BBOOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES
MORENO PIEROBON, MATTEO VIALEA
BSTRACT . Boolean valued models for a signature L are generalizations of L -structures in whichwe allow the L -relation symbols to be interpreted by boolean truth values; for example for elements a, b ∈ M with M a B -valued L -structure for some boolean algebra B , ( a = b ) may be neither truenor false, but get an intermediate truth value in B . In this paper we expand and relate the work ofMansfield and others on the semantics of boolean valued models, and of Munro and others on theadjunctions between B -valued models and B + -presheaves for a boolean algebra B . In particular wegive an exact topological characterization (the so called fullness property ) of which boolean valuedmodels satisfy Ło´s theorem (i.e. the general form of the forcing theorem which Cohen —Scott,Solovay, Vopenka— established for the special case given by the forcing method in set theory).We also give an exact categorial characterization of which presheaves correspond to full booleanvalued models in terms of the structure of global sections of their associated ´etal´e space. To do so weintroduce a slight variant of the sheafification process of a given presheaf by means of its embeddinginto an ´etal´e space. I NTRODUCTION
Boolean valued models constitute one of the classical approaches to forcing [1, 3]. They alsogive a useful method for producing new first order models of a theory; for example the standardultrapower construction is just a special instance of a more general procedure to construct booleanvalued elementary extensions of a given structure (see [13] or [8]). Many interesting spaces offunctions (for example L ∞ ( R ) , C ( X ) etc.) can be naturally endowed of the structure of a B -valuedmodel; for example set (cid:74) f = g (cid:75) = [ { x ∈ R : f ( x ) = g ( x ) } ] MALG for f, g ∈ L ∞ ( R ) , where MALG is the complete boolean algebra given by Lebesgue measurable sets modulo Lebesgue null sets; (cid:74) f = g (cid:75) gives a natural measure of the degree of equality subsisting between f and g . In particularby means of boolean valued models one can expect to employ logical methods to analyze theproperties of these function spaces. This can be fruitful, see for example [12] or [4, 10, 11].The aim of this paper is to systematize these results and connect them to sheaf theory, in doing sowe expand on the results of [9, Thm. 5.4, Prop. 5.6]. Our long-stretched hope is that this is a firststep towards a fruitful transfer of techniques, ideas and results arising in set theory in the analysisof the forcing method to other domains where a sheaf-theoretic approach to problems is useful.This paper (while written by persons whose mathematical background is rooted in set theory) isaimed primarily at scholars with some expertise in category theory and familiarity with first orderlogic. We emphasize that no knowledge or familiarity with the forcing method as presented in [1, 3, 5] is required to follow the proofs and statements of the main results. We will sporadicallyrelate our results to the forcing machinery through some examples (see in particular Examples 2.13,4.4, 4.6) and a few comments, but that’s all. The reader unfamiliar with the forcing method cansafely skip them without compromising the comprehension of the main body of this article. We Both the authors acknowledge support from INDAM through GNSAGA. The second author acknowledges supportfrom the project:
PRIN 2017-2017NWTM8R Mathematical Logic: models, sets, computability.
MSC: . We note that we became aware of Monro’s results only after preparing the first version of this paper. We thank GretaCoraglia for bringing to our attention Monro’s work. a r X i v : . [ m a t h . L O ] J un MORENO PIEROBON, MATTEO VIALE actually believe that the paper is going to be much harder to read for a person proficient with theforcing method but completely unfamiliar with the notion of adjunction or of ´etal´e space; howeverwe provide in the preliminaries all the background needed for this second type of readers.Let us spell out more precisely the outcomes of [4, 10, 11, 12] as they will serve as a motivation(at least for set theorists) for what we will do here. [4, 10, 11] show that there is a “natural”identification between: • the unit ball of commutative C ∗ -algebras of the form C ( X ) with X compact and extremallydisconnected; • the RO ( X ) -names for the unit interval [0; 1] as computed in the forcing extension V RO ( X ) of V given by RO ( X ) (where RO ( X ) denotes the complete boolean algebra of regularopen subsets of X ).In [12] these results are applied to establish a weak form of Schanuel’s conjecture on the transcen-dence property of the exponential function over the complex numbers. One of the outcomes ofthe present paper combined with [11] will be that the sheafification of the presheaf given by theessentially bounded measurable functions defined on a measurable subset of R is exactly given bythe sheaf associated to the boolean valued model describing the real numbers which exist in V MALG (see Example 4.6). These results are just samples of the various roles boolean valued models canplay across different mathematical fields, and why we strongly believe it can be useful to develop aflexible translation tool to relate concepts expressed in rather different terminology in differentsset-up. This is more or less all we will say on the relation between the results of the present paperand forcing.Let us now focus in giving a better glimpse of our results. We start by describing informally thenotion of a B -valued model for a relational language L . In a two valued model N the interpretationof a n -ary relation symbol R ∈ L is given by a function R N : N n → ; the interpretation of a n -aryrelation symbol R ∈ L for a B -valued model M is a function R M : M n → B , a (cid:55)→ (cid:74) R ( a ) (cid:75) ∈ B .In order to respect the validity of the equality axioms for first order logic, there are certain naturalconstraints R M must satisfy; we will detail more on them later on. Once truth values (cid:74) R ( a ) (cid:75) ofatomic formulae with parameters in M are assigned, the boolean operations ∧ B , ∨ B , ¬ B allow toassign a truth value in B to formulae built up using the corresponding logical connectives, while (cid:74) ∃ xφ ( x ) (cid:75) := (cid:87) B { (cid:74) φ ( a ) (cid:75) : a ∈ M} . In this set up it is natural to formulate Ło´s Theorem: it statesthat M / G | = φ if and only if (cid:74) φ (cid:75) M B ∈ G . However, Ło´s Theorem is not true for every B -valuedmodel M . Fullness describes a natural condition on a boolean valued model exactly describingwhen Ło´s Theorem holds for it. Unfortunately, to check whether a B -valued model is full, one hasto consider potentially all the formulae in the language. In most cases a stronger condition - the mixing property - is true for the boolean valued model M , a condition which is easier to check,since it is a property of the map interpreting in M the equality symbol.The aim of this work is to spread some light on the difference between the mixing property andthe fullness property, recasting the basic results on the semantics of boolean valued models in thelanguage of sheaves. To reach this objective we endow each B -valued M for a first order language L with a natural presheaf structure F M on the topological space St( B ) . Our translation identifiesthe stalks of this presheaf with the quotients M / G as G ranges on the ultrafilters of B . These arethe main results of the paper: • We show that there exists an adjunction of categories between the category of B -valuedmodels with B -morphisms and the category of separated B + -presheaves (Thm. 3.3);moreover this adjunction specializes to an equivalence of categories between the family of Where M / G is the quotient of M induced by the equivalence relation τ ∼ σ if and only if (cid:74) τ = σ (cid:75) ∈ G , see Def.2.6 for details. OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 3 extensional B -valued models and the separated B + -presheaves in which all local sectionsare restrictions of global sections (Cor. 3.3); note that (up to B -isomorphism) every B -valued model is extensional . • We show that the presheaf associated to a B -valued model M is a sheaf exactly when M satisfies the mixing property (Theorem 4.1). • We rephrase with a simple topological property the characterization of which booleanvalued models satisfy Ło´s Theorem (Thm. 2.8). • We introduce a variant of the sheafification process by means of ´etal´e spaces described in[7, Section II.5] (see Def. 1.15) which corresponds to the one described in [7, Section II.5]exactly when it is applied to a presheaf of open sets on a compact extremally disconnectedtopological space (Thm. 1.16). We also relate our sheafification process to standardcompletions processes performed in functional/analysis general topology (Examples 1 and4.6). • We characterize the fullness property elaborating more on our correspondence between B -valued models and presheaves, and taking into account the notion of ´etal´e space; inparticular we will use the following characterization of sheaves: a fixed presheaf F ona topological space ( X, τ ) (i.e. a contravariant functor from τ \ {∅} with values in thecategory of sets) is a sheaf if and only if the global sections of the associated ´etal´e spaceare exactly described by the elements of F ( X ) [7, Section II.5, Theorem 1].Given a B -valued model M for a first order language L , to every L -formula ϕ ( x ) weassociate a presheaf G ϕ M on a topological subspace N b ϕ of St( B ) (where b ϕ is the booleanvalue given to the existential formula ∃ xϕ ( x ) , and N b is the clopen subset of St( B ) givenby the ultrafilters to which b belongs).We show that M is full exactly when, for every formula ϕ , the ´etal´e space associated to G ϕ M has at least one global section (Theorem 5.1).We will also separate the fullness and the mixing properties by exhibiting a booleanvalued model which has the first but not the latter property (Example 2.13).Familiarity with the basic results on sheaves and category will help the reader, however to make thepaper self-contained, we will cover the needed background on boolean valued semantics, categories,sheaves, ´etal´e spaces.We consider the main contribution of the present paper that of giving a smooth, unified, andelegant presentation of results sparse in the literature, which are also formulated in rather differentterminology according to the author’s backgrounds. While our presentation is guided by a takeon the onthology of mathematics grounded in set theory, most of our results are formulated in thecategory theoretic terminology. For this reason we believe they will be accessible to scholars withvery different backgrounds.Let us now introduce informally an example which outlines why we will pay some special atten-tion to introduce certain basic concepts of sheaf theory, and why boolean valued models give us agood set-up where to analyze certain completion processes of interest in general topology/functionalanalysis. We will use this example to clarify the analysis of the notion of ´etal´e space we will pursue. Example 1.
Consider the presheaf (see Def. 1.9) F of continuous real valued functions defined onsome open interval of R with bounded range. This is not a sheaf as the collation of a matchingfamily of such functions on an open cover of R could result in an unbounded continuous function.The standard sheafification process of this presheaf consists in taking the ´etal´e space Λ ( F ) associated to this presheaf (see Def. 1.13 and Thm. 1.16); Λ ( F ) can be identified with the germs This is actually the special case of [9, Thm. 5.4] for boolean algebras. This is actually the special case of [9, Prop. 5.6] for complete boolean algebras.
MORENO PIEROBON, MATTEO VIALE of continuous functions f : U → R at points x ∈ R with U an open neighborhood of x and f ofbounded range. Λ ( F ) is also in natural correspondance with the sheaf of continuous functionsdefined on some open subset of R (now with no restriction on the range of f being bounded) viathe identification of sections of Λ ( F ) defined on an open set A with the continuous (and total)functions f : A → R .Consider now the family of partial constant functions f n for n ∈ N with f being constant withvalue and domain ( −∞ ; 0) , f constant with value on (1; + ∞ ) and each other f n for n > defined on (1 /n ; 1 / ( n − with constant value /n . This gives a specific family of elements of F which can be collated. Their collation f = (cid:83) n ∈ N f n however is defined only on the dense opensubset R \ ( { } ∪ { / ( n + 1) : n ∈ N } ) and does not have a germ at , because it does not admita continuous extension to any open neighborhood of . In particular it does not identify a globalsection of Λ ( F ) . This is a function regular enough to be of interest in our analysis, hence wewant to be able to include it as part of the sheafification process of F . To reach this objective weconsider the presheaf F as an A -presheaf, where A is the family of non-empty open intervals of R . The associated ´etal´e bundle Λ ( F ) (see Def. 1.15) is now given by the germs of functions in F induced by maximal filters of open intervals, or equivalently by ultrafilters on the complete booleanalgebra RO ( R ) given by the regular open subsets of R . In this case one can check that Λ ( F ) isidentified with the space of continuous functions from St( RO ( R )) to [0; 1] and f can be identifiedwith the global section of this ´etal´e space/bundle given by the continuous function: G (cid:55)→ if (1 , ∞ ) ∈ G if ( −∞ , ∈ G inf { / ( n + 1) : U ∩ (1 / ( n + 2); 1 / ( n + 1)) (cid:54) = ∅ for all U ∈ G } otherwise Our focus will be on the sheafification process which brings from F to Λ ( F ) , rather than theusual one bringing from F to Λ ( F ) . Loosely speaking the sheafification process we are interestedin will be able to comprise as global sections of the ´etal´e space of F all the pointwise limits ofcontinuous functions. To define it properly we need to pay some attention to analyze the notions ofpresheaf, sheaf, and ´etal´e bundle. Example 1 will be guiding our investigation of these concepts.C ONTENTS
Introduction 11. Basics on sheaves and boolean algebras 51.1. Complete boolean algebras and partial orders 51.2. Presheaves, Sheaves, ´etal´e spaces 71.3. Adjunctions 122. Boolean valued models 132.1. Ło´s theorem for boolean valued models and fullness 152.2. The mixing property and fullness 172.3. The category of boolean valued models for L OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 5
1. B
ASICS ON SHEAVES AND BOOLEAN ALGEBRAS
Almost all the notions introduced in this section are probably already known by most readers.We recall them also in order to fix the notation. Our reference texts for sheaves and categoriesare [6] and [7], our account of the theory of boolean valued models and boolean algebras is astreamline of the notes [14] available on the second author’s webpage. We invite the reader topay special attention to our analysis of the notion of ´etal´e space induced by a presheaf, because inthe case of interest to us our definitions are not exactly overlapping with those of [7]: there is asubstantial difference betwen the sheafification functors Λ and Λ briefly described in Example1; the functors coincide only for presheaves on topological spaces ( X, τ ) with X compact andextremally disconnected; Λ is carefully analyzed in [7], while Λ is not.1.1. Complete boolean algebras and partial orders.
Complete boolean algebras.
We recall some basic facts about complete boolean algebras. Given atopological space ( X, τ ) and A subset of X , Reg ( A ) is the interior of the closure of A in ( X, τ ) . A is regular open if A = Reg ( A ) . RO ( X ) denotes the family of regular open subsets of ( X, τ ) . CLOP ( X ) denotes the family of clopen subsets of ( X, τ ) . Clearly CLOP ( X ) ⊆ RO ( X ) .By the Stone Representation Theorem (see [14, Theorem 2.2.32]), every boolean algebra B isisomorphic to the boolean algebra CLOP (St( B )) of the clopen subsets of its Stone space St( B ) .The latter is the compact Hausdorff zero-dimensional space whose points are the ultrafilters on B ,topologized by taking as a base the family of sets N b := { G ∈ St( B ) : b ∈ G } for some b ∈ B .These sets are actually the clopen sets of St( B ) .Given a topological space ( X, τ ) , RO ( X ) is endowed of the structure of a complete boolean algebraby letting • RO ( X ) := ∅ , RO ( X ) := X , • for U, V ∈ RO ( X ) U ∨ V := Reg ( U ∪ V ) ,U ∧ V := U ∩ V, ¬ U := X \ Cl ( U ) . • for any family { U i : i ∈ I } ⊆ RO ( X ) , define (cid:95) i ∈ I U i := Reg (cid:32)(cid:91) i ∈ I U i (cid:33) , (cid:94) i ∈ I U i := Reg (cid:32)(cid:92) i ∈ I U i (cid:33) . A key result for us is the following:
Theorem 1.1.
A boolean algebra B is complete if and only if CLOP (St( B )) = RO (St( B )) .Proof. See for instance [14, Proposition 2.3.17]. (cid:3)
Remark . Let { N a : a ∈ A } be a family of basic open sets in the Stone space St( B ) of a booleanalgebra B such that (cid:87) A exists. Then (cid:83) a ∈ A N a is a dense open set in St( B ) if and only if (cid:87) A = 1 B .However, (cid:87) A = 1 B holds also in case (cid:83) a ∈ A N a (cid:54) = St( B ) . In case B is complete,Reg (cid:32) (cid:91) a ∈ A N a (cid:33) = N (cid:87) A . MORENO PIEROBON, MATTEO VIALE
Partial orders and boolean completions.
Assume ( P, ≤ ) is a pre-order, we endow it with thedownward topology: given X ⊆ P , ↓ X := { p ∈ P : there exists x ∈ X such that p ≤ x } .X ⊆ P is a down-set if X = ↓ X .The family DOWN ( P ) of the down-sets of P is a topology for P , the downward topology . Definition 1.3.
An embedding of pre-orders f : P → Q is an order and incompatibility preservingmap. f is dense if ran( f ) is dense in Q endowed with the downward topology. Definition 1.4.
A pre-order P is: • separative if, for every p, q ∈ P , p (cid:2) q implies that there exists r ∈ P such that r ≤ p and r ⊥ q (i.e. there is no s ∈ P such that s ≤ r and s ≤ q ); • upward complete if it admits suprema for all its non-empty subsets.Observe the following: • If B is a boolean algebra, then B + := B \ { B } with the induced order is a separativeordered set. • A boolean algebra B is complete if and only if B + is upward closed. • If P is a pre-order, we can always suriect it onto a separative pre-order with an order andincompatibility preserving map. Definition 1.5.
The boolean completion of a partial order ( P, ≤ ) is a complete boolean algebra B such that there exists a dense embedding e : P → B + .The boolean completion of a boolean algebra B is the boolean completion of the separative order B + . Theorem 1.6. If P is a partial order, then RO ( P ) is its boolean completion, as witnessed by themap e : P → RO ( P ) .p (cid:55)→ Reg ( ↓ p ) . Moreover the boolean completion of P is unique up to isomorphism.Proof. See for instance [14, Theorem 2.3.18]. (cid:3)
Fact 1.7.
Every boolean algebra B can be densely embedded in the complete boolean algebra RO (St( B )) via the Stone duality map b (cid:55)→ N b = { G ∈ St( B ) : b ∈ G } which identifies B with CLOP (St( B )) . The image is dense because the clopen sets form a base forthe topology on St( B ) .Moreover RO ( B + ) ∼ = RO (St( B )) via the unique extension of the map defined on B + by ↓ b (cid:55)→ N b . Definition 1.8.
Given a partial order ( P, ≤ ) , a filter on P is an upward closed subset G such thatfor any p, q ∈ G there exists r ∈ G refining p, q .The guiding example of filter on a partial order P is given by the family of open neighborhoods ofa point x ∈ X for ( X, τ ) a topological space and ( P, ≤ ) the partial given by non-empty open setsof ( X, τ ) .Observe that for any Hausdorff (actually T suffices) topological space ( X, τ ) any x ∈ X deter-mines a filter G x on O ( X ) given by its open neighborhoods. This filter in general is not a maximalfilter (consider the interval [0; 1] with the usual euclidean topology; the filter of open neighborhoodsof / is not maximal; it can be extended to a maximal filter containing [0; 1 / and to another OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 7 maximal filter containing (1 /
2; 1] ). Moreover if A is a base of non-empty open sets for X , G x ∩ A is also a filter on the partial order ( A , ⊆ ) , but in general a maximal filter G on A may not identifya point of X : the intersection of the open sets in G (or even of the closure of the open sets in G )may be empty (consider on R with the euclidean topology a maximal filter of open sets extending { ( a, + ∞ ) : a ∈ R } ). This identification of maximal filters of open sets to points of X is clearlyrelated to the existence of compactifications of X . If X is also compact any maximal filter G on ( A , ⊆ ) determines a unique point x of X with G ⊇ G x ∩ A , this point x is the unique element ofthe closed set (cid:84) { Cl ( U ) : U ∈ G } . However different maximal filters of open sets could determinethe same point of X (as shown by the case outlined above for / in [0; 1] ).In case ( X, τ ) is compact Hausdorff and -dimensional, X is the Stone space of its clopen setsalgebra, and points of X correspond bijectively to maximal filters on CLOP ( X ) [14, Proposition2.2.33].Note that these considerations hide a classical logic point of view: consider the Heyting algebraof open sets of a topological space ( X, τ ) in which negation is given by ¬ U = ( X \ Cl ( U )) ; amaximal filter G on this Heyting algebra consists of a maximal family of open subsets closed underfinite intersections (but possibly not to points of the space X , as we have already noted). Whatmatters more is that whenever U is an open subset of X , either U ∈ G or ¬ U in G , since exactlyone of the two sets has non-empty intersection with all the elements in G : otherwise there are A, B ∈ G with A ∩ U = ∅ and B ∩ ( X \ Cl ( U )) = ∅ . But then A ∩ B is open non-empty (beingin G ) and disjoint from the dense open set U ∪ ( X \ Cl ( U )) . Actually G is uniquely determinedby its restriction to the regular open sets of τ : A ∈ G if and only Reg ( A ) ∈ G , since Reg ( A ) ∩ U is non-empty if and ony if A ∩ U is non-empty for every open set U . In particular maximal filtersof open sets are uniquely determined by the boolean structure RO ( X, τ ) given by the topology τ ,and the logic determined by them validates the law of excluded middle.We will be interested in studying in detail the properties of (pre)sheaves and their associated ´etal´espaces defined on compact -dimensional spaces.1.2. Presheaves, Sheaves, ´etal´e spaces.
This section introduces familiar basic concepts of sheaftheory which in our set-up must be adressed with particular care; our aim is to elucidate and clarifythe distinctions and similarities between the two sheafification processes given by Λ ( F ) and Λ ( F ) briefly outlined in Example 1. We give a rather liberal definition of presheaf (in which weallow the domain of the presheaf to be an arbitrary partial order, rather than the family of non-emptyopen sets of a topological space) and a more restrictive definition of sheaf (in which we consideronly partial orders which are upward complete). Definition 1.9.
Let ( P, ≤ ) be a partial order. A dense open cover of p ∈ P is a downward closedset D of P (cid:22) p = { q ∈ P : q ≤ p } such that for all q ≤ p there is r ∈ D refining q . • A D -valued presheaf on P is a contravariant functor F from P to D . A presheaf on P isa Set -valued presheaf, in which case we will denote it as a P -presheaf to emphasize itsdomain. • A D -valued presheaf F is separated if for all p ∈ P and f, g ∈ F ( p ) , if the set of q ≤ p such that F ( q ≤ p )( f ) = F ( q ≤ p )( g ) is a dense cover of P below p , then f = g . • A D -valued presheaf F on P is a D -valued sheaf on P if it is separated and satisfies thefollowing condition for every p ∈ P and every dense cover D of p : Assume that for each q ∈ D , there exists f q ∈ F ( q ) such that, for s (cid:54) = q and r ≤ s, q , F ( r ≤ s )( f s ) = F ( r ≤ q )( f q ) , In the terminology of Grothendieck’s topologies, we are focusing on the presheaves defined by the dense topology.In the forcing terminology a dense cover of p is an open dense below p . MORENO PIEROBON, MATTEO VIALE then there exists f ∈ F ( p ) such that F ( q ≤ p )( f ) = f q for every q ∈ D refining p. We say that { f p : p ∈ D } is a matching family for D of elements of F and f is its amalga-mation or collation .Given a D -valued presheaf F on P , f ∈ F ( p ) , q ≤ p , we will often write f (cid:22) q instead of F ( q ≤ p )( f ) .Assume ( X, τ ) is a topological space. Then: • The partial order O ( X ) given by the non-empty open sets in τ is upward complete. In case F is a D -valued (pre)sheaf on O ( X ) , we just say that it is a D -valued (pre)sheaf on X . • The partial order
CLOP ( X ) + given by the clopen subsets of X is separative, and is a basefor τ if X is -dimensional. • The partial order RO ( X ) + given by the regular open subsets of X is separative and upwardcomplete, and is a base for τ if X is T . • X is extremally disconnected exactly when CLOP ( X ) = RO ( X ) is a base for X .We will be mostly interested in separated A -presheaves with A one of these four families of opensubsets of some topological space.Note that if P is upward complete D is an open cover of p ∈ P if and only if (cid:87) D = p and onecan reformulate the sheaf condition as: Whenever { f i : i ∈ I } is such that: • f i ∈ F ( p i ) for all i ∈ I , • (cid:87) i ∈ I p i = p , • for all i (cid:54) = j and r ≤ p i , p j f i (cid:22) r = f j (cid:22) r ,there is f ∈ F ( p ) such that f (cid:22) p i = f i for all i ∈ I . In the sequel we will mostly focus on RO ( X ) + -sheaves and use this equivalent characterization ofthe sheaf condition.Upward completeness of A is a convenient requirement we will need in order to deal withsheaves, hence our sheaves will be typically either O ( X ) -presheaves, or RO ( X ) + -presheaves. Definition 1.10.
Let F , G : P op → D be two D -valued presheaves on a separative partial order P .A morphism of presheaves from F to G is a natural transformation of functors α : F → G .It can be checked that, with this definition of morphisms, the families of D -valued (pre)sheaveson P are categories. In particular, an isomorphism of D -valued (pre)sheaves for P is an isomor-phism in the appropriate category. Moreover for P upward complete, the category of D -valuedsheaves on P is a faithful subcategory of the category of D -valued presheaves on P . See [7,Chapters II - III] for further details. Definition 1.11.
Let F : P op → D be a D -valued presheaf on a separative partial order P , and let G be a filter on P .Two elements f ∈ F ( p ) , g ∈ F ( q ) for p, q ∈ G have the same germ at G ( f ∼ G g ) if there exists r ∈ G refining p, q and such that f (cid:22) r = g (cid:22) r .The equivalence class [ f ] ∼ G is the germ of f at G .The stalk of F at G is F G := ( (cid:71) p ∈ G F ( p )) / ∼ G . OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 9
For a presheaf F : A op → Set with A a base on a topological space ( X, τ ) , we write F x for thestalk at G x ∩ A = { U ∈ A : x ∈ U } , and ∼ x rather than ∼ G x .We already remarked that ( X, τ ) is a compact -dimensional space if and only if X is the Stonespace of its clopen subsets according to τ . In this case there is a bijective correspondence betweenthe maximal filters on some topological base for ( X, τ ) and the points of X .A presheaf on a topological space X is naturally associated to its ´etal´e space over X . To constructit, we need one more definition. Definition 1.12.
Let X be a topological space. A bundle over X is a continuous map p : E → X (we will often say that E is a bundle over X , or X -bundle).A section of the bundle p : E → X is a continuous map s : X → E such that p ◦ s isthe identity of X . If U is an open set of X , a section of p over U is a section of the bundle p (cid:22) p − [ U ] : p − [ U ] → U .If p : E → X is a bundle and x ∈ X , the fiber at x is p − [ { x } ] .A bundle p : E → X is an ´etal´e space if it is a local homeomorphism in the following sense:for every e ∈ E there exists an open neighborhood V of e such that p [ V ] is open in X and p (cid:22) V : V → p [ V ] is an homeomorphism.For instance, a covering space is ´etal´e, while the converse in general does not hold.The standard procedure to associate an ´etal´e space to a presheaf on X is the following: Definition 1.13.
Let A be a base of open sets for a topological space ( X, τ ) and F : A op → Set be a presheaf. Λ F := (cid:97) x ∈ X F x = { [ f ] ∼ x : x ∈ X, f ∈ F ( U ) , x ∈ U ∈ A} ,p F :Λ F → X [ f ] ∼ x (cid:55)→ x. Each f ∈ F ( U ) determines ˙ f : U → Λ F x (cid:55)→ [ f ] ∼ x , which is a section of p F over U ( ˙ f is injective: if x (cid:54) = y , [ f ] ∼ x (cid:54) = [ f ] ∼ y , since F x , F y are disjoint). Λ F is topologized by taking as a base of open sets the family of all the image sets ˙ f [ U ] for U ∈ A and f ∈ F ( U ) .This topology on Λ F in general is not Hausdorff, but renders the map p F continuous and thesections ˙ f continuous and open. In particular, it is easy to see that p F : Λ F → X is an ´etal´e spaceover X .Conversely, any bundle p : E → X over a set X gives rise to a sheaf, the sheaf of sections of E . Itis defined as follows: Definition 1.14.
Let p : E → X be a continuous map between the topological spaces ( X, τ ) , ( E, σ ) .For U a non-empty open subset of X , let Γ p ( U ) := { s : U → E : s is a section of p over U } . For V ⊆ W non-empty open in X , the restriction Γ p ( W ) → Γ p ( V ) sends each section s over W to its restriction s (cid:22) V to V . Let us spend a few words about this correspondence between a presheaf F on a base A for ( X, τ ) and the sheaf Γ p F . Assume h : U → Λ F is a section in Γ p F ( U ) . Then, for every x ∈ U ,there is an open neighborhood x ∈ U x ⊆ U with U x ∈ A such that h ( x ) is the germ at x of someelement f x ∈ F ( U x ) , i.e. h ( x ) = [ f x ] ∼ x = ˙ f x ( x ) . Now, h is continuous and ˙ f x [ U x ] is a basicopen subset of Λ( F ) . By continuity, there exists an open subset V x ∈ A with x ∈ V x ⊆ U x suchthat h [ V x ] ⊆ ˙ f x [ U x ] . This means that h coincides with ˙ f x on V x . Therefore every section of thebundle Γ p F associated to F is locally a section induced by an element of F .Assume now the above occurs in case G : B op → Set is a sheaf with B a base for ( X, τ ) ; then ( B , ⊇ ) is upward complete, hence the family { V x : x ∈ U } has a supremum V ∈ B which mustcertainly include U since V ⊇ V x (cid:51) x for all x ∈ U . A key observation is that V (cid:54) = U iswell possible: assume B = RO ( X ) + is the family of non-empty regular open sets of a compactHausdorff space ( X, τ ) . Then it is well possible that (cid:83) { V x : x ∈ U } = U is open but not regularopen, hence V = Reg ( U ) is the supremum in B of { V x : x ∈ U } and includes strictly U . Thefamily { f x (cid:22) V x ∈ G ( V x ) : x ∈ U } is a family of pairwise compatible elements, and so being G asheaf on B there exists an amalgamation f ∈ G ( V ) such that ˙ f (cid:22) V x = ˙ f x (cid:22) V x = h (cid:22) V x for all x ∈ U . Hence, h = ˙ f (cid:22) U . We conclude that every section on some open U of the bundle Λ p G associated to a sheaf G : B op → Set is induced by some element of G ( V ) for some V ⊇ U in B .Now for any presheaf F : A op → Set with A a base for ( X, τ ) , Γ p F : O ( X ) op → Set is always asheaf and Γ p F (cid:22) A is isomorphic (in the category of A -presheaves) to F if F is a sheaf; however,in general if F is not a sheaf, Γ p F (cid:22) A op : A op → Set is not a sheaf for A , even when A is upwardcomplete. This is exactly what occurs in Example 1: A is the family of open intervals of R , F is thepresheaf of continuous functions g : A → R with bounded range and domain an open interval, and f = (cid:83) n ∈ N f n should correspond to a global section of Γ p F (cid:22) A ; this is not the case because f isdefined only on a dense subset of R and cannot be extended to a continuous function defined on allof R . This means that the matching family { f n : n ∈ N } cannot be collated inside A = RO ( R ) + .This matching family for A witnesses that Γ p F (cid:22) A is not a sheaf. On the other hand its collationon the union of the domains of each f n defines a partial section in the sheaf Γ p F defined on a denseopen subset of R .We want to investigate under which conditions on the presheaf F : A op → Set , the sheaf Γ p F issuch that Γ p F (cid:22) A remains a sheaf. There are two equivalent approaches to address this situation,and we follow both of them. • The first approach is to focus just on topological spaces where continuous functionsdefined on a dense subset admit a unique extension to a continuous function defined on thewhole space: this brings our attention to the notion of Stone-Cech compactification and ofextremally disconnected space. For extremally disconnected compact spaces ( X, τ ) , thesheafification process given by taking the ´etal´e space as described above gives rise to thedesired completion of any presheaf F : A op → Set independently of the basis A chosenfor τ (if X is extremally disconnected, then any continuous function with compact range This is exactly the case occurring in the covering given in the Example 1: the family of open sets { ( −∞ , , (1 , + ∞ ) } ∪ { (1 /n ; 1 / ( n − n ∈ N , n > } is a covering family in A with supremum (in A ) R = Reg (cid:16) ( −∞ ; 0) ∪ (1; ∞ ) ∪ (cid:91) { (1 /n ; 1 / ( n − n ∈ N : n > } (cid:17) ; on the other hand it is not a covering family in O ( R ) , since its union is a proper dense open subset of R (the key pointbeing that suprema are computed differently in O ( R ) and A ). OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 11 defined on a dense subset D of X admits a unique continuous extension to X , in particular X is the Stone-Cech compactification of any of its dense subsets). • The second approach is to give a slightly more general definition of ´etal´e space whichallow to recover rightaway the “right” sheafification of the presheaf F : A op → Set independently of the topological space ( X, τ ) for which A is a base. In this way we willobtain that the base set of the ´etal´e bundle is given by the extremally disconnected compactspace St( RO ( τ )) , and the global sections are given by the unique extensions to St( RO ( τ )) of continuous functions defined on some dense subset of X .So let us give a second definition of ´etal´e space. Definition 1.15.
Let ( P, ≤ ) be a separative partial order and F : P op → Set be a presheaf. Given G ultrafilter in St( RO ( P )) , we let ¯ G = G ∩ P and Λ F := (cid:97) G ∈ St( RO ( P )) F G = (cid:8) [ f ] ∼ ¯ G : G ∈ St( RO ( P )) , f ∈ F ( p ) , p ∈ ¯ G (cid:9) .p F :Λ F → St( RO ( P ))[ f ] ∼ ¯ G (cid:55)→ G .
Each f ∈ F ( U ) determines ˙ f : U → Λ F G (cid:55)→ [ f ] ∼ ¯ G , which is a section of p F over U .There are a few key observations to make: • For any presheaf F : P op → Set with P a separative poset Γ p F is a sheaf. • Assume A = RO ( σ ) + for some topological space ( Y, σ ) and F : A op → Set is a presheaf.Then: – Γ p F might have more stalks than Γ p F . Certain maximal filters on A do not correspondto points of Y (if Y is non-compact), or there could be points in Y whose filter ofopen neighborhoods admits different extensions to maximal filters (consider the caseof the filter of regular open neighborhoods of / for [0; 1] with euclidean topology). – A is upward complete and separative; moreover Γ p F is a sheaf whose global sectionsare uniquely determined by the collations of local sections whose union is defined juston some dense open subset of Y ; therefore Γ p F and Γ p F (cid:22) A are isomorphic. • Assume ( X, τ ) is compact extremally disconnected. Then St( RO ( X )) = St( CLOP ( X )) is homeomorphic to X via the map x (cid:55)→ G x = { U ∈ RO ( X, τ ) : x ∈ U } . In particular for A a base for ( X, τ ) , Γ p F is identified to Γ p F for any presheaf F : A op → Set via the identification of points of X with maximal filters on A . This identification ispossible because: – the space X is compact and -dimensional, hence maximal filters on A correspondsto points in X and the target space of the bundle maps p F and of p F is X ; – since X is extremally disconnected, the global sections of Λ F are uniquely determinedby the functions defined on some dense subset of X and which are continuous onthis dense subset, since any such function admits a unique extension to a continuousfunction on X .This gives that any global section of Λ ( F ) is also a global section of Λ ( F ) , and allowsto identify the two functors. • Let us come back to our introductory Example 1: recall that f = (cid:83) n ∈ N f n is continuouson the dense subset of R given by R \ ( { / ( n + 1) : n ∈ N } ∪ { } ) ; now G (cid:55)→ min { , inf { / ( n + 1) : U ∩ (1 / ( n + 2); 1 / ( n + 1)) (cid:54) = ∅ for all U ∈ G }} is a continuous extension of f to the compact extremally disconnected compact space St( RO ( R )) via any map which identifies R with a dense subset of St( RO ( R )) by assigningto some x ∈ R an ultrafilter extending the family of regular open neighborhoods of x .It is also not hard to check that the above map is the global section of Λ ( F ) given bythe collation of the local sections defined on N (1 /n ;1 / ( n − (for n > ) by G (cid:55)→ /n , on N ( −∞ ;0) by G (cid:55)→ , and on N (1;+ ∞ ) by G (cid:55)→ .We conclude this section recalling the following result, which subsumes most of the above observa-tions, extending to Λ the results obtained for Λ in [7, Chapter II.5, Theorem 1]: Theorem 1.16.
Assume ( X, τ ) is a Hausdorff topological space, A is a base of non-empty sets for X , F is an A -presheaf. Then: • The maps F ( U ) (cid:51) f (cid:55)→ ˙ f ∈ Γ p i F ( U ) for U ∈ A and i = 0 , define a natural transformation of functors η i F : F → Γ p i F . • For both i = 0 , the functors Γ p i F are sheaves . • Assume A is upward complete and F is a sheaf. Then η F : F → Λ ( F ) is always anequivalence of categories (equivalently denoted as an isomorphism of sheaves ). • Assume ( X, τ ) is compact and extremally disconnected, and A is a base for τ . Then Λ ( F ) = Λ ( F ) and Γ p F = Γ p F is isomorphic to the sheaf Γ p F (cid:22) A . In [7] the theorem is proved just in case A = O ( X ) and just for η F and Λ ( F ) ; but it isstraightforward to generalize the same proof to our more general set-up, by just modifying thedefinitions in the relevant cases. The observation we made so far allow to fill in the few missingdetails. Notation 1.17.
Uness otherwise specified, and to simplify our notation, we will from now on write η F , Λ F , Γ p F to denote respectively η F , Λ F , Γ p F . In essentially all cases of interest of us, we willbe in the set-up in which η F = η F , Λ F = Λ F , Γ p F = Γ p F . We will explicitly add the missingapexes if confusion can arise.1.3. Adjunctions.Definition 1.18.
Let C , D be two categories. An adjunction between them is a pair of functors (cid:104)F : C → D , G : D → C(cid:105) such that, for every c ∈ C , d ∈ D there exists a bijection Θ c,d : Hom D ( F ( c ) , d ) ∼ = Hom C ( c, G ( d )) , But Λ ( F ) (cid:54) = Λ ( F ) is well possible, and the sheaves Γ p F and Γ p F can be different, as outlined by Example 1. Note that if
A (cid:54) = O ( X ) , F is an A -presheaf, while η F is an O ( X ) -presheaf, hence they cannot literally beisomorphic objects being in different categories... OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 13 which is natural in c and d , i.e. such that the following diagram commutes, for every f : c → c (cid:48) in C and g : d → d (cid:48) in D : Hom D ( F ( c ) , d ) Θ c,d −−−−→ Hom C ( c, G ( d )) Hom D ( F ( f ) ,d ) (cid:121) (cid:121) Hom C ( f, G ( d )) Hom D ( F ( c (cid:48) ) , d ) −−−−→ Θ c (cid:48) ,d Hom C ( c (cid:48) , G ( d )) Hom D ( F ( c ) , d ) Θ c,d −−−−→ Hom C ( c, G ( d )) Hom D ( F ( c ) ,g ) (cid:121) (cid:121) Hom C ( c, G ( g )) Hom D ( F ( c ) , d (cid:48) ) −−−−→ Θ c,d (cid:48) Hom C ( c, G ( d (cid:48) )) where Hom D ( F ( f ) , d ) : h (cid:55)→ ( h ◦ F ( f )) and the others are defined in a similar way. F is the leftadjoint of the adjunction and G is its right adjoint.Since, for every c ∈ C , the functors Hom D ( F ( c ) , − ) , Hom C ( c, G ( − )) : D →
Set are isomor-phic, by Yoneda’s Lemma (see [6, Chapter III.2]) this isomorphism corresponds to a morphism η c : c → G ( F ( c )) , which is the image of Id F ( c ) in the isomorphism Hom D ( F ( c ) , F ( c )) ∼ = Hom C ( c, G ( F ( c ))) . Thus, we have a natural transformation η : Id C → G ◦ F , which is called the unit of the adjunction . In a similar way, we obtain a natural transformation ε : F ◦ G → Id D , the counit of the adjunction . Proposition 1.19. [6, Chapter IV.1, Theorem 2]
A pair of functors (cid:104)F : C → D , G : D → C(cid:105) is an adjunction if and only if there exist two natural transformations η : Id C → G ◦ F and ε : F ◦ G → Id D such that (1) Id G = G ε ◦ η G and Id F = ε F ◦ F η, where G ε : G ◦ F ◦ G → G ◦ Id D = G is the natural transformation obtained by the horizontalcomposition of the natural transformations Id G : G → G and η : F ◦ G → Id D , and we accordinglydefine η G : G → G ◦ F ◦ G , ε F : F ◦ G ◦ F → F , F η : F → F ◦ G ◦ F . Corollary 1.20.
Let (cid:104)F : C → D , G : D → C(cid:105) be an adjunction. Let C η be the full subcategory of C generated by the objects c ∈ C such that η c is an isomorphism. Define analogously D ε . Then theadjunction between F and G induces an equivalence of categories between C η and D ε .
2. B
OOLEAN VALUED MODELS
We will consider only boolean valued models M for relational languages as (at least for ourpurposes) there is no lack of generality in doing so, and our proofs are notationally smoother in thisset up.We start recalling basic definitions and facts about boolean valued models. Definition 2.1.
Let L = { R i : i ∈ I, c j : j ∈ J } be a relational language and let B be a booleanalgebra. A B -valued model M for L consists of:(1) a non-empty set M , called the domain of M ;(2) the boolean value of the equality symbol, which is a function = M : M → B (cid:104) σ,τ (cid:105) (cid:55)→ (cid:74) σ = τ (cid:75) M B ; (3) for each n -ary relational symbol R ∈ L , a function R M : M n → B (cid:104) σ , . . . ,σ n (cid:105) (cid:55)→ (cid:74) R ( σ , . . . , σ n ) (cid:75) M B ; (4) for each constant symbol c ∈ L , an element c M ∈ M .We require the following conditions to hold:(1) for every σ, τ, π ∈ M , (cid:74) σ = σ (cid:75) M B = 1 B , (cid:74) σ = τ (cid:75) M B = (cid:74) τ = σ (cid:75) M B , (cid:74) σ = τ (cid:75) M B ∧ (cid:74) τ = π (cid:75) M B ≤ (cid:74) σ = π (cid:75) M B ; (2) for every n -ary relational symbol R ∈ L and for every σ , . . . , σ n , τ , . . . , τ n ∈ M , (cid:16) n (cid:94) i =1 (cid:74) σ i = τ i (cid:75) M B (cid:17) ∧ (cid:74) R ( σ , . . . , σ n ) (cid:75) M B ≤ (cid:74) R ( τ , . . . , τ n ) (cid:75) M B . A B -valued model M is extensional if (cid:74) σ = τ (cid:75) M B = 1 B entails σ = τ for all σ, τ ∈ M .If no confusion can arise, we will avoid the supscript M and the subscript B . Moreover, for anelement σ , we will write equivalently σ ∈ M or σ ∈ M , identifying a boolean valued model M with its underlying set M .We now define the boolean valued semantics. By now, we have not required any condition on theboolean algebra B . For the definition of the semantic, however, we need to compute some infinitesuprema. For this reason, we will always assume in the sequel that formulae are assigned truthvalues in the boolean completion RO ( B + ) of B and we identify B + with a dense subset of thepartial order RO ( B + ) + . In most cases B is rightaway a complete booolean algebra in which case RO ( B + ) = B and this complication vanishes. Definition 2.2.
Let M = (cid:104) M, = M , R M i : i ∈ I (cid:105) be a B -valued model for the relational language L = { R i : i ∈ I } . We evaluate the formulae without free variables in the extended language L M := L ∪ { c σ : σ ∈ M } by maps with values in the boolean completion RO ( B + ) of B asfollows: • (cid:74) c σ = c τ (cid:75) M RO ( B + ) := (cid:74) σ = τ (cid:75) M B and (cid:74) R ( c σ , . . . , c σ n ) (cid:75) M RO ( B + ) := (cid:74) R ( σ , . . . , σ n ) (cid:75) M B ; • (cid:74) ϕ ∧ ψ (cid:75) M RO ( B + ) := (cid:74) ϕ (cid:75) M RO ( B + ) ∧ (cid:74) ψ (cid:75) M RO ( B + ) ; • (cid:74) ¬ ϕ (cid:75) M RO ( B + ) := ¬ (cid:74) ϕ (cid:75) M RO ( B + ) ; • (cid:74) ∃ xϕ ( x, c σ , . . . , c σ n ) (cid:75) M RO ( B + ) := (cid:87) τ ∈ M (cid:74) ϕ ( c τ , c σ , . . . c σ n ) (cid:75) M RO ( B + ) . If ϕ ( x , . . . , x n ) is any L -formula with free variables x , . . . , x n and ν is an assignment, we define (cid:74) ν ( ϕ ( x , . . . , x n )) (cid:75) M RO ( B + ) := (cid:113) ϕ ( c ν ( x ) , . . . , c ν ( x n ) ) (cid:121) M RO ( B + ) .We will often write ϕ ( σ , . . . , σ n ) rather than ϕ ( c σ , . . . , c σ n ) and (cid:74) ϕ ( τ , . . . , τ n ) (cid:75) rather than (cid:74) ϕ ( τ , . . . , τ n ) (cid:75) M RO ( B + ) if no confusion can arise.By induction on the complexity of L -formulae, it is possible to show the following: Fact 2.3.
For every σ , . . . , σ n , τ , . . . , τ n ∈ M and for every L -formula ϕ ( x , . . . , x n ) withdisplayed free variables, the following holds: (2) (cid:16) n (cid:94) i =1 (cid:74) σ i = τ i (cid:75) (cid:17) ∧ (cid:74) ϕ ( σ , . . . , σ n ) (cid:75) ≤ (cid:74) ϕ ( τ , . . . , τ n ) (cid:75) . OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 15
Definition 2.4.
A sentence ϕ in the language L is valid in a boolean valued model M for L if (cid:74) ϕ (cid:75) = 1 B .A theory T is valid in M (equivalently, M is a B -model for T ) if every axiom of T is valid in M . Theorem 2.5 (Soundness and Completeness) . Let L be a relational language. An L -formula ϕ isprovable syntatically by an L -theory T if and only if, for every boolean algebra B and for every B -valued model M for T , (cid:74) ν ( ϕ ) (cid:75) M B = 1 B for every assignment ν taking values in M .Proof. See for instance [14, Theorems 4.1.5 and 4.1.8]. (cid:3)
Ło´s theorem for boolean valued models and fullness.Definition 2.6.
Let B be a boolean algebra and let M = (cid:104) M, = M , R M i : i ∈ I (cid:105) be a B -valuedmodel for the relational language L = { R i : i ∈ I } . Let F be a filter of B . The quotient M / F isthe B / F -valued model for L defined as follows: • the domain is M/ F := { [ σ ] F : σ ∈ M } , where [ σ ] F := { τ ∈ M : (cid:74) τ = σ (cid:75) ∈ F } ; • if R ∈ L is an n -ary relational symbol and [ σ ] F , . . . , [ σ n ] F ∈ M/ F , (cid:74) R ([ σ ] F , . . . , [ σ n ] F ) (cid:75) M / F B / F := (cid:104) (cid:74) R ( σ , . . . , σ n ) (cid:75) M B (cid:105) F ∈ B / F . It is easy to check that this quotient is well-defined. In particular, if G is an ultrafilter, the quotient M / G is a two-valued Tarski structure for L . On the other hand if B is a boolean algebra and F is afilter on B , it can happen that B is complete while B / F is not, and conversely. In particular it is notclear how the semantics of formulae with quantifiers is affected by the quotient operation. We nowaddress this problem. Definition 2.7.
Given a first order signature L , a B -valued model M for L is • well behaved if: for all L -formulae φ ( x , . . . , x n ) and τ , . . . , τ n ∈ M (cid:74) φ ( τ , . . . , τ n ) (cid:75) M is in B ; • full if for all ultrafilters G on B , all L -formulae φ ( x , . . . , x n ) and all τ , . . . , τ n ∈ MM / G | = φ ([ τ ] G , . . . , [ τ n ] G ) if and only if (cid:74) φ ( τ , . . . , τ n ) (cid:75) M ∈ G. Notice that if B is complete B ∼ = RO ( B + ) hence any B -valued model M is automaticallywell-behaved. But being well-behaved does not require B to be complete, just to be able to computeall the suprema and infima required by the satisfaction clauses for ∃ , ∀ . On the other hand foratomic L M -formulae the condition expressing fullness is automatic by definition. The uniquedelicate case occurs for L M -formulae in which the principal connective is a quantifier, for exampleof the form ∃ xφ ( x ) , in which case the fullness conditions for the formula amounts to ask that sup σ ∈M (cid:74) φ ( σ ) (cid:75) M is actually a finite supremum, as we will see. We will investigate this propertyin many details in the last two sections of this paper. For the moment letting a L M -formula φ ( x , . . . , x n ) be F -full for a filter F if and only if (cid:74) φ ([ τ ] G , . . . , [ τ n ] G ) (cid:75) M / F = 1 B / F ⇐⇒ (cid:74) φ ( τ , . . . , τ n ) (cid:75) M ∈ F, we automatically have that the family of F -full L M -formulae includes the atomic formulae andis closed under conjuctions for any filter F , and under boolean combinations for any ultrafilter F .Another key observation is that if F is a filter and m (cid:95) i =1 (cid:74) φ ( σ i ) (cid:75) M = (cid:74) ∃ x φ ( x ) (cid:75) M = (cid:95) τ ∈M (cid:74) φ ( τ ) (cid:75) M , then m (cid:95) i =1 (cid:74) φ ([ σ i ] F ) (cid:75) M / F = (cid:74) ∃ x φ ( x ) (cid:75) M / F , since m (cid:95) i =1 [ (cid:74) φ ( σ i ) (cid:75) M ] F = (cid:104) m (cid:95) i =1 (cid:74) φ ( σ i ) (cid:75) M (cid:105) F ≥ F [ (cid:74) φ ( τ ) (cid:75) M ] F holds for all τ ∈ M . Hence all full-formulae pass to quotients.The notion of full B -valued model displays its full power in the following result: Theorem 2.8 (Ło´s Theorem for boolean valued models) . Let M be a well behaved B -valued modelfor the signature L . The following are equivalent: • M is full. • for all L M -formulae φ ( x , . . . , x n ) and all τ , . . . , τ n ∈ M there exist σ , . . . , σ m ∈ M such that (cid:95) τ ∈M (cid:74) φ ( τ, τ , . . . , τ n ) (cid:75) = m (cid:95) i =1 (cid:74) φ ( σ i , τ , . . . , τ n ) (cid:75) Proof.
We sketch the proof for the case of existential formulae.First, assume M to be full: M / G | = ∃ xψ ( x, [ τ ] G , . . . , [ τ n ] G ) ⇔ (cid:74) ∃ xψ ( x, τ , . . . , τ n ) (cid:75) M ∈ G. Thus for every G such that (cid:74) ∃ xψ ( x, τ , . . . , τ n ) (cid:75) M ∈ G there exists σ G ∈ M such that M / G | = ψ ([ σ G ] G , [ τ ] G , . . . , [ τ n ] G ) . Again using the hypothesis, we obtain that for every G ∈ N (cid:74) ∃ xψ ( x,τ ,...,τ n ) (cid:75) there exists σ G ∈ M such that G ∈ N (cid:74) ψ ( σ G ,τ ,...,τ n ) (cid:75) , that is: N (cid:74) ∃ xψ ( x,τ ,...,τ n ) (cid:75) ⊆ (cid:91) σ ∈M N (cid:74) ψ ( σ,τ ,...,τ n ) (cid:75) . By compactness, N (cid:74) ∃ xψ ( x,τ ,...,τ n ) (cid:75) = (cid:83) mi =1 N (cid:74) ψ ( σ i ,τ ,...,τ n ) (cid:75) , which is our thesis.Conversely, assume that for every formula φ , τ , . . . , τ n ∈ M there exist σ , . . . , σ m ∈ M suchthat (cid:95) τ ∈M (cid:74) φ ( τ, τ , . . . , τ n ) (cid:75) = m (cid:95) i =1 (cid:74) φ ( σ i , τ , . . . , τ n ) (cid:75) . By induction on the complexity of the formulae we prove that M is full. Let us consider only thenon-trivial case: ψ ( x , . . . , x n ) = ∃ xφ ( x, x , . . . , x n ) . Then M /G | = ∃ xφ ( x, [ τ ] G , . . . , [ τ n ] G ) ⇔M /G | = φ ([ σ ] G , [ τ ] G , . . . , [ τ n ] G ) for some σ ∈ M⇔ (cid:74) φ ( σ, τ , . . . , τ n ) (cid:75) ∈ G for some σ ∈ M⇒ (cid:74) ∃ xφ ( x, τ , . . . , τ n ) (cid:75) ∈ G. Conversely, if (cid:74) ∃ xφ ( x, τ , . . . , τ n ) (cid:75) ∈ G , since (cid:74) ∃ xφ ( x, τ , . . . , τ n ) (cid:75) = (cid:87) mi =1 (cid:74) φ ( σ i , τ , . . . , τ n ) (cid:75) ,there exists i ∈ { , . . . , m } such that (cid:74) φ ( σ i , τ , . . . , τ n ) (cid:75) ∈ G . By inductive hypothesis M /G | = φ ([ σ i ] G , [ τ ] G , . . . , [ τ n ] G ) and so M /G | = ∃ xφ ( x, [ τ ] G , . . . , [ τ n ] G ) . (cid:3) One recovers the standard Ło´s theorem for ultraproducts observing that any ultraproduct (cid:81) i ∈ I N i of L -structures is a full P ( I ) -valued model for L , and that the semantic of the quotient structure (cid:0)(cid:81) i ∈ I N i (cid:1) /G for G ultrafilter on P ( I ) is exactly ruled by the conditions set up in the abovetheorem. OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 17
To appreciate the fact that fullness is not automatic (and as we will see is a slight weakening ofthe notion of sheaf) we will give now a counterexample to fullness.
Example 2.9.
Let R be equipped with the Lebesgue measure µ L . Let M L ( R ) be the family ofLebesgue measurable subsets of R , which is obviously a boolean algebra. Let Null be the idealof null subsets and define the measure algebra as MALG := M L ( R ) / Null . It can be checked that
MALG is a complete boolean algebra, whose countable suprema are easily computed by taking theequivalence class of the union of representatives. For further details we refer to [14, Section 3.3.1].We make the set C ω ( R ) of analytic functions R → R the domain of a MALG -valued model for thelanguage L = { <, C } as follows. For f, g ∈ C ω ( R ) , let (cid:74) f = g (cid:75) := (cid:2) { r ∈ R : f ( r ) = g ( r ) } (cid:3) Null (cid:74) f < g (cid:75) := (cid:2) { r ∈ R : f ( r ) < g ( r ) } (cid:3) Null (cid:74) C ( f ) (cid:75) := (cid:95) { [ U ] Null ∈ MALG : f (cid:22) U is constant } . A key property of analytic functions in one variable is that they are constant on a non discrete set ifand only if they are everywhere constant.The reader can check that ( C ω ( R ) , (cid:74) − = − (cid:75) , (cid:74) − < − (cid:75) , (cid:74) C ( − ) (cid:75) ) is a MALG -valued model.Fix f ∈ C ω ( R ) and consider the sentence φ := ∃ y ( f < y ∧ C ( y )) . Let c r denote the map R → R which is constantly r and let a n := sup( f (cid:22) ( n − , n )) + 1 . Then (cid:74) ∃ y ( f < y ∧ C ( y )) (cid:75) = (cid:95) g ∈ C ω ( R ) (cid:74) f < g ∧ C ( g ) (cid:75) ≥ (cid:95) r ∈ R (cid:74) f < c r (cid:75) ∧ (cid:74) C ( c r ) (cid:75) = (cid:95) r ∈ R (cid:74) f < c r (cid:75) ≥ (cid:95) n ∈ Z (cid:74) f < c a n (cid:75) ≥ (cid:95) n ∈ Z [( n − , n )] Null = [ R ] Null . In particular, taking f = Id R , we have (cid:74) ∃ y ( Id R < y ∧ C ( y ) (cid:75) C ω ( R ) = 1 MALG . However, let G be an ultrafilter in MALG which extends the family { [( n, + ∞ )] Null : n ∈ Z } . Then,for every r ∈ R , (cid:74) ¬ ( Id R < c r ) (cid:75) = ( r, + ∞ ) ∈ G, therefore C ω ( R ) /G | = ¬∃ y ([ Id R ] G < y ∧ C ( y )) .2.2. The mixing property and fullness.
The mixing property gives us a sufficient condition forhaving the fullness property which is, usually, easier to check; as we will see later the mixingproperty characterizes those boolean valued models which are sheaves.
Definition 2.10.
Let κ be a cardinal, L be a first order language, B a κ -complete boolean algebra, M a B -valued model for L . • M satisfies the κ -mixing property if for every antichain A ⊂ B of size at most κ , and forevery subset { τ a : a ∈ A } ⊆ M , there exists τ ∈ M such that a ≤ (cid:74) τ = τ a (cid:75) for every a ∈ A . • M satisfies the < κ -mixing property if it satisfies the λ -mixing property for all cardinals λ < κ . • M satisfies the mixing property if it satisfies the | B | -mixing property.In [2] models with the < ω -mixing property are called models which admit gluing .Whether a B -valued model M has the mixing property depends only on the interpretation of theequality symbol by (cid:74) − = − (cid:75) M . Proposition 2.11.
Let B be a complete boolean algebra and let M be a B -valued model for L .Assume that M satisfies the κ -mixing property for some κ ≥ min {| B | , | M |} . Then M is full.Proof. Fix a formula φ ( x, y , . . . , y n ) in L and σ , . . . , σ n ∈ M . Fix moreover an enumeration (cid:104) τ i : i ∈ I (cid:105) of M . Since (cid:74) ∃ xϕ ( x, σ , . . . , σ n ) (cid:75) = (cid:87) i ∈ I (cid:74) ϕ ( τ i , σ , . . . , σ n ) (cid:75) , we can refine thefamily { (cid:74) ( ϕ ( τ i , σ , . . . , σ n ) (cid:75) : i ∈ I } to an antichain { a j : j ∈ J } as follows: let J := I \ i ∈ I : (cid:74) ϕ ( τ i , σ , . . . , σ n ) (cid:75) \ (cid:95) j min J , a i := (cid:74) ϕ ( τ i , σ , . . . , σ n ) (cid:75) \ (cid:95) J (cid:51) j
Example 2.13.
We stick in this example to the standard terminology on forcing as can be foundin Jech’s [3] or Bell’s [1] books. Let M ∈ V be a countable transitive model of (a sufficientlylarge fragment of) ZFC and let B ∈ M be an infinite boolean algebra which M models to be OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 19 complete. Consider the B -valued model ( M B , (cid:74) − = − (cid:75) M B , (cid:74) − ∈ − (cid:75) M B ) , which is a definableclass in ( M, ∈ ) (and a countable set in V ). Now, it can be proved (see [1, Mixing Lemma 1.25])that M | = ( M B , (cid:74) − = − (cid:75) M B ) has the mixing property In particular, M | = ( M B , (cid:74) − = − (cid:75) M B , (cid:74) − ∈ − (cid:75) M B ) is full . The notion of being full is absolute between transitive models of set theory, therefore ( V, ∈ ) | = ( M B , (cid:74) − = − (cid:75) M B , (cid:74) − ∈ − (cid:75) M B ) is full . For a complete discussion of the fullness of M B we refer to [14, Sections 5.1.2 and 5.2] and inparticular to [14, Lemma 5.2.1].We will now show that, in V , ( M B , (cid:74) − = − (cid:75) M B , (cid:74) − ∈ − (cid:75) M B ) does not satisfy the mixingproperty. Being M countable, for sure there exists a maximal antichain A of B ⊆ M suchthat A ∈ V \ M . We claim that this antichain witnesses the fact that the B -valued model ( M B , (cid:74) − = − (cid:75) M B , (cid:74) − ∈ − (cid:75) M B ) does not have the mixing property.By contradiction, let τ ∈ M B be such that (cid:74) τ = ˇ a (cid:75) ≥ a for every a ∈ A . Thus A is definable in M by the formula ϕ ( x, B ) where ϕ ( x, B ) := ( x is an antichain of B ) ∧ ( (cid:95) B x = 1 B ) ∧ ∀ y ( y ∈ x ↔ ( y ∈ B ∧ (cid:74) τ = ˇ y (cid:75) ≥ y )) . The antichain A is the unique solution of the formula ϕ . Indeed, assume C to be another solutionof ϕ . Let c ∈ C \ A . Then, by the maximality of A , there exists a ∈ A such that a ∧ c > . Being A and C solutions of ϕ , we have B < a ∧ c ≤ (cid:74) τ = ˇ a (cid:75) ∧ (cid:74) τ = ˇ c (cid:75) ≤ (cid:74) ˇ a = ˇ c (cid:75) = 0 B , where (cid:74) ˇ a = ˇ c (cid:75) = 0 B since M | = ( a (cid:54) = c ) and so M | = (cid:74) ¬ (ˇ a = ˇ c ) (cid:75) M B B = 1 B . Being A a subset of B definable in M , by the Axiom of Comprehension (which holds in the ZFC -model M ) we have that A ∈ M , against our assumption.2.3. The category of boolean valued models for L .Definition 2.14. Fix a language L . Let M be a B -valued model for L and M be a B -valuedmodel for L , where B and B are boolean algebras. A morphism of boolean valued models for L is a pair (cid:104) Φ , i (cid:105) where: • i : B → B is a morphism of boolean algebras; • Φ : M → M is an i -morphism , that is: for every n -ary relational symbol R in thelanguage and for every τ , . . . , τ n ∈ M , i ( (cid:74) R ( τ , . . . , τ n ) (cid:75) M B ) ≤ (cid:74) R (Φ( τ ) , . . . , Φ( τ n )) (cid:75) M B ,i ( (cid:74) τ = τ (cid:75) M B ) ≤ (cid:74) Φ( τ ) = Φ( τ ) (cid:75) M B . If in both these equations equality holds, we call Φ an i -embedding . If M proves that M B is full, M proves that a certain element (cid:87) mi =1 (cid:74) ϕ ( a i ) (cid:75) M B ∈ B is equal to the element (cid:74) ∃ xϕ ( x ) (cid:75) M B ∈ B , and this equality between elements of B must be true also in V . If i is an isomorphism, Φ is an i -embedding, and for all τ ∈ M there is σ ∈ M such that (cid:74) Φ( σ ) = τ (cid:75) B = 1 B , (cid:104) Φ , i (cid:105) is an isomorphism of boolean valued models .Clearly, if B = B = 2 , an Id -morphism and an Id -embedding are exactly a morphism andan embedding between two different Tarski models, respectively.It is more delicate to define the notion of elementary morphism. We defer this task to a later stage. Definition 2.15.
Let L be a relational language. Mod Bool L is the following category: • objects are pairs (cid:104)M , B (cid:105) , where B is a complete boolean algebra and M is a B -valuedmodel for L ; • if (cid:104)M , B (cid:105) and (cid:104)N , C (cid:105) are objects, a morphism between them is a morphism of booleanvalued models (cid:104) Φ , i (cid:105) : M → N ; • the composition of morphisms (cid:104) Φ , i (cid:105) : (cid:104)M , B (cid:105) → (cid:104)M , B (cid:105) and (cid:104) Ψ , j (cid:105) : (cid:104)M , B (cid:105) →(cid:104)M , B (cid:105) is the morphism (cid:104) Ψ ◦ Φ , j ◦ i (cid:105) : (cid:104)M , B (cid:105) → (cid:104)M , B (cid:105) .For a given boolean algebra B , Mod B L is the subcategoy of Mod Bool L given by B -valued models andmorphisms (cid:104) Φ , i (cid:105) between them with i the identity on B .3. T HE PRESHEAF STRUCTURE OF A BOOLEAN VALUED MODEL
We now describe the presheaf structure arising naturally from the definition of boolean valuedmodel. To this extent, let B be a boolean algebra and let M be a B -valued model. We associateto M a presheaf on St( B ) given by a contravariant functor F M from CLOP (St( B )) + ∼ = B + toMod B L .For every b ∈ B , let F b be the filter generated by b , and define F M ( N b ) := M / F b , which is a B / F b -valued model.We also have to say what the F M -images of morphisms are. The morphisms in CLOP (St( B )) arethe inclusions N b ⊆ N c . Let us now assume that N b ⊆ N c , i.e. b ≤ c . Then F c ⊇ F b , therefore thenatural map i M bc : M / F c → M / F b [ τ ] F c (cid:55)→ [ τ ] F b defines a morphism from the B / F c valued model M / F c onto the B / F b -valued model M / F b .It will be convenient to resume this information in the following definition: Definition 3.1.
Given a compact -dimensional space ( X, τ ) and a CLOP ( X ) -valued model M ,its associated presheaf F M : ( CLOP ( X ) + ) op → Mod
Bool L is such that • F M ( U ) = M / F U for any U ∈ CLOP ( X ) + ; • F M ( U ⊆ V ) for U ⊆ V ∈ CLOP ( X ) + is the map i M UV : M / F V → M / F U [ τ ] F V (cid:55)→ [ τ ] F U . Let us stress the fact that until now we have built nothing else than a separated presheaf. Finally,we want to describe the stalk of F M at G ∈ St( B ) . Translating the definition of stalk in our setting,we have that ( F M ) G := (cid:16) (cid:71) b ∈ G M / F b (cid:17) / ∼ G , where [ σ ] F b ∼ G [ τ ] F c if and only if [ σ ] F b ∧ c = [ τ ] F b ∧ c . OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 21
Now observe that for each σ ∈ M and b ∈ G [ σ ] F b = { τ ∈ M : (cid:74) σ = τ (cid:75) ≥ b } . In particular the map Θ G : σ (cid:55)→ [[ σ ] F B ] ∼ G defines a surjection of M onto ( F M ) G with Θ G ( σ ) = Θ G ( τ ) ⇐⇒ [[ σ ] F B ] ∼ G = [[ τ ] F B ] ∼ G ⇐⇒ [ σ ] F b = [ τ ] F b for some b ∈ G ⇐⇒ (cid:74) σ = τ (cid:75) ≥ b for some b ∈ G ⇐⇒ (cid:74) σ = τ (cid:75) ∈ G. This shows that Θ G ( σ ) = Θ G ( τ ) if and only if M / G | = [ σ ] G = [ τ ] G .In particular we can identify the stalk ( F M ) G with the Tarski quotient M / G of M at G .Moreover notice that, identifying a boolean valued model with its underlying set, we can consider F M as a Set -valued presheaf ( CLOP ( X ) + ) op → Set .Let us now analyze the ´etal´e space E M := Λ ( F M ) canonically associated to the presheaf F M . Itis the disjoint union of the fibers ( F M ) G for G ∈ St( B ) , which by our observation above amountsto a disjoint union of the Tarski structures M / G as G varies in St( B ) , that is: E M = { [ σ ] G : σ ∈ M, G ∈ St( B ) } . Equivalently, we can say that E M = {(cid:104) σ, G (cid:105) : σ ∈ M and G ∈ St( B ) } / R M , where R M is the equivalence relation such that (cid:104) σ, G (cid:105) R M (cid:104) τ, H (cid:105) if and only if G = H and (cid:74) σ = τ (cid:75) ∈ G .Moreover, the projection map p : E M → St( B ) sends each (cid:104) σ, G (cid:105) = [ σ ] G to G .A base for the topology of E M is the family B := { ˙ σ [ N b ] = { [ (cid:104) σ, G (cid:105) ] R M : b ∈ G } : σ ∈ M, b ∈ B } . This topology is readily Hausdorff. Indeed, if (cid:104) σ, G (cid:105) (cid:54) = (cid:104) τ, H (cid:105) , either G (cid:54) = H or G = H and ¬ (cid:74) σ = τ (cid:75) ∈ G . In the first case, being St( B ) Hausdorff, there exists an open neighborhood N b of G and an open neighborhood N c of H which are disjoint. Then the basic open sets ˙ σ [ N b ] and ˙ τ [ N c ] separate (cid:104) σ, G (cid:105) from (cid:104) τ, H (cid:105) . Otherwise, if G = H , we have that [ σ ] G (cid:54) = [ τ ] G . Let b := (cid:74) σ (cid:54) = τ (cid:75) = ¬ (cid:74) σ = τ (cid:75) ∈ G . Then, ˙ σ [ N b ] and ˙ τ [ N b ] are disjoint open neighborhoods of (cid:104) σ, G (cid:105) and (cid:104) τ, G (cid:105) , respectively. Moreover, for any open U ⊆ St( B ) and σ ∈ M , the local section ˙ σ (cid:22) U isopen and injective and so it is an homeomorphism on its image.3.1. An adjuction between boolean valued models and separated presheaves.
For the momentwe will consider boolean valued models for the language with only the equality symbol, because themain property of these structures we will now focus on is the mixing property, which is formulatedonly in terms of the boolean value of equality (in the last section we will again turn our attentionalso on boolean valued models for richer first order relational languages). However the readercan easily verify that the results of this section immediately generalize to arbitrary boolean valuedmodels for a first order signature .What we have done so far can be summarized in the following observation: let X be a compact The point is that we should spend some space to define rigorously the notion of algebraic presheaf for a givensignature τ , and then also to verify that the required functoriality properties are maintained by R X also for the symbolsof τ other than equality; for the moment we do not have any specific use for this more general framework, hence werefrain from getting into details. Hausdorff -dimensional topological space. Identifying X up to homeomorphism with the Stonespace of the boolean algebra of its clopen sets, we have defined a functor L X : Mod
CLOP ( X ) → S-Presh ( CLOP ( X ) + , Set ) , where: • Mod
CLOP ( X ) is the category of CLOP ( X ) -valued model for the empty language { = } andmorphisms between them, with the condition that, if (cid:104) Φ , i (cid:105) : M → N is a morphism, i isthe identity map Id CLOP ( X ) , while the morphism Φ need not be injective; • S-Presh ( CLOP ( X ) + , Set ) is the category of separated Set -valued presheaves on the partialorder
CLOP ( X ) \ {∅} and natural transformations between them; • if M is an object in Mod CLOP ( X ) , its image under L X is the presheaf L X ( M ) := F M ; • if (cid:104) Φ , Id (cid:105) : M → N is a morphism in Mod
CLOP ( X ) , then its image under L X is the naturaltransformation L X (Φ) = { L X (Φ) U } U ∈ CLOP ( X ) op such that L X (Φ) U := Φ / F U : M / F U → N / F U [ τ ] F U (cid:55)→ [Φ( τ )] F U , where F U is the filter given by the clopen supersets of U .It is left to the reader to check that L X (Φ) is well-defined and that the required composition andcommutativity laws hold, making L X indeed a functor.A natural question arises: can we find an inverse for L X for a given -dimensional Hausdorffcompact space ( X, τ ) ? Definition 3.2. R X : Presh ( CLOP ( X ) + , Set ) → Mod RO ( X ) is the functor defined by: • if F is any presheaf R X ( F ) := M F is the RO ( X ) -valued model whose domain is F ( X ) and such that, for f, g ∈ F ( X ) , (cid:74) f = g (cid:75) M F := Reg (cid:16)(cid:91) { U ∈ CLOP ( X ) : F ( U ⊆ X )( f ) = F ( U ⊆ X )( g ) } (cid:17) ; • if α : F → G is a natural transformation of sheaves, R X ( α ) := α X , where α X : F ( X ) →G ( X ) is the map defined by α at X .We spend a few words to justify the fact that R X is a well defined functor on the whole class of CLOP ( X ) + -presheaves.First of all M F is a RO ( X ) -valued model, because the functorial properties of a presheaf corre-spond naturally to the constraints imposed on the interpretation of the equality symbol given by (cid:74) f = g (cid:75) M F .We also need to check that R X ( α ) is an Id RO ( X ) -morphism of RO ( X ) -valued models. In particular,if f, g ∈ F ( X ) , we want that (cid:74) f = g (cid:75) M F ≤ (cid:74) α X ( f ) = α X ( g ) (cid:75) M G . Note that R X can be defined on arbitrary CLOP ( X ) + -presheaves, however the pairs (cid:104) L X , R X (cid:105) will define anadjunction only when R X is restricted to the class of separated CLOP ( X ) + -presheaves and X is compact and extremallydisconnected (see Thm. 3.3). OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 23
Remember that (cid:74) f = g (cid:75) M F = (cid:87) { b ∈ B : F ( N b ⊆ X )( f ) = F ( N b ⊆ X )( g ) } . Now, F ( N b ⊆ X )( f ) = F ( N b ⊆ X )( g ) ⇒ α N b ( F ( N b ⊆ X )( f )) = α N b ( F ( N b ⊆ X )( g )) ⇔ G ( N b ⊆ X )( α X ( f )) = G ( N b ⊆ X )( α X ( g )) ⇒ b < (cid:74) α X ( f ) = α X ( g ) (cid:75) M G . Moreover the second implication is an equivalence if α N b is injective. That is: if F and G are RO ( X ) + -presheaves, R X ( α ) is an Id CLOP ( X ) -embedding if each component of α is injective.Notice that R X ( F ) is an extensional boolean valued model for any presheaf F .Now if ( X, τ ) is compact extremally disconnected CLOP ( X ) = RO ( X ) , and R X ◦ L X is theidentity functor on the CLOP ( X ) -models that are extensional: let M be an extensional CLOP ( X ) -valued model; then ( R X ◦ L X )( M ) = R X ( F M ) = F M ( X ) = M / F X = M and, if τ, σ ∈ M , (cid:74) σ = τ (cid:75) ( R X ◦ L X )( M ) = Reg (cid:16)(cid:91) { U ∈ CLOP ( X ) : F M ( U ⊆ X )( τ ) = F M ( U ⊆ X )( σ ) } (cid:17) = (cid:95) { U ∈ CLOP ( X ) : F M ( U ⊆ X )( τ ) = F M ( U ⊆ X )( σ ) } = (cid:95) (cid:110) U ∈ CLOP ( X ) : U ≤ (cid:74) σ = τ (cid:75) M (cid:111) = (cid:74) σ = τ (cid:75) M . In general, for every
M ∈
Mod
CLOP ( X ) , ( R X ◦ L X )( M ) = M / F X . Moreover, if Φ :
M → N isa morphism of
CLOP ( X ) -valued models, ( R X ◦ L X )(Φ) = R X ( { Φ / F U } U ∈ CLOP ( X ) ) = Φ / F X . Summing up our considerations, we have all the elements to prove the following theorem : Theorem 3.3.
Assume ( X, τ ) is compact and extremally disconnected. Then the pair (cid:104) L X , R X (cid:105) is an adjuction between the categories S-Presh ( CLOP ( X ) + , Set ) of separated presheaves andMod CLOP ( X ) with L X being the left adjoint.Proof. By Proposition 1.19, we have only to find the unit and the counit of the adjunction and thento verify the identities (1).First, we define the unit η : Id Mod
CLOP ( X ) → R X ◦ L X . We have simply to take, for M ∈
Mod
CLOP ( X ) , η M : M → M / F X .τ (cid:55)→ [ τ ] F X The fact that η is a natural transformation is left to the reader. Notice that η M for M extensional isthe identity morphism.We need now to define the counit ε . Here we need to use crucially the assumption that we restrict R X to the family of separated presheaves. Towards this aim if U ⊆ V regular open (equivalentlyclopen) in X , we have that the following diagram is commutative with horizontal lines beingbijections (recall that F U is the principal filter on CLOP ( X ) given by the clopen supersets of U ):(3) ( F ( V ⊆ X ))[ F ( X )] τ (cid:22) V (cid:55)→ [ τ ] V −−−−−−→ R X ( F ) / F V τ (cid:22) V (cid:55)→ τ (cid:22) U (cid:121) (cid:121) [ τ ] FV (cid:55)→ [ τ ] FU ( F ( U ⊆ X ))[ F ( X )] −−−−−−→ τ (cid:22) U (cid:55)→ [ τ ] U R X ( F ) / F U This is [9, Thm. 5.4] for the special case of boolean algebras. We became aware of this only after completing a firstdraft of this paper. where, by our convention, F ( U ⊆ X )( τ ) is equally denoted as τ (cid:22) U .The commutativity is automatic by definition of the various maps occurring in the diagram. Tosee why the horizontal lines are bijections, observe that [ τ ] F U = [ σ ] F U for τ, σ ∈ F ( X ) and U ∈ CLOP ( X ) if and only if U ≤ (cid:74) τ = σ (cid:75) R X ( F ) if and only if F ( U ⊆ X )( σ ) = F ( U ⊆ X )( τ ) . In particular we conclude that:(4) (( L X ◦ R X )( F ))( U ) := R X ( F ) / F U ∼ = F ( U ⊆ X )[ F ( X )] ⊆ F ( U ) , and (( L X ◦ R X )( F ))( U ⊆ V ) = { [ τ ] V (cid:55)→ [ τ ] U : τ ∈ F ( X ) } ∼ = (5) ∼ = F ( U ⊆ V ) (cid:22) ( F ( U ⊆ X )[ F ( X )]) ⊆ F ( U ⊆ V ) . Now we let ε be the natural transformation which in each component F is such that: • for any U ∈ CLOP ( X ) + ε ( F )( U ) : F ( X ) / F U → F ( U ⊆ X )[ F ( X )][ τ ] U (cid:55)→ ( τ (cid:22) U ); • for any U ⊆ V ∈ CLOP ( X ) + , ε ( F )( U ⊆ V ) transfers via the horizontal lines of diagram(3) the mapping { [ τ ] V (cid:55)→ [ τ ] U : τ ∈ F ( X ) } from F ( X ) / F V to F ( X ) / F U to the mapping F ( U ⊆ V ) (cid:22) { τ (cid:22) V : τ ∈ F ( X ) } (where the latter set is exactly F ( V ⊆ X )[ F ( X )] ).The morphism ε F is a well defined natural transformation of L X ◦ R X with Id S-Presh ( CLOP ( X ) \{∅} , Set ) due to the commutativity of diagram (3).Equations (1) characterizing the unit and counit properties of η, ε with respect to L X , R X aresatisfied due to the following observations: • The unit η restricted to the image of the functor R X is the identity: the target of R X is theclass of CLOP ( X ) -valued extensional models. • The counit ε restricted to the image of the functor L X is an isomorphism with the iden-tity functor on the image of L X : for any CLOP ( X ) -valued model M , L X ( M ) is apresheaf F M in which F M ( U ) = M / F U is the surjective image of F M ( X ) = M / F X via F M ( U ⊆ X ) : [ τ ] F X (cid:55)→ [ τ ] F U ; in particular for F M the last inclusions of (4) and (5) arereinforced to equalities, and we can conclude that ε is an isomorphism appealing to the factthat the horizontal lines of the commutative diagram (3) are bijections . (cid:3) Applying Corollary 1.20 we obtain the following:
Corollary 3.4.
Let X be a compact, Hausdorff, -dimensional, extremally disconnected topologicalspace. Let • ExPresh ( CLOP ( X ) , Set ) be the full subcategory of S-Presh ( CLOP ( X ) + , Set ) given by theseparated presheaves F such that F ( U ) = F ( U ⊆ X )[ F ( X )] for all U ∈ CLOP ( X ) + , The left to right implication of this equivalence uses that F is separated, otherwise we can only assert that the setof V ⊆ U such that F ( V ⊆ X )( σ ) = F ( V ⊆ X )( τ ) is a dense cover of U , but U may not belong to this dense coverif F is not separated. Here we use essentially that R X is applied to a separated presheaf. Otherwise we could have different sections f, g in F ( X ) such that (cid:74) f = g (cid:75) R X ( F ) = X ; in which case [ f ] F X = [ g ] F X , hence the horizontal lines in (3) are notanymore bijections. It is not clear then how to select canonically the representatives in the equivalence classes of [ − ] F X so to maintain the requested naturality properties for the counit given by (4) and (5). OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 25 • ExMod
CLOP ( X ) be the full subcategory of Mod CLOP ( X ) generated by extensional models.The adjunction (cid:104) L X : Mod
CLOP ( X ) → S-Presh ( CLOP ( X ) + , Set ) , R X : S-Presh ( CLOP ( X ) + , Set ) → Mod
CLOP ( X ) (cid:105) specializes to an equivalence of categories between ExMod CLOP ( X ) and ExPresh ( CLOP ( X ) , Set ) . Note that every B -valued model M is boolean isomorphic to the extensional model M /F B .Hence the corollary amounts to say that B -valued models identify the class of separated B + -presheaves in which the local sections are always restrictions of global sections.Note also that any CLOP ( X ) + -sheaf F is such that every local section is the restriction of a globalsection: for each A ∈ CLOP ( X ) + let f A be some section in F ( A ) ; given any local section g in F ( U ) the family (cid:8) g, f X \ U (cid:9) is matching, and a collation of this family is a global section whoserestriction to U is g . In particular on the class of CLOP ( X ) + -sheaves the adjunction also restrictsto an equivalence of categories.We are now going to give a nice combinatorial characterization of the sheaf condition for booleanvalued mdoels. 4. T HE MIXING PROPERTY CHARACTERIZES SHEAVES
We can characterize the boolean valued models satisfying the mixing property in the followingway . Theorem 4.1.
Let X be an extremally disconnected compact space and M be a CLOP ( X ) -valuedmodel. Then M has the mixing property if and only if the separated presheaf F M : ( CLOP ( X ) + ) op → Set of Definition 3.1 is a sheaf.Proof.
Let U be an open set and let { U i : i ∈ I } be such that (cid:87) i ∈ I U i = U in CLOP ( X ) . Fromnow on, fix a well-order ≤ on I .Assume that M satisfies the mixing property. Let f i ∈ F ( U i ) = M / F Ui for every i ∈ I andsuppose that, if i (cid:54) = j , then(6) F M ( U i ∩ U j ⊆ U i )( f i ) = F M ( U i ∩ U j ⊆ U j )( f j ) . In particular, we may assume that, for every i ∈ I , O i = U i ∧ ¬ (cid:95) j
By induction on the well order of I , (cid:74) τ = σ i (cid:75) ≥ U i . Indeed, (cid:74) τ = σ min I (cid:75) ≥ O min I = U min I and,if we assume that (cid:74) τ = σ j (cid:75) ≥ b j for all j < i , (cid:74) τ = σ i (cid:75) ≥ O i ∨ (cid:95) j
Corollary 4.2.
Let X be compact and extremally disconnected. A CLOP ( X ) -valued model M has the mixing property if and only if every global section of the ´etal´e space Λ ( F M ) = Λ ( F M ) is a section induced by an element of M .In particular if M has the mixing property the presheaf F M is a CLOP ( X ) + -sheaf isomorphic tothe restriction to CLOP ( X ) + of the sheaf defined on the whole of O ( X ) by the local sections of Λ ( F M ) .Proof. Left to the reader. (cid:3)
Example 4.3.
Let us recall the
MALG -valued model C ω ( R ) of analytic functions introduced inExample 2.9. Let us show explicitly that its associated ´etal´e space E C ω ( R ) does have global sectionswhich are not induced by elements of C ω ( R ) . Consider a := [( −∞ , Null ∈ MALG . It is clearthat ¬ a = [(0 , + ∞ )] Null . Define s : St( MALG ) → E C ω ( R ) by s ( G ) := (cid:40) [ (cid:104) c , G (cid:105) ] R if a ∈ G , [ (cid:104) c , G (cid:105) ] R if ¬ a ∈ G (recall that c r : R → R is the constant function R (cid:51) x (cid:55)→ r ). It is immediate to see that s is a globalsection, and it is continuous from St(
MALG ) to R . However, there is no analytic function in C ω ( R ) inducing s : the candidates to induce s have to be constantly on an open interval I ⊆ ( −∞ , andhave to be constantly on an open interval I ⊆ (0 , + ∞ ) , hence they are not analytic functions. Example 4.4.
In Example 2.13 we are just saying that the continuous section h : (cid:83) a ∈ A N a → E M B such that h ( G ) = (cid:104) ˇ a, G (cid:105) if and only if a ∈ G is not induced by any element of M B .Furthermore, by Corollary 3.4 we can conclude the following. OOLEAN VALUED MODELS, PRESHEAVES, AND ´ETAL ´E SPACES 27
Corollary 4.5.
Let X be a compact Hausdorff -dimensional extremally disconnected topologicalspace. Then the equivalence of categories between ExMod CLOP ( X ) and ExPresh ( CLOP ( X ) + , Set ) induces an equivalence of categories between the full subcategory of ExMod CLOP ( X ) generatedby the models satisfying the mixing property and the full subcategory ExSh ( CLOP ( X ) + , Set ) of ExPresh ( CLOP ( X ) + , Set ) generated by the sheaves F : ( CLOP ( X ) + ) op → Set such that F ( U ) = F ( U ⊆ X )[ F ( X )] for every U ∈ CLOP ( X ) + . Example 4.6.
Let us now outline why the
MALG -names for real numbers correspond to thesheafification of the presheaf F assigning to each [ A ] ∈ MALG the space L ∞ ( A ) , for A ameasurable subset of R .It is not hard to check that the boolean valued model M = (cid:110) τ ∈ V MALG : (cid:74) τ ∈ R (cid:75) MALG = 1
MALG (cid:111) is a bolean valued model with the mixing property (for details see [11]). Let X = St( MALG ) ; then L X ( M ) is a sheaf. Let also R ∪ {∞} be the one point compactification of R . By the results of[4, 10, 11] L X ( M )( X ) ∼ = C + ( X ) = (cid:8) f : X → R ∪ {∞} : f is continuous and f − [ {∞} ] is nowhere dense (cid:9) . It is also possible to see that the Gelfand transform of L ∞ ( R ) extends to a natural isomorphism C + ( X ) ∼ = L ∞ + ( R ) = (cid:8) f : R → R ∪ {∞} : f is measurable and ν ( f − [ {∞} ]) = 0 (cid:9) , where ν is Lebesgue measure (for details see [11], natural here means that it is a MALG -isomorphismaccording to Def. 2.14 when for f, g ∈ C + ( X ) we set (cid:74) f = g (cid:75) = Reg ( { G ∈ X : f ( G ) = g ( G ) } ) and for f ∗ , g ∗ ∈ L ∞ + ( R ) (cid:74) f ∗ = g ∗ (cid:75) = [ { a ∈ R : f ∗ ( a ) = g ∗ ( a ) } ] Null ).In particular this isomorphism shows that for all A ⊆ R measurable L X ( M )([ A ] Null ) ∼ = (cid:8) f : A → R ∪ {∞} : f is continuous and f − [ {∞} ] is nowhere dense (cid:9) and that C + ( X ) is naturally identified with Λ ( F X ( M )) .5. A CHARACTERIZATION OF THE FULLNESS PROPERTY USING ´ ETAL ´ E SPACES
We are now interested to characterize the fullness property for boolean valued models in terms ofsheaf theory. The notion we found appropriate to characterize fullness is the notion of ´etal´e space.First of all we notice that a well behaved B -valued model M for the language L is full if and only if,for every L M -formula ϕ ( x , . . . , x n ) with n free variables x , . . . , x n , there exists a finite number m of n -tuples (cid:104) σ (1)1 , . . . , σ (1) n (cid:105) , . . . , (cid:104) σ ( m )1 , . . . , σ ( m ) n (cid:105) ∈ M n such that (cid:74) ∃ x , . . . , ∃ x n ϕ ( x , . . . , x n ) (cid:75) = m (cid:95) i =1 (cid:114) ϕ ( σ ( i )1 , . . . , σ ( i ) n ) (cid:122) . Now fix an L M -formula ϕ ( x , . . . , x n ) with n -free variables and let b ϕ := (cid:74) ∃ x , . . . , x n ϕ ( x , . . . , x n ) (cid:75) .We define an ´etal´e space E ϕ M over N b ϕ (which is a topological subspace of St( B ) ) by: E ϕ M := {(cid:104) σ , . . . , σ n , G (cid:105) : σ , . . . , σ n ∈ M, (cid:74) ϕ ( σ , . . . , σ n ) (cid:75) ∈ G ∈ St( B ) } / R , where R is the equivalence relation such that (cid:104) σ , . . . , σ n , G (cid:105) R (cid:104) τ , . . . , τ n , H (cid:105) if and only if G = H and (cid:74) σ i = τ i (cid:75) ∈ G for every i = 1 , . . . , n .We can equivalently say that E ϕ M = {(cid:104) [ σ ] G , . . . , [ σ n ] G (cid:105) : σ , . . . , σ n ∈ M, (cid:74) ϕ ( σ , . . . , σ n ) (cid:75) ∈ G ∈ St( B ) } . This set is mapped over N b ϕ by the function p ϕ : (cid:104) σ , . . . , σ n , G (cid:105) (cid:55)→ G ( notice that, if there aresome σ , . . . , σ n ∈ M such that (cid:74) ϕ ( σ , . . . , σ n ) (cid:75) ∈ G , then b ϕ ∈ G ). An important observation is that, in general, p ϕ is not surjective; its image is just a dense subset of N b ϕ . The topology of E ϕ M isthe one obtained as a subspace of E n M : a base for the topology of E ϕ M is B ϕ := {{(cid:104) σ , . . . , σ n , G (cid:105) : (cid:74) ϕ ( σ , . . . , σ n ) (cid:75) ∈ G ∈ N c } : σ , . . . , σ n ∈ M, c ≤ b ϕ } . It is easy to check that E ϕ M equipped with this topology renders continuous the map p ϕ : E ϕ M → N b ϕ . Moreover, p ϕ : E ϕ M → N b ϕ is the ´etal´e space associated to the presheaf G ϕ M : ( CLOP ( N b ϕ ) + ) op → Set defined as follows:For c ≤ b ϕ , G ϕ M ( N c ) := {(cid:104) [ σ ] F c , . . . , [ σ n ] F c (cid:105) : σ , . . . , σ n ∈ M and c ≤ (cid:74) ϕ ( σ , . . . , σ n ) (cid:75) } . Notice by the way that, if ϕ has one free variable and U ⊆ N b ϕ is clopen, then G ϕ M ( U ) ⊆ F M ( U ) .Moreover, each (cid:104) [ σ ] F c , . . . , [ σ n ] F c (cid:105) ∈ G ϕ M ( N c ) defines a continuous open section ( ˙ σ × · · · × ˙ σ n ) : G (cid:55)→ (cid:104) [ σ ] G , . . . , [ σ n ] G , G (cid:105) of p ϕ over N c . Theorem 5.1.
Let B be a complete boolean algebra and let M be a well behaved B -valued modelfor the language L . The following are equivalent:(1) M is full;(2) for every L M -formula ϕ ( x , . . . , x n ) such that E ϕ M is non-empty, E ϕ M has at least oneglobal section.Proof. If M is full, for every L M -formula ϕ ( x , . . . , x n ) there exist σ (1)1 , . . . , σ (1) n , . . . , σ ( m )1 , . . . , σ ( m ) n ∈ M such that n (cid:95) i =1 (cid:114) ϕ ( σ ( i )1 , . . . , σ ( i ) n ) (cid:122) = (cid:74) ∃ x , . . . , x n ϕ ( x , . . . , x n ) (cid:75) = b ϕ , i. e. N b ϕ = m (cid:91) i =1 N (cid:114) ϕ ( σ ( i )1 ,...,σ ( i ) n ) (cid:122) . Define C := N (cid:114) ϕ ( σ (1)1 ,...,σ (1) n ) (cid:122) , C i := N (cid:114) ϕ ( σ ( i )1 ,...,σ ( i ) n ) (cid:122) \ i − (cid:91) j =1 C j , i = 2 , . . . , m. The sets just defined are clopen, pairwise disjoint and they cover N b ϕ . Then the map s : N b ϕ → E ϕ M defined by s ( G ) := (cid:104) [ σ ( i )1 ] G , . . . , [ σ ( i ) n ] G (cid:105) if G ∈ C i is a global section.Conversely, fix an L M -formula ϕ ( x , . . . , x n ) . Assume that p ϕ : E ϕ M → N b ϕ has a global section.In particular, p ϕ is surjective. Thus, we have N b ϕ = (cid:91) (cid:104) σ ,...,σ n (cid:105)∈ M n N (cid:74) ϕ ( σ ,...,σ n ) (cid:75) . Being N b ϕ compact, we can refine this open cover to a finite one. (cid:3) The mixing property is strictly stronger than the fullness property. Here is a reformulation ofProposition 2.11 in the language of bundles and sheaves:
Corollary 5.2.
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