory K 0 of K such that 0 is initial in K 0 , X ∈ Ob ( σ K 0 ) , h K 0 , L i has the mixe d pushout pr op erty, and X is K 0 -inje ctive in σ ( K 0 , L ) . (c) T her e exists a sub c ate gory K 0 of K such that 0 is initial in K 0 , X ∈ Ob ( σ K 0 ) , h K 0 , L i has the mixe d pushout pr op erty, and X is a r e tr act of a F r a ¨ ıss ´ e se quenc e in K 0 . The existence of the initial ob ject in K is not essen tial, but to remo v e it w e w o uld ha v e to ma k e more tec hnical assumptions in volving the jo in t em b edding prop ert y . Pr o of. The equiv alence (b) ⇐ ⇒ (c) is contained in Theorem 2.7 . The fact that K 0 is coun table has b een used here f or the e xistence of a F ra ¨ ıss ´ e sequence. Implication (b) = ⇒ (a) is contained in Corollar y 3.10 . It remains to sho w that (a) = ⇒ (b). W e ma y assume that x 0 = 0 in the sequence X . Let S = { x m n : n 6 m, n, m ∈ ω } . Then S is a sub category of K that contains the intial ob ject 0. W e first c hec k that X is S -injectiv e. Fix a σ ( K , L )-a rro w f : x n → X and fix m > n . Since X is L -homogeneous, there ex ists a σ ( K , L )-arro w F : X → X satisfying F ◦ x ∞ n = f . Note that x ∞ n = x ∞ m ◦ x m n , therefore ( F ◦ x ∞ m ) ◦ x m n = f , which show s the S -injectivit y of X . No w let K 0 consist of all K -arrows j : c → a suc h that X is j -injectiv e in σ ( K , L ) and there exists at least one σ ( K , L )- arro w from c to X . That is, for ev ery σ ( K , L )-ar ro w f : c → X , t here exists a σ ( K , L )-arrow g : a → X satisfying g ◦ j = f . The second assumption is needed for k eeping 0 initial in K 0 , namely , X should also b e injectiv e for t he (unique) arro w 0 → a . It is clear that K 0 is a sub category of K con taining S . In particular, X ∈ Ob ( σ K 0 ). It remains to sho w that h K 0 , L i has the mixed pushout prop ert y . F or this aim, fix an K 0 -arrow j : c → b , an L -arro w p : c → a , and let k : a → w , q : b → w b e suc h that k ∈ K , q ∈ L and b q / / w c O O j O O p / / a O O k O O is a pushout square in L . F ix a σ ( K , L )-a rro w f : a → X . Since X is j -injectiv e, t here exists a σ ( K , L )-arrow g : b → X satisfying g ◦ j = f ◦ p . Both arrows f and g are factorized through some x n , namely , f = x ∞ n ◦ f ′ and g = x ∞ n ◦ g ′ for some L -arrows f ′ , g ′ . Using the pro perty of a pushout, we find a unique L -arrow h : w → x n satisfying h ◦ k = f ′ and h ◦ q = g ′ . In particular, f = x ∞ n ◦ h has the prop ert y that f ◦ k = f . This shows that X is k -injectiv e in σ ( K , L ) a nd completes the pro of . 21 Unfortunately , the result ab ov e is not fully applicable to F ra ¨ ıss ´ e classes. Namely , in cas e where K is a countable F ra ¨ ıss ´ e class, K 0 ma y not b e a full sub category of K . This is demonstrated b elo w, for t he class of finite graphs. Example 3.13. Let X be the tw o-elemen t complete graph. It is clear that X is homomorphism-homogeneous ( and also ultrahomogeneous). W e consider graphs with- out lo o ps, therefore ev ery endomorphism o f X is an auto morphism. More precisely , w e consider the pair h K , L i , where Ob ( K ) = Ob ( L ) a re all finite simple graphs, the K -arrows are em b eddings a nd the L -a rro ws are graph homomorphisms. Let K 0 b e an y sub category of K that has pushouts in L and contains all the em b ed- dings of subgraphs of X . So, Ob ( K 0 ) con tains the empt y gra ph and c omplete subgraphs of size 6 2. The pushout with embeddings of the empty graph is just the copro duct (disjoin t sum), there Ob ( K 0 ) con tains the 2-elemen t graph D with no edges. F urther- more, Ob ( K 0 ) con tains the graph G whose set of vertice s is {− 1 , 0 , 1 } and the edges are {− 1 , 0 } and { 0 , 1 } . Suc h a graph comes from the pushout of tw o em b eddings of the one- elemen t graph into X . Now consider an em b edding j : D → G suc h that j [ D ] = {− 1 , 1 } . Let f : D → X b e one-to-one. Clearly , f is a homomorphism and no homomorphism g : G → X satisfies g ◦ j = f . This sho ws tha t X is not j - injectiv e. In particular, K 0 is not a full sub category of K . 3.3 Metric spaces W e shall now discuss a concre te mo del-theoretic application of our result: Retracts of the univ ersal metric spac e of Urys ohn. Let M b e the category of finite metric spaces with isometric embeddings. The ob j ects of M are mo dels of a first-order language: F or eac h r > 0 w e can define the binary relation D r ( x, y ) ⇐ ⇒ d ( x, y ) < r , where d denotes the metric on a fixed set X . The axioms of a metric can b e rephrased in terms o f the relations D r . F or example, the triangle inequalit y follows fro m the following (infinitely man y) form ulae: D r ( x, z ) ∧ D s ( z , y ) = ⇒ D r + s ( x, y ) . Note that it suffice s to consider the relations D r with r p ositiv e rational: The metric is then defined b y d ( x, y ) = inf r ∈ Q + D r ( x, y ), where Q + denotes the set of all p ositiv e rationals. In other w ords, metric spaces can b e describ ed in a coun table langua ge. It is clear that, in this language, a homomorphism of metric spaces is a non- expansiv e map. Recall that f : X → Y is non-e xp ansive if d Y ( f ( p ) , f ( q )) 6 d X ( p, q ) for ev ery p, q ∈ X , where d X , d Y denote the metrics on X and Y resp ectiv ely . It is also p ossible to describ e a metric space b y similar relatio ns D r , no w meaning tha t the distance is 6 r . W e shall see la ter that, ev en though b oth languages describ e the same ob jects, the notion of b eing a lgebraically closed is completely differen t. Clearly , the language of metric spaces is infinite a nd there exist infinitely man y t yp es of 2-elemen t metric spaces (ev en when restricting to rational distances), therefore one 22 cannot apply Dolink a’s result here. Moreo ver, M is formally not a F ra ¨ ıss ´ e class, b ecause it con tains contin uum many pairwise non-isomorphic ob jects. It b ecomes a F ra ¨ ıss ´ e class when restricting to spaces with ratio nal distances. How ev er, in t hat case we cannot sp eak ab out complete metric space s. In an y case, our main result is applicable to the complete metric space of Urysohn, as we show b elow . The fo llo wing lemma, in a slightly differen t form, can b e found in [ 10 , Lemma 3 .5]. Lemma 3.14. L et f : X → Y b e a non-exp ansive map of nonempty finite m etric sp ac es. Assume X ∪ { a } is a metric extension of X . Then ther e exists a metric extension Y ∪ { b } of Y such that Y / / / / Y ∪ { b } X f O O / / / / X ∪ { a } g O O wher e g ↾ X = f and g ( a ) = b , is a pusho ut squar e in the c ate gory of metric sp ac es with non-e xp ansive maps. F urthermor e (M) d ( y , b ) = min x ∈ X  d ( y , f ( x )) + d ( x, a )  for every y ∈ Y . The statemen t obviously fails when X = ∅ and Y 6 = ∅ . Pr o of. W e first need to sho w that ( M ) defines a metric on Y ∪ { b } . Of course, only the triang le inequalit y requires an argumen t. F ix y , y 1 ∈ Y . Find x 1 ∈ X suc h that d ( y 1 , b ) = d ( y 1 , f ( x 1 )) + d ( x 1 , a ). Using the tria ngle inequalit y in Y , w e get d ( y , b ) 6 d ( y , f ( x 1 )) + d ( x 1 , a ) 6 d ( y , y 1 ) + d ( y 1 , f ( x 1 )) + d ( x 1 , a ) = d ( y , y 1 ) + d ( y 1 , b ) . No w find x ∈ X suc h that d ( y , b ) = d ( y , f ( x )) + d ( x, a ). Using the triangle inequalities in X and Y , and the fact that d ( f ( x ) , f ( x 1 )) 6 d ( x, x 1 ), w e obtain d ( y , b ) + d ( y 1 , b ) = d ( y , f ( x )) + d ( x, a ) + d ( y 1 , f ( x 1 )) + d ( x 1 , a ) > d ( y , f ( x )) + d ( x, x 1 ) + d ( y 1 , f ( x 1 )) > d ( y , f ( x )) + d ( f ( x ) , f ( x 1 )) + d ( y 1 , f ( x 1 )) > d ( y , y 1 ) . Th us, d defined by ( M ) fulfills the tria ngle inequality . Giv en x ∈ X , w e hav e d ( g ( x ) , g ( a )) = d ( f ( x ) , b ) 6 d ( f ( x ) , f ( x )) + d ( x, a ) = d ( x, a ). This shows that g is non-expansiv e. Finally , assume p : X ∪ { a } → W and q : Y → W are non-expansiv e maps suc h that p ↾ X = q ◦ f . W e need to sho w that there exists a unique non- expansiv e map h : Y ∪ 23 { b } → W satisfying h ◦ g = p and h ↾ Y = q . The uniqueness of h is clear, namely h ( b ) = h ( g ( a )) = p ( a ). It remains to v erify that h is no n-expansiv e. Supp ose otherwise and fix y ∈ Y su c h that d ( h ( y ) , h ( b )) > d ( y , b ). Find x ∈ X suc h that d ( y , b ) = d ( y , f ( x )) + d ( x, a ). So w e hav e (*) d ( h ( y ) , p ( a )) > d ( y , f ( x )) + d ( x, a ) . Kno wing tha t p and q are non- expansiv e, w e get (**) d ( p ( x ) , p ( a )) 6 d ( x, a ) and d ( q ( y ) , q ( f ( x )) 6 d ( y , f ( x )) . Note that q ( f ( x )) = p ( x ) and q ( y ) = h ( y ). Fina lly , ( * ) a nd ( ** ) giv e d ( h ( y ) , p ( a )) > d ( p ( x ) , p ( a )) + d ( h ( y ) , p ( x )) whic h contradicts t he triangle inequality in W . This completes the pro of. W e say that a metric space h X , d i is finitely hyp er c onvex if fo r ev ery finite fa mily of closed balls A =  B( x 0 , r 0 ) , B( x 1 , r 1 ) , . . . , B( x n − 1 , r n − 1 )  suc h that T A = ∅ , there exist i, j < n suc h that d ( x i , x j ) > r i + r j . This is a w eak ening of the notion o f a hyp er c onvex metric sp ac e , due to Aronsza jn & P anitchp akdi [ 2 ], where the fa mily ab o v e ma y b e of arbitrar y cardinality . Actually , t he authors of [ 2 ] had already considered κ -h yp ercon v ex metric spaces; finite h yp ercon vex it y corresp onds to ℵ 0 -h yp erconv exit y . A v arian t of finite h yp erconv exit y (with closed balls replaced b y op en balls) has b een recen tly studied b y Niemiec [ 33 ] in the context of top ological absolute retracts. The follow ing fa cts relate this definition to o ur main topic. The first one should b e w ell kno wn to readers f amiliar with hypercon v exit y , namely , ev ery metric space em b eds isometrically into a h yp ercon v ex one. Lemma 3.15. L et X b e a finite metric sp ac e an d let A = { B( x i , r i ) } i d ( x, x i ) + d ( y , x k ) + d ( x i , x k ) > d ( x, y ) . This shows that d defined by ( * ) satisfies the tria ngle inequality . The next lemma is a sp ecial case of tw o results of Aronsza jn & Panitc hpakdi, namely , Theorem 2 on pag e 413 and Theorem 3 o n page 415 in [ 2 ]. W e presen t the pr o of for t he sak e of completeness. Lemma 3.16. A metric sp ac e is finitely hyp er c onvex if and only if it is inje ctive with r esp e ct to is ometric emb e ddings of finite metric sp ac es. Pr o of. Let X b e a finitely h yp ercon vex metric space a nd fix a non-expansiv e map f : S → X , whe re S is a finite metric space. It suffices to show that f can b e extended to a non-expansiv e map f ′ : T → X whenev er T is a metric e xtension of S and T \ S = { a } . Fix T = S ∪ { a } and let A = { B( f ( s ) , r s ) : s ∈ S } , where r s = d ( s, a ). Giv en s, s 1 ∈ S , we ha ve that d ( f ( s ) , f ( s 1 )) 6 d ( s, s 1 ) 6 r s + r s 1 . Since X is finitely hypercon v ex, there exists b ∈ T A . This means that d ( b, f ( s )) 6 d ( s, a ) for ev ery s ∈ S . Thus, setting f ′ ( a ) = b and f ′ ↾ S = f , we obta in a non- expansiv e extension of f . This sho ws the “o nly if” part. F or the “if” part , fix a f amily A = { B( x i , r i ) } i 0, there exists a linear op erator ˜ T : Y → E suc h that ˜ T ↾ X = T and k ˜ T k 6 1 + ε . The Gur arii sp ac e [ 14 ] is a separable Banac h space G satisfying the following condition: Giv en ε > 0 a nd finite-dimensional spaces X ⊆ Y , ev ery isometric embedding e : X → G extends to an ε -isometric em b edding ˜ e : Y → G (that is, ˜ e is one-to-one and k ˜ e k 6 1 + ε , k ˜ e − 1 k 6 1 + ε ). The fact that G is unique up to a linear isometry w as pro v ed by Lusky [ 23 ]; an elemen tary arg umen t has b een found recen tly , see [ 21 ]. W e no w w o uld lik e to apply Theorem 2 .7 . The obstacle is that t he category B iso is to o big, it do es not ha v e a F ra ¨ ıss ´ e seque nce. On the other hand, giv en a countable S ⊆ B iso there exists a coun ta ble K ⊆ B iso suc h that S ⊆ K and K has pushouts in B . The 27 category K has a F ra ¨ ıss ´ e sequence. If S is “ric h enough” then this F ra ¨ ıss ´ e sequence induces the Gurarii space G . This w ay w e obtain the follo wing result, o riginally due to W o jtaszczyk [ 35 ]. Theorem 3.21. L e t E b e a sep ar able Banach sp ac e. T he fol lowing pr op erties ar e e quiv- alent. (a) E is line arly isometric to a 1 - c omplemente d subsp ac e of the Gur arii sp ac e. (b) E is almost 1 - inje ctive for fi nite-dimensional Ba nach sp ac es. (c) E is an isom etric L 1 pr e dual. Pr o of. (a) = ⇒ (b) By the mixed pushout prop ert y , it is straigh tforw a rd to see that the Gurarii space is almost 1-injectiv e. Clearly , this prop erty is preserv ed by 1-complemen ted subspaces. (b) = ⇒ (c) This is part of the main result o f Lindenstrauss [ 24 ]. In fact, it is prov ed in [ 24 , Thm. 6.1] that ( c) is equiv alen t t o a lmost 1- injectivit y f or Banach spaces of dimension 6 4. (c) = ⇒ (a) A result of Lazar & Lindenstrauss [ 22 ] says that there exists a c hain E 0 ⊆ E 1 ⊆ E 2 ⊆ . . . of finite-dimensional subspaces of E whose union is dense in E and eac h E n is isometric to some ℓ ∞ k ( n ) . In fact, due to Mic hael & Pe lczy´ nski [ 30 ], one may assume that k ( n ) = n fo r n ∈ ω , a lthough this is not needed here. Let L be a coun table sub category o f B t hat con tains all inclusions E n ⊆ E n +1 and a fixed c hain defining the Gurarii space. Enlarging L by a dding countably man y a rro ws, w e may assume that it is closed under mixed pushouts, that is, the pair h K , L i has the mixed pushout prop erty , where K = L ∩ B iso . Let G denote the Gura rii space. By the a ssumptions on L , w e hav e that b oth G and E are ob j ects of σ K . No w observ e that E is K - injectiv e in σ ( K , L ). Indeed, if f : A → E is an arrow in σ ( K , L ), where A, B ∈ Ob ( K ) are suc h that A ⊆ B , then f is an isometric em b edding o f A in to some E n (b y the definition o f arro ws b et w een sequences). It is easy and w ell kno wn that ev ery space isometric to ℓ ∞ m is 1-injectiv e for all Banac h spaces. Th us, f can b e extended to a linear isometry f : B → E n . W e actually need one more assumption on K : namely that f ∈ K whenev er f ∈ K . This can b e ac hiev ed b y a standard closing-off a rgumen t. Finally , Theorem 2.7 implies that E is isometric to a 1-complemen ted sub space of G . A non- separable vers ion of the ab ov e result is actually m uc h simpler and comes exactly as a pa rticular case of the uncountable vers ion of Theorem 2.7 : Theorem 3.22. Assume the c ontinuum hyp othesis. L et V b e the unique Banach sp ac e of d ensity ℵ 1 that is o f universal disp osition for sep ar able s p ac es. A B anach sp ac e of density 6 ℵ 1 is isom etric to a 1 -c omplemen te d subsp ac e of V if and on ly if it is 1 - sep ar ably in je ctive. 28 Some explanations are needed here. Namely , a Banac h space V is of universal disp osition for separable spaces if for eve ry separable Banac h spaces X ⊆ Y , ev ery isometric em b edding o f X in to V extends to an isometric embedding of Y in to V . Our result from [ 20 ] sa ys that, under the contin uum h yp othesis, there ex ists a unique Banac h space V of densit y ℵ 1 and of univ ersal dispo sition for separable spaces. Extensions of this result can b e found in [ 3 ], where more general constructions of spaces of univ ersal disp osition a re presen t ed. It is sho wn there that 2 ℵ 0 is the minimal densit y of a Bana c h space of univ ersal disposition for separable spaces. F inally , assuming the contin uum h yp othesis, the space V is the F ra ¨ ıss ´ e limit of separable Banac h spaces with linear isometric em b eddings. The notion of being “1-separably injectiv e” has obvious meaning; it has b een recen tly studied in [ 4 ]. In this con text, Theorem 3.22 comple men ts the results of [ 4 ]. 3.5 Linear orders Let κ b e an infinite cardinal and let L O <κ denote the class of a ll linearly o rdered sets of cardinalit y < κ . A homo morphism of linearly ordered sets will b e called an incr e asing map . As men tioned b efore, L O <ω giv es a natural example of a pair h em b L O <ω , hom L O <ω i failing the pushout prop ert y . Ho wev er, w e hav e the following Prop osition 3.23. F or every infinite c ar dinal κ , the p air h em b L O <κ , hom L O <κ i has pr op- erty ( H ) . Pr o of. Condition ( H 1 ) follows from ( H 3 ), b ecause em b L O <κ has an initial ob ject (the empt y set) and hom L O <κ has a t erminal ob ject, the 1-elemen t linearly ordered set. It remains t o show ( H 2 ) and ( H 3 ). Call an em b edding j : A → B primitive if | B \ j [ A ] | 6 1. It is clear t hat ev ery increasing em b edding is the colimit o f a transfinite sequence of primitive em b eddings. W e shall use an uncoun table v ersion o f Prop osition 3.1 , whic h can b e easily prov ed by transfinite induction, using the fact that the cat egory em b L O <κ is κ -con tinuous in hom L O <κ . Denote b y P the class of all primitive em b eddings in em b L O <κ . Let us prov e first that hP , hom L O <κ i has the amalgamated extension prop erty (condition ( H 3 )). Fix linearly ordered sets C , A, B suc h that A = C ∪ { a } and B = C ∪ { b } . Fix increasing maps f : A → L a nd g : B → L suc h t hat f ↾ C = g ↾ C . F ormally , w e ha ve to assume that a 6 = b . Let W = A ∪ B . W e let a < b if f ( a ) < g ( b ); w e let a > b otherwise. It is clear, using the compatibility of f and g , that t his defines a linear order on W , extending t he orders of A and B . The unique ma p h : W → L satisfying h ↾ A = f and h ↾ B = g is increasing. This sho ws ( H 3 ). No w fix linearly o rdered sets C , A, B suc h that A = C ∪ { a } with a / ∈ C , and fix an increasing ma p f : C → B . Let L = [ ca [ f ( c ) , → ) , 29 where ( ← , x ] = { p : p 6 x } and [ x, → ) = { p : p > x } . Note t hat B = L ∪ R and L ∩ R is either empty or a singleton. Let W = B ∪ { w } , where either w ∈ L ∩ R or w / ∈ B in case where L ∩ R = ∅ . In the latter case, define x < w and w < y for x ∈ L , y ∈ R . Define g : A → W by setting g ( a ) = w and g ↾ C = f . Clearly , g is increasing and the inclusion B ⊆ W is primitiv e. This shows ( H 2 ) and completes the pro of. Note that ev ery increasing em b edding of finite linear orders is left- in v ertible. Th us, w e immediately obtain the followin g result. Corollary 3.24. Every c ountable line ar or der is or der-isomorph ic to an incr e asing r etr act of the set of r ational numb ers. Of course, this result can b e prov ed directly , realizing that X · Q with the lexicographic ordering is isomorphic to Q , whenev er X is a coun ta ble linear o rder. Not e that this completely answ ers Question 10.6 from [ 29 ]. P assing to the unc oun ta ble case, let us note that L O <ω 1 has the F ra ¨ ıss ´ e limit if and only if the Contin uum Hypot hesis holds. D enote this F ra ¨ ıss ´ e limit by Q ω 1 . It is easy to c hec k that a linearly ordered set X of cardinalit y ω 1 is injective for countable linear orders (isomorphic t o Q ω 1 ) if and only if for ev ery coun table sets A, B ⊆ X such that a < b for a ∈ A , b ∈ B , there exists x ∈ X suc h that a 6 x 6 b ( a < x < b ) whenev er a ∈ A , b ∈ B (one of the sets A , B ma y b e empty ) . F or example, the closed unit in terv al [0 , 1 ] satisfies this condition, therefore it can b e em b edded a s an increasing retract of Q ω 1 . *** W e finish with some r emarks on rev ersed F ra ¨ ıss ´ e sequences . General theory of r ev ersed F ra ¨ ıss ´ e limits of finite mo dels (of a first-o rder language) w as dev elop ed in [ 16 ]. The idea come s just b y cons idering the oppo site category . More specifically , fix a class M of finite mo dels and consider the pair h quo M , hom M i , where quo M is the category whose ob jects are elemen ts of M and a rro ws are quotient maps. No w prop ert y ( H ) is defined b y rev ersing the arrows in all the diagrams. F or example, amalgamation is replaced b y “rev ersed a malgamation” and pushouts a re replaced by pullbac ks. Seq uences are no w con tr a v arian t functors and it is natural to consider their limits endo w ed with the top ology , inherited from the product of finite sets. It is not hard to see that precisely the con tinuous ho momorphisms are induced b y arro ws b etw een s equences. It is w o rth noting that if M is closed under finite pro ducts and substructures then quo M has pullback s in hom M . The pullbac k of t w o quotient maps f : X → Z , g : Y → Z is pro vided b y the structure w = {h s, t i ∈ X × Y : f ( s ) = g ( t ) } . Coming bac k to finite linear orders, consider the pair h quo L O <ω , hom L O <ω i . It is straigh t- forw ar d to see that quo L O <ω has no pullbac ks in hom L O <ω . O n the ot her hand, it is easy and standard to c hec k that this pair has (the rev ersed v arian t of ) prop ert y ( H ). Not e 30 that ev ery increasing quotien t of finite linearly ordered sets is righ t- in v ertible. Th us, all sequences in quo L O <ω are “finitely pro jectiv e”. It is clear that the in ve rse F ra ¨ ıss ´ e limit of L O <ω is the Can tor set endo w ed with the standard linear order. Th us, using Theorem 2.7 (or, more precise ly , Corollary 3.4 ), w e o btain the follo wing w ell kno wn fa ct whic h b elongs to the folklore. Corollary 3.25. Every c omp act m etric total l y disc onne cte d line arly or der e d sp ac e is a c ontinuous incr e asing r etr act of the standar d Cantor set. Again, it is not hard to prov e this fact directly , b y sho wing that a metric compact totally disconnected linearly o rdered space K can b e isomorphically em b edded in to t he Can tor set and constructing the r etraction “manually”. No te that the rev ersed F ra ¨ ıss ´ e theory w ould only say that K is a con tin uous increasing quotien t of the Can tor se t, how ev er not a ll contin uous increasing quotient maps of the Can tor set are right-in v ertible. Ac kno wledgmen ts The author is indebted t o the anon ymous referee for sev eral helpful remarks, in part ic- ular f or p ointing out the reference [ 29 ]. References [1] J. Ad ´ amek, H. Herrlich, J. R os ick ´ y, W. Tholen , Inje ctive hul ls ar e not natur al , Algebra Univ ersalis 48 (2002) 379–388 2 [2] N. Aronszajn, P . 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