Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory

Mo dules on in v olutiv e quan tales: canonical Hilb ert structure, appli cations to sheaf theory Hans Heymans ∗ and Isar Stubb e † W ritten: August 2008 Submitted: 10 Septem b er 2008 Revised: 18 Ma y 200 9 Abstract W e explain the precise relationship b etw een tw o module-theor etic descriptions of she aves on an in volutiv e quan tale, namely the descr iptio n via so-called Hilbert s tr uctures on modules and that via so- called principally ge nerated mo dules. F or a principa lly g e ne r ated mo dule satisfying a suitable symmetry condition we obse rve the existence of a canonical Hilbert structure. W e prov e that, when working o ver a mo dular quantal fra me, a mo dule b ears a Hilber t struc tur e if and only if it is principally ge nerated and symmetric, in which case its Hilber t structure is necess arily the canonica l one. W e indicate applications to sheaves on lo cales, on quantal frames a nd even on sites. 1 In tro duction Jan P asek a [199 9, 2002 , 2003] in trod uced the notio n of Hilb ert mo dule on an inv olutive quan tale: it is a mo du le equipp ed with an inner pr o duct . This pro vides for an order-theoretic notion of “inner p ro du ct space”, originally inte nded as a generalisation of complete lattices with a dualit y . Recen tly , P edro Resend e and E lias Rod rigues [2008] applied this d efi nition to a locale X and further d efined what it means for a Hilbert X -mo dule to ha ve a Hilb e rt b asis . These Hilb er t X -mo d ules with Hilb ert basis describ e, in a mod ule-theoretic wa y , the s hea v es on X . A t the same time, the present authors defin ed th e n otion of (lo c al ly) princip al ly gener ate d mo dule on a quan taloid [Heymans and Stubb e, 2009]. Our aim to o w as to describ e “sheav es as mo dules”, alb eit shea v es on quan taloids in the sense of [Stubb e, 200 5b]. In this form ulation the ordinary shea v es on a locale X are d escrib ed as lo cally principally generated X -mo dules wh ose lo cally principal elements satisfy an extra “op enness” condition. Whereas Hilb ert lo cale mo dules easily generalise to mo du les on in v olutiv e qu an tales, the principally generated qu antalo id mo du les straigh tforw ardly sp ecialise to inv olutive qu an tales. ∗ Department of Mathematics and Computer Science, Universit y of Ant w erp, Middelheimlaan 1, 2020 An tw er- p en, Belgi um, hans.heymans@ua.a c.be † P ostdo ctoral F ellow of th e Research F oundation Flanders (FWO), Department of Mathematics and Co mputer Science, Universit y of Antw erp , Middelheimla an 1, 2020 Ant werpen, Belgi um, isar.st ubbe@ua.ac.be 1 Th us we h a v e t wo mo dule-theoretic appr oac h es to sh eav es on inv olutive quant ales: in this n ote w e explain th e pr ecise relationship b et we en them. This work can b e summarised as follo w s : After s ome preliminary definitions w e sho w in Section 2 that an y principally generated mo du le on an in volutiv e qu an tale co mes with a c anoncial (pr e-)inner pr o duct . In Section 3 w e firs t p r esen t the notion of Hilb ert basis for mo dules on an inv olutive quantale [Resende, 2008]. Af ter introdu cing a su itable notion of s y m metry for suc h mo du les, termed princip al symmetry , w e pro ve that a mo dule is principally generated and principally symmetric if an d only if it admits a canonical Hilb ert structure (= canonical in n er pro du ct plus canonical Hilb ert basis). When wo rking o v er a mo dular quantal fr ame it is a fact, as w e p ro v e in Section 4, that a mo dule b ears a Hilb ert s tr ucture if and only if it is pr in cipally generated and pr incipally symmetric, in wh ic h case th e giv en inner pro d uct is necessarily the canonical one (admitting the canonical Hilb ert basis). That is to sa y , in this case the only p ossible (and th us the only relev ant ) Hilb ert structur e is the ca nonical one. W e illustrate all this mo du le-theory with many examples. In the final Section 5 we draw some conclusions from our w ork. W e explain all new resu lts in this pap er in a self-con tained mann er in th e language of quan tale mo dules, focus sing on the p urely order-theoretic asp ects. How eve r, in some examp les, partic- ularly those concerned with sh eaf theory in one wa y or another, w e freely u se material from the references without recalling muc h of the d etails. Thus, the r eader wh o is mainly in terested in order th eory can safely skip those examples; but the r eader who is also interested in the applications to sheaf theory will most lik ely ha v e to ha v e a quick lo ok at the cited p ap ers too, insofar as the notions inv olv ed are not already familiar to her or him. 2 Canonical inner pro duct W e b egin by r ecalling some definitions. T hroughout this pap er, Q = ( Q, W , ◦ , 1) stands for a quantale , i.e. a monoid in the monoidal ca tegory Sup of complete latti ces and maps that preserve arbitrary suprema. Explicitly , a q u an tale Q consists of a complete lattice ( Q, W ) equipp ed with a binary op eration Q × Q / / Q : ( f , g ) 7→ f ◦ g and a constant 1 ∈ Q such that f ◦ ( g ◦ h ) = ( f ◦ g ) ◦ h , 1 ◦ f = f = f ◦ 1 and ( _ i ∈ I f i ) ◦ ( _ j ∈ J g j ) = _ i ∈ I _ j ∈ J ( f i ◦ g j ) for all f , g , h, f i , g j ∈ Q . ( Some call this a unital quantale , but sin ce w e shall not encounter “non-unital quant ales” in this work w e drop that adjectiv e.) Definition 2.1 A map Q / / Q : f 7→ f o is an inv olution , and the p air ( Q, ( − ) o ) forms an in v olutiv e qu an tale , if it is or der-pr eserving ( f ≤ g ⇒ f o ≤ g o ), inv olutive ( f oo = f ) and multiplic ation-r eversing ( ( f ◦ g ) o = g o ◦ f o ). It follo ws that an in v olution is an isomorphism of complete lattices, and also unit-preserving (1 o = 1). Most often we shall simply sp eak of “a n inv olutiv e quantale Q ” and lea v e it understo o d that the inv olution is wr itten as f 7→ f o . Definition 2.2 An element f ∈ Q of an involutive quantale is symmetric if f o = f . 2 A symmetric id emp oten t element of Q ( f o = f = f ◦ f ) is sometimes called a pr oje ction . Example 2.3 Among the man y examples of inv olutive quan tales, w e p oint out some of partic- ular inte rest. 1. A qu an tale Q is comm utativ e if and only if the identit y map 1 Q : Q / / Q : q 7→ q is an in v olution. In particular is eve ry lo c ale (also calle d fr ame ) X = ( X , W , ∧ , ⊤ ) an inv olutiv e quan tale f or this trivial inv olution. 2. Let S b e a complete lattic e w ith a duality , i.e . a sup rem um-pr eserving map d : S / / S op suc h that d ( x ) = d ∗ ( x ) and d ( d ( x ) ) = x for all x ∈ S , where d ∗ is the righ t adjoin t to d (abbre- viated as d ⊣ d ∗ ) in the category Ord of ord er ed sets and order-preserving maps, explicitly , d ∗ : S op / / S : y 7→ W { x ∈ S | d ( x ) ≤ op y } . The qu an tale Q ( S ) := ( Sup ( S, S ) , W , ◦ , 1 S ) has a n atural inv olution [Mulvey and Pel letier, 1992]: for f ∈ Q ( S ) put f o := d op ◦ ( f ∗ ) op ◦ d (where f ⊣ f ∗ in Ord ). When putting f o := d op ◦ f op ◦ d , we ha v e f o ⊣ f o in Ord . 3. A mo dular quantale Q is an in v olutiv e one whic h satisfies F reyd ’s m o dular la w [F reyd and Scedro v, 1990]: ( p ◦ q ∧ r ) ≤ p ◦ ( q ∧ p o ◦ r ) for all p, q , r ∈ Q . W e follo w [Resend e, 2007] in sp eaking of a quantal f r ame when w e mean a quan tale whose un derlying lattice is a frame (= lo cale); the term mo dular quantal fr ame then sp eaks for itself. It is a matter of fact that modu lar quan tal frames are pr ecisely the one-ob ject lo cally complete distribu tiv e allego ries of P . F reyd and A. Scedro v [1990]. Allegories are closely relate d to toposes; b elo w w e shall see that mo du lar quan tal frames in particular app ear in the study of shea v es (cf. Theorem 4.1 and Example 4.7). 4. An inverse quantal fr ame is a mo du lar quanta l frame Q in w hic h ev ery elemen t is the join of so-cal led p artial units ( p ∈ Q is a partial unit if p o p ∨ pp o ≤ 1 Q ); it suffices that the top of Q is s u c h a join. Th is d efinition is equiv alent to the origi nal one given in [Resende, 2007] b ecause it is p ro v ed in that reference that inv er s e quan tal frames arise as qu otien ts (as frames and as in v olutiv e quanta les) of quan tal frames that are eviden tly mo dular. Th er e is a corr esp ondence up to isomorphism b etw een inv erse qu antal fr ames and ´ etale group oids [Resende, 2007], p ro viding a context to consid er ´ etendues in terms of quantale s. 5. (In this and the n ext example we use notions that, for a lac k of space, we cannot recall; but w e do include ample references.) A quantalo id is a Sup -enric hed category . If A is an ob ject of a q u an taloid Q , th en Q ( A, A ) is a quan tale; in particular, a quantal oid with only one ob ject is precisely a quan tale. A quantalo id Q has a direct-sum completion, wh ic h can b e describ ed as Ma tr ( Q ), the qu antalo id of matric es with elements in Q . All definitions ab o v e can straigh tforwardly b e generalised fr om q u an tales to q u an taloids. F or details, see e.g. [F reyd and Scedro v, 1990; Rosen thal, 1996; Stubb e, 2005a]. A s m all quan taloid Q is Morita e quiv alent to the qu an tale Q := Matr ( Q )( Q 0 , Q 0 ) [Mesablishvili, 2004], and it is easily seen that sev eral prop erties of Q are carr ied o v er to its Morita-equiv alen t quanta le Q : for example, if Q is inv olutiv e then so is Q . 6. F or a small site ( C , J ), i.e . C a small category and J a Grothend ieck top ology on C , the J -closed relations b etw een the repr esen tables in Set C op form a locally complete distribu tiv e 3 allego ry , i.e. a mo dular quan taloid Q whose hom-ob jects are frames [W alters, 1982; Betti and Carb oni, 1983]. It is easy to v erify that this small quantalo id’s Morita-equiv alent quan tale Q is a mo du lar quan tal frame, and that Q can b e iden tified with a sub quan taloid of the u niv ersal sp litting of the symmetric idemp otent s of Q . When we speak of a (right) Q -mo dule M we mean so in the ob vious w a y in Sup . Th at is to sa y , ( M , W ) is a co mplete lattice on which Q acts b y m eans of a function M × Q / / M : ( m, f ) 7→ m · f satisfying m · ( f ◦ g ) = ( m · f ) · g , m · 1 = m and ( _ i ∈ I m i ) · ( _ j ∈ j f j ) = _ i ∈ I _ j ∈ J ( m i · f j ) for all m, m i ∈ M and f , g , f j ∈ Q . Accordingly , a function φ : M / / N b et w een tw o Q -mo d u les is a Q - mo dule morphism if φ ( m · f ) = φ ( m ) · f and φ ( _ i ∈ I m i ) = _ i ∈ I φ ( m i ) for all m, m i ∈ M and f ∈ Q . W e shall write Mo d ( Q ) for the category of Q -mod ules and mod u le morphisms. Of course Q itself is a Q -mo d ule, with action giv en by m ultiplicatio n in Q . Definition 2.4 (P asek a, 1999) L et M b e a mo dule on an involutive quantale Q . A map M × M / / Q : ( m, n ) 7→ h m, n i is a pre-inner pr o duct if, for al l m, n ∈ M , 1. h m, −i : M / / Q is a mo dule morphism, 2. h m, n i o = h n, m i (which we r efer to as Hermitian symmetry ). It is an inner pro duct if i t mor e over satisfies 3. h− , m i = h− , n i implies m = n and it is said to b e strict if 4. h m, m i = 0 implies m = 0 . No w w e shall recall some definitions from [Heymans and Stubb e, 2009], where they are giv en for quanta loids but whic h we apply here to quan tales. Let e ∈ Q b e an idemp oten t. T he fixp oin ts of e ◦ − : Q / / Q form a Q -mo dule which w e shall write as Q e : the action of Q on Q e is give n b y multiplicat ion, so the inclusion ι e : Q e / / Q : f 7→ f is a mo dule morphism. F urther, if M is an y Q -mo dule then for an y m ∈ M the map τ m : Q / / M : f 7→ m · f 4 is a mo dule morphism. Thus also the comp osite Q e ι e @ @ @ @ @ @ @ @ ζ m / / M Q τ m ? ?         is a mo dule morphism. Essentia lly as a n applicati on of the Y oneda Lemma f or enr iched categories [Kelly , 1982] w e find the follo win g c haracterisation. Prop osition 2.5 L et Q b e a qu antale, e ∈ Q an idemp otent, and M a Q -mo dule. Ther e is a one-one c orr esp ondenc e b etwe en the fixp oints of − · e : M / / M and the mo dule morphisms fr om Q e to M . Pr o of : If ζ : Q e / / M is an y mo du le morphism, then m ζ := ζ ( e ) ∈ M satisfies m ζ · e = m ζ ; con v ersely , if m ∈ M satisfies m · e = m , then ζ m : Q e / / M : f 7→ m · f is a mo dule morph ism . This is easily seen to set up a one-one corresp ondence. ✷ In particular, suc h a map ζ m : Q e / / M b et w een complete lattices preserves sup rema; therefore it has an in fima-preserving righ t adjoin t in the cate gory of ordered sets and order-preservin g maps. Ho w ev er, in general the order-preserving righ t adjoin t n eed not b e a mo d ule morp hism, i.e. it n eed not b e right adjoin t to ζ m in the cate gory Mo d ( Q ) of Q -mo dules. Definition 2.6 (Heymans and Stubb e, 2009) L et Q b e a quantale and M a Q -mo dule. An element m ∈ M is said to b e lo cally p rincipal at an idemp oten t e ∈ Q if m · e = m and ζ m : Q e / / M : f 7→ m · f has a right adjoint in Mo d ( Q ) . Prop osition 2.7 L et Q b e a quantale, e ∈ Q an idemp otent, and M a Q - mo dule. Th er e i s a one-one c orr esp ondenc e b etwe en M ’s lo c al ly princip al elements at e and left adjoint mo dule morphism s fr om Q e to M . In wh at follo ws w e shall alw a ys write ζ ∗ : M / / Q e for th e r ight adjoin t to a giv en mo dule morphism ζ : Q e / / M , whenev er we kno w or assu m e it exists. No w we come to a tr ivial but crucial obser v ation. Prop osition 2.8 L et Q b e an i nv olutive quantale, E ⊆ Q the set of symmetric idemp otents, and M a Q - mo dule. The formula h m, n i can := _ n ( ζ ∗ ( m )) o ◦ ζ ∗ ( n )    e ∈ E , ζ : Q e / / M left adjoint in Mo d ( Q ) o defines a pr e-inner pr o duct, c al le d the canonical p re-inner pr o duct , on M . Pr o of : F or an y e ∈ E and an y left adjoint ζ : Q e / / M , the p oin t wise multiplicatio n of th e comp osite mo d ule morph ism M ζ ∗ / / Q e ι e / / Q 5 with the elemen t ( ζ ∗ ( m )) o ∈ Q giv es a mo dule morph ism ( ζ ∗ ( m )) o ◦ ζ ∗ ( − ): M / / Q. As any p oin t wise su premum of parallel mo dule morphisms is again a mod ule morphism, w e fin d that h m, −i can = _ n ( ζ ∗ ( m )) o ◦ ζ ∗ ( − )    e ∈ E , ζ : Q e / / M left adjoint in Mo d ( Q ) o is a mo du le morphism from M to Q . It is a trivialit y that the fun ction h− , − i can : M × M / / Q is symmetric. ✷ Example 2.9 W e shall co mpute some m ore explic it examples in the next section, bu t we already include the follo wing here. 1. Ev ery in v olutiv e quant ale Q , regarded as a mo dule o v er itself, has a natural in n er pro d uct [P asek a, 199 9]: for f , g ∈ Q let h f , g i := f o ◦ g . And the canonical pre-inner pro duct on a Q -mo dule M is expressed as a su premum of v alues of the natural inner pro duct on Q . 2. P articularly for a complete lattice S with dualit y d : S / / S op w e can consider the n atural inner pro d uct on the inv olutiv e qu an tale Q ( S ): we ha v e for s, t ∈ S and f , g ∈ Q ( S ) that h f , g i ( s ) ≤ t ⇐ ⇒ g ( s ) ≤ f o ( t ) . F rom this it is easy to v erify that h f , g i = 0 if and only if f an d g are disjoint : f ( s ) ≤ d ( g ( t )) for all s, t ∈ S . 3 Canonical Hilb ert basis W e s tart by recalling another definition from [Heymans and Stub b e, 2009] (where it w as actually stated more generally for mo du les on quanta loids). Definition 3.1 L et Q b e a quantale, E ⊆ Q any set of idemp otent elements c ontaining the u nit 1 , and M a Q - mo dule. If, for al l m ∈ M , m = _ n ζ ( ζ ∗ ( m ))    e ∈ E , ζ : Q e / / M lef t adjoint in Mo d ( Q ) o then M is E -principally generated (which is short for: generated by its elemen ts whic h are lo cally principal at some e ∈ E ). This Definition 3.1 r esem bles the follo wing notion, whic h w as originally defin ed by Resende and R o drigues [2008] for p re-Hilb ert mo d ules on a lo cale, but whic h can straigh tforwardly b e extrap olated to p re-Hilb ert mo d ules on an in v olutiv e qu an tale, as [Resende, 2008] do es: Definition 3.2 L et Q b e an involutive quantale, and M a Q - mo dule with pr e- inner pr o duct h− , −i . If a subset Γ ⊆ M satisfies, for al l m ∈ M , m = _ s ∈ Γ s · h s, m i 6 then it is a Hilb ert basis 1 for M . If a Q -mo dule M b ears a p re-inner pro d uct admitting a Hilb ert b asis, we sp eak of its Hilb ert structur e ; unless exp licitly stated o therwise we shall alw a ys write h− , −i for the pre-inner pr o duct and Γ for the Hilb ert b asis. As also p ointed out in [Resende, 2008] , it is trivial to c hec k that: Prop osition 3.3 If Q is an involutive quantale, and M a Q - mo dule with a pr e- inner pr o duct h− , −i admitting a Hilb ert b asis Γ , then h− , −i is in fact an inner pr o duct. Pr o of : Giv en m, n ∈ M suc h that h− , m i = h− , n i , we certainly ha v e h s, m i = h s, n i for all s ∈ Γ. Th e formula in Definition 3.2 then allo w s u s to compu te that m = _ s ∈ Γ s · h s, m i = _ s ∈ Γ s · h s, n i = n and w e are done. ✷ Both Definitions 3.1 and 3.2 sp eak of a “generating set” for Q -modu les... But already in the lo calic case these t w o d efinitions are different! Example 3.4 Let X b e a locale, view it as a quantale ( X , W , ∧ , ⊤ ) with iden tit y inv olution. The set E of symmetric id emp oten ts in X coincides with X , and it is sho wn in [Stu b b e, 2005b; Heymans and S tubb e, 2009] that an E -principally generated X -mo du le is th e same thing as an or der e d she af on X , i.e. an ordered ob ject in Sh ( X ). On the other h and, as pro v ed in [Resende and R o drigues, 2008], a pre-Hilb ert X -mo d ule with Hilb ert basis is the same thing as a she af on X . This example hin ts at the imp ortance of the in trinsic symmetry in th e notion of “pr e-Hilb ert Q -mo dule with Hilb ert basis”, i.e. the s y m metry of the inv olved p re-inner pr o duct. In deed notice that Definitions 2.4 and 3.2 ask for a mo du le on an involutive quanta le – without whic h it w ould simp ly b e imp ossible to coheren tly sp eak of symmetry – whereas Definition 3.1 has no suc h requirement at all. T o systematicall y explain the relation b et w een the tw o defin itions w e m ust therefore dev elop a suitable n otion of symmetry in the con text of E -principally generated Q -mo dules. Prop osition 3.5 L et Q b e an i nv olutive quantale, E ⊆ Q the set of symmetric idemp otents, and M a Q - mo dule. The fol lowing statements ar e e qu ivalent: 1. for any e ∈ E , any left adjoint ζ : Q e / / M and any m ∈ M : ζ ∗ ( m ) = h ζ ( e ) , m i can , 2. for any e, f ∈ E and any lef t adjoints ζ : Q e / / M , η : Q f / / M : ζ ∗ ( η ( f )) = h ζ ( e ) , η ( f ) i can , 3. for any e, f ∈ E and any left adjoints ζ : Q e / / M , η : Q f / / M : ζ ∗ ( η ( f )) = ( η ∗ ( ζ ( e ))) o . In this c ase we say that M is E -pr incipally symmetric . 1 As also remark ed in [R esende and Ro drigues, 2008], the w ord “basis” is quite d eceiving: since there is n o freeness condition, it w ould be more appropriate to sp eak of Hil b ert gener ators . How eve r, for the sake of clarity w e shall adopt the terminolog y that was in tro duced in the cited references. 7 Pr o of : The only non-trivial implication is (3 ⇒ 1). In fact, the “ ≤ ” in statemen t 1 alw a ys holds: b ecause ζ ∗ ( m ) = e ◦ ζ ∗ ( m ) = e o ◦ ζ ∗ ( m ) ≤ ( ζ ∗ ( ζ ( e ))) o ◦ ζ ∗ ( m ) ≤ h ζ ( e ) , m i can where w e used resp ectiv ely: ζ ∗ ( m ) ∈ Q e ; e = e o ; the unit of the adjunction ζ ⊣ ζ ∗ to get e ≤ ζ ∗ ( ζ ( e )) from wh ic h e o ≤ ( ζ ∗ ( ζ ( e ))) o b ecause the in v olution preserv es order; and finally the definition of the canonical pre-inn er pro du ct. Th us, assuming statemen t 3 w e m ust sho w that the “ ≥ ” in statemen t 1 holds. But w e can compute that, for an y f ∈ E and any left adjoint η : Q f / / M , ( η ∗ ( ζ ( e ))) o ◦ η ∗ ( m ) = ζ ∗ ( η ( f )) ◦ η ∗ ( m ) = ζ ∗ ◦ η ◦ η ∗ ( m ) ≤ ζ ∗ ( m ) using r esp ectiv ely: the assump tion; the fact that ζ ∗ ( η ( f )) is the represen ting ele ment for the Q -mo dule morph ism ζ ∗ ◦ η : Q f / / Q e (cf. Prop osition 2.5); and the counit of the adjunction η ⊣ η ∗ . ✷ Remark that (1 ⇒ 2 ⇒ 3) in Prop osition 3.5 holds for any pr e-inner pr o duct on M (bu t (3 ⇒ 1) do es not!): that is to sa y , if one can p ro v e the fir st or the second condition for a giv en pre- inner pro d uct on M (n ot necessarily the canonical one), then it follo ws that M is E -prin cipally symmetric. This shall b e usefu l in the p ro of of Lemma 4.5 . W e can no w pro v e a first “comparison” b et w een Defin itions 3.1 and 3.2 . Theorem 3.6 L et Q b e an involutive quantale, E ⊆ Q the set of symmetric idemp otents, and M a Q -mo dule. The f ol lowing ar e e quivalent: 1. M is E - princip al ly ge ner ate d and E - princip al ly symmetric, 2. the set Γ can := { al l e lements of M which ar e lo c al ly princip al at some e ∈ E } is a H ilb ert b asis for the c anonic al pr e-inner pr o duct on M , c al le d the canonical Hilb ert basis . In this c ase, it fol lows by Pr op osition 3.3 that the c anonic al pr e- i nner pr o duct is an inner pr o duct; we sp e ak of the canonical Hilb ert stru cture on M . Pr o of : (1 ⇒ 2) Assu m ing that M is E -p rincipally generated w e h a v e b y definition th at, for any m ∈ M , m = _ n ζ ( ζ ∗ ( m ))    e ∈ E , ζ : Q e / / M left adjoint in Mo d ( Q ) o . Assuming moreo v er that M is E -p r incipally symm etric w e can compute ζ ( ζ ∗ ( m )) = ζ ( e ◦ ζ ∗ ( m )) = ζ ( e ◦ h ζ ( e ) , m i can ) = ζ ( e ) · h ζ ( e ) , m i can using resp ectiv ely: ζ ∗ ( m ) ∈ Q e ; the first statemen t in Prop osition 3.5; and the fact that ζ is a mo dule morphism . Replacing this in the righ t hand side of the fi rst expression, we obtain m = _ n ζ ( e ) · h ζ ( e ) , m i can    e ∈ E , ζ : Q e / / M left adjoint in Mo d ( Q ) o 8 so that, if w e put Γ can := n ζ ( e )    e ∈ E , ζ : Q e / / M left adjoint in Mo d ( Q ) o , whic h w e know by Pr op osition 2.7 ind eed corresp onds to the set of elemen ts of M which are lo cally principal at some e ∈ E , we find precisely w h at w e claimed. (2 ⇒ 1) F or an y e ∈ E and left adjoint ζ : Q e / / M , there certainly is a mo dule morphism h ζ ( e ) , −i can : M / / Q . But w e can compu te that, f or an y m ∈ M , e ◦ h ζ ( e ) , m i can = h ζ ( e ) · e o , m i can = h ζ ( e ◦ e o ) , m i can = h ζ ( e ) , m i can using: the “conju gate-linearit y” of h− , m i can ; the mo dule morphism ζ ; the fact that e is a symmetric idemp oten t. T herefore, this mo d u le morph ism corestricts to h ζ ( e ) , −i can : M / / Q e . W e claim that it is righ t adjoint to ζ : Q / / M . Indeed, if for q ∈ Q e and m ∈ M we assume that q ≤ h ζ ( e ) , m i can then we can compute ζ ( q ) = ζ ( e ◦ q ) = ζ ( e ) · q ≤ ζ ( e ) · h ζ ( e ) , m i can ≤ m using: q ∈ Q e , i.e. e ◦ q = q ; ζ is a m o dule morphism; the assumed inequalit y whic h is preserved b y ζ ( e ) · − ; an d fi nally the hypothesis that M has Hilb ert basis Γ. Assuming con v ersely that ζ ( q ) ≤ m , then w e can compute q = e ◦ q ≤ h ζ ( e ) , ζ ( e ) i can ◦ q = h ζ ( e ) , ζ ( e ) · q i can = h ζ ( e ) , ζ ( e ◦ q ) i can = h ζ ( e ) , ζ ( q ) i can ≤ h ζ ( e ) , m i can using: q ∈ Q e ; the un it of ζ ⊣ ζ ∗ in e = e o ◦ e ≤ ( ζ ∗ ( ζ ( e ))) o ◦ ζ ∗ ( ζ ( e )) ≤ h ζ ( e ) , ζ ( e ) i can ; the mo dule morphism h ζ ( e ) , −i can ; the mo dule morphism ζ ; again q ∈ Q e ; and fin ally th e assumed inequalit y w hic h is preserved b y the mo du le morphism h ζ ( e ) , −i can . Hence, for an y q ∈ Q e and m ∈ M , q ≤ h ζ ( e ) , m i can ⇐ ⇒ ζ ( q ) ≤ m. Adjoin ts are u nique and so we obtain that ζ ∗ ( m ) = h ζ ( e ) , m i can for all m ∈ M . By Pr op osition 3.5 this exactly means that M is E -prin cipally symmetric. Since w e assume that Γ is a Hilb ert basis, we ha v e that m = _ n ζ ( e ) · h ζ ( e ) , m i can    e ∈ E , ζ : Q e / / M left adjoin t in Mo d ( Q ) o . But the p revious computation allo ws us to wr ite ζ ( e ) · h ζ ( e ) , m i can = ζ ( e ) · ζ ∗ ( m ) = ζ ( e ◦ ζ ∗ ( m )) = ζ ( ζ ∗ ( m )) hence w e fi nd that m = _ n ζ ( ζ ∗ ( m ))    e ∈ E , ζ : Q e / / M left adjoint in Mo d ( Q ) o as w an ted. ✷ Example 3.7 W e shall give some examples of Q -mod ules with Hilber t structure, and then mak e a commen t on the c ate gory of Q -mo du les with Hilb ert structure. 9 1. Cf. Example 2.9–1, Γ := { 1 Q } is a Hilb ert basis for the natural inner pro du ct on Q . More generally , if e ∈ Q is an idemp oten t, th en Q e is a Q -mo du le with inn er pr o duct h f , g i := f o ◦ g admitting Γ := { e } as Hilb ert b asis. 2. Let Q b e the 2-el emen t c hain 2 = { 0 < 1 } (with ∧ as multiplicat ion, trivial in v olution, etc.); b oth its elemen ts are symmetric idemp otent s. Let ( A, ≤ ) b e an ordered set and consider Dwn ( A, ⊆ ), the downclose d subs ets of A order ed b y inclusion. This is the t ypical example of an E -principally generated 2 -mo d ule [Heymans and S tubb e, 2009] and is also one of th e fundamen tal constructions in [Resende and Ro dr igues, 2008 ]. If D ∈ Dwn ( A, ⊆ ) is a lo cally principal element, then it is either the empt y do wnset D = ∅ (locally pr incipal at 0 ∈ 2 ) or a principal do wnset D = ↓ x for some x ∈ A (lo cally prin cipal at 1 ∈ 2 ). F or an y D , E ∈ Dwn ( A, ⊆ ), their canonical inner pr o duct is h D , E i can = ( 1 if D ∩ E 6 = ∅ 0 otherwise T o say that Dwn ( A, ⊆ ) is E -principally symmetric is to require that for an y x, y ∈ A : ↓ x ⊆ ↓ y ⇐ ⇒ ↓ y ⊆ ↓ x. This mak es the order ( A, ⊆ ) in realit y an equiv alence relation ( A, ≈ ). 3. The localic case: Let X b e an y lo cale and S an y set. Th en X S is an X -mo dule, with p oint wise su prema and ( f · x )( s ) = f ( s ) ∧ x , for an y f ∈ X S , x ∈ X and s ∈ S . T ake no w an X -matrix Σ: S / / S (= a family (Σ( y , x )) ( x,y ) ∈ S × S of elemen ts of X ) satisfying Σ( z , y ) ∧ Σ( y , x ) ≤ Σ( z , x ) and Σ( x, x ) ∧ Σ( x, y ) = Σ( x, y ) = Σ( x, y ) ∧ Σ ( y , y ) and consider th e X -submo dule R (Σ) of X S consisting of those functions f : S / / X satis- fying f ( s ) = _ x ∈ S Σ( s, x ) ∧ f ( x ) . In the terminology of [Stub b e, 2005b], Σ is a totally regular X -semicatego ry and R (Σ) is (up to the iden tification of X -mo du les with co complete X -categ ories [Stu bb e, 2006]) the co complete X -category of (totally) regular p reshea v es on Σ . T his is the typical example of a lo c al ly princip al ly gener ate d X - mo dule [Heymans and Stubb e, 2009 ] an d is one of th e fundamental constru ctions of [Resende and Ro drigues, 2008] too. I t is n ot to o difficult to sho w by d irect calculations, bu t it also follo ws from our fu rther results, that R (Σ ) is E - principally symmetric if and only if Σ is a symm etric X -matrix. Moreo v er, for a symmetric X -matrix Σ to satisfy the ab o v e conditions is equiv alen t to it b eing an id emp oten t, hence the mo d ule R (Σ) is E -pr incipally generated and E -principally symmetric if and only if Σ is a so-called pr oje ction matrix (with elemen ts in X ). Our up coming Theorem 4.1 sa ys that suc h str u ctures coincide in turn with X -mo du les w ith (nece ssarily canonical) Hilb ert structure. 4. The p revious example is an instance of a more general situation. W e write Hilb ( Q ) for the qu an taloid whose ob jects are Q -mo d ules with Hilb er t structure and whose morphisms 10 are mo dule morphism s. An d w e write Matr ( Q ) f or the quan taloid whose ob jects are s ets and wh ose morphisms are matric es with elements in A : such a matrix Λ: S / / T is an indexed set of element s of Q , (Λ( y , x )) ( x,y ) ∈ S × T ∈ Q . Matrices comp ose straightforw ard ly with a “linear algebra formula”, and the iden tit y matrix on a s et S h as all 1’s on the diagonal and 0’s elsewhere. This matrix construction m ak es sense for any quan tale (and ev en quantalo id), and whenever Q is in v olutiv e then so is Matr ( Q ): the in v olute of a matrix is computed element wise. No w there is an equ iv alence of quan taloids 2 Hilb ( Q ) ≃ Proj ( Q ) , where the latter is th e quantal oid obtained b y splitting the sym metric idemp oten ts in Matr ( Q ), i.e. th e quan taloi d of so-called pr oje ction matric es with element s in Q . Exp licitly , if Σ: S / / S is such a pro jection matrix, then R (Σ) := { f : S / / Q | ∀ s ∈ S : f ( s ) = _ s ∈ S Σ( s, x ) ◦ f ( x ) } is a Q -mo d ule with in ner pr o duct and Hilb ert basis resp ectiv ely h f , g i := _ s ∈ S ( f ( s )) o ◦ g ( s ) and Γ := { f s : S / / Q : x 7→ Σ( x, s ) | s ∈ S } . This ob ject corresp ondence Σ 7→ R (Σ) extends to a Sup -functor fr om Proj ( Q ) to Hilb ( Q ): it is the restriction to symm etric idemp oten t matrices of the em b eddin g of the Cauch y completion of Q qua one-o b j ect Sup -category – i.e. the quan taloid obtained b y splitting al l idemp otent s of Matr ( Q ) – in to Mod ( Q ). Conv ersely , a mod ule M with inner pro d uct h− , −i and Hilb ert basis Γ obviously determines a pro j ection matrix Σ: Γ / / Γ with elements Σ( s, t ) := h s, t i ; this easily extends to a Sup -functor fr om Hilb ( Q ) to Proj ( Q ). Th ese t w o functors set up the equiv alence. 5. A notable consequence of th e previous example is the existence of an inv olution on Hilb ( Q ), induced by the ob vious inv olution on Proj ( Q ): th e inv olute of a morphism φ : M / / N in Hilb ( Q ) is th e unique m o dule morp hism φ o : N / / M charact erised b y h φ ( s ) , t i = h s, φ o ( t ) i for all basis elemen ts s of M and t of N . In the localic case (cf. the example ab o v e) w e can moreo ver p ro v e an alternativ e form ulation of the symm etry condition in P rop osition 3.5: an “op enness” condition formulated on the p rincipal elemen ts. In the n ext example we recall and explain this. Example 3.8 Let X b e a lo cale. Every elemen t u ∈ X is a symm etric idemp oten t, and th e op en sublo cale ↓ u ⊆ X is precisely the X -mo d ule of fixp oints of u ∧ − : X / / X . If M is an X -mo d ule for wh ic h eac h left adjoin t ζ : ↓ u / / M is op en , in the sense that for all x ≤ u and m ∈ M : ζ ( x ∧ ζ ∗ ( m )) = ζ ( x ) ∧ m, 2 [Resende, 2008] also notes the ob ject correspondence, but not th e morphism corresp ondence, and thus not the equiv alence of these quantaloids. 11 then it is E -principally s ymmetric. The con verse also h olds, pro vided that M is E -principally generated, in whic h case M is an ´ eta le X -mo dule in the terminology of [Heymans and Stubb e, 2009]. Pr o of : Let ζ : ↓ u / / M and η : ↓ v / / M b e left adj oin ts, supp ose that ζ is op en : with x := u and m := η ( v ) in the ab o v e formula, it follo ws that ζ ( ζ ∗ ( η ( v ))) = ζ ( u ) ∧ η ( v ) . Applying η ∗ (whic h preserves infima) giv es ( η ∗ ◦ ζ )( ζ ∗ ( η ( v ))) = η ∗ ( ζ ( u )) ∧ η ∗ ( η ( v )) . The r igh t hand sid e equals η ∗ ( ζ ( u )) b ecause η ∗ ( η ( v )) = v (the adjunction η ⊣ η ∗ splits). T h e left hand side equals η ∗ ( ζ ( u )) ∧ ζ ∗ ( η ( v )), b ecause the X -mo d ule morphism η ∗ ◦ ζ : ↓ u / / ↓ v is represent ed by η ∗ ( ζ ( u )) ≤ u ∧ v . Thus we get η ∗ ( ζ ( u )) ∧ ζ ∗ ( η ( v )) = η ∗ ( ζ ( u )), or in other wo rds η ∗ ( ζ ( u )) ≤ ζ ∗ ( η ( v )) . Going through the same argumen t but exc hanging ζ an d η p ro v es that M is E -principally symmetric. T o prov e the conv ers e, we assume that M is E -principally generated. W e show ed in [He ymans and Stub b e, 2009, Prop. 8.2] that then necessarily M is a lo cale and that there is a lo cale morphism 3 f : M / / X suc h th at m · x = m ∧ f ∗ ( x ) for all m ∈ M and x ∈ X . It follo w s easily from this charact erisation that, for all s ∈ Γ can , M / / M : m 7→ s ∧ m is an X -mo dule morphism. But un der the h yp othesis that M is E -prin cipally symmetric, w e can comp ose the left adjoin t mo d ule m orphism ↓ h s , s i can / / M : x 7→ s · x with its righ t adjoin t M / / ↓ h s, s i can : m 7→ h s, m i can to obtain M / / M : m 7→ s · h s, m i can . W e claim that these mo d ule morph isms are equal: we shall show that they coincide on element s of Γ can , whic h suffices b ecause Γ can is a Hilb ert b asis. Ind eed, for r, t ∈ Γ can w e can compute that h r , s ∧ t i can = h r , s i can ∧ h r, t i can = h r , s i can ∧ h s , t i can = h r , s · h s, t i can i can . (The first equalit y holds b ecause h r, −i can is a right adjoin t, and th e second equalit y holds b ecause h r, s i can ∧ h r , t i can = h s, r i can ∧ h r , t i can = h s, r · h r , t i can i can ≤ h s, t i can and similarly h r , s i can ∧ h s , t i can ≤ h r , t i can .) T aking the su premum ov er all r ∈ Γ can pro v es th at s ∧ t = _ r ∈ Γ can r · h r , s ∧ t i can = _ r ∈ Γ can r · h r, s · h s, t i can i can = s · h s, t i can 3 That locale morphism satisfies some further particular prop erties, which made u s call it a skew lo c al home o- morphism in that reference. 12 as claime d. F or an y left adjoin t ζ : ↓ u / / M in M o d ( X ) we can apply the ab ov e to s := ζ ( u ) ∈ Γ can , to fi nd that ζ ( ζ ∗ ( m )) = ζ ( u ) ∧ m for an y m ∈ M . Th is allo ws us to v erify in turn that for an y x ≤ u , ζ ( x ∧ ζ ∗ ( m )) = ζ ( ζ ∗ ( m )) · x = ( ζ ( u ) ∧ m ) · x = ( ζ ( u ) · x ) ∧ m = ζ ( x ) ∧ m as w an ted. ✷ The ab o v e direct argumen t relies on elementa ry order theory . There is a shorter alternativ e, using results in the literature: an ´ e tale X -mo d ule is the same thing as lo cal homeomorphism into X [Heymans and Stubb e, 200 9, Theorem 7.12], which is th e same thing as a Hilb ert X -mo dule [Ro drigues and Resende, 2008 , Theorem 3.15], w h ic h is the same thing as an E -principally generated and E -principally symmetric X -mo d ule (by our up coming Theorem 4.1). 4 (Sometimes) all Hilb ert structure is canonical The pr evious s ection was concerned with the c anonic al Hilb ert structure on a Q -mo dule M : we sho w ed that there is a c anonic al Hilb ert basis for the c anonic al (pre-)inner pro duct on M if and only if M is E -principally generated and E -principally sym m etric, t w o n atural n otions based on the b ehaviour of certain adjunctions in Mo d ( Q ). This section is d ev oted to the p erhaps surpr ising fact that, for a certain class of quan tales (con taining many cases of inte rest), the only p ossible Hilb ert str u cture is the canonical one. Theorem 4.1 L et Q b e a mo dular quantal fr ame, E ⊆ Q the set of symmetric idemp otents, and M a Q -mo dule. If M b e ars a Hilb ert structur e, then ne c essarily M i s E -princip al ly gener ate d and E -symmetric, and the involve d inner pr o duct is the c anonic al one, which mor e over is strict (and, by The or em 3.6, admits the c anonic al Hilb ert b asis). The pro of of the theorem shall b e giv en as a series of lemmas. The first one straigh tforw ardly extrap olates a result kn o wn to [Resende and Ro drigues, 2009] in the case of mo dules on a lo cale, and app ears in [Resende, 2008] . W e r ecalled the construction of the category Matr ( Q ) of matrices with entries in Q in Example 3.7 –4, and remark ed that whenev er Q is in v olutiv e then so is M atr ( Q ). Lemma 4.2 If Q is an involutive quantale and M is a Q - mo dule with an inner pr o duct h− , −i admitting a Hilb ert b asis Γ , then the fol lowing holds for al l m, n ∈ M : _ s ∈ Γ h m, s i ◦ h s, n i = h m, n i . In p articular, (Γ , h− , −i ) is a so-c al le d pro j ection matrix : a symmetric idemp otent in the invo- lutive quantaloid Matr ( Q ) . Pr o of : In h m, n i , use n = W s ∈ Γ s · h s, n i and apply the linearit y of h m, −i . ✷ The follo w ing lemma refers to the notion of total r e gularity , w h ic h we here s tate in a bare- b ones matrix-form, but which actually has d eep connecti ons with sheaf theory; it w as in tro du ced 13 in the con text of quantal oid-enric hed categorical structures by Stubb e [2005b] and pla y ed a crucial role in [Heymans and Stub b e, 2009] too. Lemma 4.3 L e t Q b e an involutive quantale, and M a Q - mo dule with an inner pr o duct h− , −i admitting a Hilb ert b asis Γ . The fol lowing statements ar e e quivalent: 1. for al l s ∈ Γ : s = s · h s, s i , 2. for al l s ∈ Γ : s ≤ s · h s, s i , 3. the pr oje ction matrix (Γ , h− , −i ) is totally regular , i .e. for al l s, t ∈ Γ : h s, t i ◦ h t, t i = h s, t i = h s, s i ◦ h s, t i . If Q mor e over satisfies q ≤ q ◦ q o ◦ q for every q ∈ Q , then these e quivalent c onditions always hold 4 . Pr o of : Due to the Hilb ert basis, s · h s, s i ≤ W t ∈ Γ t · h t, s i = s for any s ∈ Γ and th us (2 ⇒ 1). T o see that (2 ⇒ 3), compu te for s, t ∈ Γ that h s, t i = h s, t · h t, t ii = h s , t i ◦ h t, t i . Conv ersely , (3 ⇒ 2) b ecause, fixing a t ∈ Γ we ha v e for all s ∈ Γ that h s, t i = h s, t i ◦ h t, t i = h s, t · h t, t ii ; bu t therefore t = _ s ∈ Γ s · h s, t i = _ s ∈ Γ s · h s, t · h t, t ii = t · h t, t i . No w if moreo v er eve ry elemen t q ∈ Q satisfies q ≤ q ◦ q o ◦ q then we can compute for s, t ∈ Γ that h s, t i ≤ h s, t i ◦ h s, t i o ◦ h s , t i = h s, t i ◦ h t, s i ◦ h s, t i      ≤ h s, s i ◦ h s, t i ≤ h s , t i ≤ h s, t i ◦ h t, t i ≤ h s, t i precisely as w an ted in the seco nd condition. (W e used that h r , s i ◦ h s, t i ≤ h r, t i f or any r , s, t ∈ Γ, as follo ws trivially from the form ula in Lemma 4.2.) ✷ P articularly for a mo dular quan tale Q the ab o v e result is inte resting: b ecause q ≤ q ◦ q o ◦ q holds as consequence of th e mod ular la w, it follo ws that for ev ery Q -mo d ule with Hilb ert structur e its Hilb ert basis is totally regular. Next are tw o lemmas which con tain the imp ortant (and less straigh tforw ard) matter. Lemma 4.4 If Q is an involutive quantale, and M a Q -mo dule with an inner pr o duct h− , −i admitting a Hilb ert b asis Γ satisfying the e quivalent c onditions in L emma 4.3, then for any s ∈ Γ ther e is an adjunction Q h s,s i ⊥ s · − ( ( h s, −i h h M in Mo d ( Q ) . Writing E ⊆ Q for the set of symmetric idemp otents, such an an M is always E -princip al ly gener ate d; and if M is mor e over E -princip al ly symmetric then h− , −i c oincides with the c anonic al (pr e-)inner pr o duct h− , − i can . 4 More general ly , un d er th is condition it is true that any p ro jection matrix with en tries in Q , i.e. an y symmetric idemp otent in Matr ( Q ), is totally regular. 14 Pr o of : F or s ∈ Γ, comp ose the inclusion Q h s,s i / / Q with the mo dule morphism s · − : Q / / M to obtain a m o dule morp hism ζ s : Q h s,s i / / M : q 7→ s · q . Because w e assume s = s · h s , s i it follo ws that h s, s i ◦ h s, m i = h s · h s, s i , m i = h s, m i for any m ∈ M , and therefore the ob vious mo d ule morp hism h s, −i : M / / Q co-restricts to a mo dule morphism ζ ′ s : M / / Q h s,s i : m 7→ h s, m i . W e sh all sh o w that ζ s ⊣ ζ ′ s in M o d ( Q ); in fact, it suffices to pro v e that this adjunction holds in the catego ry of ordered sets and order-pr eserving maps. Th us, consid er q ∈ Q h s,s i and m ∈ M : if s · q ≤ m then q = h s, s i ◦ q = h s, s · q i ≤ h s, m i ; con v ersely , if q ≤ h s, m i then s · q ≤ s · h s, m i ≤ W t ∈ Γ t · h t, m i = m . The mo d ule M is E -prin cipally generated b ecause for an y m ∈ M we ha v e m = _ s ∈ Γ s · h s, m i = _ s ∈ Γ ζ s ζ ∗ s ( m ) ≤ _ n ζ ( ζ ∗ ( m ))    e ∈ E , ζ : Q e / / M left adjoin t in M o d ( Q ) o ≤ m. It follo ws directly from Lemma 4.2 that h m, n i = _ s ∈ Γ ( ζ ∗ s ( m )) o ◦ ζ ∗ s ( n ) , and from the ab o v e it is clear that th is is smaller than h m, n i can = _ n ( ζ ∗ ( m )) o ◦ ζ ∗ ( n )    e ∈ E , ζ : Q e / / M left adjoin t in M o d ( Q ) o . No w sup p ose that M is E -prin cipally symmetric. Fixing a left adjoint ζ : Q e / / M in Mo d ( Q ), with e ∈ E , we can compute for any s ∈ Γ that h ζ ( e ) , s i = h s, ζ ( e ) i o = ( ζ ∗ s ( ζ ( e ))) o = ζ ∗ ( ζ s ( h s, s i )) = ζ ∗ ( s · h s, s i ) = ζ ∗ ( s ); the symmetry w as cru cially used in the th ird equalit y , an d the assump tion that the equiv alen t conditions in Lemma 4.3 hold in the last on e. But morph isms in Mo d ( Q ) with domain M are equal if they coincide on the Hilb ert b asis Γ, so for all m ∈ M w e ha v e h ζ ( e ) , m i = ζ ∗ ( m ). Therefore we find that ( ζ ∗ ( m )) o ◦ ζ ∗ ( n ) = h m, ζ ( e ) i ◦ ζ ∗ ( n ) = h m, ζ ( e ) · ζ ∗ ( n ) i = h m, ζ ( ζ ∗ ( n )) i ≤ h m, n i . T o pass from the third to the four th line we use the counit of the adjunction ζ ⊣ ζ ∗ . All this means that h m, n i can ≤ h m, n i , and we are done. ✷ It is only in the next s tatement that we require Q to b e a m o dular quanta l frame. 15 Lemma 4.5 If Q is a mo dular quantal fr ame, and M a Q -mo dule with an inner pr o duct h− , −i admitting a Hilb ert b asis Γ , then M is E -princip al ly symmetric (for E ⊆ Q the set of symmetric idemp otents). Pr o of : W e shall pro v e that the first of the equiv alen t cond itions in P r op osition 3.5 holds for the given inner pr o duct on M ; as we ha v e r emark ed r igh t after the pr o of of th at Prop osition, this suffices to infer the E -principal symmetry of M . Because h er e we assume M to ha v e a Hilb ert basis Γ, it in fact suffices to sho w that, for an y e ∈ E , an y left adjoin t ζ : Q e / / M and any s ∈ Γ: ζ ∗ ( s ) = h ζ ( e ) , s i . First remark that, with these notatio ns, ζ ( e ) · ζ ∗ ( s ) = ζ ( ζ ∗ ( s )) ≤ s trivially holds. On the other hand, u sing all assum ptions we can compute that e = e ∧ ζ ∗ ( ζ ( e )) (unit of ζ ⊣ ζ ∗ ) = e ∧ ζ ∗  _ s ∈ Γ s · h s, ζ ( e ) i  (Γ is a Hilb ert b asis) = e ∧ _ s ∈ Γ  ζ ∗ ( s ) ◦ h s, ζ ( e ) i  ( ζ ∗ is a mo dule morph ism) = _ s ∈ Γ  e ∧ ζ ∗ ( s ) ◦ h s, ζ ( e ) i  ( Q is a f r ame) ≤ _ s ∈ Γ  e ◦ h s, ζ ( e ) i o ∧ ζ ∗ ( s )  ◦ h s, ζ ( e ) i (b y the mo d ular la w) = _ s ∈ Γ  h ζ ( e ) , s i ∧ ζ ∗ ( s )  ◦ h s , ζ ( e ) i (symmetry) ≤ _ s ∈ Γ h ζ ( e ) , s i ◦ h s, ζ ( e ) i (trivially) = h ζ ( e ) , ζ ( e ) i . (b y Lemma 4.2) Hence, com bining b oth the pr evious inequalitites, ζ ∗ ( s ) = e ◦ ζ ∗ ( s ) ≤ h ζ ( e ) , ζ ( e ) i ◦ ζ ∗ ( s ) = h ζ ( e ) , ζ ( e ) · ζ ∗ ( s ) i ≤ h ζ ( e ) , s i and we hav e the “ ≤ ” of the required equalit y . T o see that also “ ≥ ” holds, w e first apply the mo dularity of Q again to compu te e = _ s ∈ Γ  e ∧ ζ ∗ ( s ) ◦ h s, ζ ( e ) i  (as ab ov e) ≤ _ s ∈ Γ ζ ∗ ( s ) ◦  ( ζ ∗ ( s )) o ◦ e ∧ h s, ζ ( e ) i  (b y the mod ular la w) ≤ _ s ∈ Γ ζ ∗ ( s ) ◦ ( ζ ∗ ( s )) o . (trivial) 16 No w we combine this with the first in equalit y th at w e pr ov ed to obtain h ζ ( e ) , s i = e ◦ h ζ ( e ) , s i (trivial) ≤ _ t ∈ Γ ζ ∗ ( t ) ◦ ( ζ ∗ ( t )) o ◦ h ζ ( e ) , s i (b y the ab o v e) = _ t ∈ Γ ζ ∗ ( t ) ◦ h ζ ( e ) · ζ ∗ ( t ) , s i (“conjugate-l inearit y” of inn er pro du ct) ≤ _ t ∈ Γ ζ ∗ ( t ) ◦ h t, s i (b ecause ζ ( e ) · ζ ∗ ( t ) ≤ t ) = ζ ∗  _ t ∈ Γ t · h t, s i  ( ζ ∗ is a mo dule morph ism) = ζ ∗ ( s ) . (Γ is a Hilb ert basis) and w e are done. ✷ Relying on categorical mac hinery , the pr evious Lemma can alternativ ely b e pr o v ed as follo ws: Bearing in mind Examp le 3.7–5, the requirement that ζ ∗ ( s ) = h ζ ( u ) , s i at the end of the first paragraph of the pro of ab ov e is equiv alen t to asking for ζ ∗ = ζ o in the category Hilb ( Q ). Bu t Q b eing a mo d ular quantal fr ame is equiv alent to the matrix qu an taloid M atr ( Q ) b eing mo dular, whic h in turn implies that the quan taloid Proj ( Q ) of pro jection matrices, obtained by splitting the symm etric idemp oten ts in Matr ( Q ), is mo d ular to o. In an y mo d ular quantal oid, th e right adjoin t of a m orphism, should it exist, is necessarily its in v olute: this th us holds in Proj ( Q ), and also in its equiv alen t Hil b ( Q ). Therefore in particular ζ ∗ = ζ o for an y left adjoint ζ : Q e / / M , as w an ted. Lastly we ha v e a simp le lemma asserting the strictness of inner pro ducts in certain cases. Lemma 4.6 L e t Q b e an involutive q uantale in which q ≤ q ◦ q o ◦ q holds for any q ∈ Q . If M is a Q -mo dule with an inner pr o duct h− , −i admitt ing a Hilb ert b asis Γ , then this inner pr o duct is strict. Pr o of : Let q ∈ Q : if q o ◦ q = 0 then q ≤ q ◦ q o ◦ q = q ◦ 0 = 0 hence q = 0. No w supp ose that h m, m i = 0 for an m ∈ M . The formula in Lemma 4.2 implies that h s, m i o ◦ h s , m i = h m, s i ◦ h s, m i ≤ _ t ∈ Γ h m, t i ◦ h t, m i = 0 for all s ∈ Γ, whence h s, m i = 0 for all s ∈ Γ. But th en m = W s ∈ Γ s · h s, m i = 0 as required. ✷ Ha ving all these lemmas, w e assem ble the pro of of the statemen t in the b eginning of this section. Pr o of of The or em 4.1 : Bec ause Q is b y h yp othesis a mo du lar quan tal frame we hav e b y Lemma 4.5 that M is E -principally symm etric. It follo ws from Q ’s m o dularity and Lemma 4.3 that Lemma 4.4 applies, sho wing that M is E -principally generated. T ogether with the f act that M is E -principally symmetric th is m oreo v er en tails the equalit y of the giv en inpro du ct with the canonical one. Finally , the strictness of the (canonical ) inner p ro du ct is a consequence of Lemma 4.6. ✷ Example 4.7 W e end with examp les that refer to the cate gory Hilb ( Q ) of Hilbert mo dules, and particularly to applicatio ns in sheaf theory . 17 1. As in Example 3.7–4 we write Hilb ( Q ) for the qu an taloid of Q -mo d ules with Hilb ert s truc- ture. F or a mo d ular quant al f rame Q , Theorem 4.1 allo ws us to consid er Hilb ( Q ) as a full sub quantalo id of Mo d ( Q ): there is only one relev ant Hilb ert structure on a Q -mo d ule. Moreo ver, Lemma 4.5 implies that, whenev er φ : M / / N is a left adjoint in Hilb ( Q ), th en φ ⊣ φ o (compare with Examp le 3.7–5). Because in this case ev ery symmetric idemp oten t in Matr ( Q ) is totally regular, w e ther efore get the equiv alences of qu an taloids Hilb ( Q ) ≃ Proj ( Q ) ≃ Dist o ( Q E ) where Q E denotes the quantalo id obtained as un iv ersal s plitting of the symmetric idem- p oten ts of Q , and D ist o ( Q E ) is the full sub quantalo id of Di st ( Q E ) (= the quantalo id of Q E -enric hed categories and distributors [Stu b b e, 2005a]) determined by the symmetric Q E -c ate gories . 2. Shea v es on sites: F or a small site ( C , J ) and Q the asso ciated mo d ular quantal frame as in Example 2.3–6, the category Sh ( C , J ) is equiv alent to the category of Q -mo dules with canonical Hilb ert structure and the left adjoint mo dule morphisms b et w een them: Sh ( C , J ) ≃ M ap ( Hilb ( Q )) . With a bit more w ork th is can b e reph rased as equiv alent quan taloids: Rel ( Sh ( C , J )) ≃ Dist o ( Q ) ≃ Hil b ( Q ) . Sketch of the pr o of: Let Q b e as in Examp le 2.3–6. W alters [19 82] p ro v ed that S h ( C , J ) is equiv alent to M ap ( Dist o ( Q )), the f ull sub category of Map ( Dist ( Q )) determined by the symmetric Q -categories. In the previous example w e ind icated that Hilb ( Q ) ≃ Proj ( Q ) ≃ Dist o ( Q E ), hence it su ffi ces to pr o v e that Dist ( Q ) ≃ Dist ( Q E ). But th is f ollo w s f rom the fact that Q , regarded as a sub quanta loid of Q E , is dense in Q E : for any X ∈ Q E , id X = W i ∈ I f i ◦ f ∗ i with eac h f i : X i / / X a left adjoin t with X i ∈ Q 0 . This prop ert y is du e to the mo du larit y of Q and the coreflexiv e idemp otents ( e ≤ id dom ( e ) ) of Q (corresp onding to closed sieves on C ) b eing su prema of the form ab o v e. ✷ 3. Shea v es on an ´ etale grou p oid: W e understand that Resende [2008] defines a “sheaf ” on an inv olutiv e quanta le Q to b e a Q -mo d ule M with Hilb ert stru cture satisfying mor e over W Γ = ⊤ M , and prov es – via the corresp ondence b et we en ´ et ale group oids and in v erse quan tal frames from [Resende, 2007] – that, f or an ´ etale group oid G , the top os of G - shea v es is equiv alen t to the categ ory with as ob jects those “sh ea v es” on an in ve rse qu an tal frame O ( G ) and as m orphisms th e left adjoint O ( G )-mo du le morphisms th at hav e their in v olute as righ t adjoin t (whic h h e describ es as “direct image homomorphism s ”). This ma y app ear to b e in con tradiction with the examples ab o v e: shea v es (on a site) can b e describ ed as mo dules (on a mo du lar quanta l frame) with Hilb er t stru cture witho ut an y further conditions. Ho wev er it tur n s out that, when Q is an in v erse quanta l frame (as in the main example of [Resend e, 2008]) , then the extra condition is an ywa y a c onse q u enc e of the features of Q (see the pro of b elo w). W e further rep eat from Example 4.7–1 that, b ecause O ( G ) is a mo dular quan tal frame, an y left adjoin t in Hilb ( O ( G )) has its in v olute as 18 righ t adjoin t (but this need not b e so for inv olutiv e qu an tales in general, where we think this is an imp ortan t extra condition). Conclusivel y , b y Theorem 4.1 the top os of sh ea v es on an ´ etale group oid G is equiv alen t to Map ( Hilb ( O ( G ))). Pr o of: I f Q is an inv erse quantal f rame and M a Q -mo du le with inner p r o duct h− , −i admitting a Hilb ert basis Γ, w e may assu me withou t loss of generalit y that Γ is maximal in the follo wing w a y: Γ = n s ∈ M    ∀ m ∈ M : s · h s, m i ≤ m o . If p ∈ Q is a p artial unit and s ∈ Γ then s · p ∈ Γ: b ecause for all m ∈ M , ( s · p ) · h s · p, m i = ( s · pp o ) · h s, m i ≤ s · h s, m i ≤ m ; hence certainly s · p ≤ W Γ. Since ⊤ Q is by assumption the j oin of all p artial u nits, this implies s · ⊤ Q ≤ W Γ, wh ence ⊤ M = _ s ∈ Γ s · h s, ⊤ M i ≤ _ s ∈ Γ s · ⊤ Q ≤ _ Γ whic h pr o v es the claim. ✷ 5 Concluding remarks In this pap er w e prov ed th e follo wing results in the theory of quan tale mo du les: (1) ev ery mo d ule on an in v olutiv e quan tale Q b ears a canonical (pre-)inner pro duct; (2) th at canonical (pre-)inner pro du ct admits the canonical Hilber t b asis if and only if the mo du le is p rincipally generated and prin cipally sym m etric; an d (3) if Q is a mo dular qu antal fr ame then the only p ossible Hilb ert structur e (= inner pro du ct plus Hilb ert basis) on a Q -mo du le is the canonical one. In the examples we explained the use of these results in sheaf theory: we argued in particular that the categ ory of shea v es on a site ( C , J ) is equiv alen t to a category of quanta le mo dules with (canonical) Hilb ert structure. These resu lts are a natural con tin uation of our pr evious w ork. Whereas Stub b e [2005b] describ ed ordered shea v es on a quan taloid Q as particular Q -enric hed categorica l stru tures, Heymans and S tubb e [2009] reformulate d this – via the corresp ondence b et we en co complete Q -categ ories and Q -mo dules, and the particular role of Q -mo dules in the theory of ordered shea v es on Q [Stubb e, 2006, 2007 ] – in a mo du le-theoretic language: ordered shea v es on Q are the same thing as pr incipally generated Q -mo d ules. Th e m aterial in this pap er suggests that the “symmetrically ordered” sh ea v es (i.e. shea v es tout c ourt ) on an inv olutiv e quantale Q are those p rincipally generated Q -mo dules whic h are moreo v er principally symmetric. The latter in tur n coincide with mo d ules b earing a canonical Hilb ert structure (wh ic h, for mo du les on a mo dular quantal frame, is the only p ossible Hilb ert stru cture). Our curr en t researc h is concerned with a further elab oration of that nov el notion, “prin- cipal symmetry”: we extend it f rom quantale mo d u les to quantal oid mo dules, and even to quan taloid-enric hed categories. A future p ap er shall in p articular con tain all remaining d etails from Examples 2.3–6 and 4.7–2. 19 References [1] [Renato Betti and Aur elio C arb oni, 1983] Notio n of top ology for b icatego ries, Cahiers T op. et G´ eo m. Diff. 24 , pp. 19–22. [2] [Peter J. 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