Characterizations of categories of commutative C*-subalgebras

CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS CHRIS HEUNEN Abstract. W e aim to c ha racterize the cat egory of injectiv e ∗ -homomorphisms betw een comm utativ e C*-subalgebras of a give n C*-algebra A . W e r educe this problem to finding a weakly terminal comm utativ e subalgebra of A , and solve the latter for v arious C*-algebras, including all commutat ive ones and all type I von Neumann algebras. This addresses a natural generalization of the Mac k ey–Piron programme: which lattices are those of closed subspaces of Hilb ert space? W e also discuss the wa y this categorified generalization differs from the original question. 1. Intr oduction The collection C ( A ) of co mm utative C*-subalgebr as of a fix ed C*-alge br a A ca n be made into a catego ry under v arious choices of mor phis ms . Tw o natural ones are inclusions and injective ∗ -homomor phisms, resulting in categor ies C ⊆ ( A ) and C ֌ ( A ), res pe c tively . The g oal of this article is to character ize these ca teg ories. Categorie s bas e d on C ( A ) are interesting for a num b er of rea sons. A first motiv a - tion to study such categ ories is the hop e that they could lea d to a nonco mm utative extension of Gelfand duality . It is known that C ⊆ ( A ) determines A as a partial C*- algebra [2]. Except when A ∼ = C 2 or A ∼ = M 2 ( C ), equiv alently C ⊆ ( A ) determines precisely the quasi-J ordan str ucture of A [11, 12]. Th us, C ( A ) in itself is alrea dy an int eresting inv a riant of A . Moreover, structures based on C ( A ) circumv ent obstr uc- tions to a no ncommutativ e Gelfand duality that afflict ma ny other ca ndidates [1]. Indeed, for C*-alg ebras A with eno ug h pro jections, adding a little mo re str ucture to C ( A ) fully determines the alg ebra structur e of A [18, 1 7]. T o get a full nonco m- m utative Gelfand duality for such a lgebras, it suffices to characterize the structures based o n C ( A ) that arise this wa y; an impor tant step is clearly to characterize categorie s of the form C ( A ). Second, there is a physical per sp ective on C ( A ). The underlying idea, due to Bohr, is that o ne can only empirically a ccess a q uantum mechanical system, who se observ a bles are mo deled by a (nonco mmutative) C* -algebra , thr ough its classical subsystems, a s mo deled by c o mmut ative C*-s ubalgebras [15]. Categ o ries based on C ( A ) are of paramo unt imp ortance in the recent uses of topo s theor y in research Date : October 19, 2018. 1991 Mathematics Subje ct Classific ation. 46L35, 18F99, 06B75, 81P10. Key wor ds and phr ases. C*-algebra, m axim al abelian subalgebra, i njectiv e ∗ -homomorphism, category , Grothendiec k construction. The author is grateful to Jonathon F unk for helping him understand and in fact suggesting large parts of App endix A, and to Manny Reyes, for many interesting discussions, esp ecially ab out functoriality . The author was supported by the Netherlands Organisation for Scientific Researc h and the Office of N av al Research. 1 2 CHRIS HEUNEN in fo unda tions o f physics base d on this idea that prop os es a new for m of quantum logic [7, 16]. Knowing which categories ar e of the form C ( A ) also characterizes which to po ses ar e of the form studied in that pr ogramme. This should incre ase insight into the in trinsic structure o f such top o ses, and hence shed light on the foundations o f quantum physics such top oses aim to desc r ib e log ically . Third, more generally , a characterization of C ( A ) sa tisfactorily addresses a ge n- eral theme in resear ch in foundations of quantum mechanics. F or example, it ad- dresses (a categor ification of ) the Mack ey– Piron progra mme. This prog r amme asks the q ue s tion: which or thomo dular lattices are those of closed subspaces o f Hilb ert space? (See [24, 2 8, 22].) A characterizatio n of C ( A ) would provide an answer, b e- cause choosing a co mmutative C*- subalgebra of the matr ix algebra M n ( C ) amounts to choosing an o rthonorma l subset and hence a clo sed subspace of C n , a nd an ap- propriate genera lization to infinite dimens io n holds a s well (see also Theorem 2.5 below and [14]). Similarly , a characterization of C ( A ) ha s consequences in the study of test spaces . These are defined as collec tions of orthog onal subs ets of a Hilb ert space s atisfying some conditions, and hav e b een pr op osed as axio ms for op erationa l quantum mechanics. One of the ma jor questions there is a gain which test spaces arise fro m prop os itions o n Hilb ert spaces [30]. Our ma in result is to reduce characterizing C ֌ ( A ) to finding a weakly terminal commutativ e subalgebra of A . This is closely related to a nalyzing all maximal ab elian subalgebr as (masas). Explicating the structure o f masas of C*-alg ebras in g eneral is a har d problem, and not muc h seems to be known systematica lly outside of the case of factor s of type I a nd type II 1 ; see [5, 2 7]. F or tuna tely , finding a weakly terminal co mm utative subalgebr a is generally ea sier tha n finding all masas. W e prove that the following classes o f C* -algebr as A po ssess weakly terminal co mm utative s ubalgebras , and therefore we find a full characterization o f C ֌ ( A ) for: • type I von Neumann algebr as, including all finite-dimensiona l C*-a lgebras ; • co mm utative C* -algebr a s. The strategy b ehind our characterization is a s follows. The key insig ht is to recognize C ֌ ( D ) fo r a commutativ e C*- algebra D as the Grothendieck constructio n of an actio n of a monoid M on a partially ordered set P . W e characterize such so-called amalga mations. Next, we use known results to c haracteriz e the pa r tially ordered set P = C ⊆ ( D ), c onsisting of partitions of the Gelfand spec tr um of D . Then, we show that C ֌ ( A ) is eq uiv alent to C ֌ ( D ) for a weakly terminal ob ject D in C ֌ ( A ). Finally , we establish such a weakly ter minal ob ject D fo r the v ar ious t yp es of C*- algebra s A mentioned, finishing the characteriza tion. This las t step is the only one limiting our characterization to C*-a lgebras A with weakly terminal commutativ e s ubalgebras . Summarizing : (1) show that a C*-alge br a A has a weakly terminal ab elian suba lgebra D ; (2) show that C ֌ ( A ) is equiv alent to C ֌ ( D ); (3) show that C ֌ ( D ) is equiv alent to P ( X ) ⋊ S ( X ), with X the sp ectrum of D ; (4) characterize P ( X ) ⋊ S ( X ) in terms o f P ( X ) and S ( X ); (5) a characterization of P ( X ) exists; (6) in the cases in ques tion, X , and hence S ( X ), is easy to characteriz e . Thu s we addres s the Mackey–Piron progr amme in a differ ent wa y than the the- orems of Piro n [24] and So l` er [2 8], which together for m the only characteriz a tion of the la ttice of closed subspaces of a Hilb ert space we are aw a re o f. Piron’s theor em CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 3 states that the lattice should be complete, a tomic, irr educible, or tho mo dular, and satisfy the cov er ing law, from which it follows that it must be the lattice of closed subspaces of some Euclidean spac e ov er a skew field. Sol` er ’s theor em says that if additionally this Euclidea n space is infinite-dimensiona l and has the prop erty that any closed subspace is a dir ect s ummand, then the s kew field mu st b e the r eals, complexes or qua ter nions, and the space m us t b e a Hilber t space. Both Sol ` er’s direct summand condition and Pir on’s lattice-theor etic axioms r elate to our use o f partition lattices P ( X ), but instead of orthomo dula rity we use the action of S ( X ). Int erestingly , our res ults apply to ar bitrary Hilb ert s pa ces, whereas Sol` er’s theore m only ho lds for infinite-dimensional ones . The pap er is structured as follows. W e s tart with Sectio n 2, which intro duces the po set C ⊆ ( A ) and the ca tegory C ֌ ( A ) and discusses their ba sic pro p erties and moti- v ation. A mor e in-depth ana lysis o f the relationship betw een the t w o, aga in dep end- ing on the Grothendieck cons truction, is made later, in Se c tion 7. Our ma in results are present ed in b etw een. T o aid int uition, we fir st co ver the finite-dimensional case, and only then incor po rate the subtleties of the infinite-dimensiona l case . Sec- tion 3 c haracter izes a malgamations of groups and p o sets, which is then used in Section 4 to establis h the character ization in the finite-dimensional case. Then, Section 5 refines the ear lier analysis to characterize amalga mations of monoids and po sets. This is used in Section 6 to es tablish the c haracter ization in the infinite- dimensional case. Appendix A recor ds some in termediate re s ults of indep endent int erest. In par ticular, it discusses an alterna tive wa y to in vestigate the rela tionship betw een C ֌ ( A ) and C ⊆ ( A ). T o end this int ro duction let us briefly indicate the differences b e t ween C ֌ ( A ) and C ⊆ ( A ). This will be dis cussed in more depth in Se c tio n 7, but it might b e helpful to men tion them now to set the sc e ne. An y morphis m in C ֌ ( A ) factor s uniquely as a ∗ -isomor phism followed by a mor phis m in C ⊆ ( A ). If C ֌ ( A ) ∼ = C ֌ ( B ) ar e isomor- phic categories, then C ⊆ ( A ) ∼ = C ⊆ ( B ) ar e iso morphic posets. Therefore, as discus sed ab ov e, bo th categ o ries C ֌ ( A ) and C ⊆ ( A ) are invari ants of the C*- a lgebra A , in the sense that b oth determine the (quasi- )Jordan s tructure of A , and are hence resp ected b y (quasi-)Jor dan homomorphisms. W e will mostly be interested in a coarser notion of in v ariant, namely equiv alence of categ ories, r a ther than isomor - phism of categ ories. F or p o setal categor ies like C ⊆ ( A ), isomorphis m and equiv alence coincide, but for C ֌ ( A ) this ma kes a differe nc e : C ֌ ( A ) ≃ C ֌ ( B ) need not imply C ⊆ ( A ) ∼ = C ⊆ ( B ) (and cer tainly not A ∼ = B ). It tur ns out that C ֌ ( A ) ≃ C ֌ ( B ) are equiv alent categor ies pr ecisely when C ⊆ ( A ) and C ⊆ ( B ) are Morita - equiv alent, in the sense that they hav e equiv alent pr esheaf catego ries PSh( C ⊆ ( A )) ≃ P Sh( C ⊆ ( B )). This explains why equiv a lence of categ ories is a more natural inv a riant from the po int of view of categor y theory and top os theory . 2. Motiv a tion W e do not require C*-algebr as to hav e a unit, and write Cs tar for the catego ry of C*- a lgebras and ∗ -homomo rphisms. Definition 2.1. W rite C ( A ), or simply C , for the collection of nonzero commutativ e C*-subalge br as C of a C*-a lgebra A . T his set of ob jects c a n b e made into a category by v a rious choices of morphisms, such as: • inclusio ns C ֒ → C ′ , given by c 7→ c , yielding a (p o setal) ca tegory C ⊆ ( A ); • injective ∗ -mor phisms C ֌ C ′ , giving a (left-cancellative) categor y C ֌ ( A ). 4 CHRIS HEUNEN These tw o catego ries are interesting for t wo related reasons. First, they form a ma jor ingredient in a new attack on a noncommutativ e extension of Gelfand duality [2, 1, 18]. Essentially , o ne could think o f them as inv a r iants o f a C*- algebra. Second, they play a n impo rtant r ole in the recent use o f top o s theory in the foundations of quantum ph y sics. F rom this per s pe c tive, one co uld think of them as encoding the logic o f a quantum-mec ha nical system whose o bserv ables are mo deled by the C*-alge bra A . W e will discuss these tw o p ersp ectives in turn, but firs t we consider functoriality of the constructio n A 7→ C ( A ). Section 7 b elow discusses the r elationship betw een the t wo choices of mo rphisms, C ֌ ( A ) or C ⊆ ( A ) in more detail. F unctorialit y. The a ssignment A 7→ C ⊆ ( A ) extends to a functor: given a ∗ - homomorphism ϕ : A → B , direct images C 7→ ϕ ( C ) form a mo rphism C ⊆ ( A ) → C ⊆ ( B ) of p osets, for if C ⊆ C ′ , then ϕ ( C ) ⊆ ϕ ( C ′ ). W e ll- definedness re lies on the following fundamental fact, that we recor d a s a lemma for future reference. Lemma 2.2 . The set-t he or etic image of a C*-algebr a u nder a ∗ - homomorph ism is again a C*-algebr a. Pr o of. See [20, Theorem 4.1.9].   The assignment A 7→ C ֌ ( A ) has to b e adapted to be made functorial. Either w e only consider injective ∗ -homomo rphisms A ֌ B , or w e restrict the target category C ֌ ( A ) as follows. W rite Cat for the catego ry of s mall categor ies a nd functor s . Lemma 2.3. Ther e is a fun ctor Cstar → Cat , sending A to t he sub c ate gory of C ֌ ( A ) with morphisms those i : C → C ′ satisfying i − 1 ( I ∩ C ′ ) = I ∩ C for al l close d (two-side d) ide als I of A . Pr o of. Let ϕ : A → B b e a ∗ -ho momorphism, and let i b e as in the statement of the lemma. Then i induces a well-defined injective ∗ -homomor phism ϕ ( C ) → ϕ ( C ′ ) precisely when ϕ ( c 1 ) = ϕ ( c 2 ) ⇐ ⇒ ϕ ( i ( c 1 )) = ϕ ( i ( c 2 )). Since ϕ and i are linea r, this c omes down to ϕ ( c ) = 0 ⇐ ⇒ ϕ ( i ( c )) = 0, i.e. ker( ϕ ) ∩ C = ker( ϕ ◦ i ). Setting I = ker ( ϕ ), this b e comes I ∩ C = { c ∈ C | ϕ ( c ) = 0 } = { c ∈ C | ϕ ( i ( c )) = 0 } = i − 1  { c ′ ∈ C ′ | ϕ ( c ′ ) = 0 }  = i − 1 ( I ∩ C ′ ) and is therefo r e satisfied.   Notice that ∗ - homomorphisms satisfying the condition of the previous lemma are automa tically injective, a s is seen by taking I = { 0 } . Notice also that when A is a topolo gically simple C*-a lgebra, s uch as the a lgebra M n ( C ) of n -by- n complex matrices , then the sub catego ry of the previous lemma is actually the who le categ ory C ֌ ( A ). CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 5 In v ariants. Let us tempo rarily consider v on Neumann algebr a s A and their von Neumann subalgebr as V ( A ), g iv ing categ ories V ⊆ and V ֌ . W e will show that V ⊆ contains ex actly the same information a s the la ttice Pro j( A ) of pro jections of A , in the technical sense that they are functors with equiv alent images. This lattice has b een studied in depth, so from the p oint o f view o f (new) inv ariants of A , the category V ֌ is more in teresting. See also Remark 7.8 b elow. By ex tens ion, C ֌ is po ssibly more int eresting a s an inv ar iant than C ⊆ , be c a use C ( A ) a nd V ( A ) coincide for finite-dimensio nal C* -algebra s A . Denote the category o f von Neumann a lgebras a nd unital nor mal ∗ - homomor- phisms by Neumann , and write cNeumann for the full sub ca tegory of commuta- tive (unital von Neumann) algebra s. Denote the ca tegory of o rthomo dular la ttices and lattice mo rphisms preserving the or tho c o mplement by Ortho . The functor Pro j : Neumann → Ortho takes A to { p ∈ A | p 2 = p = p ∗ } under the or dering p ≤ q iff pq = p . On morphisms f : A → B it a cts as p 7→ f ( p ). Recall tha t the essential image of a functor F is the smallest sub categor y of the targ et catego ry containing all isomor phisms and a ll mor phisms of the for m F ( f ). Denote the ess en- tial image of Pro j by D ; traditional quantum logic is the study of this sub categor y of Ortho [25]. Denote by Poset [ cNeumann ] the following categor y : o b jects ar e sets of co m- m utative v on Neumann algebr as partia lly or dered by inclusion ( i.e. C ≤ C ′ iff C ⊆ C ′ ); morphisms are mono tonic functions. W e ma y reg ard V ⊆ as a functor Neumann → P oset [ cNeumann ]. Denote the es sential image o f V ⊆ by C ; this is a sub categ ory of P oset [ cNeumann ]. W e now define tw o new functor s, F : C → D and G : D → C . The functor F acts on an ob ject V ⊆ ( A ) as follows. F or each C ∈ V ⊆ ( A ), w e know that P ro j( C ) is a Bo olean algebra [25, 4.16 ]. Beca use additio nally the hypothesis of Kalmbac h’s Bundle lemma, is sa tisfied, these Bo olean alg e br as unite int o an orthomo dular lattice F ( V ⊆ ( A )). This assignment extends na turally to morphisms . Lemma 2. 4 (Bundle lemma) . L et { B i } b e a family of Bo ole an algebr as s uch that ∨ i = ∨ j , ¬ i = ¬ j , and 0 i = 0 j on int erse ctions B i ∩ B j . If ≤ on S i B i is tr ansitive and makes it into a lattic e, then S i B i is an orthomo dular lattic e. Pr o of. See [21, 1.4.22 ].   The functor G acts on the pro jection lattice L of a von Neumann algebra as follows. Consider all complete Bo olea n sublattices B of L as a po set under inclusion. F or each B , the contin uous functions on its Stone sp ectr um for m a commut ative von Neumann alg ebra. Thus we obtain an o b ject G ( L ) in C , and this assignment extends natur a lly to morphisms. Theorem 2. 5. The functors F and G form an e quivalenc e, and make the fol lowing diagr am c ommute. Neumann V ⊆ w w ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ Pro j ' ' P P P P P P P P P C F / / ≃ D G o o Pr o of. F ollows directly fro m the definitions and the pr evious lemma.   Indeed, b oth V ⊆ ( A ) and Pro j( A ) capture the Jo r dan a lgebra structure of A [13], excepting the cas e where A has summands of type I 2 . 6 CHRIS HEUNEN Returning to the setting of C* - algebra s, notice that the previous theorem fails, bec ause ther e ar e C*-alg ebras without any non trivial pro jections. But every C*- algebra has many comm utative C*-subalgebr as: every self-adjo int element gener- ates one, and every element o f a C*-a lgebra is a complex linear com bination of self-adjoint element s. F or C*-alg e br as, C ⊆ ( A ) captures pr ecisely the pseudo-Jorda n algebra structure o f A [11, 12]. In this re g ard, it is also worth remarking that the functor C ⊆ : Cstar → Poset [ cCstar ] factors through the categor y of pa r tial C*-algebr as [2]. T op oses in found atio ns of ph ysics. The main theo rem in the application of top os theory to foundations of quantum physics is the following. The ta uto logical functor C 7→ C is a n internal (p os sibly nonunital) C*-alg ebra [16, Theor em 6 .4.8]. It holds in b o th top oses Set C ⊆ and Set C ֌ bec ause o f the fundamental Le mma 2.2 ab ov e. Catego rically , C ֌ is a mor e natura l choice than C ⊆ . But to c haracteriz e a preshea f catego ry is the sa me as c haracterizing the ca teg ory it is base d on, by Mo r ita equiv alence; s e e als o Section 7 and Appendix A b elow. Thu s, our main r esults a lso characterize top oses o f the form Set C ֌ . F or a mor e or less pr actical acco unt of the ab ov e folklore knowledge we refer to [4]. 3. P oset-gr oup-amalgama tions This section recalls the Gr othendieck constructio n, fo cus ing on the sp ecial case of an action of a group on a p os et. W e will call the resulting categor ies poset- group-a malgamations . The goa l of this section is to characterize such ca tegories . This is in teresting in its own right, but even more so b eca us e in Section 6 w e will see that C ֌ is of this for m. F or that reason, we prefer a pr actical characteriza tion. Therefore, w e will not pursue the highest p ossible level of genera lity: the discussion in this s ection is in elementary terms sp elling out w ha t is probably folk lo re knowl- edge. In particular , the c haracter ization in this sectio n can b e ex tended to p oset- category -amalga mations, a nd p e r haps even to a characterization of Gro thendieck constructions of arbitrar y indexed ca tegories, but we will not pursue this here. W e will use the Grothendieck constr uc t, also called the categor y of e le men ts, ag ain in Section 7, where it is discussed more abstractly . The main idea in this section is to factor out symmetries into a monoid action, leaving just the partial order. Definition 3.1. An action of a mo noid M (in the ca teg ory of se ts) on a catego r y C is a functor F : M → Cat ( C , C ). W rite mx for the action of F m on an ob ject x o f C , and mf for the action of F m o n a morphism f of C . Definition 3. 2 . If a mono id M a c ts o n a c a tegory C , then we ca n per fo rm the Gr othendie ck c onstruction : we can ma ke a new category C ⋊ M whose ob jects are those of C , and whose mo r phisms x → y ar e pairs ( m, f ) such that dom( f ) = x and co d( f ) = my . Compos ition and identities are inherited from M and C . Ex plicitly , id x = (1 , id x ), a nd ( n, g ) ◦ ( m, f ) = ( mn, ( mg ) f ). If the category C in the previous definition is a partially or dered set P , then P ⋊ M has as ob jects p ∈ P , and mor phisms p → q are m ∈ M such that p ≤ mq , with unit and comp osition from M op . An illustr a tive exa mple to keep in mind is the following. Let M b e the group of unitary n -by- n matrices . Let P b e the lattice of subspaces o f C n , or dered by inclusion. Then M acts on P b y U V = { U ( v ) | v ∈ V } for U ∈ M and V ∈ P . CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 7 Morphisms in P ⋊ M b etw een s ubspaces V ⊆ C n and W ⊆ C n are unitary matrices U suc h that U − 1 ( v ) ∈ W for all v ∈ V . This sectio n characterizes ca tegories of the for m P ⋊ G for an a ction o f a g roup G on a pos e t P with a le ast elemen t. Our characterization will rely o n weakly initial ob jects to recover P from P ⋊ G . Categoric a lly , this is trivial, but as we will see in Sections 4 and 6, it is a very imp orta nt step in our application. An ob ject 0 is we akly initial when for any o b ject x there ex ists a (not necessar ily unique) morphism 0 → x ; no tice that s uch an o b ject is not necessar ily unique up to isomorphism, as an initial ob ject would b e. I f a category A has a w eak initial ob ject 0, we ca n r egard the endoho ms et mono id A (0 , 0) as a o ne - ob ject categor y . Recall that a r et r action of a functor is a left-inv erse. Lemma 3.3. If a c ate gory A has a we ak initial obje ct 0 and a faithful r etr action F of the inclusion A (0 , 0) ֒ → A , then its obje cts ar e pr e or der e d by x ≤ y ⇐ ⇒ ∃ f ∈ A ( x, y ) . F ( f ) = 1 . Pr o of. Clea rly ≤ is reflexive, b ecause F (id x ) = 1 . It is also transitive, for if x ≤ y and y ≤ z , then there are f : x → y and g : y → z with F ( f ) = 1 = F ( g ), s o that g ◦ f : x → z satisfies F ( g ◦ f ) = F ( g ) ◦ F ( f ) = 1 ◦ 1 = 1 and x ≤ z .   Thu s we ca n recov er the gr oup G from A = P ⋊ G by lo o king at A (0 , 0). W e can also re c over the pose t P from A by the previous lemma. What is left is to reconstruct the action of G on P given just A . F or m ∈ G and p ∈ P , we can access the ob ject mq through the morphisms m : p → q in A . There is always at least o ne such morphism, namely m : mq → q , b ecause triv ially mq ≤ mq . In fact, this is alwa ys an iso morphism. W e will no w use this fact to recover the action of G o n P fro m A . W e w ill call this an amalgamation by analog y with the use of the term in alge br a. Definition 3.4. A categ ory A is called a p oset-gr oup-amalgamatio n when ther e exist a partial order P and a gro up G such that: (A1) ther e is a weak initial ob ject 0, unique up to isomorphism; (A2) ther e is a faithful retra ction F of the inclusion A (0 , 0) ֒ → A ; (A3) ther e is an iso morphism α : A (0 , 0 ) → G op of monoids; (A4) ther e is an equiv alence ( A , ≤ ) β / / P β ′ o o of preo rders; (A5) fo r ea ch ob ject x there is an isomorphism f : x → β ′ ( β ( x )) with αF ( f ) = 1; (A6) fo r ea ch y and m there is an isomo rphism f : x → y with αF ( f ) = m . Example 3 . 5. If P is a partial o rder with least element, a nd G is a g r oup a cting on P , then P ⋊ G satisfies (A1)–(A6). Pr o of. The least element 0 of P is a weak initial ob ject, sa tisfying (A1). Conditions (A2)–(A4) are satisfied b y definition, and (A5) is v acuo us. T o verify (A6) for q ∈ P and m ∈ G , notice that mq ≤ mq , so f = 1 : mq → mq is an is o morphism with αF ( f ) = 1.   Lemma 3.6. If A satisfies (A1)–(A6), then it induc es an action of G on P given by mp = β ( x ) if f : x → β ′ ( p ) is an isomorphism with α ( F ( f )) = m . Pr o of. Firs t, notice tha t for any p ∈ P and m ∈ G there exists an isomor phism f : x → β ′ ( p ) with α ( F ( f )) = m by (A6). If there is another iso morphism f ′ : x ′ → β ′ ( p ) w ith α ( F ( f ′ )) = m , then their comp ositio n gives x ∼ = x ′ , a nd therefor e 8 CHRIS HEUNEN β ( x ) ∼ = β ( x ′ ). But b eca use P is a par tial order , this means β ( x ) = β ( x ′ ). Th us the action is well-defined on ob jects. T o see that it is well-defined on morphisms, supp ose that p ≤ q . Then ther e is a morphism f : β ′ ( p ) → β ′ ( q ) with F ( f ) = 1. F or any m : 0 → 0, axiom (A6) provides isomorphisms f p : x p → β ′ ( p ) and f q : x q → β ′ ( q ) with α ( F ( f p )) = m = α ( F ( f q )). Then f = f − 1 q f f p : x p → x q is an isomo rphism satisfying αF ( f ) = mm − 1 = 1 . So mp ≤ mq . Next, we verify that this assignment is functoria l G → Cat ( P, P ). Cle a rly id β ′ ( p ) is an isomor phis m x → β ′ ( p ) with F (id β ′ ( p ) ) = 1. Therefor e 1 p = β ( β ′ ( p )) = p . Finally , for m 2 , m 1 ∈ M a nd p ∈ P , we hav e m 1 p = β ( x 1 ) where f 1 : x 1 → β ′ ( p ) is an is omorphism with α ( F ( f 1 )) = m 1 . So m 2 ( m 1 p ) = β ( x 2 ) where f 2 : x 2 → β ′ ( β ( x 1 )) is an isomor phism with α ( F ( f 2 )) = m 2 . By (A5), there is an isomorphism h : x 1 → β ′ ( β ( x 1 )) with F ( h ) = 1. So f = f 1 h − 1 f 2 is an isomorphis m x 2 → β ′ ( p ) with α ( F ( f )) = m 2 m 1 . Thu s ( m 2 m 1 ) p = β ( x 2 ) = m 2 ( m 1 p ).   Theorem 3.7. If A satisfies (A1)–(A6), t hen ther e is an e quivalenc e A → P ⋊ G given by x 7→ β ( x ) on obje cts and f 7→ α ( F ( f )) on morphisms. Pr o of. Firs t we verify that the assignment of the statement is well-defined, i.e. that α ( F ( f )) is indee d a mor phism of P ⋊ G . Giv en f : x → y , w e need to show that β ( x ) ≤ α ( F ( f )) · β ( y ). Unfolding the definition of action, this means finding an isomorphism k : x ′ → β ′ ( β ( y )) with α ( F ( k )) = α ( F ( f )) and β ( x ) ≤ β ( x ′ ). Unfolding the definitio n of the preo rder, the la tter means finding a mo rphism h ′ : β ′ ( β ( x )) → β ′ ( β ( x ′ )) with F ( h ′ ) = 1. By (A5), it suffices to find h : x → x ′ with F ( h ) = 1 instead. B ut (A6) provides an is omorphism k : x ′ → β ′ ( β ( y )) with α ( F ( k )) = α ( F ( f )). By (A5) a gain, there exis ts an iso morphism l : y → β ′ ( β ( y )) with α ( F ( l )) = 1. Finally , we can take h = k − 1 l f : x → x ′ . This mo rphism indeed satisfies α ( F ( h )) = α ( F ( f )) · α ( F ( l )) · α ( F ( k )) − 1 = α ( F ( k )) · α ( F ( k )) − 1 = 1. F unctoria lity follows directly from the previous lemma, so indee d we have a well- defined functor A → P ⋊ G . Mo reov er, our functor is essentially surjective beca use β is an equiv alence, and it is faithful b ecause F is faithful. Finally , to prove fullness, let m : β ( x ) → β ( y ) b e a morphism in P ⋊ G . This means that β ( x ) ≤ mβ ( y ), which unfolds to: there ar e a mor phism f : x → z and an iso mo rphism h : z → β ′ ( β ( y )) in A with α ( F ( f )) = 1 and α ( F ( h )) = m . By (A5), this is equiv alent to the existence of a mor phism f : x → z with α ( F ( f )) = 1 and an isomorphism h : z → y in A with α ( F ( h )) = m . Now take k = hf : x → y in A . Then α ( F ( k )) = α ( F ( hf )) = α ( F ( f )) · α ( F ( h )) = 1 · m = m. Hence our functor is full, and we conclude that it is (half of ) an eq uiv alence.   4. The finite-dimensional case This section uses po set-gro up-amalga mations to completely c haracteriz e the cat- egory C ֌ ( A ) for finite-dimensional C*-algebr as A . En p assant , we will also char- acterize the p os et catego ry C ⊆ ( C ) for commutativ e finite-dimensional C*-a lgebras C . Finite part ition lattices. W e start with iden tifying the appr o priate p oset P . Recall that a p artition p o f { 1 , . . . , n } is a family o f disjoint subsets p 1 , . . . , p k of { 1 , . . . , n } whos e union is { 1 , . . . , n } . P artitions a re or dered by r efi nement : p ≤ q CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 9 whenever each p i is cont ained in a q j . Ordered this wa y , the partitions o f { 1 , . . . , n } form a lattice, called the p artition lattic e , that w e denote b y P ( n ). It is k nown when a lattice is (iso mo rphic to) the pa rtition la ttice P ( n ). W e r ecall such a characterization be low, but first we briefly hav e to recall some terminolog y . Recall that a lattice is semimo du lar if a ∨ b cov er s b whenever a covers a ∧ b . A finite lattice is ge ometric when it is atomic and semimo dular. Any geometric lattice has a well-defined r ank function: rank( x ) is the leng th of a(n y) chain from 0 to x in L . An elemen t x in a lattice is mo dular when a ∨ ( x ∧ y ) = ( a ∨ x ) ∧ y for a ll a ≤ y . The M¨ obius fun ction of a finite lattice is the unique function µ : L → Z satisfying P y 0; see [3]. The char acteristic p olynomial of a finite lattice L is P x ∈ L µ ( x ) · λ rank(1) − rank( x ) . Finally , we write ↑ x for the principa l ideal { z ∈ L | x ≤ z } o f x ∈ L . Theorem 4.1. A lattic e L is isomorphic to P ( n + 1) if and only if: (P1) it is ge ometric; (P2) if rank( x ) = rank( y ) , then ↑ x ∼ = ↑ y ; (P3) it has a mo dular c o atom; (P4) its char acteristic p olynomia l is ( λ − 1) · · · ( λ − n ) . Pr o of. See [31].   This immediately extends to a characterization of C ⊆ ( A ) for finite-dimensional commutativ e C* -algebr as A (which ar e alwa ys unital). Corollary 4.2. A lattic e L is isomorphic t o C ⊆ ( A ) op for a c ommutative C*-algebr a A of dimension n + 1 if and only if it satisfies (P1)–(P4). Pr o of. The lattice C ⊆ ( A ) is that of subob jects o f A in the category of finite- dimensional commutativ e C * -algebr as and unital ∗ -homomo rphisms. Recall that a sub obje ct is an equiv alence cla ss o f monomor phisms into a given ob ject, wher e tw o monics a re identified when they factor through one a no ther by an isomor phism. The dual notion is a quotient : an equiv alence class of epimorphis ms out of a g iven ob ject. By Gelfand duality , C ⊆ ( A ) is isomor phic to the o pp o site of the lattice of quotients of the discrete to po logical s pace Sp ec( A ) with n + 1 p o ints. But the latter is precis ely P ( n + 1) op .   Symmetric group actions. The a ppropria te gro up to co ns ider is the sy mmetric group S ( n ) of all p ermutations π of { 1 , . . . , n } . The g r oup S ( n ) acts on P ( n ). Explicitly , π p = ( π p 1 , . . . , π p k ) for p = ( p 1 , . . . , p k ) ∈ P ( n ) and π ∈ S ( n ), where π p l = { π ( i ) | i ∈ π l } . That is, one works in the quotient g roup of S ( n ) by the Y oung subgr o ups S ( n 1 ) × · · · × S ( n k ), where the n l ares the cardinality of the par ts p l of the par tition p . The following lemma might b e conside r ed the main insight of this article . Lemma 4.3. If A is a c ommutative C*-algebr a of dimension n , then ther e is an isomorphi sm C ֌ ( A ) op ∼ = P ( n ) ⋊ S ( n ) of c ate gories. Pr o of. W e may ass ume that A = C n . Ob jects C of C ֌ ( A ) then ar e of the form C = { ( x 1 , . . . , x n ) ∈ C n | ∀ k ∀ i, j ∈ p k : x i = x j } for some par tition p = ( p 1 , . . . , p l ) of { 1 , . . . , n } . But these ar e precisely the o b jects of P ( n ), a nd hence of P ( n ) ⋊ S ( n ). If f : C ′ → C is a morphism of C ֌ ( A ), i.e. an injective ∗ -homomor phis m, then f ( C ′ ) ⊆ C is a C*-subalg ebra. Say C ′ = { x ∈ C n | ∀ k ∀ i, j ∈ p ′ k : x i = x j } 10 CHRIS HEUNEN for a partition p ′ = ( p ′ 1 , . . . , p ′ l ′ ). Then w e see that f m ust be induced b y an injectiv e function { 1 , . . . , n } → { 1 , . . . , n } , which we ca n ex tend to a p ermutation π ∈ S ( n ). Then C ′ → C means that π p ′ ≤ p . But this is pre cisely a mo r phism in ( P ( n ) ⋊ S ( n )) op .   T erminal subalgebras. A maximal a b elia n subalgebr a D of a C*-a lg ebra A is a maximal elemen t in C ⊆ ( A ). If A is finite-dimensio nal, such D ar e unique up to conjugation with a unitary . The prime example is the fo llowing: if A is the C* -algebr a M n ( C ) of n -by- n com- plex matrice s, then maxima l ab elian subalgebra s D are pr ecisely the subalg ebras consisting of all matrices that are diag onal in some fixed basis. In finite dimension, maximal elements of C ⊆ ( A ) are the sa me as terminal ob jects of C ֌ ( A ). F or the following lemma, weakly terminal ob jects o f C ⊆ ( A ) a re in fact enough. Recall that an ob ject D is w eakly ter minal when every ob ject C allows a morphism C → D . Lemma 4.4 . If C ֌ ( A ) has a we ak terminal obje ct D , then ther e is an e quivalenc e C ֌ ( A ) ≃ C ֌ ( D ) of c ate gories. Pr o of. Clea rly the inclusio n C ֌ ( D ) ֒ → C ֌ ( A ) is a full and faithful functor, so it suffices to prove that it is es s entially s ur jective. Let C ∈ C ֌ ( A ). Then there exists an injectiv e ∗ -homomor phism f : C → D b ecaus e D is weakly terminal. Hence C ∼ = f ( C ) ∈ C ֌ ( D ).   The c haracterizat ion. W e can now bring all the pieces together. Theorem 4.5. F or a c ate gory A , the fol lowing ar e e qu ivalent: • t he c ate gory A is e quivalent to C ֌ ( M n ( C )) op ; • t he c ate gory A is e quivalent to P ( n ) ⋊ S ( n ) ; • A satisfies (A1)–(A6), and ( A , ≤ ) satisfies (P1)–(P4) for n − 1 , and A (0 , 0) op is isomorphic to t he symmetric gr oup on n elements. Pr o of. Co mbine the pr evious tw o lemmas with Theorem 3 .7 and Theo rem 4.1.   W e can actually do better than c haracterizing factors A = M n ( C ) of type I n : the nex t theorem characterizes C ֌ ( A ) for any finite-dimensional C*-alg ebra A . Lemma 4. 6 . If C ֌ ( A i ) has a we ak terminal obje ct D i for e ach i in a set I , then the C*-dir e ct su m L i ∈ I D i is a we ak terminal obje ct in C ֌ ( L i ∈ I A i ) . Pr o of. Let C ∈ C ( L i ∈ I A i ). Then C is contained in the commutativ e subalg ebra L i ∈ I π i ( C ) o f L i ∈ I A i . Beca use each D i is weakly terminal, there exist morphisms f i : π i ( C ) → D i . Ther e fo re L i ∈ I f i is a mor phism L i ∈ I π i ( C ) → L i ∈ I D i , and th us the latter is weakly ter minal in C ֌ ( L i ∈ I A i ).   Theorem 4.7. A c ate gory A is e quivalent to C ֌ ( A ) op for a finite-dimensional C*-algebr a A if and only if ther e ar e n 1 , . . . , n k ∈ N such that: • A satisfies (A1)–(A5) and (A6’); • ( A , ≤ ) satisfies (P1)–(P4) for ( P k i =1 n i ) − 1 ; • A (0 , 0) op is isomorphic to t he symmetric gr oup on P k i =1 n i elements; • P k i =1 n 2 i = dim ( A ) . CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 11 Pr o of. E very finite-dimensio na l C*-alg e bra A is isomo rphic to a matrix rea lization of the form L k i =1 M n i ( C ) with n = P k i =1 n 2 i [6, Theorem I I I.1.1]. By Lemmas 4.4 and 4.6, we hav e C ֌ ( A ) ≃ C ֌ ( k M i =1 C n i ) ∼ = C ֌ ( C ( P k i =1 n i ) ) . So by Lemma 4.3, C ֌ ( A ) op ≃ P ( m ) ⋊ S ( m ) for m = P k i =1 n i . Now the statement follows from Theorem 4.5.   Notice that by L e mma 4.6 we may indeed use the whole partition lattice P ( m ) in the previous theorem instead of the trunca ted one P ( n 1 ) × · · · × P ( n k ); this is one of the consequence s of working with equiv alences of ca tegories instead of isomorphisms. 5. P oset-monoid-amalgama tions The main idea of o ur characterization of C ֌ ( A ) for finite-dimensio na l C*-alg ebras A holds unabated in the infinite-dimensiona l case . How e ver, the technical imple- men tation of the idea nee ds s ome adapting. F or example, the a ppropriate monoid is no longer a group. Ther efore, we will hav e to refine axiom (A6) into (A6’) and (A7’) a s follows. W e r e-list the other axio ms for conv enience. Definition 5.1. A ca tegory A is called a p oset-monoid-amalgamation when ther e exist a partial order P and a monoid M such tha t: (A1’) ther e is a weak initial ob ject 0, unique up to isomorphism; (A2’) ther e is a faithful retra ction F of the inclusion A (0 , 0) ֒ → A ; (A3’) ther e is an iso morphism α : A (0 , 0 ) → M op of monoids; (A4’) ther e is an equiv alence ( A , ≤ ) β / / P β ′ o o of preo rders; (A5’) fo r ea ch ob ject x there is an isomorphism f : x → β ′ ( β ( x )) with αF ( f ) = 1; (A6’) fo r ea ch ob ject y and m : 0 → 0, there is f : x → y such tha t F ( f ) = m , that is univ ersal in the s e nse that f ′ = f g w ith F ( g ) = 1 for an y f ′ : x ′ → y with F ( f ′ ) = m ; (A7’) if F ( f ) = m 2 m 1 for a morphism f , then f = f 1 f 2 with F ( f i ) = m i . The idea behind axiom (A6’) is that in P ⋊ M , we can acce ss the ob ject mq through the morphisms m : p → q . Ther e is a lwa y s at least one s uch mo rphism, namely m : mq → q , b eca use trivially mq ≤ mq . This might no t b e an isomorphis m, but it still has the universal prop er ty that all other morphisms m : p → q factor through it. W e can rephrase this universality as follo ws: for each ob ject y o f P ⋊ M and m ∈ M , there is a maxima l element of the set { f : x → y | α ( F ( f )) = m } , preorder ed by f ≤ g iff f = hg for some morphism h satisfying α ( F ( h )) = 1. x f / / y z g ? ? ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ h O O ✤ ✤ ✤ Also, notice the swap in (A7’). It is caused by the contrav a riance in the comp os ition of P ⋊ M and (A3’), a nd is not a mistake, as the following example shows. Example 5.2. If P is a partial order with le ast elemen t, and M is a monoid acting on P , then P ⋊ M is a p ose t-monoid-amalg amation. 12 CHRIS HEUNEN Pr o of. The least element 0 of P is a weak initial o b ject, satisfying (A1’). Conditions (A2’)–(A4’) are satisfied by definition, and (A5’) is v acuo us. T o verify (A6’) for q ∈ P and m ∈ M , no tice tha t mq ≤ m q , and if p ≤ mq , then c ertainly p ≤ 1 m q . Finally , (A7’) means that if p ≤ m 2 m 1 r , we should b e able to provide q such that p ≤ m 2 q and q ≤ m 1 r ; taking q = m 1 r suffices.   Lemma 5.3. If A satisfies (A1’)–(A7’), then it induc es an action of M on P given by pm = β ( x ) if f : x → β ′ ( p ) is a maximal element with α ( F ( f )) = m . Pr o of. Firs t, notice that for a ny p ∈ P and m ∈ M there exists a maximal f : x → β ′ ( p ) with α ( F ( f )) = m by (A6’). If there is another maxima l f ′ : x ′ → β ′ ( p ) with α ( F ( f ′ )) = m , then there are morphisms g : x → x ′ and g ′ : x ′ → x with F ( g ) = 1 = F ( g ′ ). Hence F ( g g ′ ) = 1 = F ( g ′ g ), and becaus e F is faithful, g is a n isomorphism with g ′ as inv erse. So x ∼ = x ′ , and therefore β ( x ) ∼ = β ( x ′ ). But becaus e P is a par tial order, this means β ( x ) = β ( x ′ ). Thus the a ction is well-defined on ob jects. T o see that it is well-defined on morphisms, supp ose that p ≤ q . Then ther e is a morphism f : β ′ ( p ) → β ′ ( q ) with F ( f ) = 1 . F or any m : 0 → 0, w e can find maximal f p : x p → β ′ ( p ) with F ( f p ) = m , and maxima l f q : x q → β ′ ( q ) with F ( f q ) = m . Now f f p : x p → β ′ ( q ) has F ( f f p ) = m . Because f q is a maximal such mo rphism, f f p factors thro ugh f q . That is, ther e is h : x p → x q with f q h = f f p and F ( h ) = 1. So mp ≤ mq . Next, we verify that this a ssignment is functor ial M → Cat ( P, P ). Clear ly id β ′ ( p ) is maximal among morphisms f : x → β ′ ( p ) with F ( f ) = 1. Therefore 1 p = β ( β ′ ( p )) = p . F or m 2 , m 1 ∈ M and p ∈ P , we hav e m 1 p = β ( x 1 ) where f 1 : x 1 → β ′ ( p ) is maximal such that α ( F ( f 1 )) = m 1 . Therefore m 2 ( m 1 p ) = β ( x 2 ), where the morphism f 2 : x 2 → β ′ ( β ( x 1 )) is maximal such that α ( F ( f 2 )) = m 2 . By axio m (A5’), there is an isomorphism h : x 1 → β ′ ( β ( x 1 )) with F ( h ) = 1. This gives f = f 1 h − 1 f 2 : x 2 → β ′ ( p ) with α ( F ( f )) = α ( F ( f 2 )) · α ( F ( h )) − 1 · α ( F ( f 1 )) = m 2 m 1 . W e will now show that f is universal w ith this prop erty . If g : y → β ′ ( p ) als o has α ( F ( g )) = m 2 m 1 , then (A7’) pr ovides g 2 : y → z a nd g 1 : z → β ′ ( p ) with g = g 1 g 2 and α ( F ( g i )) = m i . x 2 f 2 / / f   β ′ ( β ( x 1 )) h − 1 / / x 1 f 1 / / β ′ ( p ) z hk f f ◆ ◆ ◆ k < < ③ ③ ③ g 1 ❤ ❤ ❤ ❤ ❤ ❤ ❤ 3 3 ❤ ❤ ❤ ❤ ❤ ❤ ❤ y l O O ✤ ✤ ✤ ✤ g 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ 2 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ g C C By maximality of f 1 , there ex is ts k with g 1 = f 1 k and α ( F ( k )) = 1. And by maximality of f 2 , ther e is exis ts l with hk g 2 = f 2 l and α ( F ( l )) = 1. Hence g = g 1 g 2 = f 1 k g 2 = f 1 h − 1 hk g 2 = f 1 h − 1 f 2 l = f l . So f is maximal with F ( f ) = m 2 m 1 . Thu s ( m 2 m 1 ) p = β ( x 2 ) = m 2 ( m 1 p ).   Theorem 5.4. If A satisfies (A1’)–(A7’), then ther e is an e quivalenc e A → P ⋊ M given by x 7→ β ( x ) on obje cts and f 7→ α ( F ( f )) on morphisms. CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 13 Pr o of. Firs t, it follows from (A6’) that the assignment of the statemen t is well- defined, i.e. that α ( F ( f )) is indeed a mor phism of P ⋊ M . Indeed, if f : x → y , then we need to show that β ( x ) ≤ α ( F ( f )) · β ( y ). Unfolding the definition o f the action, this mea ns we need to find a maximal k : x ′ → β ′ ( β ( y )) with F ( f ) = F ( k ), such that β ( x ) ≤ β ( x ′ ). Unfolding the definition of the preorder, this mea ns we need to find a mo rphism h ′ : β ′ ( β ( x )) → β ′ ( β ( x ′ )) with F ( h ′ ) = 1. By (A5’), it suffices to find h : x → x ′ with F ( h ) = 1 instead. But by (A6’), there exis ts maximal k : x ′ → β ′ ( β ( y )) with F ( k ) = F ( f ). By its maximality , there exists h : x → x ′ with F ( h ) = 1 and f = k h . In particular , β ( x ) ≤ β ( x ′ ). F unctoria lity follows directly from the previous lemma, so indee d we have a well- defined functor A → P ⋊ M . Mor eov er, our functor is essentially surjective b ecause β is an equiv alence, and it is faithful b ecause F is faithful. Finally , to prove fullness, let m : β ( x ) → β ( y ) b e a morphism in P ⋊ M . This means that β ( x ) ≤ β ( y ) m , which unfolds to: there are mor phisms f : x → z and h : z → β ′ ( β ( y )) with F ( f ) = 1 and h maximal with α ( F ( h )) = m . By (A5’), this is equiv alent to the existence o f a morphism f : x → z with F ( f ) = 1 and a morphism h : z → y max ima l with α ( F ( h )) = m . Now ta ke k = hf : x → y . Then α ( F ( k )) = α ( F ( hf )) = α ( F ( f )) α ( F ( h )) = 1 · m = m. Hence our functor is full, and we conclude that it is (half of ) an eq uiv alence.   6. The infinite-dimensional case T o a dapt Theor em 4 .5 to the infinite-dimensional case, we hav e to make three more a daptations. First, the p oset P now b ecomes a lattice of partitions of a (lo cally) co mpa ct Hausdor ff space. Seco nd, the symmetric group gets replaced by symmetric monoids o n (lo cally) compact Hausdorff spa ces. Thir d, w e hav e to b e more car eful ab out maximal ab elian subalgebra s. Infinite partition lattices. F o r ar bitrary (lo cally) co mpact Hausdo r ff spaces, it is more convenien t to talk ab out equiv alence relations than abo ut partitions . An equiv alence rela tio n ∼ o n a (lo cally) compac t Hausdorff spa ce X is called close d when the s e t { x ∈ X | ∃ u ∈ U . x ∼ u } is c lo sed for every closed U ⊆ X . Closed equiv alence relations on X are also called p artitions , and form a partial o rder P ( X ) under r efinement : ∼ ≤ ≈ ⇐ ⇒  ∀ x, y ∈ X . x ∼ y = ⇒ x ≈ y  . Notice that quotients of a (lo ca lly) compact Hausdo r ff space by an equiv alence relation ar e again (lo cally) compact Hausdor ff if and only if the equiv alence relation is clos ed. F or tuna tely , a characterization of P ( X ) is known, due to Firby [8, 9]. This als o gives a characterization o f C ⊆ ( A ) for commut ative C*-alge bras A . As in Sec tio n 4, we firs t brie fly recall the nece ssary terminolog y . An element b of a lattice is calle d b ounding when (i) it is zer o or a n atom; o r (ii) it cov er s an atom and dominates exactly three ato ms ; or (iii) for distinct ato ms p, q there exists a n atom r ≤ b such that there are exactly three a toms less than r ∨ p and exa ctly three atoms less than r ∨ q . A collection of atoms of a lattice with at least four elemen ts is called single when it is a maximal c ollection o f atoms of whic h the join of any t w o domina tes exactly three atoms (not necess arily in the collection). A collection B of no nzero bo unding elements of a lattice is called a 1-p oint when (i) its atoms form a single 14 CHRIS HEUNEN collection; and (ii) if a is bo unding and a ≥ b ∈ B , then a ∈ B ; and (iii) any a ∈ B dominates a n atom p ∈ B . Theorem 6.1. A lattic e L with at le ast four element s is isomorphic to P ( X ) for a c omp act H au s dorff sp ac e X if and only if: (P1’) L is c omplete and atomic; (P2’) the interse ction of any two 1-p oints c ontains exactly one atom, and any atom b elongs t o exactly two 1-p oints; (P3’) for b ounding a, b ∈ L that ar e c ontaine d in a 1-p oint, { p ∈ Atoms( L ) | p ≤ a ∨ b } = { p ∈ Atoms( L ) | if x is a 1-p oint with p ∈ x then a ∈ x or b ∈ x } ; for b ounding a, b ∈ L that ar e not c ontaine d in a 1-p oint, { p ∈ Atoms( L ) | p ≤ a ∨ b } = { p ∈ A toms( L ) | p ≤ a or p ≤ b } ; (P4’) for 1-p oints x 6 = y ther e ar e b oun ding a, b with a 6∈ x , b 6∈ y , and a ∨ b = 1 ; (P5’) joins of b ounding elements ar e b ounding; (P6’) for n onzero a ∈ L , the c ol le ction B of b ounding elements e qual to or c over e d by a is t he un ique one satisfying: • W B = a ; • no 1-p oint c ontains two memb ers of B ; • if c is b ounding, b 1 ∈ B , and no 1-p oint c ontains b 1 and c , then ther e is a b oun ding b ≥ c s uch that (i) ther e is no 1-p oint c ontaining b oth b and b 1 , and (ii) wheneve r ther e is a 1-p oint c ontaining b oth b and b 2 ∈ B , then b ≥ b 2 ; (P7’) any c ol le ction of nonzer o b ounding elements that is not c ontaine d in a 1- p oint has a finite su b c ol le ction that is not c ontaine d in a 1-p oint; and X is (home omorphic to) the set of 1-p oints of L , wher e a su bset is close d if it is a singleton 1-p oint or it is t he set of 1-p oints c ontaining a fix e d b ounding element. Pr o of. See [9].   Remark 6 .2. The axiom resp ons ible for co mpactness of X is (P 7’). The previous theorem ho lds for lo cally compact Hausdorff spaces X when we replac e (P7’) by (P7”) every 1 -p oint cont ains a b ounding b such that { l ∈ L | l ≥ b } satisfies (P7’). Indeed, bec ause (P1’)–(P6’) alrea dy guar a ntee Hausdo r ffness, w e may take lo cal compactness to mea n that ev ery point ha s a compac t neig hbourho o d that is c losed. And closed sets corr esp ond to sets of 1-p oints cont aining a fixed b ounding element. As befo re, this dire c tly leads to a c haracteriza tion of C ⊆ ( A ) for comm utative C*-algebr as A . Corollary 6.3. A lattic e L is isomorphic t o C ⊆ ( A ) op for a c ommutative C*-algebr a A of dimension at le ast thr e e if and only if it satisfies (P1’)–(P6 ’) and (P7”). The C*-algebr a A is unital if and only if L additional ly satisfies (P7’). Pr o of. The la ttice C ⊆ ( A ) is that of sub ob jects of A in the categ ory of commutativ e (unital) C*- algebra s and (unital) nondege na rate ∗ -ho momorphisms. Recall that a sub obje ct is an equiv alence cla ss o f monomor phisms into a given ob ject, wher e tw o monics a re identified when they factor through one ano ther by an iso morphism. The dual notio n is a quotient : an eq uiv alence class o f epimorphisms out of a given ob ject. CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 15 By Gelfand duality , C ⊆ ( A ) is isomorphic to the o ppo site of the lattice of quotients of X = Sp ec( A ). But the latter is pr ecisely P ( X ) op , b ecause categ orical q uo tient s in the catego r y of (lo cally) compact Ha us dorff spa ces a re quotient spaces .   Symmetric monoid actions. W e write S ( X ) for the monoid of contin uous func- tions f : X ։ X with dense image on a lo cally compact Hausdorff space X , called the symm et ric monoid on X . The monoid S ( X ) acts on P ( X ), as describ ed in the following lemma. W e stick to the nota tion mx for the action of an element m of a monoid M on an ob ject x of a categor y as in Definition 3 .1. F or f ∈ S ( X ) and ∼ ∈ P ( X ) this b eco mes f ∼ . Prop ositi o n 6.4. F or any lo c al ly c omp act Hausdorff sp ac e X , t he monoid S ( X ) acts on P ( X ) by ( f ∼ ) = ( f × f ) − 1 ( ∼ ) . Pr o of. Firs t o f all, notice tha t f ∼ is reflexive, sy mmetric and transitive, so indeed is a well-defined eq uiv alence relation on X , whic h is closed b ecause f is contin uous . Concretely , x ( f ∼ ) y if and only if f ( x ) ∼ f ( y ). Moreo ver, c learly id ∼ = ∼ , and g ( f ∼ ) = ( g f ) ∼ , s o the ab ove is a g enuine ac tio n.   As b efore, this directly leads to a characterization of C ֌ ( A ) for co mm utative C*-algebr as A . Lemma 6.5. If A = C ( X ) for a lo c al ly c omp act Hausdorff sp ac e X , t her e is an isomorphi sm C ֌ ( A ) op ∼ = P ( X ) ⋊ S ( X ) of c ate gories. Pr o of. By definition, ob jects C of C ֌ ( A ) are sub o b jects of C ( X ) in the category o f commutativ e C*-a lg ebras. B y Gelfand duality , these corr e sp ond to quo tient s of X in the categor y of lo cally c o mpact Hausdorff spaces. But thes e, in turn, co rresp ond to closed equiv a lence relations on X , establishing a bijection betw een the ob jects of C ֌ ( A ) and P ( X ). Through Gelfand duality , a mor phism C ֌ C ′ in C ֌ ( A ) co rresp ondsto an epimorphism g : Y ′ ։ Y betw een the corres po nding spec tr a. W riting the quotients as Y = X/ ∼ and Y ′ = X/ ≈ for clos ed equiv alence rela tions ∼ and ≈ , we find that g co r resp onds to a co ntin uous f : X ։ X with dense image r esp ecting equiv alence: x ≈ y = ⇒ f ( x ) ∼ f ( y ) . But this just means ≈ ≤ ( f ∼ ). In other words, this is precis e ly a morphism f : ≈ → ∼ in P ( X ) ⋊ S ( X ).   W eakly termi nal subalgebras. In the infinite-dimensio na l c a se, it is no longer true that all ma ximal ab elian suba lg ebras of a C*-alge bra A are isomorphic. Ho w- ever, it suffices if there exists a commutativ e subalgebra into which all other s em bed by an injective ∗ -homomo rphism. T o b e precise, a co mmutative C*-s ubalgebra D of a C* -algebr a A is we akly terminal when eac h C ∈ C ( A ) allows an injective ∗ - homomorphisms C → D (that is not necessarily an inc lus ion, and not necessarily unique). Equiv alent ly , every mas a is isomorphic to a subalgebra of D . W eakly terminal co mmu tative subalgebra s D are maximal up to isomor phism, in the sense that if D can b e extended to a la rger commutativ e C*-subalg ebra E , then D ∼ = E . This do es not imply tha t D = E , i.e. that D is maximal. F o r a counterexample, take A = E = C ([0 , 1]) and D = { f ∈ E | f constant on [0 , 1 2 ] } . Then D ( E , but D ∼ = C ([ 1 2 , 1]) ∼ = E . 16 CHRIS HEUNEN Lemma 6.6. If A = B ( H ) for an infin ite-dimensional sep ar able Hilb ert sp ac e H , then C ֌ ( A ) has a we ak t erminal obje ct, ∗ -isomorphic to L ∞ (0 , 1) ⊕ ℓ ∞ ( N ) . Pr o of. Let C ∈ C ֌ ( A ). By Zorn’s lemma, C is a C*-s ubalgebra of a maximal element o f C ⊆ ( A ). A maximal element in C ⊆ ( A ) for a v on Neumann algebra A is itself a von Neumann alg ebra, bec a use it must eq ual its weak closur e. It is known that maxima l ab elian von Neumann subalg ebras of A = B ( H ) for an infinite- dimensional separa ble Hilb ert space H are unitarily equiv alent to one of the fol- lowing: L ∞ (0 , 1), ℓ ∞ ( { 0 , . . . , n } ) for n ∈ N , ℓ ∞ ( N ), L ∞ (0 , 1) ⊕ ℓ ∞ ( { 0 , . . . , n } ) for n ∈ N , or L ∞ (0 , 1) ⊕ ℓ ∞ ( N ) (se e [20, T he o rem 9.4 .1 ]). Each of these a llows an injectiv e ∗ -homomorphis m into the latter one, D = L ∞ (0 , 1) ⊕ ℓ ∞ ( N ). Therefore, there exists a morphis m C → D in C ֌ ( A ) for each C in C ֌ ( A ), so that D is weakly terminal in C ֌ ( A ).   If dim( H ) is uncountable, the situation b ecomes a bit more inv olved. A complete classification o f (maximal) ab elian subalg ebras of B ( H ) is known [26, 2 7], and we will use this to establish a weakly terminal commutativ e s ubalgebra in the following lemma. Before doing so, let us ex plain the intuitition b ehind the use of c ardinal nu mbers α and β in the statemen t. F or a ny car dinal num b er α , the C*-alg e bra B ( H ) has a commutativ e subalgebra L ∞ (0 , 1) α that needs to b e a ccounted for in a weakly terminal commutativ e subalgebra , as in the previo us lemma. Because there are dim( H ) many o f those , a sum ov er a sec o nd car dinal β ≤ dim( H ) is called for . Lemma 6.7. If A = B ( H ) for an infi n ite-dimensional Hilb ert sp ac e H , then C ֌ ( A ) has a we ak terminal obje ct, ∗ - isomorphic to L α,β ≤ di m( H ) L ∞  (0 , 1) α  , wher e α, β ar e c ar dinals, and (0 , 1) α is the pr o duct me asur e sp ac e of L eb esgue unit intervals. Pr o of. Max imal ab elia n subalgebra s C o f B ( H ) a re isomorphic to direct sums of L ∞  (0 , 1) α  ranging ov er cardinal n umbers α [26]. W e must s how that D = L α,β ≤ di m( H ) L ∞  (0 , 1) α  can b e identified with a subalgebra of B ( H ) that em- beds any suc h C . A commutativ e a lgebra L ∞  (0 , 1) α  acts on the Hilb ert spa ce L 2  (0 , 1) α  . Observe that L 2 (0 , 1) is se pa rable. Hence dim  L 2  (0 , 1) α  = max( α, ℵ 0 ) unless α = 0, in which case the dimens ion v anishes. Therefor e dim  L 2  (0 , 1) α  ≤ dim( H ) if a nd only if α ≤ dim( H ). Because H is infinite-dimensio na l, we hav e the equation dim( H ) = dim( H ) 3 of ca rdinal num b er s. Thus any maximal ab elian subalgebra C embeds into D , and D itself embeds as a ma ximal ab elian subalgebra of B ( H ).   The following infinite-dimensio nal analog ue of Lemma 4.6 will allo w us to de- duce from the previo us lemma that a rbitrar y type I von Neumann a lgebras p osses s weakly termina l commutativ e subalgebra s. (F or direct in tegrals, see [20, Chap- ter 14 ].) Lemma 6.8. L et ( M , µ ) b e a me asure sp ac e, and for e ach t ∈ M let A t b e a von Neumann algebr a. If C ֌ ( A t ) has a we akly terminal obje ct D t for almost every t , then the dir e ct inte gr al R ⊕ M D t d µ ( t ) is a we ak t erminal obje ct in C ֌ ( R ⊕ M A t d µ ( t )) . Pr o of. Let C ∈ C ֌ ( R ⊕ M A t d µ ( t )). Supposing A t acts on a Hilber t spa ce H t , then C is contained in R ⊕ M C t d µ ( t ), where C t is the von Neumann subalgebr a of B ( H t ) generated b y { a t | a ∈ C } . But b ecause almos t every D t is weakly terminal, this in turn embeds into R ⊕ M D t d µ ( t ), which is therefore weakly terminal.   CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 17 Corollary 6. 9 . Every typ e I von N eumann algebr a A p ossesses a we akly terminal c ommutative sub algebr a D . Mor e pr e cisely: if A = R ⊕ M A t d µ ( t ) for a me asure sp ac e ( M , µ ) and typ e I factors A t acting on Hilb ert sp ac es H t , then D is ∗ -isomorphic to Sp ec  R ⊕ M L α,β ≤ di m( H t ) L ∞ ((0 , 1) α )d µ ( t )  . Pr o of. E very type I von Neuma nn algebr a is a direct integral of t yp e I factors [20, Section 14.2]. Since the latter hav e weakly terminal comm utative subalg ebras by Lemma 6.7, we can deduce that the or iginal algebra has a weakly terminal commu- tative suba lgebra by Lemma 6.8.   Much le ss is known ab out the structure o f (max imal) ab elia n subalgebra s of von Neumann a lgebras o f type I I and I I I; see [5, 27]. The r e sults of [29] indicate that the previo us lemma might extend to show that C ֌ ( A ) has a weak terminal o b ject for any v on Neumann algebr a A . It would also be interesting to see if the pre vi- ous cor ollary implies that t ype I C*- algebras hav e weakly termina l comm utative subalgebra s. The char acterization. W e now arr ive at o ur main result: the characteriza tion C ֌ for infinite-dimensional type I von Neumann algebr as. Theorem 6.10. F or a c ate gory A and an infinite-dimensional typ e I von Neumann algebr a A = R ⊕ M B ( H t )d µ ( t ) for a me asur e sp ac e ( M , µ ) and H ilb ert sp ac es H t , the fol lowing ar e e quivalent: • t he c ate gory A is e quivalent to C ֌ ( A ) op ; • t he c ate gory A is e quivalent to P ( X ) ⋊ S ( X ) , wher e X is the top olo gic al sp ac e Spec  R ⊕ M L α,β ≤ di m( H t ) L ∞ ((0 , 1) α )  ; • A satisfies (A1’)–(A7’), and ( A , ≤ ) satisfies (P1’)–(P6’), (P7”), giving a top olo gic al sp ac e X , and A (0 , 0) op is isomorphic to t he monoid S ( X ) , and X is home omorphic to Sp ec  R ⊕ M L α,β ≤ di m( H t ) L ∞ ((0 , 1) α )d µ ( t )  . When A = B ( H ) for an infinite-dimensional Hilb ert sp ac e H , the sp ac e X simplifies to F α,β ≤ di m( H ) Spec  L ∞  (0 , 1) α  . When H is s ep ar able, X further simplifies to Spec ( L ∞ (0 , 1)) ⊔ Sp ec( ℓ ∞ ( N )) . Pr o of. Co mbine the pre v ious four lemmas with Theore ms 5.4 and 6 .1. F or the last condition, remember that Gelfand duality turns dir ect sums o f co mmu tative C*-algebr as into copr o ducts of Hausdorff spaces.   The Gelfand sp ectrum of ℓ ∞ ( N ) is the Stone- ˇ Cech compactifica tion of the dis- crete to po logy of N . In other w o rds, Spec ( ℓ ∞ ( N )) consists o f the ultr a filters on N . A top ologica l space is homeomo rphic to Sp ec( L ∞ (0 , 1)) if and only if it is compact, Hausdorff, totally dis connected, and its clope n subsets are is omorphic to the Bo olea n a lgebra of (Borel) measurable subsets of the interv al (0 , 1 ) mo dulo (Lebe s gue) negligible ones. Since b oth spaces are compact, w e could hav e used (P7’) instead of (P7”) in the previous theorem for the case A = B ( H ) with H separable. 7. I n cl usions versus injections This section co mpares C ⊆ to C ֌ . W e will show for C*-algebr as A and B that: • if C ֌ ( A ) and C ֌ ( B ) ar e isomo rphic, C ⊆ ( A ) and C ⊆ ( B ) ar e isomo rphic; 18 CHRIS HEUNEN • if C ֌ ( A ) and C ֌ ( B ) are equiv a le nt , C ⊆ ( A ) and C ⊆ ( B ) are Mo rita-equiv alent . Here we call tw o categor ies C and D Mo rita-equiv alent when they hav e equiv alent presheaf categ ories PSh( C ) ≃ PSh( D ). F or any catego ry C , reca ll that the c ategory R C P of elements of a presheaf P ∈ PSh( C ) is defined as follows. Ob jects are pairs ( C, x ) of C ∈ C and x ∈ P ( C ). A mo r phism ( C, x ) → ( D , y ) is a mor phism f : C → D in C satisfying x = P ( f )( y ). Recall that, for any presheaf P ∈ PSh( C ), ob jects o f the slic e catego ry PSh( C ) /P are natura l trans formations α : Q ⇒ P fr om some pr esheaf Q ∈ PSh( C ) to P . Lemma 7. 1. F or any P ∈ PSh( C ) , the top oses P Sh( C ) /P and P Sh( R C P ) ar e e quivalent. Pr o of. See [23, Exe rcise II I.8 (a)]; we write out a pro of for the sake of explicitness . Define a functor F : P Sh( C ) /P → PSh( R C P ) by F  Q α ⇒ P  ( C, x ) = α − 1 C ( x ) , F  Q α ⇒ P  ( C, x ) f → ( D , y )  = Q ( f ) , F ( Q β ⇒ Q ′ ) ( C,x ) = β C . Define a functor G : PSh( R C P ) → P Sh( C ) /P by G ( R ) = ( Q α ⇒ P ) where Q ( C ) = a x ∈ P ( C ) R ( C, x ) , Q ( C f → D ) = R  ( C, P ( f )( y )) f → ( D , y )  , α C ( κ x ( r )) = x, where κ x : R ( C , x ) → ` x ∈ P ( C ) R ( C, x ) is the co pro duct injection. The functor G acts o n morphisms as G ( R β ⇒ R ′ ) C = a x ∈ P ( C ) β ( C,x ) . Then one finds that GF ( Q α ⇒ P ) = ( Q α ⇒ P ), and F G ( R ) = ˆ R , w her e ˆ R ( C, x ) = { x } × R ( C, x ) , ˆ R  ( C, x ) f → ( D , y )  = id × R  ( C, P ( f )( y )) f → ( D , y )  . Thu s there is a na tural iso morphism R ∼ = ˆ R , and F and G form an equiv alence.   Definition 7.2. Define a pr esheaf Aut ∈ PSh( C ֌ ) by Aut( C ) = { i : C ∼ = → C ′ | C ′ ∈ C } , Aut  C k ֌ D  j : D ∼ = → D ′  = j   k ( C ) ◦ k : C ∼ = → j ( k ( C )) = D ′ . Notice that Aut( C ) co nt ains the automor phism gr oup of C . Also, a n y a utomor- phism o f A induces an e lement o f Aut( C ). The c ategory R C ֌ Aut of elements of Aut unfolds explicitly to the fo llowing. Ob jects are pairs ( C, i ) of C ∈ C and a ∗ -iso morphism i : C ∼ = → C ′ . A morphism ( C, i ) → ( D, j ) is an injective ∗ -homomo rphism k : C ֌ D such that i = j ◦ k . Prop ositi o n 7 . 3. The c ate gories C ⊆ and R C ֌ Aut ar e e qu ivalent. CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 19 Pr o of. Define a functor F : C ⊆ → R C ֌ Aut by F ( C ) = ( C, id C ) on ob jects and F ( C ⊆ D ) = ( C ֒ → D ) on mor phisms. Define a functor G : R C ֌ Aut → C ⊆ by G ( C, i ) = i ( C ) = co d( i ) on ob jects and G  k : ( C, i ) → ( D , j )  = ( i ( C ) ⊆ j ( D )) o n morphisms. Then GF ( C ) = C , and F G ( C, i ) = ( i ( C ) , id i ( C ) ) ∼ = ( C, i ), so that F and G implement an e q uiv alence.   Theorem 7.4. The t op oses P Sh( C ⊆ ) and PSh( C ֌ ) / Aut ar e e quivalent. Pr o of. Co mbining the previo us tw o lemmas, the equiv a lence is implemented e x plic- itly by the functor F : P Sh( C ֌ ) / Aut → P Sh( C ⊆ ) defined by F  P α ⇒ Aut  ( C ) = α − 1 C (id C ) F  P α ⇒ Aut  ( C ⊆ D ) = P ( C ֒ → D ) and the functor G : PSh( C ⊆ ) → PSh( C ֌ ) / Aut defined by G ( R ) =  P α ⇒ Aut  , P ( C ) = a i : C ∼ = → C ′ R ( i ( C )) , P  C k ֌ D  = a j : D ∼ = → D ′ R  j ( k ( C )) ⊆ j ( D )  , α C ( κ i ( r )) = i. This proves the theorem.   Hence the top os T = PSh( C ֌ ) is an ´ etendue , which means it is “lo cally like a space”; more precisely , it co nt ains an o b ject E such that the unique map from E to the terminal ob ject is an epimo rphism and the slice catego ry T /E is (eq uiv alent to) a lo ca lic top os. In this case, the ob ject E is the presheaf Aut. Lemma 7.5. If F : C → D is (half of ) an e quivalenc e, X is any obje ct of C and Y ∼ = F ( X ) , then the slic e c ate gories C / X and D / Y ar e e quivalent. Pr o of. Let G : D → C be the o ther half of the g iven equiv alence, and pic k an isomorphism i : Y → F ( X ). Define a functor H : C /X → D / Y by H ( a : A → X ) = ( i ◦ F a : F A → Y ) a nd H ( f : a → b ) = F f . Define a functor K : D / Y → C /X by K ( a : A → Y ) = ( η − 1 X ◦ Gi ◦ Ga : GA → X ) a nd K ( f : a → b ) = Gf . By naturality of η − 1 we then have K H ( a ) ∼ = a . And bec ause Gε = η − 1 we also have H K ( a ) ∼ = a .   Lemma 7.6. If the c ate gories C and D ar e e qu ivalent, then the top oses PSh( C ) and PSh( D ) ar e e quivalent. Pr o of. Given functors F : C → D and G : D → C that form an equiv alence, one directy verifies that ( − ) ◦ G : P Sh( C ) → PSh( D ) and ( − ) ◦ F : PSh( D ) → P Sh( C ) also form an equiv a le nc e .   Theorem 7.7. If C ֌ ( A ) and C ֌ ( B ) ar e e quivalent c ate gories, t hen C ⊆ ( A ) and C ⊆ ( B ) ar e Morita-e quivalent p osets, i.e. the top oses PSh( C ⊆ ( A )) and PSh( C ⊆ ( B )) ar e e quivalent. 20 CHRIS HEUNEN Pr o of. If C ֌ ( A ) ≃ C ֌ ( B ), then PSh( C ֌ ( A )) ≃ PSh( C ֌ ( B )) b y Lemma 7 .6. Moreov er, the ob ject Aut B is (iso mo rphic to) the image of the o b ject Aut A un- der this equiv alence. Hence PSh( C ⊆ ( A )) ≃ P Sh( C ֌ ( A )) / Aut A ≃ PSh( C ֌ ( B )) / Aut B ≃ P Sh( C ⊆ ( B )) by Theor em 7.4.   Remark 7.8. Hence C ֌ ( A ) is an inv a riant of the to po s PSh( C ⊆ ( A )) as well as of the C*-alg ebra A . It is not a co mplete inv ariant for the latter, how ever, as shown by Lemma 4.4. F or example, C ֌ ( M n ( C )) ≃ C ֌ ( C n ), but C ⊆ ( M n ( C )) 6 ∼ = C ⊆ ( C n ), and cer tainly M n ( C ) 6 ∼ = C n . W e ha ve r elied heavily on equiv alences of categories, and indeed a log ical formula holds in the top os PSh( C ) if a nd only if it holds in P Sh( D ) for equiv alent ca te- gories C a nd D . Ther efore one might a r gue that C ֌ has to o ma ny mor phisms, as compared to C ⊆ , for top oses ba sed o n it to hav e internal logics that a re interesting from the p oint o f view of foundatio ns of qua nt um mec hanics. Instead of equiv- alences, o ne could consider isomorphisms of categ o ries. This also resembles the original Mack ey–Piron question more clos ely . After all, a n equiv alence of partia l orders is automa tically a n isomorphism. The fo llowing theore m s hows that C ֌ ( A ) is a weak er inv ariant of A than C ⊆ ( A ), in this se nse. Theorem 7 . 9. I f C ֌ ( A ) and C ֌ ( B ) ar e isomorphic c ate gories, t hen C ⊆ ( A ) and C ⊆ ( B ) ar e isomorphic p osets. Pr o of. Let K : C ֌ ( A ) → C ֌ ( B ) be an isomor phism. Supp ose that C, D ∈ C ֌ ( A ) satisfy C ⊆ D . Conside r the sub categ o ry C ֌ ( D ) of C ֌ ( A ). On the o ne hand, by Lemma 6 .5 it is iso morphic to P ( X ) ⋊ S ( X ) for X = Spec ( D ), and therefore has a faithful re traction F A of the inclusion C ֌ ( D ) → C ֌ ( D )(0 , 0) by Theo rem 5 .4. On the other ha nd, K maps it to C ֌ ( K ( D )), which is iso morphic to P ( Y ) ⋊ S ( Y ) for Y = Sp ec( K ( D )), and therefo r e similarly has a retraction F B . Mor eov er, we have K F A = F B K . Now, by Theorem 5 .4, inclusio ns in C ֌ are characterized among a ll morphisms f b y F ( f ) = 1. Hence F B ( K ( C ֒ → D )) = K F A ( C ֒ → D ) = K (1) = 1, and therefor e K ( C ) ⊆ K ( D ).   It remains ope n whether existence of an isomorphis m C ⊆ ( A ) ∼ = C ⊆ ( B ) implies existence of an isomo rphism C ֌ ( A ) ∼ = C ֌ ( B ). This question ca n b e reduced as follows, a t lea st in finite dimension, be c a use every injective *-mor phism factor s uniquely as a ∗ -isomor phism followed by an inclusio n. W rite C ∼ = ( A ) for the cate- gory with C ( A ) for ob jects and ∗ - isomorphisms as morphisms. Supposing an iso- morphism F : C ⊆ ( A ) → C ⊆ ( B ), we have C ֌ ( A ) ∼ = C ֌ ( B ) if a nd only if there is an isomorphism G : C ∼ = ( A ) → C ∼ = ( B ) that coincides with F on o b jects. No w, in case A is (isomorphic to) M n ( C ), (so is B , and) if C, D ∈ C ( A ) ar e iso morphic then so are F ( C ) and F ( D ): if C ∼ = D , then dim( C ) = dim( D ), so dim( F ( C )) = dim( F ( D )) bec ause F preserves max imal chains, and hence F ( C ) ∼ = F ( D ). How ev er, it is no t clear whether this b ehaviour is functorial, i.e. extends to a functor G , or generalizes to infinite dimension. Appendix A. Inverse semigroups and ´ etendues The direct pro of of Theorem 7.4 follows from [19, A.1.1.7 ], but it c a n also b e a r- rived at through a detour via inv er se semigro ups, based on results due to F unk [10]. CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 21 This appendix descr ibes the latter intermediate res ults, which might b e of indepen- dent in terest. F or the rest of this app endix, we fix a unital C*-a lgebra A , and may therefore write C ⊆ for C ⊆ ( A ) and C ֌ for C ֌ ( A ). Definition A.1. Define a set T with functions T × T · → T a nd T ∗ → T by: T =  C i ֌ A | C ∈ C , i is an injective ∗ -homo morphism  , ( C ′ i ′ ֌ A ) · ( C i ֌ A ) = ( i − 1 ( C ′ ) i ′ ◦ i ֌ A ) , ( C i ֌ A ) ∗ = ( i ( C ) i − 1 ֌ A ) . The multiplication is well-defined, b ecaus e the inv erse image of a *- algebra under a ∗ -homo morphism is a gain a * -algebra , and the inv erse image o f a clo sed set is again a c lo sed set, so that i − 1 ( C ) is indeed a commutativ e C*-algebr a . The op eration * is well-defined b e c ause o f Le mma 2 .2; a nd o n the ima g e, i − 1 is a well- defined injective ∗ - homomorphism. One c a n verify that together, these da ta form an in verse semigro up; that is, mult iplication is asso cia tive, and i ∗ is the unique element with ii ∗ i = i and i ∗ ii ∗ = i ∗ . Lemma A.2. F or ( C i ֌ A ) ∈ T , we have i ∗ i = ( C ֒ → A ) and ii ∗ = ( i ( C ) ֒ → A ) . Pr o of. F or the former claim: ( C i ֌ A ) ∗ · ( C i ֌ A ) = ( i ( C ) i − 1 ֌ A ) · ( C i ֌ A ) = ( i − 1 ( i ( C )) i − 1 ◦ i ֌ A ) = ( C ֒ → A ) . F or the latter claim: ( C i ֌ A ) · ( C i ֌ A ) ∗ = ( C i ֌ A ) · ( i ( C ) i − 1 ֌ A ) = (( i − 1 ) − 1 ( C ) i ◦ i − 1 ֌ A ) = ( i ( C ) ֒ → A ) . This proves the lemma.   Definition A.3. F or a ny in verse semigro up T , one ca n define the group oid G ( T ) whose ob jects a r e the idempotents o f T , i.e. the elemen ts e ∈ T with e 2 = e . A morphism e → f is an element t ∈ T satisfying e = t ∗ t a nd tt ∗ = f . Prop ositi o n A. 4. The gr oup oids G ( T ) and C ∼ = ar e isomorph ic. Pr o of. An element ( C i ֌ A ) of T is idemp otent when i − 1 ( C ) = C and i 2 = i on C . That is, the o b jects of G ( T ) are the inclusions ( C ֒ → A ) of commutativ e C*-subalge br as; we ca n identify them with C . A morphism C → C ′ in G ( T ) is an element ( D j ֌ A ) of T such tha t ( C ֒ → A ) = j ∗ j = ( D ֒ → A ) and ( C ′ ֒ → A ) = j j ∗ = ( j ( D ) ֒ → A ), i.e. C = D and C ′ = j ( D ). That is, a morphism C → C ′ is an injective ∗ -homomor phism j : C ֌ C ′ that satisfies j ( D ) = C ′ , i.e. that is als o surjective. In other words, a morphism C → C ′ is a ∗ -iso morphism C → C ′ .   Definition A.5. F or any inverse semigr oup T , one can define a partia l o rder on the s et E ( T ) = { e ∈ T | e 2 = e } of idemp otents by e ≤ f iff e = f e . In fa c t, G ( T ) is an ordere d g roup oid, with G ( T ) 0 = E ( T ). Prop ositi o n A. 6. The p osets E ( T ) and C ⊆ ar e isomorph ic. 22 CHRIS HEUNEN Pr o of. As with G ( T ), ob jects of E ( T ) ca n b e identified with C . Moreov er, there is an ar row C → C ′ if a nd only if ( C ֒ → A ) = ( C ′ ֒ → A ) · ( C ֒ → A ) = ( C ∩ C ′ ֒ → A ) , i.e. when C ∩ C ′ = C . That is, there is an a rrow C → C ′ iff C ⊆ C ′ .   Also, G ( T ) is alwa ys a sub categor y of the following categ ory L ( T ). Definition A.7. F or a ny inv er se s emigroup T , o ne can define the le ft-ca ncellative category L ( T ) whose ob jects a re the idempotents of T . A morphism e → f is a n element t ∈ T sa tisfying e = t ∗ t a nd t = f t . Prop ositi o n A. 8. The c ate gories L ( T ) and C ֌ ar e isomorph ic. Pr o of. As with G ( T ), ob jects of L ( T ) can b e identifi ed with C . A morphism C → C ′ in L ( T ) is an element ( j : D ֌ A ) of T such that ( C ֒ → A ) = j ∗ j = ( D ֒ → A ) and ( D j ֌ A ) = ( C ′ ֒ → A ) · ( D j ֌ A ) = ( j − 1 ( C ′ ) j ֌ A ) . That is, a morphis m C → C ′ is an injective ∗ -homo mo rphism j : C ֌ A suc h that C = j − 1 ( C ′ ). Hence we can iden tify mo r phisms C → C ′ with injective ∗ - homomorphisms j : C ֌ C ′ .   Every ordered gr oup oid G has a class ifying top os B ( G ). W e now descr ibe the top os B ( G ( T )) explicitly , unfolding the definitions on [10, pa ge 4 87]. F or a presheaf P : C op ⊆ → S e t , define another preshea f P ∗ : C op ⊆ → S e t by P ∗ ( C ) = { ( j, x ) | j ∈ C ∼ = ( A )( C, C ′ ) , x ∈ P ( C ′ ) } . On a morphism C ⊆ D , the presheaf P ∗ : P ∗ ( D ) → P ∗ ( C ) a cts as ( k : D ′ ∼ = → D , y ∈ P ( D ′ )) 7− →  k   C : C ∼ = → k ( C ) , P ( k ( C ) ⊆ D ′ )( y )  . An o b ject of B ( G ( T )) is a pa ir ( P, θ ) o f a pr e s heaf P : C op ⊆ → Se t a nd a natural transformatio n θ : P ∗ ⇒ P . A morphism ( P, θ ) → ( Q, ξ ) is a natural tra ns formation α : P ⇒ Q satisfying α ◦ θ = ξ ◦ α ∗ , where the natura l transforma tion α ∗ : P ∗ ⇒ Q ∗ is defined by α ∗ C ( j, x ) = ( j, α C ( x )). Lemma A.9. The top oses P Sh( C ֌ ) and B ( G ( T )) ar e e quivalent. Pr o of. Co mbine P rop osition A.8 with [10, Prop osition 1.1 2]. Explicitly , ( P, θ ) in B ( G ( T )) gets ma ppe d to F : C ֌ ( A ) op → S e t defined by F ( C ) = P ( C ) and F ( k : C ֌ D )( y ) = θ C ( k : C ∼ = → k ( C ) , P ( k ( C ) ⊆ D )( y )) . Conv e r sely , F in PSh( C ֌ ) g ets mapp ed to ( P, θ ), where P ( C ) = F ( C ) , P ( C ⊆ D ) = F ( C ֒ → D ) , θ C ( j : C ∼ = → C ′ , x ∈ F ( C ′ )) = F ( C ′ j − 1 → C ⊆ D )( x ) . This finishes the pro o f.   There is a cano nical ob ject S = ( S, θ ) in B ( G ( T )), defined as fo llows. S ( C ) = { i : C ֌ A } , S ( C ⊆ D )( j : D ֌ A ) = ( j   C : C ֌ A ) . CHARA CTERIZA TIONS OF CA TEGORIES OF COMMUT A TIVE C*-SUBALGEBRAS 23 In this case S ∗ bec omes S ∗ ( C ) = { ( j, i ) | j : C ∼ = → C ′ , i : C ′ ֌ A } , S ∗ ( C ⊆ D )( j, i ) = ( j | C : C ∼ = → j ( C ) , i   j ( C ) : j ( C ) ֌ A ) . Hence we can define a na tural tr ansformatio n θ : S ∗ ⇒ S by θ C ( j, i ) = i ◦ j. The equiv alence of the previous lemma maps S in B ( G ( T )) to D in PSh( C ֌ ): D ( C ) = { i : C ֌ A } , D ( k : C ֌ D )( j : D ֌ A ) = ( j ◦ k : C ֌ A ) . T echnically , the top os B ( G ( T )) is an ´ etendue: the unique morphism fro m s ome ob ject S to the terminal ob ject is epic, and the slice top os B ( G ( T )) / S is (equiv alent to) a lo ca lic top os. The following lemma makes the latter equiv alence ex plic it. Lemma A.10. The top oses B ( G ( T )) / S and PSh( C ⊆ ) ar e e quivalent. Pr o of. Co mbine Pro p o sition A.6 with equation (1) in [10, page 488].   Combining the prev ious tw o lemmas , we find: Theorem A.11. The t op oses PSh( C ֌ ) / D and PSh( C ⊆ ) e quivalent.  In our sp ecific applicatio n, we ha ve more information and it is helpful to r e- formulate things slightly . By Le mma 2.2, giving an injective ∗ -homomor phism i : C ֌ A is the same as g iving a ∗ - isomorphism C ∼ = C ′ for some C ′ ∈ C (by taking C ′ = i ( C )). Hence S is isomorphic to the ob ject Aut = (Aut , θ ) in B ( G ( T )) with θ C ( j, i ) = i ◦ j . This leads to T he o rem 7.4. References [1] Benno v an den Berg and Chris Heunen. No-go theo rems for functo rial lo calic sp ectra of noncomm utativ e rings. Ele ctr onic Pr o c e e dings in The or etic al Computer Scienc e , 95:21–25, 2012. [2] Benno v an den Berg and Chr is Heunen. Noncomm utativit y as a colimit. Applie d Cate goric al Structur es , 20(4):393–414, 2012. [3] Andr eas Blass and Br uce E. Sagan. M¨ obius functions of lattices. A dvanc es in Mathematics , 127:94–123, 1997. [4] M arta Bunge. In ternal preshea ves toposes. Cahiers de top olo gie et g´ eom ´ etrie diff´ er entiel le c at´ egoriques , 18(3) :291–330, 1977. [5] Donald Bures. A b elian sub algebr as of von Neumann algebr as . Number 110 in M emoirs. Amer- ican Mathematical So ciety , 1971. [6] Kenneth R. Davidson. C*-algebr as by example . American Mathematical Society , 1991. [7] Andr eas D¨ oring and Christopher J. Isham. De ep Be auty , ch apter T op os methods in the foundations of ph ysics. Cam bridge Universit y Press, 2011. [8] Paul A. Fir b y . Lattices and compactificat ions I. Pr o ce e dings of the Lond on Mathematic al So ciety , 27:22–50, 1973. [9] Paul A. Fi rby . Lattices and compactific ations I I. Pr o c e e dings of the Lo ndon Mathematic al So ciety , 27:51–60, 1973. [10] Jonathon F unk. Semigroups and toposes. Se migr oup forum , 75:480–519, 2007. [11] Jan Hamhalter. Isomorphisms of ordered structures of ab elian C*-subalgebras of C*-algebras. Journal of Mathematic al Anal ysis and A pplic ations , 383:391–399, 2011. [12] Jan Hamhalter and Ek aterina T ur i lov a. Structure of associative subalgebras of Jordan oper- ator algebras. Q uarterly Journal of Mathematics , 64(2):397–408, 2013. [13] John Har ding and Andreas D¨ oring. Ab eli an subalgebras and the Jordan structure of a von Neumann algebra. Houston Journal of Mathematics , 2014. 24 CHRIS HEUNEN [14] Chri s Heunen. Compl ementarit y in categorical quan tum mec hanics. F oundations of Physics , 42(7):856– 873, 2012. [15] Chri s Heunen, Nicolaas P . Landsman, and Bas Spitters. A topos for algebraic quantu m theory . Communic ations in Mathematic al Physics , 291:63–110, 2009. [16] Chri s Heunen, Nicolaas P . Landsman, and Bas Spitters. De ep Be auty , chapte r Bohrification. Camb ridge Univ ersity Press, 2011. [17] Chri s Heunen and M an uel L. Reyes. Diagonalizing m atri ces ov er A W*-algebras. Journal of F unctional Analysis , 264(8) :1873–1898, 2013. [18] Chri s Heunen and Manuel L. Reye s. Active lattices determine A W*-algebras. Jou rnal of Mathematic al Analysis and Applic ations , 2014. [19] Peter T. Johnstone. Sketches of an elephant: A top os the ory c omp endium . Oxford Univ ersity Press, 2002. [20] Ri chard V. Kadison and John R. Ringrose. F undamentals of the the ory of op er ator algebr as . Academic Press, 1983. [21] Gudrun Kalmbac h. Orthomo dular L attic e s . Academic Pr ess, 1983. [22] Gudrun Kalmbac h. Me asur e s and Hilb ert lattic es . W or l d Scien tific, 1986. [23] Saunders Mac Lane and Iek e Mo erdijk. She aves in geo metry and lo gic . Springer, 1992. [24] Constantin Pir on. F oundations of quantum physics . Number 19 in Mathematical P hysics Monographs. W.A. Benjamin, 1976. [25] Mi kl´ os R´ edei. Quantum Lo gi c in Algebr aic Appr o ach . Kluw er, 1998. [26] Irving Segal. De c omp ositions of op er ator algebr as II: multiplicity the ory . Num ber 9 in Mem- oirs. American Mathematical So ciety , 1951. [27] Al lan M . Sinclair and Roger R. Smi th. Finite von Neumann algebr as and masas . Number 351 in London M athematical Society lecture notes. Cambridge Universit y Pr ess, 2008. [28] Mar ia Pia Sol` er. Characterization of Hilb ert spaces b y orthomodular s paces. Communic ations in Algebr a , 23(1):219–243, 1995. [29] Jun T omiyama. On s ome types of maximal ab elian subalgebras. Journal of funct ional anal- ysis , 10(373– 386), 1972. [30] Al exander Wilce. Handb o ok of quantum lo gic , volume II, c hapter T est spaces. Elsevier, 2008. [31] Y oung-Jin Y oon. C haracterizations of partition l attices. Bul letin of the K or e an Mathematic al So ciety , 31(2):237–242, 1994. University of Oxford E-mail addr ess : heunen@cs.ox .ac.uk