Atomic toposes and countable categoricity
We give a model-theoretic characterization of the class of geometric theories classified by an atomic topos having enough points; in particular, we show that every complete geometric theory classified by an atomic topos is countably categorical. Some…
Authors: Olivia Caramello
A tomi top oses and oun table ategoriit y Olivia Caramello DPMMS, Univ ersit y of Cam bridge, Wilb erfore Road, Cam bridge CB3 0WB, UK O.Caramellodpmms.am.a.uk Otob er 27, 2018 Abstrat W e giv e a mo del-theoreti haraterization of the lass of geometri theories lassied b y an atomi top os ha ving enough p oin ts; in partiular, w e sho w that ev ery omplete geometri theory lassied b y an atomi top os is oun tably ategorial. Some appliations are also disussed. 1 Some results on atomi top oses In this setion w e presen t some results on atomi top oses whi h are relev an t to our haraterization theorem in the seond setion. Let us reall the follo wing standard denition. Denition 1.1. Let E b e a top os. An ob jet A ∈ E is said to b e an atom of E if the only sub ob jets of A (up to isomorphism) are the iden tit y arro w 1 A : A → A and the zero arro w 0 A : 0 → A , and they are distint from ea h other. The follo wing prop osition desrib es the b eha viour of asso iated sheaf funtors with resp et to atoms. Prop osition 1.2. L et E b e a top os and j a top olo gy on it with asso iate d she af funtor a j : E → sh j ( E ) . If A is an atom of E then a j ( A ) is an atom of sh j ( E ) , pr ovide d that it is non-zer o. 1 Pro of Giv en a monomorphism m : C → a j ( A ) in sh j ( E ) , m is a monomorphism also in E sine the inlusion i : sh j ( E ) ֒ → E preserv es monomorphisms (ha ving a left adjoin t). No w, denoted b y η the unit of the adjution a j ⊣ i , onsider the pullba k C ′ m ′ / / A η A C m / / a j ( A ) in E . The arro w m ′ is a monomorphism in E , b eing the pullba k of a monomorphism, so, sine A is an atom of E w e dedue that m ′ is either (isomorphi to) the iden tit y arro w on A or the zero arro w 0 A . No w, b y applying a j to the pullba k ab o v e w e obtain a pullba k in sh j ( E ) (as a j preserv es pullba ks); but a j ( η A ) ∼ = 1 a j ( A ) , so m ∼ = a j ( m ′ ) and m is either (isomorphi to) the iden tit y or the zero arro w on a j ( A ) ; of ourse, if a j ( A ) ≇ 0 sh j ( E ) these t w o arro ws are distint from ea h other. W e reall that an atomi top os is an elemen tary top os E whi h p ossesses an atomi geometri morphism E → Set . W e refer the reader to setion C3.5 in [6℄ for a omprehensiv e treatmen t of the topi of atomi top oses. Here w e limit ourselv es to remarking the follo wing fats. Prop osition 1.3. L et E b e a Gr othendie k top os. Then (i) E is atomi if and only if it has a gener ating set of atoms; (ii) if { a i | i ∈ I } is a gener ating set of atoms for E then the atoms of E ar e exatly the epimorphi images of the atoms in the gener ating set; in p artiular, E has only a set of (isomorphism lasses of ) atoms. Pro of (i) Supp ose that E is atomi. Then all the sub ob jet latties in E are atomi Bo olean algebras (fr. p. 685 [6℄) and hene ev ery ob jet of E an b e written as a disjoin t opro dut of atoms; on the other hand, there an b e only a set of atoms (up to isomorphism) in E , b y the argumen t at the top of p. 690 [6℄. Con v ersely , if E has a generating set of atoms then the full sub ategory C of E on it satises the righ t Ore ondition and E ∼ = Sh ( C , J at ) , where J at is the atomi top ology on C (fr. the disussion p. 689 [6℄); so it is atomi (b y Theorem C3.5.8 [ 6℄). (ii) This w as remark ed p. 690 [6 ℄. 2 As a onsequene of Prop ositions 1.2 and 1.3(i), w e ma y dedue that an y subtop os of an atomi Grothendie k top os E is atomi; indeed, the images of the atoms in a generating set of E via the orresp oning asso iated sheaf funtor learly form a generating set for the subtop os. In fat, this prop ert y holds more generally at the elemen tary lev el (i.e. ev ery subtop os of an atomi top os is atomi), b y the follo wing argumen t. Let E b e an atomi top os; then, E b eing Bo olean, ev ery subtop os F of E is op en (b y Prop osition A4.5.22 [6℄) and hene the inlusion of F in to E is an atomi morphism (b y Prop osition A4.5.1 [5℄); this implies that the geometri morphism F → Set is atomi, b eing the omp osite of t w o atomi morphisms (the inlusion F ֒ → E and the morphism E → Set ); so F is atomi. In terms of sites, if E ∼ = Sh ( C , J C at ) (where C satises the righ t Ore ondition and J C at is the atomi top ology on it) then the subtop oses of it an b e desrib ed as follo ws. Let F b e a subtop os of E ; as w e ha v e already remark ed, F m ust b e op en, that is of the form E /U ֒ → E for a subterminal ob jet U in E . No w, b y Remark C2.3.21 [6℄, U an b e iden tied with a J at -ideal on C , that is with a olletion of ob jets C ′ of C with the prop ert y that for an y arro w f : a → b in C , a ∈ C ′ if and only if b ∈ C ′ . If w e regard C ′ as a full sub ategory of C then Sh ( C , J C at ) /U ∼ = Sh ( C ′ , J C ′ at ) (where J C ′ at is the atomi top ology on C ′ ). Indeed, w e ma y dene an equiv alene as follo ws. Giv en a ob jet G → U in Sh ( C , J C at ) /U , for ev ery c ∈ C whi h do es not b elong to C ′ w e m ust ha v e G ( c ) = ∅ , sine w e ha v e an arro w G ( c ) → U ( c ) and U ( c ) = ∅ ; so w e asso iate to it the restrition G | C ′ , whi h is a J C ′ at -sheaf sine J C ′ at learly oinides with the Grothendie k top ology indued b y J C at on C ′ . It is no w lear that this assigmen t denes an equiv alene b et w een our t w o ategories. So w e ha v e pro v ed that the subtop oses of Sh ( C , J C at ) are exatly those of the form Sh ( C ′ , J C ′ at ) where C ′ is a full sub ategory of C with the prop ert y that for an y arro w f : a → b in C , a ∈ C ′ if and only if b ∈ C ′ . Also, sine the assigmen t sending a subterminal ob jet in E to the orresp onding op en subtop os of E is a lattie isomorphism from Sub E (1) to the lattie of op en subtop oses of E , t w o su h subtop oses of Sh ( C , J C at ) are equiv alen t if and only if the orresp onding ategories are equal (as sub ategories of C ). Next, let us onsider a general ategory C . W e kno w that, pro vided that C satises the righ t Ore ondition, one an dene the atomi top ology on C as the top ology ha ving as o v ering siev es exatly the non-empt y ones. Su h a top ology do es not exist on a general ategory C but, b y analogy with it, w e ma y dene the atomi top ology J C at on C as the smallest Grothendie k top ology on C su h that all the non-empt y siev es are o v ering; of ourse, this denition sp eializes to the w ell-kno wn one in the ase C satises the righ t Ore ondition. As stated in follo wing prop osition, the orresp onding ategory of shea v es is an atomi top os. 3 Prop osition 1.4. L et C b e a ate gory and J C at the atomi top olo gy on it. Then Sh ( C , J C at ) is an atomi top os. Pro of Let C ′ b e the full sub ategory of C on the ob jets whi h are not J C at -o v ered b y the empt y siev e. Then, b y the Comparison Lemma, w e ha v e that Sh ( C , J C at ) ∼ = Sh ( C ′ , J C at | C ′ ) . W e no w pro v e that C ′ satises the righ t Ore ondition and J C at | C ′ = J C ′ at , that is for ev ery siev e R in C ′ , R 6 = ∅ if and only if R is J C at | C ′ -o v ering; from this our thesis will learly follo w. In one diretion, supp ose that R 6 = ∅ . Then the siev e R generated b y R in C is ob viously non-empt y and, C ′ b eing a full sub ategory of C , w e ha v e that R ∩ ar r ( C ′ ) = R ; so R is J C at | C ′ -o v ering b y denition of indued top ology . Con v ersely , supp ose that R is a J C at | C ′ -o v ering siev e on an ob jet c ∈ C ′ . Then there exists a J C at -o v ering siev e H on c in C su h that H ∩ ar r ( C ′ ) = R . Supp ose R b e empt y; then for ev ery arro w f in H w e ha v e ∅ ∈ J C at ( dom ( f )) . But H is J C at -o v ering so from the transitivit y axiom for Grothendie k top ologies it follo ws that ∅ ∈ J C at ( c ) , on tradition sine c ∈ C ′ . So w e onlude that R is non-empt y , as required. Remark 1.5. By the transitivit y axiom for Grothendie k top ologies, the sub ategory C ′ in the pro of of the prop osition ab o v e satises the prop ert y that for an y arro w f : a → b in C , a ∈ C ′ if and only if b ∈ C ′ ; in other w ords, C ′ is a union of onneted omp onen ts of C . In partiular, if C ′ 6 = C (i.e. C do es not satisfy the righ t Ore ondition) and C is onneted then C ′ = ∅ , that is the top os Sh ( C , J C at ) is trivial. The follo wing result generalizes the prop osition ab o v e. Prop osition 1.6. L et E b e a Gr othendie k top os with a gener ating set L and j b e an elementary top olo gy on E suh that al l the monomorphisms a → b in E wher e a ≇ 0 and b ∈ L ar e j -dense. Then sh j ( E ) is an atomi top os. Pro of By Prop osition 1.2 , it is enough to pro v e that the images of the ob jets of L via the asso iated sheaf funtor a j form a generating set of ob jets of sh j ( E ) whi h are either zero or atoms. Our argumen t follo ws the lines of the pro of of Prop osition 1.2 . Giv en an ob jet b ∈ L and a monomorphism m : a → a j ( b ) in sh j ( E ) , onsider the pullba k a ′ m ′ / / b η b a m / / a j ( b ) 4 in E . The arro w m ′ is a monomorphism in E , b eing the pullba k of a monomorphism, so, if a ′ ≇ 0 then m ′ is j -dense b y our h yp otheses, that is a j ( m ′ ) is an isomorphism. But a j preserv es pullba ks, from whi h it follo ws that m is an isomorphism. If instead a ′ ∼ = 0 then a ∼ = a j ( a ′ ) ∼ = a j (0) = 0 sh j ( E ) so m is the zero arro w on a j ( b ) . Remark 1.7. W e note that Prop osition 1.4 is the partiular ase of Prop osition 1.6 when E is a presheaf top os [ C op , Set ] , L is the olletion of all the represen tables on C and j is the elemen tary top ology on [ C op , Set ] orresp onding to the atomi top ology on C ; indeed, the siev es in C on an ob jet c ∈ C an b e iden tied with the sub ob jets in [ C op , Set ] of the represen table C ( − , c ) . No w, let us briey onsider another approa h for obtaining an atomi top os starting from a general one, based on the onsideration of the atoms of the giv en top os. Prop osition 1.8. L et E b e a Gr othendie k top os and L a ol le tion of atoms of E , r e gar de d as a ful l sub ate gory of E . Then, if J E can is the anoni al top olo gy on E , the top os Sh ( L , J E can | L ) is atomi. Pro of Ob viously , sine ev ery arro w in L is an epimorphism in E , w e ha v e J L at ⊆ J E can | L so Sh ( L , J E can | L ) is a subtop os of the top os Sh ( L , J L at ) . But Sh ( L , J L at ) is atomi b y Prop osition 1.4 , hene Sh ( L , J L at ) is atomi b y the disussion follo wing the pro of of Prop osition 1.3. Let us no w haraterize the atoms of the top os Sh ( C , J C at ) , where C is a ategory satisfying the righ t Ore ondition. Prop osition 1.9. L et Sh ( C , J ) b e a lo al ly onne te d top os, and a J : [ C op , Set ] → Sh ( C , J ) b e the asso iate d she af funtor. Then al l the funtors a J ( C ( − , c )) ar e onne te d obje ts of Sh ( C , J ) if and only if al l the onstant funtors C op → Set ar e J -she aves. Pro of Consider the diagram Sh ( C , J ) p % % J J J J J J J J J i / / [ C op , Set ] q y y s s s s s s s s s Set of geometri morphisms in the 2-ategory of Grothendie k top oses, where p and q are the unique geometri morphisms resp etiv ely from Sh ( C , J ) and 5 [ C op , Set ] to Set . Both these geometri morphisms are essen tial, that is their in v erse image funtors ha v e left adjoin ts, whi h w e indiate resp etiv ely b y p ! and q ! ; indeed, p is essen tial b eause b y h yp othesis Sh ( C , J ) is lo ally onneted, while q is essen tial b y Example A4.1.4 [ 5 ℄. It is w ell-kno wn that the represen tables in [ C op , Set ] are all indeomp osable, so q ! ( C ( − , c )) = 1 for ea h c ∈ C . No w, the ondition that all the onstan t funtors C op → Set are J -shea v es is learly equiv alen t to demanding that q ∗ = i ◦ p ∗ where i is the inlusion Sh ( C , J ) ֒ → [ C op , Set ] or, passing to the left adjoin ts, that q ! = p ! ◦ a (of ourse, the equalities here are in tended to b e isomorphisms); but, sine all these funtors preserv e olimits (ha ving righ t adjoin ts) and ev ery funtor in [ C op , Set ] is a olimit of represen tables, the equalit y ab o v e holds if and only if 1 = q ! ( C ( − , c )) = p ! ( a J ( C ( − , c )) , that is if and only if the a J ( C ( − , c )) are all onneted ob jets of Sh ( C , J ) . Remark 1.10. W e note that for a general Grothendie k site ( C , J ) , the onstan t funtor ∆ ∅ : C op → Set is a J -sheaf if and only if ev ery J -o v ering siev e is non-empt y , and all the onstan t funtors ∆ L : C op → Set for a non-empt y set L ∈ Set are J -shea v es if and only if for ea h ob jet c ∈ C , all the J -o v ering siev es on c are empt y or onneted as full sub ategories of C /c ; in partiular, the onjution of these t w o onditions implies, b y Theorem C3.3.10 [ 6℄, that the top os Sh ( C , J ) is lo ally onneted. As a onsequene of Prop osition 1.9 and Remark 1.10 , w e dedue that if C is a ategory satisfying the righ t Ore ondition and J is a Grothendie k top ology on C su h that ev ery J -o v ering siev e is non-empt y , then all the funtors a ( C ( − , c )) are onneted ob jets of the lo ally onneted top os Sh ( C , J ) . In partiular, if J C at is the atomi top ology on C then the a ( C ( − , c )) are all atoms of the atomi top os Sh ( C , J C at ) (sine in an atomi top os the atoms are preisely the onneted ob jets, fr. p. 685 [ 6 ℄); sine they also form a generating set for the top os Sh ( C , J C at ) , w e dedue from Prop osition 1.3(ii) that the atoms of Sh ( C , J C at ) are exatly the epimorphi images of the funtors of the form a ( C ( − , c )) . By using Y oneda's lemma, one an easily rephrase this ondition as follo ws: a J C at -sheaf F is an atom of Sh ( C , J C at ) if and only if there exists an ob jet c ∈ C and an elemen t x ∈ F ( c ) with the prop ert y that ev ery natural transformation α from F to an y J C at -sheaf G is uniquely determined b y its v alue α ( c )( x ) at x . 6 2 The haraterization theorem In this setion w e pro v e our main haraterization result onerning the geometri theories lassied b y an atomi top os with enough p oin ts. Let us rst in tro due the relev an t denitions and establish some basi fats. F or the general ba kground w e refer the reader to [ 6℄. Conerning notation, for on v eniene signatures are supp osed to b e one-sorted throughout the whole setion, but all the argumen ts an b e easily adapted to the general man y-sorted ase. Denition 2.1. Let T b e a geometri theory . T is said to b e atomi if its lassifying top os Set [ T ] is an atomi top os. Denition 2.2. Let T b e a geometri theory o v er a signature Σ . T is said to ha v e enough mo dels if for ev ery geometri sequen t σ o v er Σ , M σ for all the T -mo dels M in Set implies that σ is pro v able in T . Note that sine the soundness theorem for geometri logi alw a ys holds (see for example Prop osition D1.3.2 p. 832 [6℄), the lass of theories with enough mo dels is exatly the lass of geometri theories for whi h `the' ompleteness theorem holds. Prop osition 2.3. L et T b e a ge ometri the ory over a signatur e Σ . Then T has enough mo dels if and only if its lassifying top os Set [ T ] has enough p oints. Pro of By denition, Set [ T ] has enough p oin ts if and only if the in v erse image funtors f ∗ of the geometri morphisms f : Set → Set [ T ] are join tly onserv ativ e. No w, sine the geometri morphism f M : Set → Set [ T ] orresp onding to a T -mo del M in Set satises f ∗ ( M T ) = M (where M T is the univ ersal mo del of T lying in Set [ T ] ) then it follo ws from Lemma D1.2.13 p. 825 [6℄ that if a geometri sequen t σ o v er Σ is satised in ev ery T -mo del M in Set then σ is satised in M T , equiv alen tly it is pro v able in T . Con v ersely , supp ose that T has enough mo dels. Then it is easily seen, b y using an argumen t analogous to that emplo y ed in the pro of of Prop osition D3.3.13 p. 915 [6℄, that Set [ T ] has enough p oin ts. Denition 2.4. Let T b e a geometri theory o v er a signature Σ . T is said to b e omplete if ev ery geometri sen tene φ o v er Σ is T -pro v ably equiv alen t to ⊤ or ⊥ , but not b oth. Remark 2.5. F rom the top os-theoreti p oin t of view, a geometri theory is omplete if and only if its lassifying top os is t w o-v alued (to see this, it 7 sues to onsider the syn tati represen tation for the lassifying top os as the ategory of shea v es on the geometri syn tati ategory of the theory with resp et to the `syn tati top ology' on it); moreo v er, if T is atomi then its lassifying top os is t w o-v alued if and only if it is (atomi and) onneted (fr. the pro of of Theorem 2.5. [2℄). Giv en a geometri theory T o v er a signature Σ , from no w on w e will denote the relation of T -pro v able equiv alene of geometri form ulas o v er Σ in the same on text b y T ∼ . Denition 2.6. Let T b e a geometri theory o v er a signature Σ . T is said to b e Bo olean if it lassifying top os is a Bo olean top os. Remark 2.7. W e reall from [3℄ that a geometri theory T o v er a signature Σ is a Bo olean if and only if for ev ery geometri form ula φ ( ~ x ) o v er Σ there exists a geometri form ula ψ ( ~ x ) o v er Σ in the same on text, denoted ¬ φ ( ~ x ) , su h that φ ( ~ x ) ∧ ψ ( ~ x ) T ∼ ⊥ and φ ( ~ x ) ∨ ψ ( ~ x ) T ∼ ⊤ . F rom this riterion, it follo ws that if T is a Bo olean then ev ery innitarily disjuntiv e rst-order form ula o v er Σ (i.e. an innitary rst-order form ula o v er Σ whi h do not on tain innitary onjuntions) is T -pro v ably equiv alen t using lassial logi to a geometri form ula in the same on text; indeed, this an b e pro v ed b y an indutiv e argumen t as in the pro of of Theorem D3.4.6 p. 921 [6℄. Denition 2.8. Let T b e a geometri theory o v er a signature Σ . T w o T -mo dels (in Set ) M and N are said to b e geometrially equiv alen t if and only if for ev ery geometri sen tene φ o v er Σ , M φ if and only if N φ . Let us reall that a mo del M of a geometri theory T o v er a signature Σ is said to b e onserv ativ e if M σ for ev ery geometri sequen t σ o v er Σ implies σ pro v able in T . The follo wing result represen ts the geometri analogue of the w ell-kno wn haraterization of ompleteness of a rst-order theory in mo del theory . Belo w, b y a trivial geometri theory w e mean a geometri theory in whi h ⊥ is pro v able. Prop osition 2.9. L et T b e a non-trivial Bo ole an ge ometri the ory with enough mo dels. Then the fol lowing ar e e quivalent: (i) T is omplete; (ii) for every ge ometri senten e φ , either φ T ∼ ⊤ or ¬ φ T ∼ ⊤ ; (iii) every two T -mo dels in Set ar e ge ometri al ly e quivalent; (iv) every T -mo del M in Set is onservative. 8 Pro of (i) ⇔ (ii) is ob vious. (i) ⇒ (iii) F or an y geometri sen tene φ o v er Σ , either φ T ∼ ⊤ , and hene M φ for all the T -mo dels, or φ T ∼ ⊥ , and hene M 2 φ for all T -mo dels; so (iii) immediately follo ws. (iii) ⇒ (i) Giv en a geometri sen tene φ o v er Σ , sine T has enough mo dels, if φ T ≁ ⊤ then there exists a T -mo del M in Set su h that φ do es not hold in M ; then φ do es not hold in an y T -mo del in Set , these mo dels b eing all geometrially equiv alen t. This preisely means that the geometri sequen t φ ⊢ [] ⊥ holds in ev ery T -mo del in Set , that is, T ha ving enough mo dels, φ T ∼ ⊥ . (iii) ⇒ (iv) Giv en a geometri sequen t φ ⊢ ~ x ψ o v er Σ , it is lear that for an y T -mo del M , φ ⊢ ~ x ψ holds in M if and only if the innitarily disjuntiv e rst-order sen tene ∀ ~ x ( φ → ψ ) holds in M . But, b y Remark 2.7, this form ula is T -pro v ably equiv alen t using lassial logi to a geometri sen tene; so w e onlude that if a geometri sequen t is satised in a T -mo del M then it is satised in ev ery T -mo del in Set and hene, T ha ving enough mo dels, it is pro v able in T . (iv) ⇒ (iii) is ob vious. Remarks 2.10. (a) As it is lear from the pro of, the equiv alene (i) ⇔ (iii) in the prop osition ab o v e holds in general for an y geometri theory with enough mo dels. (b) Sine ev ery Bo olean top os ha ving enough p oin ts is atomi (Corollary C3.5.2 p. 685 [ 6℄), the impliation (i) ⇒ (iv) in the prop osition ab o v e an b e seen, in view of Remark 2.5, as the logial v ersion of the top os-theoreti fat that ev ery p oin t of a onneted atomi top os is a surjetion (fr. Prop osition C3.5.6(ii) [6 ℄). Denition 2.11. Let T b e a geometri theory o v er a signature Σ . A t yp e-in-on text (or, more briey , a t yp e) of T is an y set of geometri form ulas o v er Σ in the same on text of the form { φ ( ~ x ) | M φ ( ~ a ) } , where M is a mo del of T in Set and ~ a is a tuple of elemen ts of (the underlying set of ) M ; the t yp e { φ ( ~ x ) | M φ ( ~ a ) } will b e denoted b y S T ( M , ~ a ) . A t yp e of T is said to b e omplete if it is maximal (with resp et to the inlusion) in the set of all t yp es of T . A t yp e S of T is said to b e prinipal if there exists a form ula φ ( ~ x ) ∈ S su h that for an y geometri form ula ψ ( ~ x ) o v er Σ in the same on text, φ ( ~ x ) T -pro v ably implies ψ ( ~ x ) if (and only if ) ψ ( ~ x ) ∈ S ; the form ula φ ( ~ x ) is said to b e a generator of the t yp e S . 9 Remark 2.12. Note that, b y Prop osition 2.9, the notion of omplete geometri theory in tro dued ab o v e rewrites in terms of t yp es as follo ws: a non-trivial geometri theory T ha ving enough mo dels is omplete if and only if for an y t w o T -mo dels M and N in Set , S T ( M , []) = S T ( N , []) . Denition 2.13. Let Σ b e a signature, M a Σ -struture and N a substruture of M . Then N is said to b e a geometri substruture of M if, for ev ery geometri form ula φ ( ~ x ) o v er Σ and an y tuple of elemen ts ~ a (of the same length as ~ x ) from N , M φ ( ~ a ) if and only if N φ ( ~ a ) ; equiv alen tly , S ∅ ( M , ~ a ) = S ∅ ( N , ~ a ) for an y tuple ~ a of elemen ts of N (where ∅ denotes the empt y geometri theory o v er Σ ). Remark 2.14. It is easy to pro v e b y indution on the struture of geometri form ulas that ev ery geometri form ula is equiv alen t in geometri logi to an innitary disjuntion of geometri form ulas whi h do not on tain innitary disjuntions; sine these latter form ulas are in partiular rst-order, w e ma y dedue that if N is an elemen tary substruture of M then N is a geometri substruture of M ; moreo v er, giv en a geometri sequen t φ ( ~ x ) ⊢ ~ x ψ ( ~ x ) , if this sequen t holds in M then it also holds in N . Indeed, for ev ery tuple ~ a of elemen ts in N (of the same length as ~ x ), N φ ( ~ a ) implies M φ ( ~ a ) , whi h in turn implies M ψ ( ~ a ) and hene N ψ ( ~ a ) (where the rst and third impliations follo w from the fat that N is a geometri substruture of M ). W e note that this remark justies the use of the do wn w ard Lö w enheim-Sk olem theorem in the on text of geometri logi; more preisely , giv en a geometri theory T o v er a signature Σ of ardinalit y | Σ | , if T has a mo del M su h that | M | ≥ | Σ | then T has a mo del of ardinalit y | Σ | . Belo w b y `oun table' w e mean either nite or den umerable. Denition 2.15. Let T b e a geometri theory . Then T is said to b e oun tably ategorial if an y t w o mo dels of T in Set of oun table ardinalit y are isomorphi. W e remark that, b y our denition, an y geometri theory ha ving no mo dels in Set is (v aously) oun tably ategorial. The follo wing denition is the geometri equiv alen t of the notion of atomi mo del in lassial mo del theory . Denition 2.16. Let T b e a geometri theory o v er a signature Σ . A mo del M of T in Set is said to b e atomi if for an y tuple of elemen ts ~ a of M , the t yp e S T ( M , ~ a ) is prinipal and omplete. 10 Let us reall from [3 ℄ that a geometri theory o v er a signature Σ is Bo olean if and only if ev ery geometri form ula φ ( ~ x ) o v er Σ whi h is stably onsisten t with resp et to T (i.e. su h that φ ( ~ x ) ∧ ψ ( ~ x ) T ≁ ⊥ for ev ery geometri form ula ψ ( ~ x ) o v er Σ in the same on text) is pro v able in T ; let us also reall from [6℄ that a geometri theory T is atomi if and only if all the sub ob jet latties in the geometri syn tati ategory C T of T are atomi Bo olean algebras (this also follo ws from the results in the rst setion b y using the fat that ev ery sub ob jet in the lassifying top os Set [ T ] of T of an ob jet in C T lies in C T ). W e will mak e use of these haraterizations in the pro of of the theorem b elo w. Theorem 2.17. L et T b e a omplete ge ometri the ory having a mo del in Set . Then the fol lowing ar e e quivalent: (i) T is ountably ate gori al and Bo ole an (ii) T is atomi (iii) every T -mo del in Set is atomi Pro of (i) ⇒ (ii) By Prop osition 2.9 , an y Bo olean omplete geometri theory with a mo del in Set has enough mo dels; so the thesis follo ws from the fat that ev ery Bo olean top os with enough p oin ts is atomi (Corollary C3.5.2 p. 685 [6℄). (ii) ⇒ (iii) Let M b e a T -mo del in Set and ~ a b e a tuple of elemen ts of M ; w e w an t to pro v e that S T ( M , ~ a ) is prinipal and omplete. Consider the sub ob jet lattie Sub C T ( { ~ x . ⊤} ) in the geometri syn tati ategory C T of T , where ~ x is a set of v ariables of the same length as ~ a . Sine Sub C T ( { ~ x . ⊤} ) is an atomi Bo olean algebra, w e an write { ~ x . ⊤} as a disjution of atoms of Sub C T ( { ~ x . ⊤} ) ; so, sine { ~ x . ⊤} ob viously b elongs to S T ( M , ~ a ) , there exists exatly one atom of Sub C T ( { ~ x . ⊤} ) (up to T -pro v able equiv alene) whi h b elongs to S T ( M , ~ a ) ; then it is lear that this atom generates the t yp e S T ( M , ~ a ) . So w e ha v e pro v ed that all the t yp es of T are prinipal; it remains to v erify that they are also omplete. T o this end, let us rst observ e that T is Bo olean (sine ev ery atomi top os is Bo olean). So, giv en an inlusion S T ( M , ~ a ) ⊆ S T ( N , ~ b ) of t yp es of T , this inlusion m ust b e an equalit y b eause if there w ere a form ula φ ( ~ x ) ∈ S T ( N , ~ b ) \ S T ( M , ~ a ) then, b y denition of ¬ φ ( ~ x ) , w e w ould ha v e ¬ φ ( ~ x ) ∈ S T ( M , ~ a ) and hene ¬ φ ( ~ x ) ∈ S T ( N , ~ b ) , a on tradition. (iii) ⇒ (ii) Let us rst pro v e that T is Bo olean, that is ev ery form ula φ ( ~ x ) whi h is stably onsisten t with resp et to T is pro v able in T . Giv en a T -mo del M and a tuple ~ a of elemen ts of M of the same length as ~ x , let ψ ( M , ~ a ) b e a generator of the t yp e S T ( M , ~ a ) . As w e ha v e already observ ed, under our h yp otheses T has enough mo dels so, sine φ ( ~ x ) ∧ ψ ( M , ~ a ) T ≁ ⊥ , 11 there exists a T -mo del N and a tuple ~ b of elemen ts of it (of the same length as ~ x ) su h that φ ( ~ x ) and ψ ( M , ~ a ) b oth b elong to S T ( N , ~ b ) . No w, sine ψ ( M , ~ a ) generates the t yp e S T ( M , ~ a ) , it follo ws that S T ( M , ~ a ) ⊆ S T ( N , ~ b ) and hene, sine all the t yp es of T are omplete, S T ( M , ~ a ) = S T ( N , ~ b ) . This in turn implies that φ ( ~ x ) ∈ S T ( M , ~ a ) , that is M φ ( ~ a ) . Sine the T -mo del M and the tuple ~ a are arbitrary , w e onlude, again b y in v oking the fat that T has enough mo dels, that φ ( ~ x ) is pro v able in T , as required. No w that w e ha v e pro v ed that T is Bo olean, to sho w that T is atomi, it remains to v erify that all the Bo olean sub ob jet latties in the geometri syn tati ategory C T of T are atomi, equiv alen tly for ev ery form ula φ ( ~ x ) T ≁ ⊥ there exists an atom b elo w it in the Bo olean algebra Sub C T ( { ~ x . ⊤} ) . If φ ( ~ x ) T ≁ ⊥ then, sine T has enough mo dels, there exists a T -mo del M and a tuple ~ a of elemen ts of it (of the same length as ~ x ) su h that φ ( ~ x ) ∈ S T ( M , ~ a ) . It is no w enough to he k that the generator ψ ( M , ~ a ) of the t yp e S T ( M , ~ a ) is an atom of Sub C T ( { ~ x . ⊤} ) ; this follo ws similarly as ab o v e b y using the fat that T has enough mo dels and the t yp es of T are omplete. (ii) ⇒ (i) Being atomi, T is Bo olean, as ev ery atomi top os is Bo olean. T o pro v e that T is oun tably ategorial, let us distinguish t w o ases: either T has a nite mo del in Set or all the mo dels of T are innite. Let us supp ose that all the mo dels of T are innite. W e ha v e to pro v e that an y t w o den umerable mo dels of T are isomorphi. W e will onstrut expliitly su h an isomorphism as in the pro of of Theorem 7.2.2 p. 336 [4℄. Let M and N b e t w o mo dels of T of ardinalit y ℵ 0 . Then, T b eing omplete, w e ha v e S T ( M , []) = S T ( N , []) b y Remark 2.12 . Let us rst pro v e b y indution on k ∈ N the follo wing fat: giv en tuples ~ a and ~ b of length k resp etiv ely in M and N su h that S T ( M , ~ a ) = S T ( N , ~ b ) , and an elemen t d ∈ N there exists an elemen t c ∈ M su h that S T ( M , ~ a,c ) = S T ( N , ~ b,d ) (and, symmetrially , giv en an elemen t c ∈ M there exists an elemen t d ∈ N su h that S T ( M , ~ a,c ) = S T ( N , ~ b,d ) ). Consider the t yp e S T ( N , ~ b,d ) ; this is prinipal, b y our h yp otheses (ha ving already pro v ed the impliation (ii) ⇒ (iii) in the theorem), so it is generated b y a form ula ψ ( ~ x, y ) . No w, N ( ∃ y ψ ( ~ x, y ))( ~ b ) so sine S T ( M , ~ a ) = S T ( N , ~ b ) w e dedue that there exists c ∈ M su h that M ψ ( ~ a, c ) ; but ψ ( ~ x, y ) is a generator of S T ( N , ~ b,d ) and all the t yp es of T are omplete b y our h yp othesis, so w e onlude that S T ( M , ~ a,c ) = S T ( N , ~ b,d ) , as required. No w, sine M and N are geometrially equiv alen t b y Prop osition 2.9, an ob vious ba k-and-forth argumen t yields t w o sequenes ( m 0 , m 1 , . . . , m k , ... ) and ( n 0 , n 1 , . . . , n k , ... ) en umerating resp etiv ely M and N , su h that for ea h k ∈ N S T ( M ,m 0 ,m 1 ,...,m k ) = S T ( N ,n 0 ,n 1 ,...,n k ) ; then the map 12 f : M → N sending ea h m k to n k is an isomorphism of T -mo dels, as it is a bijetion preserving the in terpretation of all the atomi form ulas. Let us instead supp ose that T has a nite mo del M in Set of ardinalit y n . Consider the geometri sequen ts (o v er Σ ) ⊤ ⊢ [] ∃ x 1 . . . ∃ x n ( ∧ 1 ≤ i
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