A characterization theorem for geometric logic

A  haraterization theorem for geometri logi Olivia Caramello Cen tro di Riera Matematia Ennio De Giorgi Suola Normale Sup eriore Piazza dei Ca v alieri, 3, 56100 Pisa, Italy olivia.aramellosns.it Mar h 29, 2018 Abstrat W e establish a riterion for deiding whether a lass of strutures is the lass of mo dels of a geometri theory inside Grothendie k top oses; then w e sp eialize this result to obtain a  haraterization of the inni- tary rst-order theories whi h are geometri in terms of their mo dels in Grothendie k top oses, solving a problem p osed b y Iek e Mo erdijk in 1989. 1 In tro dution In a letter to Mi hael Makk ai of 1989, Iek e Mo erdijk pro v ed the follo wing result: Let Σ b e a signature, and let Σ - str ( E ) denote the ategory of Σ -strutures in a Grothendie k top os E . Then a nitary rst-order theory T o v er Σ an b e axiomatized b y oheren t sequen ts o v er Σ if and only if (i) for an y geometri morphism f : F → E b et w een Grothendie k top oses, if M ∈ Σ - str ( E ) is a mo del of T then f ∗ ( M ) is a mo del of T ; (ii) for an y surjetiv e geometri morphism f : F → E b et w een Grothendi- e k top oses and an y M ∈ Σ - str ( E ) , if f ∗ ( M ) is a mo del of T then M is a mo del of T . His pro of of this result in v olv ed mo del-theoreti as w ell as top os-theoreti argumen ts, and hea vily relied on the ompatness theorem. 1 In the same letter, Mo erdijk ask ed for a pro of of his onjeture that this result ould b e extended to the innitary on text i.e. that the v ersion of it obtained b y replaing `nitary rst-order' with `innitary rst-order' and `oheren t' b y `geometri' also hold. This question remained unansw ered for the past t w en t y y ears; in fat, the diult y lies in the fat that, sine in the innitary on text one an no longer rely on the ompatness theorem, one annot hop e to pro v e the onjeture b y extending the argumen t giv en in the nitary ase. In this pap er, w e pro v e the onjeture b y adopting the p oin t of view of lassifying top oses. W e start b y establishing some fats that will b e useful for our analysis; then, in the third setion, w e pro v e our main theorem giving a seman ti  haraterization of the lasses of strutures whi h arise as the olletion of mo dels in Grothendie k top oses of a geometri theory . In the last setion, w e deriv e Mo erdijk's onjeture as an appliation of our riterion in the ase of the lass of mo dels of an innitary rst-order theory , and w e sho w that (a stronger v ersion of ) Mo erdijk's result also follo ws as a onsequene of our theorem. Before pro eeding further, I w ould lik e to express m y gratitude to Iek e Mo erdijk for bringing m y atten tion to his onjeture at a reen t onferene; it is also a pleasure to thank him, as w ell as P eter Johnstone, for their useful remarks on a preliminary v ersion of this pap er. 2 Join tly surjetiv e families of geometri morphisms Reall from [4℄ that a geometri morphism of (elemen tary) top oses is surje- tiv e if its in v erse image funtor is onserv ativ e i.e. it is faithful and reets isomorphisms; more generally , a family { f i : E i → E | i ∈ I } of geometri morphisms with ommon o domain is said to b e join tly surjetiv e if and only if the in v erse image funtors f ∗ i are join tly onserv ativ e. Note that if C and D are ategories with equalizers and F : C → D is a funtor preserving equalizers then F is onserv ativ e if and only if it reets isomorphisms; indeed, t w o arro ws with ommon domain and o domain are equal if and only if their equalizer is an isomorphism. In partiular, a family of geometri morphisms is join tly surjetiv e if and only if the family formed b y their in v erse image funtors join tly reets isomorphisms. Giv en a olletion {E i ֒ → E | i ∈ I } of subtop oses of a giv en elemen tary top os E , w e denote b y ∪ i ∈ I E i ֒ → E the smallest subtop os of E on taining all the E i , pro vided that it exists; reall from [2 ℄ that if E is a Grothendie k 2 top os then there is only a set of (equiv alene lasses of ) subtop oses of E , and arbitrary unions of subtop oses alw a ys exist. The follo wing lemma giv es a  haraterization of join tly surjetiv e families of geometri morphisms. Lemma 2.1. L et { f i : E i → E | i ∈ I } b e a family of ge ometri morphisms of elementary top oses with  ommon  o domain E . Then { f i : E i → E | i ∈ I } is jointly surje tive if and only if E = ∪ i ∈ I E ′ i , wher e for e ah i ∈ I , E i ։ E ′ i ֒ → E is the surje tion-inlusion fatorization of f i . Pro of It is lear that { f i : E i → E | i ∈ I } is join tly surjetiv e if and only if the family {E ′ i ֒ → E | i ∈ I } of subtop oses of E is join tly surjetiv e. F or an y i ∈ I , let j i denote the lo al op erator on E orresp onding to the subtop os E ′ i of E and let a j i : E → E ′ i b e the orresp onding asso iated sheaf funtor. Let us supp ose that {E ′ i ֒ → E | i ∈ I } is join tly surjetiv e; w e w an t to pro v e that E = ∪ i ∈ I E ′ i . Giv en a lo al op erator j on E whi h is smaller than ea h of the j i , w e w an t to pro v e that j is the smallest lo al op erator on E . No w, for an y arro w f in E , if a j ( f ) is an isomorphism then a j i ( f ) is an isomorphism for ea h i , and hene, b y our h yp othesis, f is an isomorphism; this pro v es our laim. Con v ersely , let us supp ose that E = ∪ i ∈ I E ′ i ; w e ha v e to pro v e that {E ′ i ֒ → E | i ∈ I } is join tly surjetiv e i.e. for an y arro w f in E , if a j i ( f ) is an isomorphism for ev ery i ∈ I then f is an isomorphism. No w, for a xed arro w f in E , onsider the smallest lo al op erator k on E su h that the orresp onding asso iated sheaf funtor a k sends f to an isomorphism (fr. Example A4.5.14() [4℄). By our h yp othesis, k ≤ j i for ea h i and hene k is the smallest lo al op erator, whi h implies that f is an isomorphism, as required.  Remark 2.2. If all the top oses in the statemen t of the lemma are Grothendi- e k top oses and I is a set then the lemma admits the follo wing 2 -ategorial in terpretation. Reall that, for an y set-indexed olletion {E i | i ∈ I } of Grothendie k top oses, there exists the opro dut (Grothendie k) top os ` i ∈ I E i . No w, it is immediate to see, b y using the argumen ts in the pro of of the lemma, that, giv en a family { f i : E i → E | i ∈ I } of geometri morphisms with ommon o domain, the surjetion-inlusion fatorization of the indued opro dut map f : ` i ∈ I E i → E is giv en b y its fatorization through the inlu- 3 sion ∪ i ∈ I E ′ i ֒ → E ; in partiular, { f i : E i → E | i ∈ I } is join tly surjetiv e if and only if f is surjetiv e. 3 The  haraterization theorem All the top oses in this setion will b e Grothendie k top oses. Let Σ b e a signature. Let us denote b y O Σ the empt y (geometri) theory o v er Σ and b y Set [ O Σ ] its lassifying top os. Note that the O Σ -mo dels in an y Grothendie k top os E are preisely the Σ -strutures in E . Th us, for an y Grothendie k top os E , geometri morphisms E → Set [ O Σ ] orresp ond to Σ - strutures in E ; the geometri morphism orresp onding to a Σ -struture M will b e denoted b y f M (note that if U is a univ ersal mo del of O Σ in Set [ O Σ ] then M ∼ = f ∗ M ( U ) ). Let us denote b y Σ - str ( E ) the ategory of Σ -strutures in a top os E , as in the in tro dution ab o v e. Theorem 3.1. L et Σ b e a signatur e and S b e a  ol le tion of Σ -strutur es in Gr othendie k top oses lose d under isomorphisms of strutur es. Then S is the  ol le tion of al l mo dels in Gr othendie k top oses of a ge ometri the ory over Σ if and only if it satises the fol lowing two  onditions: (i) for any ge ometri morphism f : F → E , if M ∈ Σ - str ( E ) is in S then f ∗ ( M ) is in S ; (ii) for any (set-indexe d) jointly surje tive family { f i : E i → E | i ∈ I } of ge ometri morphisms and any Σ -strutur e M in E , if f ∗ i ( M ) is in S for every i ∈ I then M is in S . Pro of The `only if ' part of the theorem is w ell-kno wn. Let us pro v e the `if ' part. Let us onsider the olletion of geometri morphisms to Set [ O Σ ] of the form f M for M in S ; let E ֒ → Set [ O Σ ] b e the subtop os of Set [ O Σ ] giv en b y the union of all the subtop oses of Set [ O Σ ] arising as the inlusion parts of the surjetion-inlusion fatorizations of these geometri morphisms, and let a : Set [ O Σ ] → E b e the orresp onding asso iated sheaf funtor. W e kno w from [2℄ (Theorem 3.6) that the subtop os E ֒ → Set [ O Σ ] of Set [ O Σ ] orresp onds to a (unique up to syn tati equiv alene) geometri quotien t T of O Σ su h that if U O Σ is a univ ersal mo del of O Σ in Set [ O Σ ] then U T := a ( U O Σ ) is a univ ersal mo del of T in E . W e will sho w that T axiomatizes our lass of strutures S . Let M ∈ Σ - str ( E M ) b e a struture in S . The subtop os E ′ M ֒ → Set [ O Σ ] arising in the surjetion-inlusion fatorization of f M : E M → Set [ O Σ ] fators as the inlusion E ֒ → Set [ O Σ ] omp osed with the anonial inlusion l M : 4 E ′ M ֒ → E . No w, if w e omp ose this latter inlusion with the surjetion part of the surjetion-inlusion fatorization of f M , w e obtain a geometri morphism h M : E M → E su h that the omp osite of E ֒ → Set [ O Σ ] with h M is equal to f M . But M ∼ = f ∗ M ( U O Σ ) , from whi h it follo ws that h ∗ M ( U T ) ∼ = M , and hene that M is a mo del of T . This sho ws that ev ery struture in S is a mo del of T . T o pro v e the on v erse, w e note that, b y Lemma 2.1, the family of geometri morphisms h M for M in S is join tly surjetiv e; hene, under assumption (ii), U T lies in S . No w, sine (b y the univ ersal prop ert y of the lassifying top os E of T ) ev ery mo del N of T in a Grothendie k top os F is of the form g ∗ ( U T ) for some geometri morphism g : F → E , ondition (i) implies that an y T -mo del in a Grothendie k top os lies in S . This onludes the pro of of the theorem. Note in passing that T an b e desrib ed as the olletion of all the geo- metri sequen ts o v er Σ whi h are v alid in ev ery struture M of S (fr. also Theorem 9.1 [ 2℄).  It is natural to w onder if one an supp ose the set I in the statemen t of the theorem to b e a singleton without loss of generalit y; in fat, w e no w sho w that this is not p ossible. Giv en a lass S of Σ -strutures in Grothendie k top oses, w e an expliitly desrib e the smallest lass ˜ S of Σ -strutures on taining S whi h is losed under (i) and the v ersion of (ii) obtained b y requiring I to ha v e ardinalit y 1 . Indeed, with the notation used in the pro of of the theorem, onsider, for an y M in S , the struture ˜ M = i ∗ M ( U O Σ ) where i M : E ′ M → Set [ O Σ ] is the inlusion part of the surjetion-inlusion fatorization of f M ; then ˜ S is equal to the olletion R of all the Σ -strutures of the form g ∗ ( ˜ M ) for some geometri morphism g . T o pro v e this, w e argue as follo ws. Clearly , R is on tained in ˜ S and is losed under (i), so it remains to pro v e that it is losed under the v ersion of (ii) obtained b y requiring I to ha v e ardinalit y 1 . Let N b e a Σ -struture in a Grothendie k top os F and p : G → F b e a surjetiv e geometri morphism su h that p ∗ ( N ) is in R ; w e w an t to pro v e that N is in R . Sine p ∗ ( N ) is in R , there exists a Σ -struture M in S su h that p ∗ ( N ) = g ∗ ( ˜ M ) for some geometri morphism g : G → E ′ M . Then, b y the univ ersal prop ert y of the lassifying top os for O Σ , the geometri morphisms i M ◦ g and f N ◦ p are isomorphi. Let G p g ։ U g ′ ֌ E ′ M and F p f N ։ F ′ f ′ N ֌ Set [ O Σ ] b e resp etiv ely the surjetion-inlusion fatorization of g and of f N ; then G p g ։ U i M ◦ g ′ ֌ Set [ O Σ ] and G p f N ◦ p ։ F ′ f ′ N ֌ Set [ O Σ ] are resp etiv ely the surjetion-inlusion fatorization of i M ◦ g and of f N ◦ p . Then, b y the uniqueness (up to equiv alene) of the surjetion-inlusion fatorization of a geometri morphism, the geometri morphisms i M ◦ g ′ and f ′ N are isomorphi, 5 from whi h it follo ws that N is in R . This ompletes the pro of of the equalit y R = ˜ S . Let us no w sho w that it is not true in general that ˜ S is axiomatized b y a geometri theory o v er Σ . F or a oun terexample, tak e t w o subtop oses i 1 : E 1 ֒ → Set [ O Σ ] and i 2 : E 2 ֒ → Set [ O Σ ] of Set [ O Σ ] whi h are not on tained in ea h other, and tak e S to onsist of the t w o mo dels M 1 := i ∗ 1 ( U O Σ ) and M 2 := i ∗ 2 ( U O Σ ) ; if ˜ S w ere axiomatized b y a geometri theory o v er Σ then, b y Lemma 2.1, the Σ -struture i ∗ ( U O Σ ) , where i : E 1 ∪ E 2 ֒ → Set [ O Σ ] is the union of the subtop oses E 1 and E 2 of Set [ O Σ ] , w ould lie in ˜ S , and w e an sho w this to b e imp ossible. Indeed, if i ∗ ( U O Σ ) w ere in ˜ S then there w ould b e a geometri morphism g : E 1 ∪ E 2 → E 1 (or g : E 1 ∪ E 2 → E 2 ) su h that i ∼ = i 1 ◦ g (or i ∼ = i 2 ◦ g ); but the existene and uniqueness of the surjetion-inlusion fatorizations of a geometri morphism ensure that g is an equiv alene, whi h on tradits our assumption that E 1 and E 2 b e not on tained in ea h other. Remark 3.2. In view of Remark 2.2, ondition (ii) in the statemen t of the theorem an b e rephrased as follo ws: (i) for an y surjetiv e geometri morphism f : F → E and an y M ∈ Σ - str ( E ) , if f ∗ ( M ) is in S then M is in S ; (ii) for an y set-indexed family { M i | i ∈ I } of strutures in top oses E i all of whi h are in S , the struture in the opro dut top os ` i ∈ I E i whose i th o ordinate is M i is also in S . 4 Appliations Let T b e an innitary rst-order theory o v er a giv en signature Σ and S T b e the olletion of its mo dels inside Grothendie k top oses. Clearly , S T satises ondition (ii) of Remark 3.2, so it is axiomatizable b y geometri sequen ts o v er Σ if and only if it satises ondition (i) of Theorem 3.1 and ondition (i) of Remark 3.2. Note that, b y Prop osition D1.3.2 [5℄ and Corollary 3.4 [1℄, t w o innitary rst-order theories o v er the same signature are dedutiv ely equiv alen t (relativ e to the in tuitionisti pro of system of innitary rst-order logi of setion D1.3 [ 5℄) if and only if they ha v e the same mo dels in ev- ery Grothendie k top os. Hene w e ha v e pro v ed the onjeture b y Mo erdijk men tioned in the in tro dution of this pap er. If T is a nitary rst-order theory satisfying the onditions of the  har- aterization theorem, it is natural to w onder whether the geometri theory axiomatizing T pro vided b y the theorem is in fat oheren t. As realled in the in tro dution ab o v e, it w as already pro v ed b y Mo erdijk in his letter that if T satises ondition (i) of Theorem 3.1 and ondition (i) of Remark 3.2 6 then T is axiomatizable o v er Set b y oheren t sequen ts o v er its signature. In fat, it will follo w diretly from our theorem that this is true not only o v er Set but o v er ev ery Grothendie k top os, one w e ha v e sho wn that if T is a nitary rst-order theory o v er a signature Σ and T ′ is a geometri theory o v er Σ ha ving the same mo dels in Grothendie k top oses as T then T ′ is oheren t. T o pro v e this, w e argue as follo ws. By using Theorem 3.5 [3℄, w e are redued to v erify that for an y oheren t form ula { ~ x . φ } o v er Σ , for an y family { ψ i ( ~ x ) | i ∈ I } of oheren t (equiv a- len tly , geometri) form ulae in the same on text, if φ ⊢ ~ x ∨ i ∈ I ψ i is pro v able in T ′ (using geometri logi) then φ ⊢ ~ x ∨ i ∈ I ′ ψ i is pro v able in T ′ (using geometri logi) for some nite subset I ′ of I . W e an supp ose, without loss of generalit y , ~ x to b e the empt y string; indeed, if ~ c is a string of new onstan ts of the same length and t yp e as ~ x , a geometri sequen t χ ⊢ ~ x ξ o v er Σ is pro v able in T ′ if and only if the sequen t χ [ ~ c /~ x ] ⊢ [] ξ [ ~ c /~ x ] is pro v able in T ′ , regarded as a theory o v er the signature Σ ∪ { ~ c } . If φ ⊢ ~ x ∨ i ∈ I ψ i is pro v able in T ′ then ev ery mo del in Set of the theory T ∪ {¬ ψ i | i ∈ I } is a mo del of ¬ φ . Sine the theory T ∪ {¬ ψ i | i ∈ I } is nitary rst-order, this ondition is equiv alen t to sa ying that ¬ φ is pro v able in the theory T ∪ {¬ ψ i | i ∈ I } (using lassial nitary rst-order logi), from whi h it follo ws, b y the niteness theorem in lassial Mo del Theory , that ¬ φ is pro v able in T ∪ {¬ ψ i | i ∈ I ′ } for some nite subset I ′ of I i.e. φ ⊢ [] ∨ i ∈ I ′ ψ i is pro v able in T (using lassial nitary rst-order logi). Th us φ ⊢ ~ x ∨ i ∈ I ′ ψ i is v alid in ev ery mo del of T (equiv alen tly , of T ′ ) in Bo olean Grothendie k top oses and hene, b y Prop osition D3.1.16 [ 5℄, it is pro v able in T ′ , as required. 7 Referenes [1℄ C. Butz and P . T. Johnstone, Classifying top oses for rst-order theories, Ann. Pure Appl. Logi 91 (1998), 33-58. [2℄ O. Caramello, Latties of theories (2009), arXiv:math.CT/0905.0299 . [3℄ O. Caramello, One top os, man y sites (2009), arXiv:math.CT/0907.2361 . [4℄ P . T. Johnstone, Skethes of an Elephant: a top os the ory  omp endium. V ol.1 , v ol. 43 of Oxfor d L o gi Guides (Oxford Univ ersit y Press, 2002). [5℄ P . T. Johnstone, Skethes of an Elephant: a top os the ory  omp endium. V ol.2 , v ol. 44 of Oxfor d L o gi Guides (Oxford Univ ersit y Press, 2002). [6℄ I. Mo erdijk, Letter to Mi hael Makk ai (1989). 8