Noncommutativity as a colimit

Noncomm utativit y as a colimit Benno v an den Berg ∗ and Chris Heunen † Septem b er 23, 2018 Abstract W e giv e sub stance to the motto “ev ery partial algebra is the colimit of its total sub algebras” by proving it for p artial Boolean algebras (includ in g orthomodu lar la ttices), the new n otion of partial C*-algebras (including noncommutativ e C*-algebras), and v ariations such as partial complete Boolean algebras and partial A W*-algebras. Both pairs of results are related by t aking pro jections. As corollari es w e find extensions of Stone duality and Gelfand duality . Finally , we in vestigate the extent to whic h the Bohrification construction [9], that w orks on partial C*-algebras, is functorial. 1 In tro duction This pap er is intended as a contribution to the Bohrification pro gramme [9], which tries to give a mathematically precis e expression to Bohr’s do ctrine of classical concepts, sa ying that a quantum mec hanical s y stem is to b e understo o d through its classica l frag men ts. On Bohr’s view, as unders too d within this progra mme, quantum mechanical systems do not allow a ‘g lobal’ interpretation as a classica l system, but they do so ‘lo cally ’. The progra mme essentially makes t wo claims ab out these classical ‘snapshots’. Firstly , it is only thro ugh these snapshots that the behaviour of a system and physical reality can b e understo o d. Secondly , these snaps ho ts cont ain all the informatio n about the system tha t is ph ys ic ally relev ant. The main r esult of [9] (inspired b y ea rlier work o f Butterfield, Isham and D¨ oring [5]) is that the collection o f these classica l s napshots can b e seen as for m- ing a single classical system in a suitable to pos – its s o-called Bohr ific a tion; we will briefly recall the details of this construction in Section 7. The implication of this result is that a quantum mechanical system can be seen as a class ical one, if one ag rees that nothing physically relev ant is lost b y co nsidering c la ssical ∗ T ec hnische Universit¨ at Darmstadt, F ach ber eic h Mathematik. † Oxford Universit y Computing Laboratory , supported by the Netherlands Organisation for Scien tific Research (NWO) . Pa rt of this wo r k was p erformed while the author visi ted the IQI at Caltec h. 1 snapshots and if one is willing to change the logic fro m a classical in to a n intu- itionistic one. The question is le ft o pen as to how stro ng these premises ar e: ho w m uch, if anything, of the information ab out the original q ua n tum mechanical system is lost by conceding in this wa y to consider it as a clas sical system? This article inv estiga tes ho w muc h information ab out a quantum mechanical system can b e recons tructed fro m its Bo hrification by means of colimits. The fact that quantum mechanical systems can be mo delled as no nco mm utative algebr a s, and classical systems as commutativ e ones, explains the title “nonco mm utativity as a colimit”. Our conceptual co n tributions to the B ohrification pr ogramme a re tw ofold. First, we contend that the prog ramme is mo st na turally see n as concer ned with p artial algebr as , i.e. sets o n which algebraic structure is only partially defined. Such partia l a lgebras ar e e quipped with a binar y (‘commeasurability’) relatio n, which holds b et ween tw o elements whenev er they can b e element s of a sing le classical snapshot. In this setting, commeasura bilit y is often also called com- m utativity , s ince in typical examples commeasurability mea ns c o mm utativity in a totally defined algebraic structure . W e will show that a mea ning ful theory of such partial algebras can be developed. In particular, a nd this is o ur s econd contribution, we will indicate why Bo hrification makes sense for par tial a lgebras and use this to inv estigate the functorial a spects o f this constructio n. The idea to consider partial algebras is not new. In fact, in their clas sic pap er [14], Ko chen a nd Sp eck er use the languag e of partial a lgebras to state their famous result, s aying that many algebra s o ccur ring in quantum mechanics cannot b e embedded (as pa r tial algebra s) into commutativ e ones (see also [17]). Their interpretation of this fact as excluding a hidden v a riable interpretation o f quantum mechanics has r emained s omewhat controv ersial. In the B o hrification progra mme, this result is taken just as a mathema tical confir ma tion of the view that qua n tum mec hanica l systems do no t allow for g lobal interpretations as classical systems. The res ult by Ko chen and Spe cker, to gether with the fact that Bo hrification works for partial algebra s, indicates that the Bohrifica tion progr amme is most naturally dev elop ed in the context of partial algebras. This led us to develop a new notion of ‘par tial C*-a lg ebra’, to the s tudy of which most of this pa per is devoted. Our main technical result is that such a partial C*-algebr a is the col- imit of its to tal (co mmutative) s ubalgebras, which explains the rela tion b et ween a partial algebra and its classica l snapshots in categorical terms. In more detail, the c o n tents of this paper ar e as follows. First we cons ider Ko chen and Sp eck er’s notion of a partial Bo olea n alg ebra, as a kind of toy ex a m- ple, and prov e our main result for them in Section 2, as well as for the v ariation of par tial complete Boo lean algebr as. This a lso ma k es precise the widespread int uition that an orthomo dular lattice is an amalg amation of its B o o lean blo cks. In this simple setting the theorem alrea dy ha s interesting coro llaries, as it allows us to derive an adjunction extending Stone duality in Section 3. But o ur ma in int er e st lies with par tia l C* -algebra s , which Section 4 studies; we als o prove the main result for v aria tions, suc h as A W*-algebra s and Rick art C*-algebra s in tha t section. This e na bles an adjunction extending Gelfa nd dua lity and puts 2 the Ko chen-Specker theorem in ano ther light in Section 5. The tw o pa rallel set- tings of Bo olea n algebra s and C*-algebr as are related by ta king pro jections, as Section 6 discusse s. Finally , Section 7 shows that the Bohr ification construction works for partia l C*-a lgebras a s w ell: we inv estig a te its functor ia l prope r ties, and conclude that its essence is in fact (a tw o -dimensional version of ) the colimit theorem. 1 2 P artial Bo olean algebras W e start with recalling the definition of a partial Bo olean a lg ebra, a s introduced by Ko chen and Speck er [14]. Definition 1 A p artial Bo ole an algebr a consists of a set A with • a reflexive a nd symmetric binary ( c omme asura bility ) relatio n ⊙ ⊆ A × A ; • elements 0 , 1 ∈ A ; • a (total) unary op eration ¬ : A → A ; • (partia l) binary op eratio ns ∧ , ∨ : ⊙ → A ; such that ev er y set S ⊆ A of pair wise commeasurable elements is contained in a s et T ⊆ A , whose elements are also pairwise commeasurable, and on which the above o per ations determine a Bo olean algebra structure. 2 A morphism of partial Bo olea n algebr as is a function tha t preserves com- measurability and all the a lgebraic struc tur e, whenever defined. W e write PBo olean for the resulting category . Clearly , a partial B oo lean alg ebra whose commea surability relation is total is nothing but a Bo olean a lgebra. F or another example, if we declare t wo elements a, b of an orthomo dular lattice to be commeas ur able when a = ( a ∧ b ) ∨ ( a ∧ b ⊥ ), as is standard, any o rthomo dular lattice is seen to b e a par tial Bo olean alg ebra. In this case the ab ov e o bserv atio n a bout totality beco mes a k no wn fact: an orthomo dular lattice is a Bo olean algebra if and only if any pair of elemen ts is commeasura ble [13]. W e introduce s ome notation and terminology . If A is a par tial Bo olean alge- bra, then a s ubset T of pair wise commeasura ble elements which is clos ed under all the algebraic op eratio ns of A will b e called a c omme asur able or t otal sub- algebra. Clearly , a co mmea surable subalgebra ha s the struc tur e of a Bo olean algebra. Note that if A is a par tial Bo olean algebra and S is subset of pairw is e commeasura ble elements, then ther e m ust b e a smal lest commeasur able s ubal- gebra T that contains S : it has to cons ist of the v alues of Bo o lean expressions built from elements of S . W e denote it b y A h S i . 1 Both the categ ori es of Bo olean algebras and of comm utative C*-algebras are algebraic, i.e. monadic o ve r the category of sets [10]. W e exp ect that the definitions and results of the presen t article can b e extended to a more general theory of partial algebra, but r efrain fr om doing so b ecause the tw o categories mentione d are our main motiv ation. 2 Note that this means that T m ust cont ain 0 and 1 and has to b e closed under ¬ , ∧ and ∨ . 3 Given a partial Bo olean a lgebra A , the collection of its commeasur able subalgebra s C ( A ) is partia lly or dered b y inclusio n. In fact, C is a functor PBo olean → POrder to the category of pos ets and monotone functions. Re- garding p osets as categ ories, C ( A ) gives a diagr am in the category PBo ol ean (in fact, it also defines a dia g ram in the category Bo olean of Bo ole an algebr as). The fo llowing prop osition lists some ea sy pr op erties of this diagr a m. Prop osition 2 L et A b e a p artial Bo ole an algebr a. (a) The le ast element of the p oset C ( A ) is A h 0 i = A h 1 i = { 0 , 1 } . (b) The atoms of C ( A ) ar e A h a i = { 0 , a, ¬ a, 1 } for nontrivial a ∈ A (an element p of a p oset with le ast element 0 is an atom when ther e ar e no elements x such that 0 < x < p ). (c) Two total su b algebr as S and T have a c ommon upp er b ou n d in C ( A ) if and only if al l elements of S ar e c omme asur able with al l t he elements of T . (d) A is a (total) Bo ole an algebr a if and only if the p oset C ( A ) is filter e d (me aning that any two elements have an upp er b oun d). In that c ase, A is the lar gest element of the p oset C ( A ) . Proof P arts (a) and (b) are easy to s how and therefore w e omit their pro ofs. T o see (c), observe that if total s ubalgebras S and T hav e a co mmon upp er bo und U , then all elements of S ar e co mmeasurable with all elemen ts in T , bec ause all elements of S a nd T b elong to the co mmea surable subalgebra U . Conv ersely , if all elements of S are commeasur able with those of T , then A h S ∪ T i is an upper bo und (in fact, the least uppe r bound) in C ( A ) of S a nd T . If A is total, then A is the top elemen t of C ( A ) and hence C ( A ) is filter ed. If, on the other hand, C ( A ) is filtered, then fo r any tw o elements a, b ∈ A the total subalgebras A h a i a nd A h b i must hav e an upp er b ound, which implies (b y (c)) that a a nd b are commeasura ble. This shows (d).  Remark 3 One ca n s how that C ( A ) is a directed complete partial o rder (dcpo), which is algebra ic, and is such that for every co mpact ele ment x the downset ↓ x is dually is omorphic to a finite partition lattice. The main r esult of the pap er [6] suggests that every such dcp o is the C ( A ) o f a unique partial Bo olean algebr a A . Whether similar results ho ld for partial C*-alg ebras remains to be seen. W e a r e now ready to prove the first version of our main result. Theorem 4 Every p artial Bo ole an algebr a is a c olimit of its (finitely gener ate d) total su b algebr as. Proof Let A be a partial Bo olean a lgebra, and consider its diagram of (finitely generated) commeas urable subalgebra s C . Define functions i C : C → A by the inclusions; these a re morphis ms of PBo olean b y construction. Moreov er, they form a coco ne ; w e w ill prov e that this co cone is univ ersal. If f C : C → B is another co co ne, define a function m : A → B by m ( a ) = f A h a i ( a ). It now 4 follows from the ass umption that the f C are mo r phisms of PBo olean that m is a well-defined morphism, to o. T o see this, w e need to show that m pr eserves commeasura bilit y and the alg ebraic op erations of A . W e chec k that m preserves commeasura bilit y , omitting a very similar v erifica tion tha t it also pr eserves the algebraic op eratio ns. So, if a ⊙ b , then a lso m ( a ) ⊙ m ( b ), s ince A h a, b i is a to ta l subalgebra and m ( a ) = f A h a i ( a ) = f A h a,b i ( a ) , m ( b ) = f A h b i ( b ) = f A h a,b i ( b ) , bec ause the f C form a co cone. O ne eas ily verifies that f C = m ◦ i C , and that m is the unique such mor phism.  Kalmbac h’s “Bundle Lemma” [13] gives sufficient conditio ns for a family of Bo olean algebr as to combine in to a partia l B oo lean algebra , so that it could b e regar ded as a conv erse of the previo us theorem. Notice that the morphisms of PBo ol ean are the weak est o nes for w hich the previous theorem holds. F or example, even when A and B are or thomo dular la t- tices, the mediating morphis m m : A → B in the pro of of the previous theo r em need not be a homomorphism of orthomo dular lattices. F or a co un terex ample, consider the function 1 ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧   ✾ ✾ ✾ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ a ¬ a b ¬ b 0 ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❃ ❃ ❃ ✆ ✆ ✆ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ m − → 1 ✆ ✆ ✆ ❃ ❃ ❃ c ¬ c 0 ✾ ✾ ✾    given by m (0) = 0 , m ( a ) = m ( b ) = c, m ( ¬ a ) = m ( ¬ b ) = ¬ c, m (1) = 1. It preserves 0, 1 , ¬ and ≤ . The only co mmeasurable suba lgebras of the do main A are A h 0 i , A h a i and A h b i , and m preserves ∧ when r e stricted to those. Ho wever, m ( a ∧ b ) = m (0) = 0 6 = c = c ∧ c = m ( a ) ∧ m ( b ). (Of course, { 0 , a, b, 1 } is a Bo olean algebr a, but it is not a commea surable subalgebr a, as it do es not hav e the same negation ¬ as A ; see also fo otnote 2.) It follows from the pre vious theorem that the par tial Bo olean alg e br a with a prescrib ed pos et of total subalgebras is unique up to isomorphism (of partial Bo olean alg ebras). T o actually reconstruct a partial Bo olea n algebra fro m its total subalg ebras, this should b e c omplemen ted by a description of c olimits in the category PBo o lean , a s we now discuss. The copro duct of a fa mily A i of pa r tial Bo o lean algebr as is g ot by tak- ing their disjoint union, ide ntifying all the ele ments 0 i , and iden tifying all the elements 1 i . Notice that elements fro m different summands A i are never com- measurable in the copro duct. In par ticular, the initial ob ject 0 is the partial Bo olean a lgebra { 0 , 1 } with tw o distinct elemen ts. Incident a lly , PBo ol e an is complete. Pro ducts and eq ua lizers of partial Bo olean alg e bras are constr ucted as in the catego ry of sets; pro ducts hav e com- measurability and alg e braic structure defined compo nen twise, and equa liz ers 5 hav e subalg ebra s tr ucture. Hence the limit of a diagram of Bo o lean a lgebras is the sa me in the catego ries of Bo o lean algebra s a nd of partial Bo olea n algebr as. In particula r , the terminal ob ject 1 is the partial Bo o lean algebr a with a single element 0 = 1. Co equalizers are har der to describ e constructively , but the following theorem prov es they do exist. Theorem 5 The c ate gory PBo olean is c omplete and c o c omplete. Proof W e are to show that PBo olean has co equalizers, i.e. that the diago nal functor ∆ : PBo olean → PBo olean ( • ⇒ • ) has a left adjoint (where • ⇒ • is the free categor y gener ated by the gra ph consisting o f tw o vertices and tw o para llel arrows b et ween them). Since we alr eady know that PBoo lean is complete a nd ∆ pr eserves limits, F reyd’s adjoint functor theo r em shows that it suffices if the following solution set co ndition is satisfied [15, V.6]. F o r each f , g : A → B in PBo olean there is a set-indexed family h i : B → Q i such that h i f = h i g , and if hf = hg then h factorizes through some h i . T ak e the colle c tion of h i : B → Q i to co mprise all ‘quotients’, i.e. (isomor - phism class es o f ) sur jectio ns h i of par tial Bo o le an algebra s such that h i f = h i g . This collection is in fa ct a s e t. The pro of is finishe d by observing that every mor- phism h : B → Q of partia l Bo olean algebras factors thr o ugh (a surjectio n onto) its s et-theoretical image , whic h is a partial Boo lean subalgebra of Q , inheriting commeasura bilit y fr om B and algebraic opera tions from Q .  2.1 V ariations Results similar to those ab ov e hold for many classe s of Bo olean algebr as, such as complete or c o un tably co mplete Bo olea n algebras. F or example, the former v ar iation can b e defined as follows. Definition 6 A p artial c omplete Bo ole an algebr a consists of a partia l Bo olean algebra together with a (par tia l) op era tio n _ : { X ⊆ A | X × X ⊆ ⊙} → A such that ev er y set S ⊆ A of pair wise commeasurable elements is contained in a s et T ⊆ A , whose elements are also pairwise commeasurable, and on which the above o per ations determine a complete Bo olean algebra structure. 3 A morphism of par tial complete Bo olea n a lgebras is a function that preserves commeasura bilit y and all the alg e br aic structure, including W , whenever defined. W e wr ite PCBo o l ean for the resulting categor y . A version of our main theorem als o holds for such par tia l complete Bo o lean algebras , when w e define a total subalgebra of a partial complete Boo lean alge- bra to be a total subalgebra of the underlying partial Bo o lean a lg ebra that is additionally closed under W . 3 So T is not only closed under ¬ , ∧ and ∨ , but also under W . 6 Theorem 7 Every p artial c omplete Bo ole an algebr a is a c olimit of its total sub algebr as. Proof Completely analog ous to Theorem 4.  3 Stone dualit y The full s ubcateg ory of PBo o l ean consisting of (total) Bo o le an algebras is just the categor y Boo lean of Boo lean algebras and their homo morphisms. This category is dual to the ca tegory of Stone spaces a nd contin uous functions via Stone dualit y [10]: Bo olean Σ / / ∼ Stone op , Lo c ( − , { 0 , 1 } ) o o (1) where Σ( A ) is the Sto ne sp ectrum of a Bo olean alge bra A . The dualizing ob ject { 0 , 1 } is b oth a loca le and a (partial) Boo lean a lgebra; r e c all that it is in fact the initial partial Bo olea n a lg ebra 0 . One might exp ect that the catego r y of partial Bo olean alg ebras e nters Stone duality (1), a nd indeed the colimit theo rem, Theor em 4, enables us to pr ove the following extension. Prop osition 8 Ther e is a r efle ction PBo olean K / / ⊥ Stone op , Lo c ( − , { 0 , 1 } ) o o in which the fun ctor K is determine d by K ( A ) = lim C ∈C ( A ) op Σ( C ) . Proof Let A b e a partial Bo olean alge br a and X a Stone space . Then there are bijective corresp ondences: f : K ( A ) = lim C ∈C ( A ) op Σ( C ) → X (in Stone op ) ∀ C ∈C ( A ) . f C : Σ( C ) → X (in Stone op ) ∀ C ∈C ( A ) . g C : C → Lo c ( X , { 0 , 1 } ) (in Bo olean ) g : A → Lo c ( X , { 0 , 1 } ) (in PBo olean ). The first cor r esp ondence ho lds by definition of limit, the middle corr esp ondence holds b y Stone duality (1), a nd the la st corres p ondence holds by Theore m 4. Since all corre spo ndenc e s ar e natural in A and X , this establis hes the a djunction K ⊣ Lo c ( − , { 0 , 1 } ). Finally , since a Bo olean algebra is triv ia lly a colimit of itself in PBo olean , the adjunction is a reflection.  7 Theorem 9 The r efle ction K ⊣ Lo c ( − , { 0 , 1 } ) extends Stone duality, i.e. the fol lowing diagr am c ommutes (serial ly). Bo olean i I w w ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ Σ # # ∼ PBo olean K . . ⊥ Stone op Lo c ( − , { 0 , 1 } ) n n Lo c ( − , { 0 , 1 } ) c c Proof If A is a Bo o le an a lgebra, it is the initial ele men t in the diagr am C ( A ) op by Prop osition 2(b). Hence K ( A ) = lim C ∈C ( A ) op Σ( C ) = Σ( A ).  Corollary 10 Bo ole an algebr as form a r efl e ctive ful l sub c ate gory of the c ate gory of p artial Bo ole an algebr as, i.e. the inclusion Bo ole an ֒ → PBo olean has a left adjoint L : PBo olean → Bo ol ean . Proof The adjunctions of the previous theor em compo se, giving the requir ed left adjoin t as L = Lo c ( − , { 0 , 1 } ) ◦ K .  4 P artial C*-algebras The definitions of pa rtial C*-alg e br as and their morphisms closely r esemble those of partial B o o lean algebras . Indeed, bo th ar e instances of the pa rtial algebras of Ko chen a nd Sp eck er, ov er the fields Z 2 and C , resp ectiv ely . How ever, partial C* -algebra s also hav e to account for a nor m and in volution, calling for some c hang e s that w e now sp ell out. Definition 1 1 A p artial C*-algebr a consists of a set A with • a reflexive a nd symmetric binary ( c omme asura bility ) relatio n ⊙ ⊆ A × A ; • elements 0 , 1 ∈ A ; • a (total) inv olution ∗ : A → A ; • a (total) function · : C × A → A ; • a (total) function | |−| | : A → R ; • (partia l) binary op eratio ns + , · : ⊙ → A ; such that ev er y set S ⊆ A of pair wise commeasurable elements is contained in a s et T ⊆ A , whose elements are also pairwise commeasurable, and on which the above o per ations determine the structure of a comm utative C* -algebra . 4 4 This en tails that T contains 0 and 1, is closed under al l algebraic operations, and is norm-complete. 8 It follows from the last condition in the definition of a partial C*-alg ebra that commeas urable element s in a par tia l C*-a lg ebra have to commute. In fact, as with partial B oo lean algebr as, partia l C*-algebra s whose commeasurability relation is total ar e no thing but c ommu tative C*-algebra s. W e again define the notion of a c omme asur able or c ommutative subalgebra of a partial C * -algebra A in the ob vio us wa y as a subset T of A of pairwise commeasura ble elements o n which the op erations of A deter mine a co mmutative C*-algebr a structure . Also, if S is subset o f pairwise commeasura ble elements, then a gain there mu st b e a smal lest commeasurable subalg ebra T that contains S : simply take the in ters ection of all such subalgebras T . Alternatively , one can construct it as the s et o f thos e e lemen ts in A that are limits of seq ue nc e s whose terms are algebraic expressions inv olving the e lemen ts o f S . W e denote it b y A h S i . The rea der ma y be tempted to b elieve tha t every nonco mm utative C*- algebra ca n b e r egarded as a partial C*-algebra by declaring that a ⊙ b holds whenever a and b commu te, but tha t would b e incor rect. The reason for this is that we hav e r e q uired ⊙ to b e reflexive, so that aa ∗ = a ∗ a holds for every element a in a partial C*-a lgebra. Now, an element a such that aa ∗ = a ∗ a holds is ca lled normal and it is not the case that every element in a C*-algebra is normal. What is true, howev er, is that one may regar d the collection of no rmal elements of a C*- algebra as a par tial C* - algebra by declaring tw o elements to be commeasur able whenev er they commute. In fact, taking no rmal elements is pa rt of a functor, if w e consider the following class o f morphisms of pa rtial C*-algebr as. Definition 1 2 A p artial *-morphi sm is a (total) function f : A → B b et ween partial C*-algebras s uc h that: • f ( a ) ⊙ f ( b ) for commea s urable a, b ∈ A ; • f ( ab ) = f ( a ) f ( b ) for commeasurable a, b ∈ A ; • f ( a + b ) = f ( a ) + f ( b ) for commeasurable a, b ∈ A ; • f ( z a ) = z f ( a ) for z ∈ C and a ∈ A ; • f ( a ) ∗ = f ( a ∗ ) for a ∈ A ; • f (1) = 1 . Partial C*-algebr as and par tia l *-mor phisms orga niz e themselves into a catego ry denoted b y PCstar . 5 Before w e embark on proving that taking normal parts pr ovides a functor from the categor y of C*-alg e br as to the categ ory o f par tial C*- algebras, re call that a n element a o f a C*- a lgebra is ca lled self-a djoin t when a = a ∗ , and that any element ca n b e wr itten unique ly as a linear co m bination a = a 1 + ia 2 of t wo s elf-adjoint element s a 1 = 1 2 ( a + a ∗ ) and a 2 = 1 2 i ( a − a ∗ ). 5 Most results hold for nonunital C*-algebras, but for con venienc e we consider unital ones. 9 Prop osition 13 Ther e is a functor N : Cs tar → PCstar which sends every C*-algebr a to it s normal p art N ( A ) = { a ∈ A | aa ∗ = a ∗ a } , which c an b e c onsider e d as a p artial C*-algebr a by saying that a ⊙ b holds when- ever a and b c ommute. Mor e over, N is faithful and r efle cts isomorphisms and identities. Proof The action of N is well-defined on ob jects, since a subalgebra of a C*- algebra ge nerated by a set S is commutativ e iff the elemen ts of S are normal and comm ute pairwise. On mo rphisms, N acts by res tr iction and cor estriction. T o see that it has the pr oper ties the prop osition claims it has, one uses the identit y a = a 1 + ia 2 . F or exa mple, suppose N ( f ) is surjective for a *-morphism f : A → B and let b ∈ B . Then there are a 1 , a 2 ∈ N ( A ) with f ( a i ) = b i . Hence f ( a 1 + ia 2 ) = b , so that f is surjective. Similarly , if N ( f ) is injective, supp ose that f ( a ) = f ( a ′ ). Then f ( a i ) = f ( a ′ i ), and hence a i = a ′ i , so that a = a ′ and f is injectiv e. Now, isomorphisms in ( P ) Cstar are bijective (partial) *-mor phisms. Hence f is an isomorphism when N ( f ) is.  So therefore one wa y of thinking ab out a partial C*-a lgebra is as axio matiz- ing the normal part of a C* -algebra . O f course, w e could hav e decided to drop the requirement tha t ⊙ is reflexive, so tha t every C*- algebra w ould also be a partial C*- algebra, with commeasur a bilit y g iven by commut a tivity . W e hav en’t done this for v arious reaso ns. First o f all, w e would like to ha ve a notion given in terms of the ph ysically r elev ant data. On Bohr’s philosophy , which we ado pt in this paper , the physically r elev ant infor ma tion is c o n tained in the normal part of a C*-a lgebra. Related to this is the fact that the Bohr ification functor (whic h we will study in Section 7) only tak es the normal par t of a C*- algebra int o account. Secondly , we wis h to hav e a result saying how a partia l C*-alg ebra is determined by its commutativ e subalgebr as analogo us to o ur result for partial Bo olean a lgebras. With our present definition w e do indeed have this result (it is Theorem 1 5 below), as we will now explain. Denote b y C : PCstar → POrder the functor assigning to a partia l C* - algebra A the collection of its commeasur able ( i.e. comm utative tota l) subal- gebras C ( A ), partially or dered by inclusion. One immediately derives similar prop erties fo r the dia gram C ( A ) in the category PCstar as for partial Bo olean algebras . Prop osition 14 L et A b e a p artial C*-algebr a. (a) The le ast element of the p oset C ( A ) is A h 0 i = A h 1 i = { z · 1 | z ∈ C } . (b) The p oset C ( A ) is fi lter e d if and only if A is a c ommutative C*-algebr a. In t hat c ase, A is t he lar gest element of t he p oset C ( A ) .  W e now pr ov e the C*-a lg ebra version of our main result. 10 Theorem 15 Every p artial C*-algebr a is a c olimit of its (finitely gener ate d) c ommu t ative C*-sub algebr as. Proof Let A b e a partial C*-a lgebra, and consider its diagram C ( A ) of (finitely generated) commutativ e C* -subalgebra s C . Defining functions i C : C → A by the inclusio ns yields a co cone in PCstar ; we will prov e that this co cone is universal. If f C : C → B is another co cone, define a function m : A → B b y m ( a ) = f A h a i ( a ). P recisely as in Theorem 4, it now follows from the assumption that the f C are morphisms of PCstar that m is a well-defined morphism, to o. One easily verifies that f C = m ◦ i C , and that m is the unique such morphism.  T ogether, Theorem 15 ab ov e a nd Theor em 17 b elow show that, up to par- tial *-isomo r phism, every C*-alge br a can b e reco nstructed from its commut a tive C*-subalge br as, lending fo r ce to the Bohrification pro gramme. In this light The- orem 15 could b e said to embo dy a categor ical crude version of complementarit y . Remark 16 W e has ten to p oint out that Theorem 15 o nly works becaus e of the way we have set things up. In par ticular, it would fail if we would drop the requirement that ⊙ is reflexive. Also, it do es not say that every C*- a lgebra is the colimit o f its comm utative suba lg ebras in the category Cstar , whic h would be fals e . The reaso n why Theorem 15 cannot b e changed in these wa ys is that are nonisomorphic v on Neumann alg ebras A and B for which C ( A ) and C ( B ) ar e isomorphic. (This follows fro m the w or k of Connes in [2]. F or the exp erts, the argument is this: in [2] it is shown that there is a v on Neumann algebra A that is not an ti-iso morphic to itself; since this A is in standard form, i.e. has a separating cyclic vector, it follows tha t A is not isomo r phic to its commutan t A ′ . But T omita-T ak esa ki theory sho ws that an y such von Neumann algebra A is ant i-is omorphic to its commutan t A ′ , whence C ( A ) ∼ = C ( A ′ ).) Note tha t this, combined with Theorem 15, implies that there is a pa rtial *-isomor phism N ( A ) → N ( B ) tha t is not of the form N ( f ) for some f : A → B . T o put it another w ay , the faithful functor N is not full. Because of this the relev ance of the current work to the theory of C*-algebra s prop er is rather limited. But that is not our immediate aim: this pap er pri- marily wishes to gain a better conceptual under standing o f the Bohrification progra mme; and, as we hav e a r gued ab ov e, fro m that persp ective the w ay we hav e set things up is very na tural and Theor em 1 5 is a step forw ar d. W e close this s ection with a discussion of completeness and co completeness prop erties of PCstar . It is known that the categor y of C*-a lgebras is b oth complete and co complete (for copr oducts , see [16], and for co equalize r s, s ee [8]). As it turns out, also PCstar is both complete and co complete. PCstar is co mplete, as it has b o th equa lizers and ar bitr ary pro ducts. Equal- izers of par tia l C*-a lgebras a r e cons tr ucted as in Set , ha ving inherited com- measurability and subalg e bra str ucture. Pr oducts a re g iven by Q i A i = { ( a i ) i | a i ∈ A i , sup i k a i k < ∞} , w ith comp onent wis e commea surability and alge braic 11 structure. In particular, the terminal ob ject 1 is the 0-dimensio nal (par tia l) C*-algebr a { 0 } , confusingly sometimes a ls o de no ted by 0. The copro duct of a family A i of par tial C*-alg ebras is got by taking their disjoint union, identifying for every z ∈ C the elements o f the form z 1 i . Notice that elements from differ en t s ummands A i are never commeasurable in the copro duct. In particular , the initial ob ject 0 is the 1-dimensional (partial) C*- algebra C , which is co nfusing ly so metimes a lso denoted by 1. Co equalizers are harder to descr ibe constructively , but they do exist. Theorem 17 The c ate gory PCstar is c omplete and c o c omplete. Proof T o show that PCstar has co eq ualizers, the s a me strategy as in the pro of of Theorem 5 applies, becaus e for every pa rtial C* -algebra A the collec tio n of isomorphism classes of par tial * -maps f : A → B such that f ( B ) is dense in A form a set and every partial C*-algebr a map with domain A facto rs through a map of this for m.  4.1 V ariations Theorem 15 holds for ma n y v ar ie ties of (partial) C*-alg ebras, a s its pro of only depe nds on (partial) algebr a ic prop erties. Let us co ns ider (partial) Ric k art C* - algebras as a n e x ample. Recall that a commutativ e C*-algebra A is Rick art when every a ∈ A has a unique pro jection RP( a ) ∈ A such that (1 − RP( a )) · A is the right a nnihilator { b ∈ A | ab = 0 } . W e ca ll a pa rtial C*-alg ebra A tog e ther with a tota l map RP : A → A a p artial Ri ckart C*-algebr a when every pairwise co mmeasurable S ⊆ A is contained in a pair wise commeasur able T ⊆ A on which the op erations of A y ield a commutativ e Rick art C*-a lgebra structure with RP ’s giv en by the function ab ov e. Deno te the subcateg ory of PCstar whose o b jects a re partial Ric k art C*-algebr as and who se morphism a re partial *-mo rphisms that preserve RP by PRic k art . Theorem 18 Every p artial Ricka rt C*-algebr a is t he c olimit of its c ommut ative Ric kart C*-sub algebr as. Proof The pro o f of Theorem 15 holds verbatim when ev ery refere nce to (par - tial) C*-algebr as is repla ced by (par tial) Rick art C*-alg ebras.  If Ric k art is the s ubcateg ory of Cstar consisting of Ric k a rt C*-algebras and *-morphisms pr eserving RP , then there is a functor N : R ic k art → PRi c k art sending every Rick art C*- algebra A to its normal part; this follows fro m [1, Prop osition 4.4]. Similar r esults hold for any type of C*-algebr a that is defined by algebra ic pr op e r ties, such as A W*-algebras a nd sp ectral C*-a lgebras (see [9, 5.1]). W e will come back to A W*-alg ebras in Section 6 below. 12 The distinguishing fea ture of von Neumann a lgebras amongst C*-a lg ebras, in contrast, is to polo gical in nature. This makes it ha rder to come up with a notion of partial von Neumann algebra: the ob vious definition – a pa rtial C*- algebra A in which every subset of commeasurable ele ments is contained in a von Neumann algebra – has the dr awbac k that it is not clear if N ( A ) w ould be a partial von Neumann algebra g iv en a von Neumann algebra A . W e c a n, how ever, still obtain the following. Let A b e a von Neumann alg ebra. Without loss of generality , we may assume that A a cts on a Hilb ert space H . Denote the von Neumann suba lgebra of a von Neumann alg ebra A ge ner ated by a subset S ⊆ A b y A h h S i i . It is the closure of the C*- algebra A h S i in the weak op erator top ology , and by von Neumann’s double comm utant theorem [12, Theorem 5.3.1], it equals A h S i ′′ . Lemma 19 If a C*-sub algebr a C of a von Neumann algebr a A is c ommutative, then so is its von Neumann envelop e A h h C i i . Henc e if a ∈ A is normal, then A h h a i i is c ommutative. Proof Let a, b ∈ C ′′ . Since C ′′ is the (weak o per ator) closure of C , w e can write b as a (weak op erator ) limit b = lim n b n for b n ∈ C . Then: ab = a (lim n b n ) = lim n ab n (by [12, 5.7.9(i)]) = lim n b n a (sinc e a ∈ C ′′ and b n ∈ C ⊆ C ′ ) = (lim n b n ) a (by [12, 5.7.9(ii)]) = ba.  Theorem 20 L et A b e a von Neumann algebr a acting on a Hilb ert sp ac e. Then N ( A ) is a c olimit in PCstar of the (finitely gener ate d) c ommutat ive von Neu- mann sub algebr as of A . Proof U sing Le mma 19, the pro of of Theorem 15 holds verbatim when every o ccurence o f A h S i is repla ced by A h h S i i .  5 Gelfand dualit y The full sub categor y of PCstar consis ting of commutativ e C*-a lgebras is just the ca tegory cCstar of comm utative C*- a lgebras and *-morphisms. This cate- gory is dual to the ca tegory of compact Hausdorff spaces and contin uous func- tions via Gelfand duality [10]. Constructively , the la tter categ ory is replaced by that of co mpact co mpletely regular lo cales [3]: cCstar Σ / / ∼ KRegLo c op , Lo c ( − , C ) o o (2) where Σ( A ) is the Gelfand sp ectrum of a commutativ e C*-alge br a A . The dualizing ob ject C is b oth a lo cale and a (partial) C*- algebra; recall tha t it is in fact the initial partia l C*-alg ebra 0 . 13 The colimit theorem, Theorem 15, toge ther with the fa c t that the catego ries in (2 ) are co complete and complete, enables us to prov e the following extension of Ge lfa nd dualit y . Prop osition 21 Ther e is a r efle ction PCstar K / / ⊥ KRegLo c op , Lo c ( − , C ) o o in which the fun ctor K is determine d by K ( A ) = lim C ∈C ( A ) op Σ( C ) . Proof Let A be a partia l C*-algebr a and X a compact completely regula r lo cale. Then ther e a re bijective corresp ondences: f : K ( A ) = lim C ∈C ( A ) op Σ( C ) → X (in KRegLo c op ) ∀ C ∈C ( A ) . f C : Σ( C ) → X (in KRegLo c op ) ∀ C ∈C ( A ) . g C : C → Lo c ( X, C ) (in cCstar ) g : A → Lo c ( X, C ) (in PCstar ). The first cor r esp ondence ho lds by definition of limit, the middle corr esp ondence holds by Gelfand dua lit y (2), and the la st corresp ondence holds by Theo rem 15. Since all corre spo ndenc e s ar e natural in A and X , this establis hes the a djunction K ⊣ Lo c ( − , C ). Finally , since a comm utative C*-a lgebra is trivially a colimit of its e lf in PCstar , the adjunction is a reflection.  Theorem 22 The r efle ction K ⊣ Lo c ( − , C ) ex t ends Gelfand duality, i.e. the fol lowing diagr am c ommutes (serial ly). cCstar j J x x ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ Σ $ $ ∼ PCstar K . . ⊥ KRegLo c op Lo c ( − , C ) m m Lo c ( − , C ) c c Proof If A is a c omm utative C*-a lgebra, it is the initial element in the diag ram C ( A ) op by Prop osition 14(b). Hence K ( A ) = lim C ∈C ( A ) op Σ( C ) = Σ( A ).  Corollary 23 Commutative C*-algebr as form a r efle ctive ful l sub c ate gory of p artial C*-algebr as, i.e. the inclusion cCstar ֒ → PCstar has a left adjoint L : PCstar → cCstar . Proof The adjunctions of the previous theor em compo se, giving the requir ed left adjoin t as L = Lo c ( − , C ) ◦ K .  14 This means that for a partial C*-alg ebra A one has PCstar ( A, C ) ∼ = cCstar ( L ( A ) , C ) . In o ther words, mu ltiplicative q uasi-states of A that are m ultiplicative on com- m utative subalge br as precisely co rresp ond to states of L ( A ). Th us these quasi- states hav e go o d (categorica l) be ha vio ur. How ever, things are not as interesting as they may seem. By the K ochen-Sp ec ker theorem, no von Neumann alg ebra A without factors of type I 1 or I 2 can have such states ([4], see also [17]). It follows that K ( A ) = 0 and hence L ( A ) = 1 for s uch algebras. More gener- ally , let us call a par tial C*-algebra A Ko chen-Sp e cker when L ( A ) = 1 . Any such algebra A ha s no quasi-sta tes: PCstar ( A, C ) ∼ = cCstar ( 1 , C ) = ∅ . Also, Ko chen-Speck er partia l C*-algebr as ar e a ‘copro duct-ideal’ in a sense that w e now make precise. F o r X ∈ K R egLo c we have 0 × X = 0 , so b y Gelfand dua l- it y (2), we hav e 1 + C = 1 for a co mm utative C* -algebra C . So if A ∈ PCstar is Ko chen-Speck er, and B ∈ PCstar ar bitrary , then also A + B is Ko chen-Sp e cker: L ( A + B ) = L ( A ) + L ( B ) = 1 + L ( B ) = 1 . The first equality holds b ecause L , b eing a left adjoint, pres erves colimits. Nev- ertheless, the reflectio n of Theor e m 22 is still interesting. Even though it do es not teach muc h ab out the theory of C*- algebras prop er, it is an imp orta n t s tep in seeing ho w far A can b e r econstructed from C ( A ) (or its Bohrification A , see Section 7). See a lso Remar k 1 6. 6 Pro jections, partial A W*-algebras and tensor pro duc ts This section disc usses a functor PCstar → PBo olean , rela ting Sections 2 a nd 3 to Sections 4 and 5. 6.1 Pro ject ions and partial A W*-algebras An element p of a pa rtial C*-a lgebra A is called a pr oje ction w he n it sa tisfies p ∗ = p = p 2 . The elements 0 ∈ A a nd 1 ∈ A ar e trivially pro jections; other pro jections are called nontrivial. Lemma 24 Ther e is a functor Pro j : PCstar → PBo ole an wher e Pro j( A ) is the set of pr oje ctions of A . Proof First, Pr o j( A ) is indeed a par tial Bo olean algebra. Commeas urability is inherited from A . One easily chec ks that ¬ p = 1 − p is a pro jectio n when p is. If p, q ar e commeasur able in P ro j( A ) , then they commute, whence the pro jection p ∧ q = pq is also in A [17, 4.14]. This mak es Pro j( A ) into a partial Bo olea n algebra. Finally , morphisms of partia l C*-alg ebras are e a sily s e e n to preserve pro jections, making the as signment A 7→ Pro j( A ) functorial.  15 F or the following class of partial C*-a lg ebras we get stronger results. Definition 2 5 A partial Rick art C*-alg ebra A is a p artial A W *-algebr a , if it comes equipped with an op eration _ : { X ⊆ Pro j( A ) | X × X ⊆ ⊙} → Pro j( A ) , in such a way that each pair wise commeasurable S ⊆ A is contained in a pair- wise commeasura ble T ⊆ A on which the op erations determine a commutativ e A W*-alg ebra str ucture ( i.e. the structure is that of a comm utative Rick art C*- algebra, whose pr o jections form a complete Bo olean alg e bra with s uprema given by the oper ation W ab ov e). Denote the sub categ ory of PCstar whose ob jects are pa rtial A W *- algebras and whose mor phisms are partial * -morphisms which preserve RP and W by P A Wstar . Lemma 26 The functor Pr o j r estricts to a functor P A Wstar → PCBo olean . Proof Clear from the definition of a pa rtial A W* - algebra.  Remark 27 It w ould be interesting to see whether this functor is part o f an equiv ale nce , like in the total case, where it is one side of an equiv alence of categorie s b etw een cA Wstar and CBo olean . Prop osition 28 The funct ors P ro j and C c ommut e for p artial A W*-al gebr as: writing C ′ for t he functor P A Ws tar → [ POrder , cA Wstar ] , and C for t he functor PBo ol e an → POrder , we have C ◦ Pro j = Pro j ◦ C ′ . Explicitly, { Pro j ( C ) | C ∈ C ( A ) } = C (Pro j( A )) for every p artial A W*-algebr a A . Proof This follows from the co m bination of Stone and Gelfand dualit y , which yields an equiv a lence betw een cA Wstar and CBo ol ean . One directio n of the equiv ale nce is obtained by taking pro jections a nd the other is obtained by taking C ( X ) where X is the Stone space asso ciated to the (co mplete) Bo olean algebra . F or the pur pos es of the pro of, we will denote the latter comp osite functor by F . In particular, the pro jections P ro j( C ) of a comm utative A W*-a lgebra C form a complete Boo lean a lgebra and the left-hand side is cont ained in the r ight-hand side. F or the converse, let B be a complete Bo olean la ttice of pro jections in A . Then the pro jections in B c o mm ute pairwis e, and hence genera te a commutativ e A W*-subalg ebra C = A h B i . W e o b vio usly hav e a n inc lus ion i : B ⊆ Pro j( C ). But then the co mpos ite F B / / F i / / F P ro j( C ) η − 1 C ∼ = / / C, 16 where η is the unit of the adjunction Pro j ⊣ F , sho ws that F B is isomorphic to a commutativ e A W*- subalgebra of A contained in C . This commutativ e subalgebra also c o n tains B , b ecause the diagram Pro j( F B ) / / Pro j( F i ) / / ∼ = ǫ B   Pro j( F Pr o j( C )) ∼ = ǫ Proj ( C ) =Pro j( η − 1 C )   B / / i / / Pro j( C ) commutes by naturality of the co unit ǫ : P r o j( F − ) ⇒ 1. Since C = A h B i , this implies that F i is an isomor phism. B ut then s o is i and ther efore B = Pr o j( C ).  As a cor ollary to the previous pr op o sition, we can extend P rop osition 14 with atoms to mirror Prop ositio n 2. Keep in mind that the following corollar y do es no t en tail that C ( A ) is atomic. Corollary 29 F or a p artial A W*-algebr a A , the atoms of the p oset C ( A ) ar e A h p i for nontrivial pr oje ct ions p .  6.2 T ensor pro ducts It is c lear from the description of copro ducts in PCstar and PBo olean that the funct o r Pr o j prese r ves c o pro ducts. Recall that in a copro duct of par tial Bo olean or C*-a lgebras, nontrivial e lemen ts from differen t summands are never commeasura ble. Theo rems 4 and 15 provide the option of defining a tensor pro duct s atisfying the a dv er s e universal prop erty . Definition 3 0 Let A a nd B be a pair of partial B oo lean alg e bras (par tial C* - algebras ). Define A ⊗ B = colim { C + D | C ∈ C ( A ) , D ∈ C ( B ) } , where C + D is the co pr o duct in the category of Bo olean algebr a s (commutativ e C*-algebr as). There are ca no nical morphisms κ A : A → A ⊗ B and κ B : B → A ⊗ B as follows. By definition, A ⊗ B is the co limit of C + D for C ∈ C ( A ) and D ∈ C ( B ). Pre compo sing with the copro duct injections C → C + D gives a co cone C → A ⊗ B on C ( A ). By the colimit theorem, A is the c o limit of C ( A ). Hence ther e is a media ting morphism κ A : A → A ⊗ B . The unit element for b oth the tenso r pro duct and the copr o duct is the initial ob ject 0 . The big difference b etw een A ⊗ B and the copr o duct A + B is that elements κ A ( a ) and κ B ( b ) are alwa ys c o mmeasurable in the for mer, but ne ver in the latter. I ndee d, this universal pr o per t y characterizes the tensor product. Prop osition 31 L et f : A → Z and g : B → Z b e morphisms in the c ate gory PBo olean ( PCstar ). The c otuple [ f , g ] : A + B → Z factorizes t hr ough A ⊗ B if and only if f ( a ) ⊙ f ( b ) for al l a ∈ A and b ∈ B . 17 Proof By co nstruction, giving h : A ⊗ B → Z amo unts to giving a co cone C + D → Z for C ∈ C ( A ) a nd D ∈ C ( B ). Because C + D is totally defined, any morphism C + D → Z must a lso b e total. But this holds (for all C and D ) if and only if f ( a ) and g ( b ) are commeas urable for all a ∈ A and b ∈ B , for (only ) then can o ne take h to b e the cotuple of the core strictions of f and g .  The tenso r pro ducts of Definition 30 makes Pr o j : PCstar → PBo o lean a monoidal functor: the natural transfor mation Pr o j( A ) ⊗ Pr o j( B ) → Pr o j( A ⊗ B ) is induced by the cotuples P ro j( C ) + Pr o j( D ) → Pro j( A ⊗ B ) of Pro j( C ) Pro j( C ֒ → A ) / / Pro j( A ) Pro j( κ A ) / / Pro j( A ⊗ B ) . Prop osition 32 The functor Pro j : PCstar → PBo o lean pr eserves c opr o d- ucts and is monoidal.  W e end this section b y discussing the relation b etw een the tensor pr o ducts of partial Bo olea n a lgebras a nd those of Hilb ert s paces, describing co mpound quan- tum sys tems. Le t Hilb b e the categor y of Hilbert spaces and contin uo us line a r maps, and let B : Hi lb → PCstar denote the functor B ( H ) = H ilb ( H, H ) acting on mor phisms as B ( f ) = f ◦ ( − ) ◦ f † where f † is the adjoint of f . The definition of the tenso r pro duct in PCstar as a colimit yields a natura l trans- formation B ( H ) ⊗ B ( K ) → B ( H ⊗ K ), induced b y morphis ms C → B ( H ⊗ K ) for C ∈ C ( B ( H )) given by a 7→ a ⊗ id K . Initiality o f the tenso r unit 0 gives a mo r phism 0 → B ( C ), and these data sa tisfy the cohere nc e require- men ts. Hence the functor B is mo no idal, a nd ther efore also the comp osite Pro j ◦ B : Hilb → PBo ole an is a monoidal functor. 7 F unctorialit y of B ohrification The so-called Bohrificatio n construction (see [9], whose notation we adopt) as- so ciates to ev ery C*-algebr a A an internal commutativ e C*-algebra A in the top os [ C ( A ) , Set ], giv en b y the tautological functor A ( C ) = C . Gelfand dualit y then yields an internal lo cale, which can in turn b e externaliz ed. As it happ ens this construction works equally well for par tia l C*- a lgebras, so that B ohrifica- tion for ordinary C* -algebras can b e seen as the comp osition of the functor N from P rop osition 13 with Bohrification for par tial C*-alge bras. Thus a lo cale is asso ciated to ev ery ob ject o f PCstar . In this final section we consider its functorial asp ects. It tur ns out that the whole construction summarized ab ov e can b e made in to a functor from partial C*-a lgebras to lo cales by restricting the morphisms of the former. Bohrification does not just assign a topos to each (partial) C* -algebra, it assigns a top os with an internal C*-alge br a. T o r eflect this, w e define catego ries of to pos es eq uipped with internal structure s. Definition 3 3 The category RingedT opos has as ob jects pairs ( T , R ) of a top os T and an in ternal ring ob ject R ∈ T . A mo rphism ( T , R ) → ( T ′ , R ′ ) 18 consists of a g eometric mo rphism F : T ′ → T and an internal ring mor phis m ϕ : R ′ → F ∗ ( R ) in T . By CstaredT op os we denote the subcateg ory o f RingedT op os of ob jects ( T , A ) where A is an internal C*- algebra in T and morphisms ( F, ϕ ) where ϕ is an in ternal *-ring morphism. Notice that the dire c tion of mo rphisms in this definition is opp osite to the customary one in alg ebraic g eometry [7, 4.1]. First of a ll, any functor D → C induces a geo metric morphism [ D , Set ] → [ C , Set ], of which the inv erse pa r t is given by preco mpos ition (see [11, A4.1.4 ]). W e hav e alr eady s een that C is a functor PCstar op → POrder op . Addition- ally , restricting a mo rphism f : B → A of partia l C*-alg ebras to D ∈ C ( B ) and corestricting to C f ( D ) gives a morphism of commutativ e C*-alg ebras. Hence we o btain a geometric morphism o f top oses [ C ( B ) , Se t ] → [ C ( A ) , Set ] as well as an in ternal mor phis m of commutativ e *-ring s. T he la tter is a natur al transfor- mation whose comp o nen t at D is B ( D ) → (( C f ) ∗ A )( D ) = A ( C f ( D )). In other words, we hav e a functor PCstar op → RingedT op os . In genera l, (inv ers e parts of ) geo metric mo r phisms do not preserve internal C*-algebr as. But in this particular c a se, ( C f ) ∗ A is in fact an int er nal C*- algebra in [ C ( B ) , Set ]. The pro of is contained in [9, 4.8], which essentially shows that a n y functor fr om a p oset P to the categor y of C*-a lgebras is alwa ys an internal C*-algebra in the top os [ P, Se t ]. Ther efore, we r eally hav e a functor PCstar op → CstaredT opo s , as the following prop osition records. Prop osition 34 Bohrific ation is functorial PCstar op → CstaredT op os .  Next, w e can apply constructive Gelfand dualit y Σ int er na lly . Definition 3 5 The ca tegory Lo caledT opos has as o b jects pairs ( T , L ) of a top os T and a n internal lo cale ob ject L . A mo rphism ( T , L ) → ( T ′ , L ′ ) consists of a g eometric morphism F : T ′ → T a nd an internal lo cale morphism ϕ : F ∗ ( L ) → L ′ in T ′ . Prop osition 36 Bohrific ation is functorial PCstar op → Lo caledT op os . Proof It is clear from the description in [3 ] that the cons tr uction of the g en- erating la ttice of the in ternal Gelfand s p ectrum is geometr ic. Therefore it is preserved by (inv erse ima g e par ts of ) geometr ic mor phisms. Hence taking the int er na l Gelfand sp ectrum of a commutativ e C*-alg ebra commut es with in verse image functors of g eometric morphisms. The pro of is now finished by co mpos - ing the functor o f Pro pos ition 34 with the following: on o b jects, ( T , A ) maps to ( T , Σ ( A )), and on morphisms, ( F , ϕ ) maps to ( F , Σ( ϕ )). The latter morphism Σ( ϕ ) : Σ( B ) → F ∗ (Σ( A )) is indeed w ell-defined via F ∗ (Σ( A )) ∼ = Σ ( F ∗ ( A )).  How ever, this is where the curr en t line o f reaso ning stops: there is no clea r functor Lo caledT op os → Lo c . The mo st natural wa y to g et ahead is to restrict the morphisms of PCstar as follows. 19 Lemma 37 F or morphisms f : A → B of PCstar , the fol lowing ar e e qu ivalent: (a) if C f ( C ) ≤ D and C f ( C ′ ) ≤ D for C, C ′ ∈ C ( A ) and D ∈ C ( B ) , t hen ther e is C ′′ ∈ C ( A ) such that C ≤ C ′′ and C ′ ≤ C ′′ and C f ( C ′′ ) ≤ D ; (b) a ⊙ a ′ when f ( a ) ⊙ f ( a ′ ) . Proof First assume (a) and supp ose f ( a ) ⊙ f ( a ′ ). T ake C = A h a, a ∗ i , C ′ = A h a ′ , ( a ′ ) ∗ i , and D = B h f ( a ) , f ( a ′ ) , f ( a ) ∗ , f ( a ′ ) ∗ i . The n C f ( C ) ≤ D and C f ( C ′ ) ≤ D . Hence there is C ′′ with C ≤ C ′′ and C ′ ≤ C ′′ . So a, a ′ are bo th elemen ts of the comm utative alg ebra C ′′ , so a ⊙ a ′ . Conv ersely , assuming (b) and supposing C f ( C ) ≤ D and C f ( C ′ ) ≤ D , for all a ∈ C and a ′ ∈ C ′ we ha ve f ( a ) , f ( a ′ ) ∈ D , so that f ( a ) ⊙ f ( a ′ ). But that means that C and C ′ are commuting commutativ e suba lgebras of A . Hence we can tak e C ′′ = A h C, C ′ i .  W e say that mo rphisms satisfying the conditions in the previous lemma r efle ct c omme asur ability . Notice that this class of morphisms excludes the type of counterexample discusse d after Theo rem 4. T o show how the as signment of a lo ca le to a partial C*-alge br a b ecomes functorial with these morphisms, let us switc h to its external description [9, 5.16 ]: S ( A ) = { F : C ( A ) → Set | F ( C ) op en in Σ( C ) , F monotone } . (3) F or A a partia l C*-algebr a, S ( A ) is a lo ca le. W e wan t to extend this to a functor S : PCstar op → Lo c , or equiv alently , a functor S : PCstar → F rm . Let f : A → B b e a morphism of partial C*-alg ebras, F ∈ S ( A ), and D ∈ C ( B ). If C ∈ C ( A ) satisfies C f ( C ) ≤ D , then we have a mo rphism C → D given by the comp osition C f → C f ( C ) ≤ D . Its Gelfand transform is a frame mor phis m Σ( C f → C f ( C ) ≤ D ) : Σ( C ) → Σ( D ). So, since F ( C ) ∈ Σ( C ), we get an op en in Σ( D ). The fact that Σ( D ) is a locale allows us to take the join ov er all such C , ending up with the candidate action on morphisms S f ( F )( D ) = _ C ∈C ( A ) C f ( C ) ≤ D Σ( C → C f ( C ) ≤ D )( F ( C )) . (4) Theorem 38 Bohrific ation gives a funct or S : PCstar op rc → Lo c , wher e the domain is the opp osite of the sub c ate gory of PCstar of morph isms re fle cting c omme asur ability. Proof One quickly verifies that S f ( F ), as given in (4), is monotone, and hence a well-defined element of S ( B ), and that S f prese rves suprema. The g reatest element 1 ∈ S ( A ) is also preserved by S f : S f (1)( D ) = _ C ∈C ( A ) C f ( C ) ≤ D Σ( C → D )(1) = _ C ∈C ( A ) C f ( C ) ≤ D 1 = 1 . 20 The last equality ho lds b ecause the join is no t tak en ov er the empty set: there is alw ays C ∈ C ( A ) with C f ( C ) ≤ D , namely C = 0 . T o finish well-definedness and show that S f is a frame mo rphism, we need to s how that it pres erves binary meets. Reca ll that any frame satisfies the infinitary distributive law ( W i y i ) ∧ x = W i ( y i ∧ x ). It fo llows that one always has ( W i y i ) ∧ ( W j x j ) ≥ W k ( y k ∧ x k ). A s ufficien t (but not necessary) condition for equality to hold would b e if for all i, j there ex ists k such that y i ∧ x j ≤ y k ∧ x k . Expanding the definition of S f and wr iting x C = Σ( C → D )( F ( C )) and y C = Σ( C → D )( G ( C )) g iv es precisely this situation: S f ( F ∧ G )( D ) = _ C f ( C ′′ ) ≤ D x C ′′ ∧ y C ′′ , ( S f ( F ) ∧ S f ( G ))( D ) =  _ C f ( C ) ≤ D x C  ∧  _ C f ( C ′ ) ≤ D y C ′  . So, by Lemma 37, S f will pr eserve binar y meets if f reflects commeasur a bilit y . Finally , it is easy to see that S (id ) = id and S ( g ◦ f ) = S g ◦ S f .  Let us conclude w ith four remar ks concer ning the last theorem. • F rom the a bove pro of it additionally follows that this choice o f mo rphisms is the largest for whic h the theorem ho lds: PCstar rc is the largest sub- category of PCs tar for which (4 ) giv es a w ell-defined frame morphism. • Replacing the Gelfand sp ectrum by the Stone sp ectrum yields a similar functor PBo olean op rc → Lo c . • The categ ory Lo c is PO rder - e nr ich ed, and hence a 2-ca teg ory . W e remark that the functor S given b y (3) and (4) shows that the exter- nalization of Bohrification is a tw o-dimensiona l colimit (in Loc ) of the Gelfand sp ectra o f commeasura ble subalgebra s . 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