A Topological Completeness Theorem 1 Introduction - CiteSeerX
A Topological Completeness Theorem Carsten Butz September 10, 1996 Abstract
We prove a topological completeness theorem for in nitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected.
1 Introduction Sheaf models of in nitary rst order theories were studied intensively in [9]. They are models in Grothendieck toposes and subsume Heyting valued models, boolean valued models, Kripke models, and permutation models. Of particular interest are so called geometric theories T (see De nition 2.1). To these theories one can associate a particular topos B(T) together with a generic (term) model U in B(T). The model U is generic since there is an equivalence between the category of models of T in a topos F , and the category of geometric morphisms from F to B(T). Given such a geometric morphism ': F ! B(T) the corresponding model is obtained as ' U (see below). There are usually not enough models in Sets to test semantically whether a given geometric sequent holds in all models of T in all Grothendieck toposes. If this is the case we say that T satis es the completeness theorem. We will prove the following topological completeness theorem:
Theorem Let T be a geometric theory satisfying the completeness theorem. There exists a topological space X and an OX {valued model of T (a model of T in Sh(X) corresponding to a geometric morphism ': Sh(X) ! B(T)) such that (i) for any rst order formula we have j= if and only if U j= (the geometric morphism ' is an open surjection). (ii) Any map in Sh(X) between rst order de nable sheaves is geometrically de nable (functional
completeness). (iii) The inverse image ' preserves function types of rst order de nable types.
Part (ii) and (iii) of the theorem extend a well known result by Makkai and Reyes, stating that any topos with enough points has an open spatial cover ([9], Theorem 6.3.3). This result appears here as part (i) of the theorem. Part (iii) of the theorem needs some explanation. Even if the language has already function types there is no guaranty that these function types are interpreted by exponentials of sheaves. But one can always add new function types and interpret them in the universal model in the standard way, namely by exponentials. These new function types are interpreted in by exponentials of sheaves too. The space X will contain the models of T in Sets, each model equipped with some extra structure. 1
We should remark that the theorem states only those properties of the geometric morphism ': Sh(X) ! B(T) which have a clear logical meaning. For those familiar with topos theory we remark that ' is connected, locally connected, and acyclic. In particular, ' induces isomorphisms in cohomology. Details about these aspects can be found in [3] and will appear in a joined paper with I. Moerdijk [4]. The properties connected and locally connected are closely related to the properties stated as part (ii) and (iii) of the theorem. We start reviewing sheaf models and xing our notation. Then we de ne the space X, the model , and take a closer look at the topology of the spaces involved. The rest of the paper is occupied by the proof of the theorem. Acknowledgements: I would like to thank Ieke Moerdijk for his encouragement and many stimulating discussions. This research was supported by a grant of the state Baden-Wurttemberg, Germany, by a DAAD/NUFFIC grant, by a scholarship of the University of Utrecht, and by a grant from a project funded by the Netherlands Organization for Scienti c Research NWO.
2 Review of Sheaf Models We assume that the reader is familiar with (Grothendieck) toposes, see for example [9], [6] or [8]. For sheaves over topological spaces we refer as well to [10]. In particular, we assume that the reader has seen the basic de nition of a site (C ; J), sheaves on a site and maps between sheaves. These data together yield the Grothendieck topos Sh(C ; J). Maps between Grothendieck toposes are called geometric morphisms . A geometric morphism f: Sh(D ; K) ! Sh(C ; J) is a pair of functors f : Sh(D ; K)
/
o
Sh(C ; J): f ;
f left adjoint to f , and f is moreover left exact (preserves nite limits). f is called the inverse image and f the direct image . Maps between geometric morphisms are natural transformations. In this way we get the category of geometric morphisms from Sh(D ; K) to Sh(C ; J), denoted by Hom(Sh(D ; K); Sh(C ; J)). A Grothendieck topos E has a rich structure. In particular we can speak about models of an in nitary rst order theory T in the topos E as follows (cf. [8], Section X.2 and [9], Chapter 2): An interpretation M of a language assigns to each sort Y an object Y (M) in E , to each constant c of sort Y a global element c(M) : 1 ! Y (M) , to each function symbol f: Y ! Y an arrow f (M) : Y (M) ! Y (M) and similar for relation symbols R. (Here we introduce the convention that Y stands for a sequence Y1; : : :; Yn of sorts and Y (M) for Y1(M) Yn(M) .) If we have a one sorted language we simply write M for the underlying object. Interpreting the logical connectives and quanti ers in the canonical way we get for each formula (y: Y ) a subobject fy j (y)g(M) ,! Y (M) : For a closed formula we say that M is a model of (M j= ) if f j g(M) (which is a subobject of 1E ) is equal to 1E . In the same way M j= T means M j= for all 2 T. With the natural de nition of morphisms between models we get the category Mod(T; E ) of models of T in E . If E = Sh(X), the topos of sheaves over a topological space, we speak about topological models. The category of models in the topos Sh(H) of sheaves over a (complete) Heyting algebra H is naturally equivalent to the category of H{valued models, see [5]. Models in the topos BG of G{sets (G a group) are sometimes refered to as permutation models. We are mainly interested in geometric theories:
De nition 2.1 A formula (y) is W called geometric if it is built up by atomic formulas, nite conjunction ^, arbitrary disjunction , the truth values true > and false ?, and existential quanti cation 9. A theory T is called geometric if all its axioms are (equivalent to formulas) of the form 8y( 1 (y) ! 2(y)) for 1 and 2 geometric. 2
Examples are given by the standard axiomatizations of linear orders with or without endpoints, groups or rings. The inverse image f of a geometric morphism f: F ! E induces for any theory T a functor f : Mod(T; F ) ! Mod(T; E ) (see [8], X.3.6). With this observation the main theorem relating topos theory and logic becomes as follows:
Theorem 2.2 For any in nitary geometric theory T there exists a unique Grothendieck topos B(T) such that for all Grothendieck toposes F there is an equivalence ' Hom(F ; B(T)); Mod(T; F ) ! natural in F . Conversely, any Grothendieck topos E is of the form B(T) for some geometric
2 Uniqueness in this theorem is meant as \unique up to equivalence". We say that B(T) is the classifying topos of the theory T. The theory classi ed by a topos is not unique. In the equivalence of the theorem the identity geometric morphism B(T) ! B(T) corresponds to a model U of T in B(T), called the generic model (or universal model) of T. The model U is uniquely determined up to isomorphism. It is universal in the sense that for any model M of T in F there exists a geometric morphism f: F ! B(T) such that f U ' M. The inverse image of a geometric morphism does usually not preserve the full internal rst order logic of a topos. f this is true the geometric morphism is called open (see [7], [8]). A geometric morphism is called surjective if the inverse image is faithful. All these geometrical notions can be characterized model theoretically. In the following proposition we denote by Thg (?) the set of all geometric sequents 8y( 1 (y) ! 2 (y)) for 1 ; 2 geometric formulas which are true in a model. Similar, Thfo (?) denotes the set of all closed rst order formulas which are true in a model. Proposition 2.3 Let T be a geometric theory, ': F ! B(T) a geometric morphism with corresponding model in F . Let U denote the universal model in B(T). (i) ' is surjective if and only if Thg (U) = Thg (). (ii) ' is open if and only if Thfo (U) Thfo (). (iii) ' is an open surjection if and only if Thfo (U) = Thfo (). Proof. This is implicitely in [9]. An explicit proof is given in [2]. 2 g g If Th (U) = Th () we say that is a conservative model of T. We remind the reader of the provability relation ` given in [9]. For a geometric theory T and geometric formulas 1 (y) and 2 (y) we have T ` 8y( 1 (y) ! 2(y)) if and only if U j= 8y( 1 (y) ! 2 (y)). theory T.
De nition 2.4 A geometric theory T is said to satisfy the completeness theorem if T `Sets 8y( 1 (y) ! 2 (y)) implies T ` 8y( 1(y) ! 2 (y)). Thus, a theory T satis es the completeness theorem if there are enough models in Sets to test semantically whether a geometric sequent 8y( 1(y) ! 2 (y)) is derivable or not. It is well understood that T satis es the completeness theorem if B(T) has enough points: The geometric morphisms p: Sets ! B(T) form a jointly surjective family. Most toposes arising in practice have enough points. We mention the toposes Sh(X) for X a topological space, BG for G a discrete or
continuous group, or { as a less trivial example { the etale topos Sh(Xet ) associated to a scheme X in algebraic geometry (see [1]). We sketch the construction of B(T) for a geometric theory T formulated in L! . For simplicity we assume to have a one{sorted language. A syntactic site Syn(T) for B(T) can be de ned in terms of the provability relation `. 3
| Objects of Syn(T) are equivalence classes [ (y)] of geometric formulas in L! . Two such formulas 1 (y 1 ) and 2 (y 2 ) are equivalent if T ` 8y( 1(y) $ 2 (y)). | Arrows from [(w)] to [ (y)] are equivalence classes [(w; y)] of geometric formulas in L! such that is provable functional: ? T ` 8w8y (w; y) ! (w) ^ (y) ? T ` 8w (w) ! 9y(w; y) ? T ` 8w8y 8z (w; y) ^ (w; z) ! y = z | A family f[i(wi ; y)]:[i(wi )] ! [ (y)]gi2I is a cover if it is a provable cover: ? of arrows W T ` 8y (y) ! I 9wi i(wi ; y) . The category Syn(T) has all nite limits and the Grothendieck topology is subcanonical, see [8] or [9] for more details. B(T) is now de ned to be Sh(Syn(T)). The Yoneda map y: Syn(T) ! B(T) gives U as follows: U = y([y = y]) were y is a free variable; f (U) : U ! U = y([f(z) = y]): y([z = z]) ! y([y = y]) R(U) ,! U = y([R(y)] ,! y([y = y]): For every geometric formula (y) there is an isomorphism fy j (y)g(U) ' y([ (y)]). Given a geometric morphism ': F ! B(T) and corresponding model of T in F we have for each geometric formula (y) canonical isomorphisms fy j (y)g() ' ' fy j (y)g(U) ' ' y([ (y)]): The universal model U has the property that any subobject S ,! U is geometrically de nable: there exists a geometric formula (y) such that S ' fy j (y)g(U) . (But note that if the theory T is formulated in L! such geometric formulas (y) may only exist in L0 ! for 0 > .)
3 De nition of the Space of Models Let T be a geometric theory satisfying the completeness theorem. We assume for simplicity that T is formulated in a one{sorted language. We denote by Sthe xed set of function and relation symbols. Constants are treated as 0{ary functions. We always suppress the arity of these symbols to keep notations simple. For a cardinal let Pts (T) denote a xed set of representatives of isomorphism classes of models such that the cardinality of the model is bounded by . By a downward Lowenheim{Skolem argument there exists a cardinal such that the models M 2 Pts (T) form a set of enough models: For two geometric formulas 1(y) and 2 (y) we have T ` 8y( 1 (y) ! 2(y)) i for all M 2 Pts (T): M j= 8y( 1 (y) ! 2 (y)): We x such a cardinal . The space of models we are going to de ne will consist of all models M of T, bounded by and equipped with some extra structure: De nition 3.1 A {to{one enumeration of a model M is a map : K ! M, K , such that for each a 2 M the set ?1(a) is co nal in . We write : ; M for such a map. Note that we do not distinguish between a model and its underlying set. If is a regular cardinal the de nition is equivalent to say that each set ?1 (a) has cardinality . Since we use only the partial order of we denote the elements of by the letters k; l; m; : : :. 4
Let X be the space with elements pairs (M; ) where M 2 Pts (T) and : ; M. If necessary we write such an element as p = (Mp ; p). The topology on X is generated by the following subbase: Uk;l = f(M; ) j M j= (k) = (l)g for k; l 2 , (M) for ki ; l 2 , f 2 S, Uk;l;f = f(M; ) j M j= f ( (k)) = (l)g Uk;R = f(M; ) j M j= R(M) ( (k))g for ki 2 , R 2 S. Here (k) stands for the tuple ( (k1); : : :; (kn)) and it is part of the de nitions that the (k); : : : are de ned. Since expressions like M j= f (M) ( (k)) = (l) are long and hard to read we write them mostly as M j= f(k) = l. This makes sense since the elements k 2 where (k) is de ned are so to say names for the elements in M. To construct a geometric morphism ': Sh(X) ! B(T) wePhave to give an interpretation of thePlanguage in Sh(X): The underlying object is the set p2X Mp with canonical projection : p2X Mp ! X. We generate a topology on by ?1 (U) is open for U X open, Vk = f p(k) j p (k) de nedg is open. Lemma 3.2 is a sheaf over X ( is a local homeomorphism, or an etale map). S Proof. is continuous by de nition and maps Vk bijectively onto Uk;k . Moreover = k2 Vk . To prove that is etale it is hence enough to show that the inverse of restricted to Vk , which is the map (?) (k): Uk;k ! Vk , is continuous. By de nition of the topology of a typical open subset W of Vk; is of the form T W = Vk \ ?1 (U) \ i=1;:::;n Vki = fa 2 Mp j p 2 U and a = p (k) = p (k1) = = p (kn)g for some ki 2 , U X open. It follows that \ Uk;ki [ (?)(k)]?1(W) = U \ i=1;:::;n
and (?)(k) is continuous, so is a local homeomorphism. 2 In what follows, a product symbol between sheaves always means a product of sheaves, which is computed brewise: [ = = Mp Mp p2X
with the appropriate topology. From this remark we see at once that for every function symbol P f in S there is a canonical map f () : ! , namely the sum of all the f (Mp ) (i.e., f () = p2X f (Mp ) ). The same applies to relation symbols Rp in S: There is a canonical subset R() , namely the sum (or disjoint union) of all the R(M ) . Lemma 3.3 (i) Every f () is a local homeomorphism. (ii) Every R() is a subsheaf of . Proof. We leave the details to the reader and prove only the simplest case, that of a unary relationSsymbol we have to show that R() is open in . This follows since i h R. In that case ? 1 () 2 R = k2 (Uk;R ) \ Vk . The two lemmas above prove that is indeed an interpretation of the language in Sh(X). Each point p 2 X de nes a geometric morphism p: Sets ! Sh(X) with inverse image the usual stalk map. In particular, p carries the interpretation to an interpretation p , which is isomorphic to Mp . Since Sh(X) has enough points we get for geometric formulas 1(y) and 2 (y) that 5
j= 8y( 1 (y) ! 2 (y)) i fy j 1(y)g() ,! fy j 2 (y)g() i p fy j 1(y)g() ,! p fy j 2 (y)g() for all p 2 X i fy j 1(y)g(p ) ,! fy j 2(y)g(p ) for all p 2 X i p j= 8y( 1 (y) ! 2(y)) for all p 2 X i Mp j= 8y( 1(y) ! 2 (y)) for all p 2 X. i T ` 8y( 1 (y) ! 2 (y)) Thus, is a conservative model of the theory T and corresponds to a surjective geometric morphism ': Sh(X) ! B(T). For the record we state Proposition 3.4 is a conservative model of T. 2
Remark 3.5 (Notational conventions.) In the sequel we will avoid superscripts as often as possible. But sometimes they are necessary or useful. For example, writing p = (Mp ; p) allows to de ne the sections (?) (k): Uk;k ! Vk .
There is another situation where we will use them: Consider two points p1 = (M; 1) and p2 = (M; 2) of X, that is, 1 and 2 are {to{one enumerations of one T{model M. Moreover, M corresponds to a geometric morphism (denoted by the same letter) M: Sets ! B(T). For a representable sheaf fy j (y)g(U) in B(T) there are canonical isomorphisms (fy j (y)g() )p ' fy j (y)g(p ) ' M fy j (y)g(U) ' fy j (y)g(M) ' : 1
1
It is sometimes necessary to indicate in which bre an element lives. We use for a 2 fy j (y)g(M) the notation ap for \the tuple a in the (isomorphic) bre (fy j (y)g())p ". Adding or dropping the superscript (?)p means applying an isomorphism of sets and moving an element (or tuple) to a particular bre. 1
1
1
Remark 3.6 There are several variations of the construction described in this section, which will
yield the same theorem. It is possible to start with an arbitrary family Pts(T) of models of T such that for geometric formulas 1(y) and 2 (y) we have
T ` 8y( 1 (y) ! 2 (y)) i for all M 2 Pts(T): M j= 8y( 1 (y) ! 2(y)): (We say that Pts(T) is a family of jointly conservative models.) Then has to be chosen such that the cardinality of all models in Pts(T) is less or equal to . For example, the nite linear orders f0; : : :; ng for n 2 N are a family of jointly conservative models for the theory of non{empty linear orders. In this case, can be chosen as @0 . Instead of {to{one enumerations one can use (partial) enumerations of models : K ! M, K , such that for each a 2 M the set ?1 (a) is in nite.
4 De nable Open Sets Let (y) be a geometric formula with free variables exactly the listed tuple y. We de ne for a tuple k the open set Uk; (y) = U (k) of X by induction: U? = ; U> = X Uk=l = \i=0;:::;n Uki ;li UR(k) = Uk;R Uf (k)=l = Uk;l;f S U( ^ )(k) = U (k) \ U (k) UWI i (k) = I U i (k) W U9y (y;k) = l2 U (l;k ) 1
2
1
2
6
W
In the de nition of the cases ^ and we mean on the right side the tuples k restricted to the free variables actually occurring in 1 ; 2 and in the i . Lemma 4.1 If (y) is a geometric formula and k a tuple of elements of then U (k) = f(M; ) j 2 M j= (k)g. As above the notation implies that (ki) is de ned. The collection of all open sets U (k) forms obviously a base for the topology on X. In general it is not possible to assign to an arbitrary open set U of X a geometric formula (w) and a tuple k so that U = U(k) . This is only true for basic open sets. But we can do slightly better. Lemma 4.2 (i) Let U X be a nonempty basic open set. There exists a geometric formula (w) and a tuple k such that: | U = U(k) . | For every N 2 Pts (T) and for every tuple b 2 N such that N j= (b) there exists a {to{one enumeration of N so that (k) = b and hence (N; ) 2 U(k) . (ii) Let U Un;n be an open set and p a point in U . There exists a geometric formula (w; y) and a tuple k such that: | p 2 U(k;n) U . | For every N 2 Pts (T) and for all tuples b 2 N , c 2 N such that N j= (b; c) there exists a {to{one enumeration of N so that (k) = b, (n) = c and hence (N; ) 2 U(k;n) . Proof. Ad (i). Let 0(w) be a geometric formula and k a tuple such that U = U0 (k) . The formula
(w) 0(w) ^
^
ki =kj
wi = wj
and the tuple k will do the job. Obviously U = U0 (k) = U(k) . To prove the second part take an arbitrary model N 2 Pts (T) and a tuple b 2 N such that N j= (b). Take an arbitrary {to{ one enumeration 00 of N and take out of the domain of 00 the nite set k to get the {to{one enumeration 0 . To get extend 0 to the set k by (k) = b. The fact that N j= (b) ensures that this is well{de ned (compare the construction of ). Moreover (N; ) 2 U(k) . Ad (ii). Let U 0 be a basic open set such that p 2 U 0 U. Choose a formula 0(w) and a tuple k according to (i). The geometric formula (w; y) 0(w) ^
^
ki =nj ki2k; nj 2n
wi = y j
will do the job. 2 Next we state several lemmas which summarize the information we get from a section (?) (n). To state the lemmas we rst need a de nition: De nition 4.3 We say that a formula (y) is GT if it is built up by formulas 1 (y) ! 2(y) for V 1 ; 2 geometric, arbitrary meets , and universal quanti cation 8. The abbreviation GT is used to remind the reader that this class of formulas \is" the class of geometric theories, any theory put together into one axiom. The following two lemmas are proved by induction on : 7
Lemma 4.4 Let (y) be a geometric formula and (y) a GT{formula. Then fy j (y)g(U) ,! fy j (y)g(U) 2 if and only if for all M 2 Pts (T): M j= (y) ! (y). Lemma 4.5 Let s: U ! be a section (a continuous map such that s(p) 2 Mp for each p 2 U ). Let (y) be a GT{formula with associated subsheaf fy j (y)g() . Then im(s) fy j (y)g() if and only if for all points p in U : Mp j= (s(p)). 2 The next lemma combines Lemma 4.5, which always holds, with the special situation we are dealing with. The lemma is frequently used later.
Lemma 4.6 Let U Un;n be open sets and (y) a GT{formula. Suppose that the image of (?) (n)U has nonempty intersection with the sheaf fy j (y)g() , say for p 2 U that p (n) 2 fy j (y)g(). Then there exists a geometric formula (w; y) and a tuple k such that (i) p 2 U(k;n) U . (ii) (?)(n)U(k;n) maps into fy j (y)g() . In particular for all p0 2 U(k;n) the model Mp0 satis es (n). (iii) All models N 2 Pts (T) satisfy N j= 9w(w; y) ! (y). (Note that if is geometric this is equivalent to say that T ` 9w(w; y) ! (y).) (iv) For every N 2 Pts (T) and for tuples b 2 N , c 2 N such that N j= (b; c) there exists a {to{one enumeration of N so that (k) = b, (n) = c and hence (N; ) 2 U(k;n) .
Proof. If we write U 0 = [ (?)(n)]?1 fy j (y)g() \ U we are in the situation of Lemma 4.2 and
nd a formula (w; y) and a tuple k such that (i), (ii) and (iv) hold. To prove (iii) take an arbitrary N 2 Pts (T) and a tuple c 2 N such that N j= 9w(w; c). There is thus a tuple b such that N j= (b; c). By (iv) there exists a {to{one enumeration such that (k) = b; (n) = c; and (hence) (N; ) 2 U(k;n) : 2 Since (N; ) 2 U(k;n) we can apply (ii) to get N j= (n), i.e. N j= (c).
5 Preservation of Universal Quanti ers We will show the following proposition:
Proposition 5.1 The inverse image ' preserves the interpretation of rst order formulas: For each rst order formula (y) there is an isomorphism fy j (y)g() ' ' fy j (y)g(U) : The following two corollaries are immediate:
Corollary 5.2 The models and U satisfy the same rst order formulas: If is any rst order formula, U j= if and only if j= . Proof. Take an arbitrary formula . The objects f j g() and ' f j g(U) are isomorphic by Proposition 5.1 and the corollary follows since ' preserves and re ects isomorphisms.
2
Corollary 5.3 For each rst order formula (y) there exists a geometric formula (y) such that fy j (y)g() ' fy j (y)g(): 8
Proof. Given there exists a geometric formula such that fy j (y)g(U) ' fy j (y)g(U): Applying
' to both sides together with Proposition 5.1 yields the result. 2 We rst proof Proposition 5.1 for some special cases: Lemma 5.4 The proposition holds for GT{formulas (y). Proof. It is straightforward to check that for any GT{formula (y) the inequality of subobjects ' fy j (y)g(U) fy j (y)g() always holds. For the other inequality take a in fy j (y)g(), say a 2 Mp for p = (Mp ; p) in X. Fix a tuple n such that p (n) = a. Let U be the open set Un;n \ [ (?)(n)]?1(fy j (y)g() ) which contains p. We apply Lemma 4.6 to nd a geometric formula (w; y) and a tuple k such that p 2 U(k;n) and for all M 2 Pts (T): M j= 9w(w; y) ! (y). Since p 2 U(k;n) it follows that a 2 fy j 9w(w; y)g() . Since (y) is GT we can apply Lemma 4.4 and get fy j 9w(w; y)g(U) ,! fy j (y)g(U) and hence a 2 ' fy j (y)g(U). But a was arbitrary. 2 Proof of Proposition 5.1. The proof is by induction on . For atomic formulas this is trivial. The W cases ^, and 9 follow since the p 2 X form together a jointly surjective family of geometric morphisms Sets ! Sh(X). VThe details are standard and left to the reader. From the remaining three interesting cases !, and 8 we treat one in detail: Let 1(y) and 2 (y) be rst order formulas and assume fy j i (y)g() ' ' fy j i (y)g(U) for i = 1; 2. (1) Since fy j i(y)g(U) ,! U is geometrically de nable there exist geometric formulas i (y) so that fy j i (y)g(U) ' fy j i (y)g(U) : (2) Now (by de nition) fy j 1(y) ! 2 (y)g() ' fy j 1(y)g() ! fy j 2(y)g() (U) (U) ' ' fy j 1 (y)g ! ' fy j 2 (y)g (by (1)) ' ' fy j 1 (y)g(U) ! ' fy j 2 (y)g(U) (by (2)) ' fy j 1 (y)g() ! fy j 2 (y)g() (the i are geometric) ' fy j 1 (y) ! 2 (y)g() (by de nition) (U) ' ' fy j 1 (y) ! 2 (y)g (by Lemma 5.4) ' ::: ' ' fy j 1 (y) ! 2 (y)g(U) The other two cases are similar using the same trick. 2
6 Functional Completeness In this section we present a result about functional completeness which we state as follows: Proposition 6.1 (Functional completeness) Let (y) and (z) be geometric formulas and g: fy j (y)g() ! fz j (z)g() a map of sheaves. Then g is de nable: There exists a geometric formula (y; z) such that graph(g) ' f(y; z) j (y; z)g() . Equivalently, de nes in Syn(T) a map [ (y; z)]: [ (y)] ! [(z)]; and for arbitrary p, for all tuples b 2 fy j (y)g(Mp ) and c 2 fz j (z)g(Mp ) g(b) = c i Mp j= (b; c): 9
Note that by Corollary 5.3 the proposition extends automatically to the case where and are rst order formulas. Of course we could assume that (z) z = z. The proof of Proposition 6.1 needs some preliminary remarks. We leave most of their proofs to the reader. Fix a model M in Pts (T). Thus M corresponds to a geometric morphism (again denoted by M, since there is no confusion here) M: Sets ! B(T). Let E(M) be the topological space with elements {to{one enumerations of M, i.e., maps : ; M. The topology on E(M) is generated by the open sets Wk;a = f j (k) = ag for k 2 , a 2 M arbitrary. For tuples k and a we use as well Wk7!a for the open set \i=1;:::;n Wki ;ai . There is a canonical map i = iM : E(M) ! X sending to (M; ). We will often identify E(M) along i with a subset of X. Lemma 6.2 (i) The map iM is continuous. (ii) The space E(M) and each basic open set Wk7!a are connected. Proof. As an example we show the second part. Suppose E(M) = U1 [ U2 for U1 , U2 nonempty, say for example i: Ki ! M in Ui . Each i is in some basic open set Wi Ui and there are only nitely many elements of occurring in the de nition of W1 (and of W2 ). We rst consider de ned by (k); if k occurs in the list de ning W1, (k) = 21(k); otherwise. is obtained from 2 by changing the enumeration 2 in a nite number of places. is clearly {to{one and 2 W1. After renaming we may thus assume that 1 and 2 di er only in a nite number of places. Next consider the map
: k 7! 1(k) i 1(k) = 2(k): By construction, is {to{one. But has the additional property that 2 W open implies 1 2 W and 2 2 W. Thus either 2 2 U1 or 1 2 U2 , i.e., either 1 or 2 is contained in U1 \ U2 and E(M) is connected. The same argument applies locally, that is, if we prescribe the enumerations 1 and 2 in a nite number of places (they are then prescribed at the same elements k 2 ). Hence E(M) is as well locally connected. 2 The map i induces a geometric morphism i: Sh(E(M)) ! Sh(X) with inverse image the pullback functor: i (E) = i?1 (E) E /
E(M)
i X Applying the functor i to we get the T{model i . There is another way to get a model of T: The projection = M : E(M) ! 1 induces the geometric morphism Sh(E(M)) ! Sets = Sh(1) with inverse image the constant sheaf
/
(S) = E(M) S; where S in Sets is viewed as a space with discrete topology. Applied to the model M we get the model M.
10
Lemma 6.3 The models i and M are isomorphic. Equivalently, the following diagram of toposes commutes:
i
Sh(E(M))
Sh(X) /
'
Sets
B(T)
M /
Proof. Among other things one has to show that the canonical isomorphism of sets in the diagram
below is in fact a homeomorphism: M
M E(M) ' o
P
(M ; ) M
/
/
1
E(M)
E(M)
o
X
i /
The details are straightforward. 2 The following two lemmas are the ingredients in the proof of Proposition 6.1: Lemma 6.4 Let (y) and (z) be two geometric formulas and g: fy j (y)g() ! fz j (z)g() a map of sheaves. Then the stalk maps
gp : (fy j (y)g() )p ! (fz j (z)g())p depend only on the underlying model Mp , and not on the enumeration p . Being more precise the lemmastates that for such a map g, for a model M and two enumerations 1 and 2 of M which give rise to the points p1 = (M; 1) and p2 = (M; 2) of X, the following diagram in Sets commutes: gp :(fy j (y)g() )p 1
(fz j (z)g() )p /
1
'
'
gp :(fy j (y)g() )p
2
1
(fz j (z)g() )p
/
2
2
Proof. Fix M and consider i g: i fy j (y)g() ! i fz j (z)g(). Since we have a canonical isomorphism fy j (y)g() ' ' fy j (y)g(U) and since i ' ' M the map i g can be written up to isomorphism as i g: E(M) fy j (y)g(M) ! E(M) fz j (z)g(M) : But E(M) is connected, and i g is of the form id gM for a unique map gM : fy j (y)g(M) ! fz j (z)g(M) . 2 Lemma 6.5 Let V be a subsheaf of the de nable sheaf = fy j y = yg() . If the stalks of V depend only on the di erent models M and not on the enumeration of M (i.e., for two enumerations 1 and 2 of M and the two points p1 = (M; 1) and p2 = (M; 2) of X the stalks Vp and Vp are canonically isomorphic), then V is de nable: There exists a geometric formula (y) such that V ' fy j (y)g(). 2
1
Another formulation of the assumption of this lemma is if a 2 M and ap 2 V then ap 2 V . Proof. Let ap 2 Vp be arbitrary. Fix a tuple n such that p (n) = ap and consider the open set [ (?)(n)]?1 (V ) X. By Lemma 4.6 there exists a geometric formula (w; y) and a tuple k such that (i) p 2 U(k;n) [ (?)(n)]?1 (V ). 1
2
(ii) For every N 2 Pts (T) and for all tuples b 2 N, c 2 N such that N j= (b; c) there exists a {to{one enumeration of N so that (k) = b, (n) = c and hence (N; ) 2 U(k;n) . 11
We claim that ap 2 fy j 9w(w; y)g() V which will prove the lemma: ap was arbitrary and V is the union of de nable sheaves, hence de nable. To prove the claim note that by (i) the tuple ap is in fy j 9w(w; y)g(). To show the inclusion take cq 2 fy j 9w(w; y)g() arbitrary for q = (N; ). Fix b such that N j= (b; c). By (ii) there exists a {to{one enumeration 0 of0 N such that 0 (n) = c and 0 (k) = b. The point q0 = (N; 0 ) 0 q 2 is in U(k;n) , thus (by (i)) c = q (n) is in V , as is cq by the assumption on V . Proof of Proposition 6.1: Consider g: fy j (y)g() ! fz j (z)g(). By Lemma 6.4 the stalk maps gp of g depend only on the underlying model Mp , and not on the enumeration p . Equivalently, graph(g) as a subobject of satis es the assumption of Lemma 6.5 and is de nable. 2
7 Preservation of Function Types
In this section we prove that the geometric morphism ': Sh(X) ! B(T) preserves exponentials, i.e. ' (RP ) ' ' R' P for P; R 2 B(T), at least if P and R are representable sheaves. To simplify notation we use the internal Hom (Hom (?; ?)) to denote both the exponential in B(T) and in Sh(X). Proposition 7.1 The model preserves function types: For any two rst order formulas (y) and (z) there is an isomorphism ' Hom (fy j (y)g(U); fz j (z)g(U) ) ' Hom (' fy j (y)g(U) ; 'fz j (z)g(U)): Since every object of the form fz j (z)g(U) is geometrically de nable we can assume that and are in fact geometric formulas. There is a canonical map (3) ' Hom (fy j (y)g(U) ; fz j (z)g(U)) ! Hom (fy j (y)g(); fz j (z)g() ); which is the transposed of the evaluation map. It is a formal consequence of Corollary 5.2 and Proposition 2.3 that this map is mono (see [7]). To show that it is epi (locally surjective) we have to describe this map explicitly. In what follows we identify sheaves E over X with their section functors ?(E; ?): OX op ! Sets. Following [10], p. 70, the right side of (3) is described by ?(Hom (fy j (y)g() ; fz j (z)g() ); U) o n = f: fy j (y)g() U ! fz j (z)g() U f continuous and f = for U X open. Here is the canonical etale map fz j (z)g() ! X and similar for . For V U there are canonical restriction maps. To describe the left side of (3) we have to go through Chapter VII of [8]. We rst de ne a presheaf '~ Hom (fy j (y)g(U); fz j (z)g(U) ) on OX by '~ Hom (fy j (y)g(U) ; fz j (z)g(U) )(U) n = ( ; s) [(w)] an object in Syn(T); : fw j (w)g(U) fy j (y)g(U) ! fz j (z)g(U); o s: U ! fw j (w)g() = : The equivalence relation is generated by ( ((U) id); s0 ) ( ; () s0 ) for []: [(w0)] ! [(w)] an arrow in Syn(T), : fw j (w)g(U) fy j (y)g(U) ! fz j (z)g(U) a map in B(T), and s0 : U ! fw0 j 0 (w0 )g() a section. For V U there are canonical restriction 12
maps. The sheaf ' Hom (fy j (y)g(U) ; fz j (z)g(U)) is the associated sheaf of the presheaf just de ned. There is a canonical comparison map com of presheaves '~Hom (fy j (y)g(U); fz j (z)g(U)) ! Hom (fy j (y)g() ; fz j (z)g() );
(4)
which sends the equivalence class [ ; s] to the map
fy j (y)g() U ! fz j (z)g() U a 2 fy j (y)g(Mp ) for p 2 U 7! ()(s(p); a): The shea cation of the map com is the map of (3). By [8], Corollary III.7.6, the map of (3) is locally surjective if and only if the comparison map com is locally surjective. This we will deduce from the following lemma:
Lemma 7.2 Let U be a basic open set of X and P; R representable sheaves in B(T). There exists a geometric formula (w) and a tuple k such that U = U(k) and each map f: (' P)U ! (' R)U can be lifted to a continuous map of sheaves f:^ fw j (w)g() ' P ! ' R so that for all p 2 U : ^ p (k); ?) = fp as maps (' P)p ! (' R)p . f( Proof. Let (w) and k as in Lemma 4.2 and x a map f: ' P U ! ' RU. To show the existence
of f^ note that such a map is uniquely determined by maps f^(M) : fw j (w)g(M) M P ! M R for M 2 Pts (T) (this is Lemma 6.4). We use this to de ne f^ as follows: x b 2 fw j (w)g(M) . By Lemma 4.2 there exists a {to{one enumeration of M such that (k) = b and p = (M; ) 2 U(k) = U. fp Now M P ' p ' P ! p ' R ' M R gives a map M P ! M R and we do this for each tuple (M) b 2 fw j (w)g to get f^(M) . This construction is independent of the chosen : If 1 and 2 are two enumerations which map k to b then 1 ; 2 2 Wk7!b i (U(k) ) = E(M) \ U(k) : But Wk7!b is connected, and i f restricted to this set comes from a unique map M P ! M R, viz. i f Wk7!b i (' RU)Wk7!b i (' P U)Wk7!b /
'
'
^
Wk7!b M P _ _ _id_f _ _ _ _ Wk7!b M R It remains to show that the f^(M) glue together to a continuous map. Thus to complete the proof of this lemma we have to show that for any section t: O ! fw j (w)g() ' P the composition f^ t is continuous. To this end x t, which comes from two sections (M)
/
t : O ! fw j (w)g()
and
t0 : O ! ' P:
Take q 2 O arbitrary. Locally (say on V 3 q) we can assume that t is the section (?) (l) for a tuple l 6= k. Consider the bijection of 8
We prove a topological completeness theorem for in nitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected.
1 Introduction Sheaf models of in nitary rst order theories were studied intensively in [9]. They are models in Grothendieck toposes and subsume Heyting valued models, boolean valued models, Kripke models, and permutation models. Of particular interest are so called geometric theories T (see De nition 2.1). To these theories one can associate a particular topos B(T) together with a generic (term) model U in B(T). The model U is generic since there is an equivalence between the category of models of T in a topos F , and the category of geometric morphisms from F to B(T). Given such a geometric morphism ': F ! B(T) the corresponding model is obtained as ' U (see below). There are usually not enough models in Sets to test semantically whether a given geometric sequent holds in all models of T in all Grothendieck toposes. If this is the case we say that T satis es the completeness theorem. We will prove the following topological completeness theorem:
Theorem Let T be a geometric theory satisfying the completeness theorem. There exists a topological space X and an OX {valued model of T (a model of T in Sh(X) corresponding to a geometric morphism ': Sh(X) ! B(T)) such that (i) for any rst order formula we have j= if and only if U j= (the geometric morphism ' is an open surjection). (ii) Any map in Sh(X) between rst order de nable sheaves is geometrically de nable (functional
completeness). (iii) The inverse image ' preserves function types of rst order de nable types.
Part (ii) and (iii) of the theorem extend a well known result by Makkai and Reyes, stating that any topos with enough points has an open spatial cover ([9], Theorem 6.3.3). This result appears here as part (i) of the theorem. Part (iii) of the theorem needs some explanation. Even if the language has already function types there is no guaranty that these function types are interpreted by exponentials of sheaves. But one can always add new function types and interpret them in the universal model in the standard way, namely by exponentials. These new function types are interpreted in by exponentials of sheaves too. The space X will contain the models of T in Sets, each model equipped with some extra structure. 1
We should remark that the theorem states only those properties of the geometric morphism ': Sh(X) ! B(T) which have a clear logical meaning. For those familiar with topos theory we remark that ' is connected, locally connected, and acyclic. In particular, ' induces isomorphisms in cohomology. Details about these aspects can be found in [3] and will appear in a joined paper with I. Moerdijk [4]. The properties connected and locally connected are closely related to the properties stated as part (ii) and (iii) of the theorem. We start reviewing sheaf models and xing our notation. Then we de ne the space X, the model , and take a closer look at the topology of the spaces involved. The rest of the paper is occupied by the proof of the theorem. Acknowledgements: I would like to thank Ieke Moerdijk for his encouragement and many stimulating discussions. This research was supported by a grant of the state Baden-Wurttemberg, Germany, by a DAAD/NUFFIC grant, by a scholarship of the University of Utrecht, and by a grant from a project funded by the Netherlands Organization for Scienti c Research NWO.
2 Review of Sheaf Models We assume that the reader is familiar with (Grothendieck) toposes, see for example [9], [6] or [8]. For sheaves over topological spaces we refer as well to [10]. In particular, we assume that the reader has seen the basic de nition of a site (C ; J), sheaves on a site and maps between sheaves. These data together yield the Grothendieck topos Sh(C ; J). Maps between Grothendieck toposes are called geometric morphisms . A geometric morphism f: Sh(D ; K) ! Sh(C ; J) is a pair of functors f : Sh(D ; K)
/
o
Sh(C ; J): f ;
f left adjoint to f , and f is moreover left exact (preserves nite limits). f is called the inverse image and f the direct image . Maps between geometric morphisms are natural transformations. In this way we get the category of geometric morphisms from Sh(D ; K) to Sh(C ; J), denoted by Hom(Sh(D ; K); Sh(C ; J)). A Grothendieck topos E has a rich structure. In particular we can speak about models of an in nitary rst order theory T in the topos E as follows (cf. [8], Section X.2 and [9], Chapter 2): An interpretation M of a language assigns to each sort Y an object Y (M) in E , to each constant c of sort Y a global element c(M) : 1 ! Y (M) , to each function symbol f: Y ! Y an arrow f (M) : Y (M) ! Y (M) and similar for relation symbols R. (Here we introduce the convention that Y stands for a sequence Y1; : : :; Yn of sorts and Y (M) for Y1(M) Yn(M) .) If we have a one sorted language we simply write M for the underlying object. Interpreting the logical connectives and quanti ers in the canonical way we get for each formula (y: Y ) a subobject fy j (y)g(M) ,! Y (M) : For a closed formula we say that M is a model of (M j= ) if f j g(M) (which is a subobject of 1E ) is equal to 1E . In the same way M j= T means M j= for all 2 T. With the natural de nition of morphisms between models we get the category Mod(T; E ) of models of T in E . If E = Sh(X), the topos of sheaves over a topological space, we speak about topological models. The category of models in the topos Sh(H) of sheaves over a (complete) Heyting algebra H is naturally equivalent to the category of H{valued models, see [5]. Models in the topos BG of G{sets (G a group) are sometimes refered to as permutation models. We are mainly interested in geometric theories:
De nition 2.1 A formula (y) is W called geometric if it is built up by atomic formulas, nite conjunction ^, arbitrary disjunction , the truth values true > and false ?, and existential quanti cation 9. A theory T is called geometric if all its axioms are (equivalent to formulas) of the form 8y( 1 (y) ! 2(y)) for 1 and 2 geometric. 2
Examples are given by the standard axiomatizations of linear orders with or without endpoints, groups or rings. The inverse image f of a geometric morphism f: F ! E induces for any theory T a functor f : Mod(T; F ) ! Mod(T; E ) (see [8], X.3.6). With this observation the main theorem relating topos theory and logic becomes as follows:
Theorem 2.2 For any in nitary geometric theory T there exists a unique Grothendieck topos B(T) such that for all Grothendieck toposes F there is an equivalence ' Hom(F ; B(T)); Mod(T; F ) ! natural in F . Conversely, any Grothendieck topos E is of the form B(T) for some geometric
2 Uniqueness in this theorem is meant as \unique up to equivalence". We say that B(T) is the classifying topos of the theory T. The theory classi ed by a topos is not unique. In the equivalence of the theorem the identity geometric morphism B(T) ! B(T) corresponds to a model U of T in B(T), called the generic model (or universal model) of T. The model U is uniquely determined up to isomorphism. It is universal in the sense that for any model M of T in F there exists a geometric morphism f: F ! B(T) such that f U ' M. The inverse image of a geometric morphism does usually not preserve the full internal rst order logic of a topos. f this is true the geometric morphism is called open (see [7], [8]). A geometric morphism is called surjective if the inverse image is faithful. All these geometrical notions can be characterized model theoretically. In the following proposition we denote by Thg (?) the set of all geometric sequents 8y( 1 (y) ! 2 (y)) for 1 ; 2 geometric formulas which are true in a model. Similar, Thfo (?) denotes the set of all closed rst order formulas which are true in a model. Proposition 2.3 Let T be a geometric theory, ': F ! B(T) a geometric morphism with corresponding model in F . Let U denote the universal model in B(T). (i) ' is surjective if and only if Thg (U) = Thg (). (ii) ' is open if and only if Thfo (U) Thfo (). (iii) ' is an open surjection if and only if Thfo (U) = Thfo (). Proof. This is implicitely in [9]. An explicit proof is given in [2]. 2 g g If Th (U) = Th () we say that is a conservative model of T. We remind the reader of the provability relation ` given in [9]. For a geometric theory T and geometric formulas 1 (y) and 2 (y) we have T ` 8y( 1 (y) ! 2(y)) if and only if U j= 8y( 1 (y) ! 2 (y)). theory T.
De nition 2.4 A geometric theory T is said to satisfy the completeness theorem if T `Sets 8y( 1 (y) ! 2 (y)) implies T ` 8y( 1(y) ! 2 (y)). Thus, a theory T satis es the completeness theorem if there are enough models in Sets to test semantically whether a geometric sequent 8y( 1(y) ! 2 (y)) is derivable or not. It is well understood that T satis es the completeness theorem if B(T) has enough points: The geometric morphisms p: Sets ! B(T) form a jointly surjective family. Most toposes arising in practice have enough points. We mention the toposes Sh(X) for X a topological space, BG for G a discrete or
continuous group, or { as a less trivial example { the etale topos Sh(Xet ) associated to a scheme X in algebraic geometry (see [1]). We sketch the construction of B(T) for a geometric theory T formulated in L! . For simplicity we assume to have a one{sorted language. A syntactic site Syn(T) for B(T) can be de ned in terms of the provability relation `. 3
| Objects of Syn(T) are equivalence classes [ (y)] of geometric formulas in L! . Two such formulas 1 (y 1 ) and 2 (y 2 ) are equivalent if T ` 8y( 1(y) $ 2 (y)). | Arrows from [(w)] to [ (y)] are equivalence classes [(w; y)] of geometric formulas in L! such that is provable functional: ? T ` 8w8y (w; y) ! (w) ^ (y) ? T ` 8w (w) ! 9y(w; y) ? T ` 8w8y 8z (w; y) ^ (w; z) ! y = z | A family f[i(wi ; y)]:[i(wi )] ! [ (y)]gi2I is a cover if it is a provable cover: ? of arrows W T ` 8y (y) ! I 9wi i(wi ; y) . The category Syn(T) has all nite limits and the Grothendieck topology is subcanonical, see [8] or [9] for more details. B(T) is now de ned to be Sh(Syn(T)). The Yoneda map y: Syn(T) ! B(T) gives U as follows: U = y([y = y]) were y is a free variable; f (U) : U ! U = y([f(z) = y]): y([z = z]) ! y([y = y]) R(U) ,! U = y([R(y)] ,! y([y = y]): For every geometric formula (y) there is an isomorphism fy j (y)g(U) ' y([ (y)]). Given a geometric morphism ': F ! B(T) and corresponding model of T in F we have for each geometric formula (y) canonical isomorphisms fy j (y)g() ' ' fy j (y)g(U) ' ' y([ (y)]): The universal model U has the property that any subobject S ,! U is geometrically de nable: there exists a geometric formula (y) such that S ' fy j (y)g(U) . (But note that if the theory T is formulated in L! such geometric formulas (y) may only exist in L0 ! for 0 > .)
3 De nition of the Space of Models Let T be a geometric theory satisfying the completeness theorem. We assume for simplicity that T is formulated in a one{sorted language. We denote by Sthe xed set of function and relation symbols. Constants are treated as 0{ary functions. We always suppress the arity of these symbols to keep notations simple. For a cardinal let Pts (T) denote a xed set of representatives of isomorphism classes of models such that the cardinality of the model is bounded by . By a downward Lowenheim{Skolem argument there exists a cardinal such that the models M 2 Pts (T) form a set of enough models: For two geometric formulas 1(y) and 2 (y) we have T ` 8y( 1 (y) ! 2(y)) i for all M 2 Pts (T): M j= 8y( 1 (y) ! 2 (y)): We x such a cardinal . The space of models we are going to de ne will consist of all models M of T, bounded by and equipped with some extra structure: De nition 3.1 A {to{one enumeration of a model M is a map : K ! M, K , such that for each a 2 M the set ?1(a) is co nal in . We write : ; M for such a map. Note that we do not distinguish between a model and its underlying set. If is a regular cardinal the de nition is equivalent to say that each set ?1 (a) has cardinality . Since we use only the partial order of we denote the elements of by the letters k; l; m; : : :. 4
Let X be the space with elements pairs (M; ) where M 2 Pts (T) and : ; M. If necessary we write such an element as p = (Mp ; p). The topology on X is generated by the following subbase: Uk;l = f(M; ) j M j= (k) = (l)g for k; l 2 , (M) for ki ; l 2 , f 2 S, Uk;l;f = f(M; ) j M j= f ( (k)) = (l)g Uk;R = f(M; ) j M j= R(M) ( (k))g for ki 2 , R 2 S. Here (k) stands for the tuple ( (k1); : : :; (kn)) and it is part of the de nitions that the (k); : : : are de ned. Since expressions like M j= f (M) ( (k)) = (l) are long and hard to read we write them mostly as M j= f(k) = l. This makes sense since the elements k 2 where (k) is de ned are so to say names for the elements in M. To construct a geometric morphism ': Sh(X) ! B(T) wePhave to give an interpretation of thePlanguage in Sh(X): The underlying object is the set p2X Mp with canonical projection : p2X Mp ! X. We generate a topology on by ?1 (U) is open for U X open, Vk = f p(k) j p (k) de nedg is open. Lemma 3.2 is a sheaf over X ( is a local homeomorphism, or an etale map). S Proof. is continuous by de nition and maps Vk bijectively onto Uk;k . Moreover = k2 Vk . To prove that is etale it is hence enough to show that the inverse of restricted to Vk , which is the map (?) (k): Uk;k ! Vk , is continuous. By de nition of the topology of a typical open subset W of Vk; is of the form T W = Vk \ ?1 (U) \ i=1;:::;n Vki = fa 2 Mp j p 2 U and a = p (k) = p (k1) = = p (kn)g for some ki 2 , U X open. It follows that \ Uk;ki [ (?)(k)]?1(W) = U \ i=1;:::;n
and (?)(k) is continuous, so is a local homeomorphism. 2 In what follows, a product symbol between sheaves always means a product of sheaves, which is computed brewise: [ = = Mp Mp p2X
with the appropriate topology. From this remark we see at once that for every function symbol P f in S there is a canonical map f () : ! , namely the sum of all the f (Mp ) (i.e., f () = p2X f (Mp ) ). The same applies to relation symbols Rp in S: There is a canonical subset R() , namely the sum (or disjoint union) of all the R(M ) . Lemma 3.3 (i) Every f () is a local homeomorphism. (ii) Every R() is a subsheaf of . Proof. We leave the details to the reader and prove only the simplest case, that of a unary relationSsymbol we have to show that R() is open in . This follows since i h R. In that case ? 1 () 2 R = k2 (Uk;R ) \ Vk . The two lemmas above prove that is indeed an interpretation of the language in Sh(X). Each point p 2 X de nes a geometric morphism p: Sets ! Sh(X) with inverse image the usual stalk map. In particular, p carries the interpretation to an interpretation p , which is isomorphic to Mp . Since Sh(X) has enough points we get for geometric formulas 1(y) and 2 (y) that 5
j= 8y( 1 (y) ! 2 (y)) i fy j 1(y)g() ,! fy j 2 (y)g() i p fy j 1(y)g() ,! p fy j 2 (y)g() for all p 2 X i fy j 1(y)g(p ) ,! fy j 2(y)g(p ) for all p 2 X i p j= 8y( 1 (y) ! 2(y)) for all p 2 X i Mp j= 8y( 1(y) ! 2 (y)) for all p 2 X. i T ` 8y( 1 (y) ! 2 (y)) Thus, is a conservative model of the theory T and corresponds to a surjective geometric morphism ': Sh(X) ! B(T). For the record we state Proposition 3.4 is a conservative model of T. 2
Remark 3.5 (Notational conventions.) In the sequel we will avoid superscripts as often as possible. But sometimes they are necessary or useful. For example, writing p = (Mp ; p) allows to de ne the sections (?) (k): Uk;k ! Vk .
There is another situation where we will use them: Consider two points p1 = (M; 1) and p2 = (M; 2) of X, that is, 1 and 2 are {to{one enumerations of one T{model M. Moreover, M corresponds to a geometric morphism (denoted by the same letter) M: Sets ! B(T). For a representable sheaf fy j (y)g(U) in B(T) there are canonical isomorphisms (fy j (y)g() )p ' fy j (y)g(p ) ' M fy j (y)g(U) ' fy j (y)g(M) ' : 1
1
It is sometimes necessary to indicate in which bre an element lives. We use for a 2 fy j (y)g(M) the notation ap for \the tuple a in the (isomorphic) bre (fy j (y)g())p ". Adding or dropping the superscript (?)p means applying an isomorphism of sets and moving an element (or tuple) to a particular bre. 1
1
1
Remark 3.6 There are several variations of the construction described in this section, which will
yield the same theorem. It is possible to start with an arbitrary family Pts(T) of models of T such that for geometric formulas 1(y) and 2 (y) we have
T ` 8y( 1 (y) ! 2 (y)) i for all M 2 Pts(T): M j= 8y( 1 (y) ! 2(y)): (We say that Pts(T) is a family of jointly conservative models.) Then has to be chosen such that the cardinality of all models in Pts(T) is less or equal to . For example, the nite linear orders f0; : : :; ng for n 2 N are a family of jointly conservative models for the theory of non{empty linear orders. In this case, can be chosen as @0 . Instead of {to{one enumerations one can use (partial) enumerations of models : K ! M, K , such that for each a 2 M the set ?1 (a) is in nite.
4 De nable Open Sets Let (y) be a geometric formula with free variables exactly the listed tuple y. We de ne for a tuple k the open set Uk; (y) = U (k) of X by induction: U? = ; U> = X Uk=l = \i=0;:::;n Uki ;li UR(k) = Uk;R Uf (k)=l = Uk;l;f S U( ^ )(k) = U (k) \ U (k) UWI i (k) = I U i (k) W U9y (y;k) = l2 U (l;k ) 1
2
1
2
6
W
In the de nition of the cases ^ and we mean on the right side the tuples k restricted to the free variables actually occurring in 1 ; 2 and in the i . Lemma 4.1 If (y) is a geometric formula and k a tuple of elements of then U (k) = f(M; ) j 2 M j= (k)g. As above the notation implies that (ki) is de ned. The collection of all open sets U (k) forms obviously a base for the topology on X. In general it is not possible to assign to an arbitrary open set U of X a geometric formula (w) and a tuple k so that U = U(k) . This is only true for basic open sets. But we can do slightly better. Lemma 4.2 (i) Let U X be a nonempty basic open set. There exists a geometric formula (w) and a tuple k such that: | U = U(k) . | For every N 2 Pts (T) and for every tuple b 2 N such that N j= (b) there exists a {to{one enumeration of N so that (k) = b and hence (N; ) 2 U(k) . (ii) Let U Un;n be an open set and p a point in U . There exists a geometric formula (w; y) and a tuple k such that: | p 2 U(k;n) U . | For every N 2 Pts (T) and for all tuples b 2 N , c 2 N such that N j= (b; c) there exists a {to{one enumeration of N so that (k) = b, (n) = c and hence (N; ) 2 U(k;n) . Proof. Ad (i). Let 0(w) be a geometric formula and k a tuple such that U = U0 (k) . The formula
(w) 0(w) ^
^
ki =kj
wi = wj
and the tuple k will do the job. Obviously U = U0 (k) = U(k) . To prove the second part take an arbitrary model N 2 Pts (T) and a tuple b 2 N such that N j= (b). Take an arbitrary {to{ one enumeration 00 of N and take out of the domain of 00 the nite set k to get the {to{one enumeration 0 . To get extend 0 to the set k by (k) = b. The fact that N j= (b) ensures that this is well{de ned (compare the construction of ). Moreover (N; ) 2 U(k) . Ad (ii). Let U 0 be a basic open set such that p 2 U 0 U. Choose a formula 0(w) and a tuple k according to (i). The geometric formula (w; y) 0(w) ^
^
ki =nj ki2k; nj 2n
wi = y j
will do the job. 2 Next we state several lemmas which summarize the information we get from a section (?) (n). To state the lemmas we rst need a de nition: De nition 4.3 We say that a formula (y) is GT if it is built up by formulas 1 (y) ! 2(y) for V 1 ; 2 geometric, arbitrary meets , and universal quanti cation 8. The abbreviation GT is used to remind the reader that this class of formulas \is" the class of geometric theories, any theory put together into one axiom. The following two lemmas are proved by induction on : 7
Lemma 4.4 Let (y) be a geometric formula and (y) a GT{formula. Then fy j (y)g(U) ,! fy j (y)g(U) 2 if and only if for all M 2 Pts (T): M j= (y) ! (y). Lemma 4.5 Let s: U ! be a section (a continuous map such that s(p) 2 Mp for each p 2 U ). Let (y) be a GT{formula with associated subsheaf fy j (y)g() . Then im(s) fy j (y)g() if and only if for all points p in U : Mp j= (s(p)). 2 The next lemma combines Lemma 4.5, which always holds, with the special situation we are dealing with. The lemma is frequently used later.
Lemma 4.6 Let U Un;n be open sets and (y) a GT{formula. Suppose that the image of (?) (n)U has nonempty intersection with the sheaf fy j (y)g() , say for p 2 U that p (n) 2 fy j (y)g(). Then there exists a geometric formula (w; y) and a tuple k such that (i) p 2 U(k;n) U . (ii) (?)(n)U(k;n) maps into fy j (y)g() . In particular for all p0 2 U(k;n) the model Mp0 satis es (n). (iii) All models N 2 Pts (T) satisfy N j= 9w(w; y) ! (y). (Note that if is geometric this is equivalent to say that T ` 9w(w; y) ! (y).) (iv) For every N 2 Pts (T) and for tuples b 2 N , c 2 N such that N j= (b; c) there exists a {to{one enumeration of N so that (k) = b, (n) = c and hence (N; ) 2 U(k;n) .
Proof. If we write U 0 = [ (?)(n)]?1 fy j (y)g() \ U we are in the situation of Lemma 4.2 and
nd a formula (w; y) and a tuple k such that (i), (ii) and (iv) hold. To prove (iii) take an arbitrary N 2 Pts (T) and a tuple c 2 N such that N j= 9w(w; c). There is thus a tuple b such that N j= (b; c). By (iv) there exists a {to{one enumeration such that (k) = b; (n) = c; and (hence) (N; ) 2 U(k;n) : 2 Since (N; ) 2 U(k;n) we can apply (ii) to get N j= (n), i.e. N j= (c).
5 Preservation of Universal Quanti ers We will show the following proposition:
Proposition 5.1 The inverse image ' preserves the interpretation of rst order formulas: For each rst order formula (y) there is an isomorphism fy j (y)g() ' ' fy j (y)g(U) : The following two corollaries are immediate:
Corollary 5.2 The models and U satisfy the same rst order formulas: If is any rst order formula, U j= if and only if j= . Proof. Take an arbitrary formula . The objects f j g() and ' f j g(U) are isomorphic by Proposition 5.1 and the corollary follows since ' preserves and re ects isomorphisms.
2
Corollary 5.3 For each rst order formula (y) there exists a geometric formula (y) such that fy j (y)g() ' fy j (y)g(): 8
Proof. Given there exists a geometric formula such that fy j (y)g(U) ' fy j (y)g(U): Applying
' to both sides together with Proposition 5.1 yields the result. 2 We rst proof Proposition 5.1 for some special cases: Lemma 5.4 The proposition holds for GT{formulas (y). Proof. It is straightforward to check that for any GT{formula (y) the inequality of subobjects ' fy j (y)g(U) fy j (y)g() always holds. For the other inequality take a in fy j (y)g(), say a 2 Mp for p = (Mp ; p) in X. Fix a tuple n such that p (n) = a. Let U be the open set Un;n \ [ (?)(n)]?1(fy j (y)g() ) which contains p. We apply Lemma 4.6 to nd a geometric formula (w; y) and a tuple k such that p 2 U(k;n) and for all M 2 Pts (T): M j= 9w(w; y) ! (y). Since p 2 U(k;n) it follows that a 2 fy j 9w(w; y)g() . Since (y) is GT we can apply Lemma 4.4 and get fy j 9w(w; y)g(U) ,! fy j (y)g(U) and hence a 2 ' fy j (y)g(U). But a was arbitrary. 2 Proof of Proposition 5.1. The proof is by induction on . For atomic formulas this is trivial. The W cases ^, and 9 follow since the p 2 X form together a jointly surjective family of geometric morphisms Sets ! Sh(X). VThe details are standard and left to the reader. From the remaining three interesting cases !, and 8 we treat one in detail: Let 1(y) and 2 (y) be rst order formulas and assume fy j i (y)g() ' ' fy j i (y)g(U) for i = 1; 2. (1) Since fy j i(y)g(U) ,! U is geometrically de nable there exist geometric formulas i (y) so that fy j i (y)g(U) ' fy j i (y)g(U) : (2) Now (by de nition) fy j 1(y) ! 2 (y)g() ' fy j 1(y)g() ! fy j 2(y)g() (U) (U) ' ' fy j 1 (y)g ! ' fy j 2 (y)g (by (1)) ' ' fy j 1 (y)g(U) ! ' fy j 2 (y)g(U) (by (2)) ' fy j 1 (y)g() ! fy j 2 (y)g() (the i are geometric) ' fy j 1 (y) ! 2 (y)g() (by de nition) (U) ' ' fy j 1 (y) ! 2 (y)g (by Lemma 5.4) ' ::: ' ' fy j 1 (y) ! 2 (y)g(U) The other two cases are similar using the same trick. 2
6 Functional Completeness In this section we present a result about functional completeness which we state as follows: Proposition 6.1 (Functional completeness) Let (y) and (z) be geometric formulas and g: fy j (y)g() ! fz j (z)g() a map of sheaves. Then g is de nable: There exists a geometric formula (y; z) such that graph(g) ' f(y; z) j (y; z)g() . Equivalently, de nes in Syn(T) a map [ (y; z)]: [ (y)] ! [(z)]; and for arbitrary p, for all tuples b 2 fy j (y)g(Mp ) and c 2 fz j (z)g(Mp ) g(b) = c i Mp j= (b; c): 9
Note that by Corollary 5.3 the proposition extends automatically to the case where and are rst order formulas. Of course we could assume that (z) z = z. The proof of Proposition 6.1 needs some preliminary remarks. We leave most of their proofs to the reader. Fix a model M in Pts (T). Thus M corresponds to a geometric morphism (again denoted by M, since there is no confusion here) M: Sets ! B(T). Let E(M) be the topological space with elements {to{one enumerations of M, i.e., maps : ; M. The topology on E(M) is generated by the open sets Wk;a = f j (k) = ag for k 2 , a 2 M arbitrary. For tuples k and a we use as well Wk7!a for the open set \i=1;:::;n Wki ;ai . There is a canonical map i = iM : E(M) ! X sending to (M; ). We will often identify E(M) along i with a subset of X. Lemma 6.2 (i) The map iM is continuous. (ii) The space E(M) and each basic open set Wk7!a are connected. Proof. As an example we show the second part. Suppose E(M) = U1 [ U2 for U1 , U2 nonempty, say for example i: Ki ! M in Ui . Each i is in some basic open set Wi Ui and there are only nitely many elements of occurring in the de nition of W1 (and of W2 ). We rst consider de ned by (k); if k occurs in the list de ning W1, (k) = 21(k); otherwise. is obtained from 2 by changing the enumeration 2 in a nite number of places. is clearly {to{one and 2 W1. After renaming we may thus assume that 1 and 2 di er only in a nite number of places. Next consider the map
: k 7! 1(k) i 1(k) = 2(k): By construction, is {to{one. But has the additional property that 2 W open implies 1 2 W and 2 2 W. Thus either 2 2 U1 or 1 2 U2 , i.e., either 1 or 2 is contained in U1 \ U2 and E(M) is connected. The same argument applies locally, that is, if we prescribe the enumerations 1 and 2 in a nite number of places (they are then prescribed at the same elements k 2 ). Hence E(M) is as well locally connected. 2 The map i induces a geometric morphism i: Sh(E(M)) ! Sh(X) with inverse image the pullback functor: i (E) = i?1 (E) E /
E(M)
i X Applying the functor i to we get the T{model i . There is another way to get a model of T: The projection = M : E(M) ! 1 induces the geometric morphism Sh(E(M)) ! Sets = Sh(1) with inverse image the constant sheaf
/
(S) = E(M) S; where S in Sets is viewed as a space with discrete topology. Applied to the model M we get the model M.
10
Lemma 6.3 The models i and M are isomorphic. Equivalently, the following diagram of toposes commutes:
i
Sh(E(M))
Sh(X) /
'
Sets
B(T)
M /
Proof. Among other things one has to show that the canonical isomorphism of sets in the diagram
below is in fact a homeomorphism: M
M E(M) ' o
P
(M ; ) M
/
/
1
E(M)
E(M)
o
X
i /
The details are straightforward. 2 The following two lemmas are the ingredients in the proof of Proposition 6.1: Lemma 6.4 Let (y) and (z) be two geometric formulas and g: fy j (y)g() ! fz j (z)g() a map of sheaves. Then the stalk maps
gp : (fy j (y)g() )p ! (fz j (z)g())p depend only on the underlying model Mp , and not on the enumeration p . Being more precise the lemmastates that for such a map g, for a model M and two enumerations 1 and 2 of M which give rise to the points p1 = (M; 1) and p2 = (M; 2) of X, the following diagram in Sets commutes: gp :(fy j (y)g() )p 1
(fz j (z)g() )p /
1
'
'
gp :(fy j (y)g() )p
2
1
(fz j (z)g() )p
/
2
2
Proof. Fix M and consider i g: i fy j (y)g() ! i fz j (z)g(). Since we have a canonical isomorphism fy j (y)g() ' ' fy j (y)g(U) and since i ' ' M the map i g can be written up to isomorphism as i g: E(M) fy j (y)g(M) ! E(M) fz j (z)g(M) : But E(M) is connected, and i g is of the form id gM for a unique map gM : fy j (y)g(M) ! fz j (z)g(M) . 2 Lemma 6.5 Let V be a subsheaf of the de nable sheaf = fy j y = yg() . If the stalks of V depend only on the di erent models M and not on the enumeration of M (i.e., for two enumerations 1 and 2 of M and the two points p1 = (M; 1) and p2 = (M; 2) of X the stalks Vp and Vp are canonically isomorphic), then V is de nable: There exists a geometric formula (y) such that V ' fy j (y)g(). 2
1
Another formulation of the assumption of this lemma is if a 2 M and ap 2 V then ap 2 V . Proof. Let ap 2 Vp be arbitrary. Fix a tuple n such that p (n) = ap and consider the open set [ (?)(n)]?1 (V ) X. By Lemma 4.6 there exists a geometric formula (w; y) and a tuple k such that (i) p 2 U(k;n) [ (?)(n)]?1 (V ). 1
2
(ii) For every N 2 Pts (T) and for all tuples b 2 N, c 2 N such that N j= (b; c) there exists a {to{one enumeration of N so that (k) = b, (n) = c and hence (N; ) 2 U(k;n) . 11
We claim that ap 2 fy j 9w(w; y)g() V which will prove the lemma: ap was arbitrary and V is the union of de nable sheaves, hence de nable. To prove the claim note that by (i) the tuple ap is in fy j 9w(w; y)g(). To show the inclusion take cq 2 fy j 9w(w; y)g() arbitrary for q = (N; ). Fix b such that N j= (b; c). By (ii) there exists a {to{one enumeration 0 of0 N such that 0 (n) = c and 0 (k) = b. The point q0 = (N; 0 ) 0 q 2 is in U(k;n) , thus (by (i)) c = q (n) is in V , as is cq by the assumption on V . Proof of Proposition 6.1: Consider g: fy j (y)g() ! fz j (z)g(). By Lemma 6.4 the stalk maps gp of g depend only on the underlying model Mp , and not on the enumeration p . Equivalently, graph(g) as a subobject of satis es the assumption of Lemma 6.5 and is de nable. 2
7 Preservation of Function Types
In this section we prove that the geometric morphism ': Sh(X) ! B(T) preserves exponentials, i.e. ' (RP ) ' ' R' P for P; R 2 B(T), at least if P and R are representable sheaves. To simplify notation we use the internal Hom (Hom (?; ?)) to denote both the exponential in B(T) and in Sh(X). Proposition 7.1 The model preserves function types: For any two rst order formulas (y) and (z) there is an isomorphism ' Hom (fy j (y)g(U); fz j (z)g(U) ) ' Hom (' fy j (y)g(U) ; 'fz j (z)g(U)): Since every object of the form fz j (z)g(U) is geometrically de nable we can assume that and are in fact geometric formulas. There is a canonical map (3) ' Hom (fy j (y)g(U) ; fz j (z)g(U)) ! Hom (fy j (y)g(); fz j (z)g() ); which is the transposed of the evaluation map. It is a formal consequence of Corollary 5.2 and Proposition 2.3 that this map is mono (see [7]). To show that it is epi (locally surjective) we have to describe this map explicitly. In what follows we identify sheaves E over X with their section functors ?(E; ?): OX op ! Sets. Following [10], p. 70, the right side of (3) is described by ?(Hom (fy j (y)g() ; fz j (z)g() ); U) o n = f: fy j (y)g() U ! fz j (z)g() U f continuous and f = for U X open. Here is the canonical etale map fz j (z)g() ! X and similar for . For V U there are canonical restriction maps. To describe the left side of (3) we have to go through Chapter VII of [8]. We rst de ne a presheaf '~ Hom (fy j (y)g(U); fz j (z)g(U) ) on OX by '~ Hom (fy j (y)g(U) ; fz j (z)g(U) )(U) n = ( ; s) [(w)] an object in Syn(T); : fw j (w)g(U) fy j (y)g(U) ! fz j (z)g(U); o s: U ! fw j (w)g() = : The equivalence relation is generated by ( ((U) id); s0 ) ( ; () s0 ) for []: [(w0)] ! [(w)] an arrow in Syn(T), : fw j (w)g(U) fy j (y)g(U) ! fz j (z)g(U) a map in B(T), and s0 : U ! fw0 j 0 (w0 )g() a section. For V U there are canonical restriction 12
maps. The sheaf ' Hom (fy j (y)g(U) ; fz j (z)g(U)) is the associated sheaf of the presheaf just de ned. There is a canonical comparison map com of presheaves '~Hom (fy j (y)g(U); fz j (z)g(U)) ! Hom (fy j (y)g() ; fz j (z)g() );
(4)
which sends the equivalence class [ ; s] to the map
fy j (y)g() U ! fz j (z)g() U a 2 fy j (y)g(Mp ) for p 2 U 7! ()(s(p); a): The shea cation of the map com is the map of (3). By [8], Corollary III.7.6, the map of (3) is locally surjective if and only if the comparison map com is locally surjective. This we will deduce from the following lemma:
Lemma 7.2 Let U be a basic open set of X and P; R representable sheaves in B(T). There exists a geometric formula (w) and a tuple k such that U = U(k) and each map f: (' P)U ! (' R)U can be lifted to a continuous map of sheaves f:^ fw j (w)g() ' P ! ' R so that for all p 2 U : ^ p (k); ?) = fp as maps (' P)p ! (' R)p . f( Proof. Let (w) and k as in Lemma 4.2 and x a map f: ' P U ! ' RU. To show the existence
of f^ note that such a map is uniquely determined by maps f^(M) : fw j (w)g(M) M P ! M R for M 2 Pts (T) (this is Lemma 6.4). We use this to de ne f^ as follows: x b 2 fw j (w)g(M) . By Lemma 4.2 there exists a {to{one enumeration of M such that (k) = b and p = (M; ) 2 U(k) = U. fp Now M P ' p ' P ! p ' R ' M R gives a map M P ! M R and we do this for each tuple (M) b 2 fw j (w)g to get f^(M) . This construction is independent of the chosen : If 1 and 2 are two enumerations which map k to b then 1 ; 2 2 Wk7!b i (U(k) ) = E(M) \ U(k) : But Wk7!b is connected, and i f restricted to this set comes from a unique map M P ! M R, viz. i f Wk7!b i (' RU)Wk7!b i (' P U)Wk7!b /
'
'
^
Wk7!b M P _ _ _id_f _ _ _ _ Wk7!b M R It remains to show that the f^(M) glue together to a continuous map. Thus to complete the proof of this lemma we have to show that for any section t: O ! fw j (w)g() ' P the composition f^ t is continuous. To this end x t, which comes from two sections (M)
/
t : O ! fw j (w)g()
and
t0 : O ! ' P:
Take q 2 O arbitrary. Locally (say on V 3 q) we can assume that t is the section (?) (l) for a tuple l 6= k. Consider the bijection of 8
