Fraisses construction from a topos-theoretic perspective

F raïssé's onstrution from a top os-theoreti p ersp etiv e Olivia Caramello DPMMS, Univ ersit y of Cam bridge, Wilb erfore Road, Cam bridge CB3 0WB, UK O.Caramellodpmms.am.a.uk Otob er 22, 2018 Abstrat W e presen t a top os-theoreti in terpretation of (a ategorial generalization of ) F raïssé's onstrution in mo del theory , with appliations to oun tably ategorial theories. 1 A ategorial generalization of F raïssé's theorem In this setion w e presen t a ategorial generalization of F raïssé's onstrution in mo del theory . Our result is te hnially similar to (though more general than) the ategorial theorem in [5℄, but follo ws as an appliation of the theory dev elop ed b y Kubi± in [10℄. First, let us in tro due the relev an t terminology . Denition 1.1. A ategory C is said to satisfy the amalgamation pr op erty (AP) if for ev ery ob jets a, b, c ∈ C and morphisms f : a → b , g : a → c in C there exists an ob jet d ∈ C and morphisms f ′ : b → d , g ′ : c → d in C su h that f ′ ◦ f = g ′ ◦ g : a g   f / / b f ′      c g ′ / / _ _ _ d 1 Notie that C satises the amalgamation prop ert y if and only if C op satises the righ t Ore ondition. So if C satises AP then w e ma y equip C op with the atomi top ology . This p oin t will b e the basis of our top os-theoreti in terpretation desrib ed in the next setion. Denition 1.2. A ategory C is said to satisfy the joint emb e dding pr op erty (JEP) if for ev ery pair of ob jets a, b ∈ C there exists an ob jet c ∈ C and morphisms f : a → c , g : b → c in C : a f      b g / / _ _ _ c Notie that if C has a w eakly initial ob jet then AP on C implies JEP on C ; ho w ev er in general the t w o notions are quite distint from one another. Denition 1.3. Giv en an em b edding i : C → D , an ob jet u ∈ D is said to b e C -homo gene ous if for ev ery ob jets a, b ∈ C and arro ws j : a → b in C and χ : a → u in D there exists an arro w ˜ χ : b → u su h that ˜ χ ◦ j = χ : a j   χ / / u b ˜ χ ? ?     u is said to b e C -ultr ahomo gene ous if for ev ery ob jets a, b ∈ C and arro ws j : a → b in C and χ 1 : a → u , χ 2 : b → u in D there exists an isomorphism ˇ j : u → u su h that ˇ j ◦ χ 1 = χ 2 ◦ j : a j   χ 1 / / u ˇ j      b χ 2 / / u u is said to b e C -universal if it is C -onal, that is for ev ery a ∈ C there exists an arro w χ : a → u in D : a χ / / _ _ _ u Remarks 1.4. It is easy to see that if u is C -ultrahomogeneous and C -univ ersal then u is C -homogeneous. Also, to v erify that an ob jet u in D is C -ultrahomogeneous one an learly supp ose, without loss of generalit y , that the arro w j in the denition is an iden tit y . 2 Let us reall the follo wing denitions from [10℄. Giv en a ategory C and a olletion of arro ws F ⊆ ar r ( C ) , F is said to b e dominating in C if the family D om ( F ) of ob jets whi h are domains of an arro w in F is onal in C and satises the follo wing prop ert y: for ev ery a ∈ D om ( F ) and ev ery arro w f : a → x in C there exists an arro w g : x → cod ( g ) in C su h that g ◦ f ∈ F . Notie that ar r ( C ) is alw a ys dominating in C , and if C ′ is a sk eleton of C , ar r ( C ′ ) is dominating in C . Giv en a ategory C and an ordinal κ > 0 , an indutiv e κ -sequene (or κ - hain) in C is a funtor ~ u : κ → C , where κ is regarded as a p oset ategory . F or i ∈ κ w e denote ~ u ( i ) b y u i and for i, j ∈ κ su h that i ≤ j w e denote ~ u ( i → j ) : u i → u j b y u j i . ~ u is said to b e a F raïssé sequene of length κ (or, briey , a κ -F raïssé sequene) in C if it satises the follo wing onditions: (1) F or ev ery a ∈ C there exists i ∈ κ and an arro w χ : a → u i in C ; (2) F or ev ery i ∈ κ and for ev ery arro w f : u i → cod ( f ) in C , there exists j ∈ κ with j ≥ i and an arro w g : cod ( f ) → u j su h that u j i = g ◦ f . ~ u is said to ha v e the extension prop ert y if it satises the follo wing ondition: F or ev ery arro ws f : a → b, g : a → u i in C where i ∈ κ , there exists j ∈ κ with j ≥ i and an arro w h : b → u j su h that u j i ◦ g = h ◦ f . Of ourse, ev ery sequene satisfying the extension prop ert y satises prop ert y (2) in the denition of F raïssé sequene. A ategory C is said to b e κ -b ounded if ev ery  hain in C of length λ < κ has a o one in C o v er it. Clearly , ev ery ategory is ω -b ounded. A κ - hain ~ u : κ → C is said to b e on tin uous if for ea h limit ordinal j ∈ κ , u j is the olimit of the j - hain obtained as the restrition of ~ u to j with univ ersal olimit arro ws giv en b y the arro ws u i → u j ( i < j ) of the  hain. Giv en an innite ardinal κ and an em b edding i : C → D , w e denote b y D κ the full sub ategory of D on the ob jets that an b e expressed as olimits of κ - hains in C and b y D c κ the full sub ategory of D on the ob jets that an b e expressed as olimits of on tin uous κ - hains in C . W e will sa y that an em b edding i : C → D is κ -on tin uous if D κ = D c κ . Ob viously , ev ery em b edding is ω -on tin uous. F ollo wing the terminology in [5 ℄, w e will sa y that an ob jet a in C is κ -small in D if the funtor H om D ( i ( a ) , − ) : D → Set preserv es all olimits of κ - hains in D ; in partiular, ev ery nitely presen table ob jet in C is κ -small. Notie that, giv en an em b edding i : C → D su h that all the ob jets in C are κ -small in D , for i to b e κ -on tin uous it sues that C is losed under olimits of λ - hains in D for ea h λ < κ ; indeed, giv en an indutiv e κ -sequene ~ u in C with olimit u w e an onstrut (b y transnite reursion) a on tin uous κ - hain ~ v in C with a univ ersal olimiting one D to u (fr. also the pro of of Lemma 1 in [12 ℄); indeed, denoted b y j i : u i → u (for 3 i < κ ) the univ ersal olimit arro ws for ~ u , w e dene ~ v as follo ws: ~ v (0 ) = ~ u (0) and D (0) = j 0 ; giv en ~ v ( i ) and D ( i ) : v i → u , v i b eing κ -small in D , there exists j > i and an arro w h : v i → u j su h that D ( i ) = j i ◦ h ; w e put ~ v ( i + 1) = u j and D ( i + 1) = h ; if i < κ is a limit ordinal then w e dene ~ v ( i ) and D ( i ) resp etiv ely as the olimit col im j k and an arro w s : v l → u j su h that s ◦ f = u j k ; w e put k (1) = j and G (0) = s . Giv en k ( i ) , k ( i + 1) , l ( i ) , F ( i ) , G ( i ) w e dene k ( i + 2) , l ( i + 1) , F ( i + 1) , G ( i + 1) as follo ws: b y ondition (2) in the 5 denition of F raïssé sequene applied to ~ v there exist j ∈ κ with j > l ( i ) and an arro w s : u k ( i +1) → v j su h that s ◦ G ( i ) = v j l ( i ) ; w e put l ( i + 1) = j and F ( i + 1) = s . Again, b y ondition (2) in the denition of F raïssé sequene applied to ~ u there exist j ′ ∈ κ with j ′ > k ( i + 1) and an arro w s ′ : v l ( i +1) → u j ′ su h that s ′ ◦ F ( i + i ) = u j ′ k ( i +1) ; w e put k ( i + 2) = j ′ and G ( i + 1) = s ′ . If i = sup j card (Σ) . L et T -mo d e b e the  ate gory of T -mo dels and elementary emb e ddings b etwe en them and i k : T -mo d κ e → T -mo d e b e the emb e dding of the ful l sub  ate gory T -mo d κ e of T -mo d e on the κ -pr esentable obje ts into T -mo d e . Then if T -mo d κ e satises AP, JEP and has a dominating family of arr ows in it of  ar dinality at most κ , T has a mo del of  ar dinality ≤ κ whih is T -mo d κ e -ultr ahomo gene ous and T -mo d κ e -universal; mor e over, a T -mo del with these pr op erties is unique (up to isomorphism) among the T -mo dels of  ar dinality ≤ κ . Pro of This immediately follo ws from Theorem 1.5, Prop osition 1 in [12℄ and the remarks preeding Theorem 1.5.  Finally , some ardinalit y onsiderations. If C is a ategory strutured o v er Set , or more generally o v er a funtor ategory [ I , Set ] (where I is a set, regarded here as a disrete ategory) via a forgetful funtor U : C → [ I , Set ] , then one an naturally dene a notion of ardinalit y for ob jets of C . Indeed, one an dene the ardinalit y of an ob jet c ∈ C b y the form ula car d ( c ) = | ` i ∈ I U ( c )( i ) | = ` i ∈ I | U ( c )( i ) | . These denitions apply for instane to the ase of mo dels of a man y-sorted (geometri) theory (in this ase C is the ategory of su h mo dels while I is the set of sorts of the theory), giving a notion of ardinalit y for su h mo dels that generalizes the denition of ardinalit y of a mo del in lassial mo del theory . Supp ose i : C → D is an em b edding as in Theorem 1.5 ; if D is strutured o v er a funtor ategory [ I , Set ] via a funtor U : D → [ I , Set ] then w e ha v e a notion of ardinalit y for ob jets of D and in partiular of C , and w e migh t w an t to estimate the ardinalit y of the ultrahomogeneous univ ersal ob jet giv en b y Theorem 1.5 in terms of the ardinalit y of the ob jets of C . This is partiularly easy to do in ase the funtor U reates olimits of κ - hains; in fat w e kno w that the olimits in [ I , Set ] are omputed p oin t wise and w e ha v e a partiularly elegan t desription of ltered olimits (in partiular olimits of κ - hains) in Set (see for example p. 77 in [2℄). Sp eially , if u = lim − → ~ u is the olimit in D of an indutiv e κ -sequene with v alues in C , w e ha v e 8 car d ( u ) = car d (lim − → D ~ u ) = car d (lim − → [ I , S et ] ( U ◦ ~ u )) = ` i ∈ I | lim − → Set ( U ◦ ~ u )( i ) | . Notie that for ea h i ∈ I , ( U ◦ ~ u )( i ) denes a κ - hain in Set . F rom this expression one an then dedue that if | I | ≤ κ and for ea h i ∈ I and j ∈ κ | ( U ◦ ~ u )( i )( j ) | ≤ κ then car d ( u ) ≤ κ . Th us for example if all the ob jets in C ha v e ardinalit y ≤ κ and | I | ≤ κ then ev ery ob jet in D κ has ardinalit y ≤ κ . This is for instane the ase of the lassial F raïssé's onstrution, where in fat the F raïssé's limit is alw a ys at most oun table. 2 The top os-theoreti in terpretation A remark on notation: all the top oses in this setion will b e Grothendie k top oses, if not otherwise stated. Let us reall that there exists an initial ob jet in the ategory of top oses and geometri morphisms, whi h is giv en b y the terminal ategory 1 ha ving just one ob jet and the iden tit y morphism on it; in fat, this ategory is a (oheren t, atomi) Grothendie k top os, b eing the ategory of shea v es on the empt y ategory with resp et to the atomi top ology on it (another presen tation of it is obtained b y taking the shea v es on 1 with resp et to the maximal Grothendie k top ology on it, that is the top ology in whi h all siev es o v er). W e will sa y that a top os E is trivial if it is naturally equiv alen t to 1 ; of ourse, this is the same as sa ying that E is degenerate, that is 0 E ∼ = 1 E . Let us reall that a top os E is said to ha v e enough p oin ts if the in v erse image funtors of the geometri morphisms Set → E are join tly onserv ativ e; ev ery oheren t top os has enough p oin ts (see for example [9℄). Lemma 2.1. L et E b e a top os with enough p oints. Then E is trivial if and only if it has no p oints. Pro of In one diretion, let us supp ose E trivial. Then E has no p oin ts b eause if f : Set → E w ere a p oin t then w e w ould ha v e 0 Set ∼ = f ∗ (0 E ) ∼ = f ∗ (1 E ) ∼ = 1 Set , whi h is absurd. Con v ersely , if E has no p oin ts then b y taking the unique arro w 0 : 0 E → 1 E in E then w e trivially ha v e that for ea h p oin t f of E f ∗ (0) is an isomorphism; from the fat that E has enough p oin ts w e an th us onlude that 0 is an isomorphism, that is E is trivial.  Lemma 2.2. L et C b e a  ate gory satisfying the right Or e  ondition, and J at the atomi top olo gy on it. Then Sh ( C , J at ) is trivial if and only if C is the empty  ate gory. 9 Pro of Reall that 1 Sh ( C ,J at ) is giv en b y the onstan t funtor ∆1 Set : C op → Set , while 0 Sh ( C ,J at ) is giv en b y the result of applying the asso iated sheaf funtor a : [ C op , Set ] → Sh ( C , J at ) to the initial ob jet of [ C op , Set ] , that is the onstan t funtor ∆ ∅ : C op → Set . But this funtor is trivially a sheaf with resp et to the atomi top ology on C , sine all its o v ering siev es are non-empt y , so a (∆ ∅ ) ∼ = ∆ ∅ . No w, learly , ∆ ∅ ∼ = ∆1 Set if and only if C is the empt y ategory .  Lemma 2.3. L et C b e a  ate gory satisfying the right Or e  ondition, and J at the atomi top olo gy on it. Then if [ C op , Set ] is  oher ent, Sh ( C , J at ) is  oher ent. Pro of F rom [1℄ w e kno w that if [ C op , Set ] is oheren t, then w e an axiomatize the theory of at funtors on C with oheren t axioms in the language of preshea v es on C . Then, to obtain a oheren t axiomatization for the theory of at J at -on tin uous funtors on C , it sues to add to these axioms, for ea h arro w f : c → d , the follo wing (oheren t) axiom: ⊤ ⊢ y ( ∃ x ∈ c )( f ( x ) = y ) .  W e reall that in [1℄ Bek e, Karazeris and Rosi ký ha v e in tro dued a notion of ategory ha ving all f nite limits and pro v ed the follo wing result: [ C op , Set ] is oheren t if and only if C has all f nite limits. Without going in to details, w e just remark that this fat an b e protably applied in onnetion with Lemma 2.3 (see for example Theorem 2.4 b elo w). W e reall that a geometri theory T is said to b e of presheaf t yp e if its lassifying top os is a presheaf top os (equiv alen tly , the top os [ C , Set ] , where C := ( f.p. T -mo d ( Set )) is the ategory of nitely presen table T -mo dels in Set ). W e will sa y that t w o geometri theories are Morita-equiv alen t if they ha v e the same ategory of mo dels - up to natural equiv alene - in to ev ery Grothendie k top os E naturally in E , equiv alen tly the same lassifying top os. W e reall from [ 3 ℄ that if T is a theory of presheaf t yp e su h that the ategory ( f.p. T -mo d ( Set )) op satises the righ t Ore ondition (equiv alen tly f.p. T -mo d ( Set ) satises AP), then the top os Sh (( f.p. T -mo d ( Set )) op , J at ) lassies the homogeneous T -mo dels. W e note that the notion of homogeneit y of a mo del of T in Set dened in [3℄ oinides with the notion of ( f.p. T -mo d ( Set )) -homogeneous ob jet of the ategory T -mo d ( Set ) with resp et to the em b edding ( f.p. T -mo d ( Set )) ֒ → T -mo d ( Set ) that w e dened in the rst setion of this pap er. 10 W e will sometimes iden tify theories with their Morita-equiv alene lasses; the theory of at J at -on tin uous funtors on ( f.p. T -mo d ( Set )) op , whi h an b e tak en as the anonial represen tativ e for the Morita-equiv alene lass of theories lassied b y the top os Sh (( f.p. T -mo d ( Set )) op , J at ) , will b e alled the theory of homogeneous T -mo dels. A geometri theory is said to b e onsisten t if it has at least one mo del in Set . The previous lemmas om bine to giv e the follo wing onsisteny result. Theorem 2.4. L et T b e a the ory of pr eshe af typ e suh that the  ate gory f.p. T -mo d ( Set ) has the amalgamation pr op erty. If the the ory of homo gene ous T -mo dels is Morita-e quivalent to a  oher ent the ory (for example when the  ate gory f.p. T -mo d ( Set ) has al l f nite  olimits) and ther e is at le ast one T -mo del in Set , then ther e exists at le ast one homo gene ous T -mo del in Set . Pro of The theory T ′ of homogeneous T -mo dels is Morita-equiv alen t to a oheren t theory if and only if its lassifying top os Sh (( f.p. T -mo d ( Set )) op , J at ) is a oheren t top os. Notie that for an y ategory C , C is empt y if and only if Ind - C is empt y; so if T is a theory of presheaf t yp e then T has a mo del in Set if and only if it has a nitely presen table mo del in Set . Then, sine f.p. T -mo d ( Set ) is not the empt y ategory , it follo ws from Lemma 2.2 that the top os Sh (( f.p. T -mo d ( Set )) op , J at ) is not trivial. Hene, b y Lemma 2.1, it has a p oin t. This p oin t orresp onds to a T ′ -mo del in Set , that is, to a homogeneous T -mo del in Set . The fat that when the ategory f.p. T -mo d ( Set ) has all f nite olimits, T ′ is Morita-equiv alen t to a oheren t theory follo ws from Lemma 2.3.  A (man y-sorted) geometri theory is said to b e atomi if it is lassied b y an atomi top os. Of ourse, the prop ert y of atomiit y for a theory is stable under Morita-equiv alene. A geometri theory T o v er a signature Σ is said to b e omplete if ev ery sen tene o v er Σ is T -pro v ably equiv alen t to ⊤ or ⊥ , but not b oth. It is w ell-kno wn that if T is atomi then T is omplete if and only if its lassifying top os Set [ T ] is onneted (equiv alen tly , t w o-v alued - see the pro of of Theorem 2.5 b elo w). Reall that if a theory is oheren t then its ompleteness implies its onsisteny (fr. for example Lemma 2.1), but this impliation do es not hold for a general geometri theory; in fat, there exist onneted atomi top oses without p oin ts (see for example [9℄). W e also remark that the prop ert y of ompleteness for a geometri theory is stable under Morita-equiv alene, b eing equiv alen t to a ategorial prop ert y (to b e t w o-v alued) of the orresp onding lassifying top os. 11 Theorem 2.5. L et C b e a  ate gory satisfying the right Or e  ondition, and J at the atomi top olo gy on it. Then the atomi top os Sh ( C , J at ) is  onne te d if and only if C is a  onne te d  ate gory. Pro of Reall that a top os E is said to b e lo ally onneted if the geometri morphism γ : E → Set is essen tial, that is the in v erse image funtor γ ∗ : Set → E has a left adjoin t γ ! : E → Set . An ob jet A of a lo ally onneted top os E is said to b e onneted if γ ! ( A ) ∼ = 1 Set . Ev ery atomi top os E is lo ally onneted (see for example p. 684 of [9℄), and the ob jets of E whi h are onneted are also alled atoms. W e observ e that an ob jet A of an atomi top os E is an atom if and only if the only sub ob jets of A in E are 0 A : 0 → A and 1 A : A → A and they are distint from ea h other. Indeed, this easily follo ws from the bijetion Sub E ( A ) ∼ = Sub Set ( γ ! ( A )) (fr. p. 685 of [ 9℄). Hene, sine ev ery atomi top os is lo ally onneted, Lemma C.3.3.3 in [9℄ giv es the follo wing  haraterization, to whi h w e refer as to ( ∗ ) : an atomi top os E is onneted if and only if the only sub ob jets of 1 E in E are 0 1 : 0 → 1 and 1 1 : 1 → 1 and they are distint from ea h other. W e use this riterion to pro v e our theorem. W e an iden tify the subterminals in Sh ( C , J at ) with J at -ideals on C (see p. 576 of [9℄). By realling (from the pro of of Lemma 2.2 ) that 0 Sh ( C ,J at ) is the onstan t funtor ∆ ∅ : C op → Set , ondition ( ∗ ) an th us b e rephrased as follo ws: C is non-empt y and ev ery non-empt y subset I ⊆ ob ( C ) whi h is a siev e (that is, for ea h arro w f : a → b in C , b ∈ I implies a ∈ I ) and satises the prop ert y ( ∀ R ∈ J at ( U )(( ∀ f i : U i → U ∈ R, U i ∈ I ) ⇒ ( U ∈ I )) is the whole of C . Being J at the atomi top ology on C , this ondition simplies to: C is non-empt y and ev ery non-empt y subset I ⊆ ob ( C ) whi h is a siev e and satises the prop ert y ∀ f : V → U in C , (( V ∈ I ) ⇒ ( U ∈ I )) is the whole of ob ( C ) ; but this is learly equiv alen t to sa ying that C is onneted.  Theorem 2.6. L et C b e a non-empty  ate gory satisfying the amalgamation pr op erty. Then C satises the joint emb e dding pr op erty if and only if it is a  onne te d  ate gory. Pro of If C satises JEP then for an y ob jets a, b ∈ C there exists an ob jet c ∈ C and morphisms f : a → c , g : b → c in C : a f   b g / / c 12 Then w e ha v e the follo wing zig-zag b et w een a and b : a 1 a           f   = = = = = = = = b g           1 b   ? ? ? ? ? ? ? ? a c b . Con v ersely , w e pro v e that for an y ob jets a, b ∈ C there exists an ob jet c ∈ C and morphisms f : a → c , g : b → c in C b y indution on the length n of a zig-zag that onnets a and b . If n = 1 then the thesis follo ws immediately from the amalgamation prop ert y . If n > 1 w e ha v e a zig-zag . . . . . . d ′ n f n } } { { { { { { { { g n # # G G G G G G G G G d 0 = a . . . d n − 1 d n = b . By applying the indution h yp othesis to the pair a, d n − 1 one gets an ob jet d ∈ C and morphisms h : a → d , k : d n − 1 → d in C . The amalgamation prop ert y applied to the pair of morphisms k ◦ f n and g n then giv es an ob jet c and t w o morphisms s : d → c and t : b → c . Then w e ha v e morphisms f := s ◦ h : a → c and g := t : b → c , as required.  F rom Theorems 2.5 and 2.6 w e th us dedue that giv en a onsisten t theory of presheaf t yp e T su h that the ategory f.p. T -mo d ( Set ) satises the amalgamation prop ert y , the ondition that f.p. T -mo d ( Set ) satises JEP is exatly what mak es the theory T ′ of homogeneous T -mo dels omplete. Indeed, T ′ is omplete if and only if Set [ T ′ ] ≃ Sh (( f.p. T -mo d ( Set )) op , J at ) is onneted, if and only if ( f.p. T -mo d ( Set )) op is onneted, if and only if ( f.p. T -mo d ( Set )) is onneted, if and only if ( f.p. T -mo d ( Set )) satises JEP . A geometri theory is said to b e oun tably ategorial if an y t w o oun table mo dels of T in Set are isomorphi (where b y `oun table' w e mean either nite or den umerable). Notie that, b y our denition, an y geometri theory ha ving no mo dels in Set is (v aously) oun tably ategorial. W e reall from [4℄ that ev ery omplete atomi geometri theory is oun tably ategorial; so, b y the remarks ab o v e, w e obtain the follo wing result. Theorem 2.7. L et T b e a  onsistent the ory of pr eshe af typ e suh that the  ate gory f.p. T -mo d ( Set ) has the amalgamation and joint emb e dding pr op erties. If T ′ is a ge ometri the ory whih is Morita-e quivalent to the the ory of homo gene ous T -mo dels then T ′ is  omplete and  ountably  ate gori al.  13 Remark 2.8. Conerning the existene of homogeneous T -mo dels in Set , w e note that if the theory T in Theorem 2.7 is oheren t then, b y Lemma 2.3 and Theorem 2.4 , there exists a homogeneous T -mo del in Set . If moreo v er the signature of T is oun table then, b y the results in [ 4℄, there is a oun table homogeneuos T -mo del in Set . The usefulness of Theorem 2.7 lies in the fat that it is generally not diult to see, giv en a theory of presheaf t yp e T , if a ertain theory is Morita-equiv alen t to the theory of homogeneous T -mo dels. In fat, one an use Corollary 4.7 in [3 ℄ and the expliit desription of the homogeneous mo dels giv en in [3℄. F or example, in [ 3℄ w e sa w that, giv en the theory T of linear orders, the dense linear orders without endp oin ts orresp onded preisely to the homogeneous T -mo dels. By using similar metho ds, one an also sho w that, giv en the theory of deidable ob jets, the innite deidable ob jets are exatly the homogeneous deidable ob jets and that, giv en the algebrai theory of Bo olean algebras, the atomless Bo olean algebras are exatly the homogeneous Bo olean algebras. This leads, via Theorem 2.7, to an alternativ e pro of that the theory of dense linear orders without endp oin ts and the theory of atomless Bo olean algebras are omplete and oun tably ategorial. Moreo v er, w e kno w from [4℄ that, under the h yp otheses of Theorem 2.7 , the Bo oleanization of the theory T axiomatizes the T -homogeneous mo dels, and hene w e ma y dedue that an y t w o oun table T -homogeneous mo dels in Set are isomorphi (fr. Theorem 3.3 [ 4℄). W e also reall from [ 4℄ that if T is an atomi, omplete oun table geometri theory with innite mo dels in Set then, denoted b y M the unique oun table mo del of T (up to isomorphism), w e ha v e the follo wing represen tation for the lassifying top os Set [ T ] of T : Set [ T ] ≃ Con t ( Aut ( M )) , where Con t ( Aut ( M ) ) is the top os of on tin uous Aut ( M ) -sets, Aut ( M ) b eing endo w ed with the top ology of p oin t wise on v ergene. Let us no w apply the ategorial theorem in the rst setion in the on text of the theories of presheaf t yp e. Theorem 2.9. L et T b e a  onsistent the ory of pr eshe af typ e suh that the  ate gory f.p. T -mo d ( Set ) satises AP, JEP. If ther e exists in f.p. T -mo d ( Set ) a dominating family of arr ows of nite or  ountable  ar dinality then ther e exists in Set a f.p. T -mo d ( Set ) -homo gene ous and ( f.p. T -mo d ( Set )) -universal T -mo del; also, given a ( f.p. T -mo d ( Set )) -homo gene ous and ( f.p. T -mo d ( Set )) -universal T -mo del M , 14 if M  an b e written as the  olimit in T -mo d ( Set ) of a ω -hain of nitely pr esentable T -mo dels (e quivalently, is ω + -pr esentable) then, pr ovide d that al l the morphisms in ( T -mo d ( Set )) ω ar e moni, M is ( f.p. T -mo d ( Set )) -ultr ahomo gene ous and unique (up to isomorphism) with this pr op erty among the ( f.p. T -mo d ( Set )) -universal and ω + -pr esentable T -mo dels. If T ′ is a ge ometri the ory whose mo dels (in any Gr othendie k top os) ar e the homo gene ous T -mo dels, then T ′ is  omplete and  ountably  ate gori al. In p artiular, if T is  ountable and has innite mo dels in Set ther e exists a unique (up to isomorphism)  ountable homo gene ous T -mo del M , and Set [ T ′ ] ≃ Sh ( ( f.p. T -mo d ( Set )) op , J at ) ≃ Cont ( Aut ( M )) , Aut ( M ) b eing endowe d with the top olo gy of p ointwise  onver gen e. Pro of This is immediate from Theorem 1.5 , the remarks follo wing it, Theorem 2.7 and the remark ab o v e.  Let us no w in tro due the follo wing notions. Giv en an em b edding i : C ֒ → D and an ob jet u ∈ D together with a  hoie of an arro w f c : c → u in D for ea h ob jet c of C , w e an onsider a ategory ˜ C , dened as the full sub ategory of ( C ↓ u ) on the arro ws f : a → u in D su h that there exists an automorphism α of u (that is, an isomorphism α : u → u in the ategory D ) su h that f = α ◦ f a . Then w e an dene a funtor χ : ˜ C op → S ubg r ( Aut ( u )) , where S u bg r ( Aut ( u )) is the olletion of the subgroups of Aut ( u ) regarded as a p oset ategory with resp et to the inlusion, in the follo wing w a y: χ sends an ob jet f : a → u in ˜ C to the subgroup Aut f ( u ) of Aut ( u ) formed b y the automorphisms α of u su h that α ◦ f = f and an arro w h : f → g in ˜ C to the inlusion Aut g ( u ) ⊆ Aut f ( u ) . If χ is full and faithful and reets iden tities (that is, for ea h pair of arro ws h, k in ˜ C op χ ( h ) = χ ( k ) implies h = k ) w e sa y that u satises the Galois prop ert y with resp et to ˜ C ; notie that if C is sk eletal and χ is full and faithful then u satises the Galois prop ert y with resp et to ˜ C . Also, w e an endo w the group Aut ( u ) with a top ology U b y sa ying that the subgroups in the image of the funtor χ form a base of neigh b ourho o ds of the iden tit y . In the on text of these notions, the follo wing prop osition holds. Prop osition 2.10. Given an emb e dding i : C ֒ → D suh that al l the arr ows f c (for c ∈ C ) ar e moni, let u b e a C -ultr ahomo gene ous obje t in D whih satises the Galois pr op erty with r esp e t to ˜ C . Then the  ate gory C satises the amalgamation pr op erty and ther e is a natur al e quivalen e Sh ( C op , J at ) ≃ Con t ( Aut ( u )) 15 wher e Con t ( Aut ( u )) is the top os of  ontinuous Aut ( u ) -sets, Aut ( u ) b eing endowe d with the top olo gy U . Pro of F rom Theorem 2 p. 154 [ 11℄ w e dedue that Con t ( Aut ( u )) is naturally equiv alen t to Sh ( S U ( Aut ( u )) , J at ) , where S U ( Aut ( u )) is the ategory ha ving as ob jets the on tin uous Aut ( u ) -sets of the form Aut ( u ) /χ ( f ) for f ∈ ˜ C and as arro ws Aut ( u ) /χ ( f ) → Aut ( u ) /χ ( g ) the osets χ ( g ) α with the prop ert y that χ ( f ) ⊆ α − 1 χ ( g ) α (see [11℄ for more details). T o pro v e our prop osition it is therefore enough to sho w that there is an equiv alene of ategories b et w een S U ( Aut ( u )) and C op . W e expliitly dene a funtor F : S U ( Aut ( u )) → C op and pro v e that it is an equiv alene of ategories. Let us rst dene F on ob jets: F sends an ob jet Aut ( u ) /χ ( f ) of S U ( Aut ( u )) to dom ( f ) ∈ C ; this is w ell-dened sine χ reets iden tities. Giv en an arro w Aut ( u ) /χ ( f ) → Aut ( u ) /χ ( g ) , represen ted b y a oset χ ( g ) α , w e ha v e that χ ( f ) ⊆ α − 1 χ ( g ) α , equiv alen tly αχ ( f ) α − 1 ⊆ χ ( g ) . This means that α ◦ β ◦ α − 1 ◦ g = g (equiv alen tly , β ◦ ( α − 1 ◦ g ) = ( α − 1 ◦ g ) ) for ea h β ∈ Aut ( u ) su h that β ◦ f = f , whi h is in turn equiv alen t to sa ying that χ ( f ) ⊆ χ ( α − 1 ◦ g ) . This implies, b y our h yp othesis that χ is full and faithful, that there exists a unique arro w z : dom ( g ) → dom ( f ) in C su h that f ◦ z = α − 1 ◦ g . W e put F ( χ ( g ) α ) = z ; this is w ell-dened sine χ ( g ) α = χ ( g ) α ′ if and only if α ◦ α ′− 1 ∈ χ ( g ) , if and only if α − 1 ◦ g = α ′− 1 ◦ g , if and only if f ◦ F ( χ ( g ) α ) = f ◦ F ( χ ( g ) α ′ ) if and only if F ( χ ( g ) α ) = F ( χ ( g ) α ′ ) , where the last equiv alene follo ws from the fat that f is moni. This also pro v es that F is faithful. F is full b eause u is C -ultrahomogeneous, and it is surjetiv e b y denition of U . Therefore, F is an equiv alene of ategories.  A  kno wledgemen ts. I am v ery grateful to m y Ph.D. sup ervisor P eter Johnstone for his supp ort and enouragemen t. Thanks also to Martin Hyland for ha ving suggested me to in v estigate F raïssé's onstrution top os-theoretially . 16 Referenes [1℄ T. Bek e, P . Karazeris and J. Rosi ký , When is atness oheren t?, Communi ations in A lgebr a 33 (2005), no. 6, 1903-1912. [2℄ F. 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