Quantifiers for quantum logic

Quan tifiers for quan tum logic Chris Heunen Marc h 18, 2022 Abstract W e consider categorical logic on the category of Hilb ert spaces. More generally , in fact, any pre-Hilb ert category suffices. W e chara cterise closed sub ob jects, and prov e that they form orthomo dular lattices. This sho ws that quantum logic is just an incarnation of categorical logi c, enabling us to establish an existential quantifier for quantum logic, and conclude that there cannot b e a u niv ersal q uan tifier. 1 In tro duction Quantum logic is the study of closed subspaces of a Hilb ert space [BV]. In- triguingly , this ‘logic’ is not distributive, but o nly sa tisfies the weaker axiom of orthomo dularity . One of the shortcoming s that has kept it fro m wide a doption is the lack o f quantifiers. In fact, it ha s b een called a ‘non-log ic’ [Abr]. On the other ha nd, categorica l logic [LS] ca n b e seen as a unified fr amew or k for any kind o f logic that deserves the na me. It is concer ned with interpreting (syntactical) logical formulae in catego r ies with enough str ucture to acco mmo- date this. An imp ortant part of it is the study o f subo b jects of a giv e n ob ject in the category at hand. Perhaps its mos t gra tif ying feature is that it g iv es a canonical prescription of what qua n tifiers should b e. The aim o f this pa per is to show that q uan tum logic is just an incarnation o f categoric al logic in ca tegories lik e tha t of Hilb e r t space s . In particular, we will establish an existential quantifier, a nd conclude that there cannot b e a universal quantifier. Section 2 first abstra cts the prop erties of the c ategory o f Hilb ert spaces that we need. This results in an axiomatisation of (pr e-)H il b ert c ate gories greatly resembling that of mo noidal Abelian categor ies. In fact, any (pre-)Hilbert cat- egory embeds into the ca tegory of (pre-)Hilb ert spa ces itself [Heu]. Next, Sec- tion 3 starts the inv estiga tion of subo b jects in Hilb e r t categ ories. It tur ns out that the natural ob jects of study ar e not the sub ob jects, but the close d sub obje cts or † -s ub obje cts . Section 4 then derives a functor that b ehav es as an ex isten tial quantifier a ccording to categorica l logic. Section 5 studies the emergent co ncept of orthogo nalit y in Hilbert ca tegories. First, it pr o ves that † -subo b jects form orthomo dular la ttices. Second, it exhibits a tight connec tion b etw een adjoint 1 morphisms in the base c a tegory and adjoint functors b et ween the la ttices of sub o b jects, the latter be ing impor ta n t in connection to qua n tifiers. Related w ork The present article should not be confused with the ‘categ orical qua n tum lo gic’ of [Dun]. That work develops a t yp e theo ry . Of course this is rela ted: “every logic is a logic over a t yp e theory” [Jac]. This pap er develops the log ic ov e r ‘the t y pe theory of Hilb ert spaces’. This paper also differs from [Har], in that the aim is explicitly a catego r- ical logic. Another difference is tha t that pape r restricts to those pro jectio ns that hav e an ortho complement, whereas we derive or thomodular it y from prior assumptions (namely † -kernels). 2 Pre-Hilb ert categories This sec tio n intro duce s the categorie s in which our study takes pla ce, somewha t concisely . F or mor e infor mation we refer to [Heu]. A functor † : H op → H with X † = X on ob jects and f †† = f on morphisms is called a † -functor ; the pair ( H , † ) is then ca lled a † -c ate gory . Such categorie s ar e automatically isomorphic to their opp osite, and the † -functor witnesses this self- duality . W e can consider coher ence of the † -functor with all sorts of structures. A mo rphism m in such a category that satisfies m † m = id is ca lled a † -mono and denoted  , 2 / / . Likewise, e is a † -epi , denoted  , 2 , when e e † = id . A mor phism is called a † -iso when it is b oth † -epi a nd † -mono. Similarly , a bipro duct on suc h a categor y is calle d a † -bipr o duct when π † = κ , where π is a pro jection and κ an injection. This is e q uiv alent to demanding ( f ⊕ g ) † = f † ⊕ g † . Also, an equaliser is ca lled a † -e qualiser when it is a † -mono, and a kernel is called a † -kernel when it is a † -mono. Finally , a † -category H is called † - monoidal when it is equipp ed with mono idal struc tur e ( ⊗ , C ) that co oper ates with the † - functor, in the sense that ( f ⊗ g ) † = f † ⊗ g † , and the coherence isomorphisms are † -isomor phisms. Definition 1 A c ate gory is c al le d a pre-Hilbe r t ca teg ory when • it has a † -funct or; • it has finite † -bipr o ducts; • it has (finite) † -e qualisers; • every † - mono is a † -kernel; and • it is symmet ric † -monoidal. Notice that a Hilb ert catego ry is self-dual (by the † -functor), and therefore that it automatica lly has all finite colimits, to o. 2 The categor y preHilb itself is a pre-Hilb ert catego ry whose monoidal unit is a simple generator , and so are its full subc ategories Hilb , and fdHi lb of finite- dimensional Hilb ert spa c es. Also, if C is a small categ o ry and H a pr e-Hilbert category , then [ C , H ] is ag ain a pr e-Hilbert c a tegory . W orking in pre- Hilbert categorie s can b e thought of as ‘natural’ or ‘basele s s’ (pre-)Hilb ert s pace theo ry . 3 Sub ob jects This section c har acterises closed sub ob jects categorically . But let us start with some easy prop erties of † -mono’s. Lemma 1 In any † -c ate gory: (a) A † -mono which is epi is a † - iso . (b) The c omp osite of † -epi’s is again a † -epi. (c) If X f / / Y g / / Z ar e such that b oth g f and f ar e † -epi, so is g . (d) If m and n ar e † - monos, and f is an iso with nf = m , then f is a † -iso. Pro of F or (a), notice that f f † = id implies f f † f = f , from which f † f = id follows fr o m the as sumption that f is epi. F or (b): g f ( gf ) † = g f f † g † = g † g = id . And for (c): g g † = g f f † g = g f ( g f ) † = id . Finally , cons ide r (d). If f is iso, in particular it is epi. If b oth nf and n are † -mono, then so is f , b y (c). Hence by (a), f is † -iso.  F r om now on, we work in an arbitr ary pr e-Hilb ert c ate gory H . Lemma 2 A morphism m is mono iff ker ( m ) = 0 . Conse quently, if mf = 0 implies f = 0 for al l f , then m is mono. Pro of Supp ose ker( m ) = 0. Let u , v satisfy mu = mv . Put q to be the † -co equaliser of u and v . Since q is a † - e pi, q = coker( w ) for some w . As mu = mv , m factors through q as m = nq . Then mw = nq w = n 0 = 0, so w factors through ker ( m ) a s w = ker( m ) ◦ p for some p . But since ker( m ) = 0, w = 0. So q is a † -iso, and in particular mono. Hence, from q u = q v follows u = v . Thus m is mono.   $ ker( m )   ? ? ? ? ? ? ? ? ? p o o _ _ _ _ w   u / / v / / m / / q _   n ? ?      Conv ersely , if m is mono, it follows from m ◦ ker ( m ) = 0 = m ◦ 0 that ker( m ) = 0. If f = 0 whenever mf = 0, then ker( m ) = 0 , so tha t m is mono.  3 3.1 F actorisation This subsection pr o ves that any mo rphism f : X → Y in a pre-Hilbe r t catego ry can be factor ised as an epi e : X → I follow ed b y a † -mono m : I → Y . (In Hilb , this is very ea sily prov ed concretely: e is simply the restric tio n o f f to I , the clo sure of its r ange, and m is the isometric inclusion of I in to Y .) Rec a ll that since a pre-Hilb ert category has † -kernels, it automatically also has † -cokernels by coker( f ) = ker ( f † ) † . Lemma 3 Any pr e-H ilb ert c ate gory has a factorisation system c onsisting of mono’s and † -epi’s. The factorisation is u nique u p to a un ique † -iso. Conse- quently, every † -epi is a † -c okernel of its † -kernel. Pro of Let a mor phis m f b e given. P ut k = ker ( f ) and e = coker ( k ). Since f k = 0 (as k = ker( f )), f factors thro ug h e (= coker ( k )) a s f = me . h   l               , 2 k / / e _   f / / g / / m ? ?            q  , 2 r O O s o o W e hav e to show that m is mono. Let g be such that mg = 0. B y Lemma 2 it suffices to sho w that g = 0 . Since mg = 0, m factor s through q = coker( g ) as m = rq . Now q e is a † -epi, b eing the composite o f tw o † -epi’s. So q e = coker( h ) for s ome h . Since f h = r qe h = r 0 = 0, h factor s through k (= ker( f )) as h = k l . Finally e h = ek l = 0 l = 0, so e factors through q e = coker( h ) as q = sqe . But since e is a ( † -)epi, this means sq = id , whence q is mo no. It fo llows from q g = 0 that g = 0, and the factorisa tio n is established. Since † -epi’s ar e regular epi’s, a nd hence strong epi’s, functoriality of the factorisatio n follows from [Bor, 4.4.5]. By Lemma 1d, the factorisa tion is unique up to a † -iso. Finally , supp ose that f is a † -epi. Then b oth the ab ov e f = m ◦ e and f = f ◦ id a re mono - † -epi fac torisations o f f . Hence f = e up to the unique mediating † -iso m , showing that f = coker(k er( f )).  W e just showed tha t any pre-Hilb ert catego ry has a factoris ation s ystem consisting of mono’s and † - epi’s. E quiv alently , it has a factor isation system of epi’s a nd † -mono ’s. Indeed, if we can factor f † as an † -epi follow ed b y a mono, then taking the dagg e r s o f those, we find that f †† = f factors as a n epi follow ed by a † -mono . The combination of both factoris ations yields that every morphism can be wr itten as a † -e pi, follow ed by a mo nic epimo rphism, follow ed by a † -mono ; this ca n b e thought of generalising p olar de c omp osition . 4 3.2 Closed sub ob jects, pullbac ks A sub obje ct of an ob ject X in a † -ca tegory is an equiv alence class o f mo no ’s m : M ֌ X , where m is equiv a len t to n : N ֌ X if there is a n isomor phism f : M → N satisfying nf = m . The class of sub ob jects o f X is denoted Sub( X ). It is partially order ed b y M ≤ N iff there is a morphism f : M → N with nf = m . It also has a la r gest element, r e pr esen ted by id X : X → X . Becaus e a pre- Hilbert category has pullbacks, Sub( X ) is in fact a meet-semilattice 1 , the meet of M and N being r e presen ted by the pullback of m and n . Mor eo ver, for ea c h f : X → Y , pullbac k along f induces a meet-preserving ma p f − 1 : Sub( Y ) → Sub( X ). Th us w e ha ve a functor Sub : H op → MeetSLat , the inverse image functor . A † - s u b obje ct is a sub ob ject that can be represented by a † -mono. W e wr ite ClSub( X ) for the class of † -sub ob jects of X . It inherits the par tial o rdering of Sub( X ). It can b e character ised precisely when a sub ob ject m is a † - subob ject, namely when there is an iso morphism ϕ such that m † m = ϕ † ϕ [Sel, 5.6]. Lemma 4 † -sub obje cts ar e stable un der pul lb acks. Pro of Let n : N  , 2 / / Y a nd f : X → Y . C o nsider their pullback ( P, p, q ). F actorise p as p = ker(cok er( p )) ◦ e with e epi. P e               p   q / / _  N _   n =ker(cok e r ( n ))   I   $ ker(cok e r ( p ))   ? ? ? ? ? ? ? g 5 5 j j j j j j j j j j X f / / Y Now coker( n ) ◦ f ◦ ker(cok er( p )) ◦ e = coker( n ) ◦ f ◦ p = coker( n ) ◦ n ◦ q = 0 ◦ q = 0 . Since e is epi, coker( n ) ◦ f ◦ ker(cok er( p )) = 0. So there is a g : I → N satisfying n ◦ g = f ◦ ker(coker ( p )). Since P is a pullback, there is a h : I → P such that ker( co k er( p )) = p ◦ h . No w ker (co k er( p )) = p ◦ h = ker(coker ( p )) ◦ e ◦ h and ker( co k er( p )) is mono, so e ◦ h = id P . Also p = ker(cok er( p )) ◦ e = p ◦ h ◦ e and p is mono, so h ◦ e = id I . Hence e is iso , and p = f − 1 ( n ) ∈ ClSub( X ).  1 W e dis regard si ze issues here. A † -category is called † -wel l-p ower e d if ClSub( X ) is a set for all ob j ec ts X in i t. Since C lSub( X ) for X ∈ Hilb is the set of closed subspaces of X , Hilb is † -well-p o wered. 5 Hence every morphism f : X → Y induces a meet-preserving map f − 1 : ClSub( Y ) → ClSub ( X ). Thus we hav e a functor ClSub : H op → Mee tSLa t , that we als o call the inverse image functor with abuse of ter minology . Recall that a closur e op er ation [Bir] consis ts in giving for every m ∈ Sub( X ) a m ∈ Sub( X ), satisfying (i) m ≤ m , (ii) m ≤ n ⇒ m ≤ n , and (iii) m = m . Lemma 5 m 7→ ker(cok er( m )) is a closur e op er ation. Pro of F or (i): coker ( m ) ◦ m = 0 , so m ≤ ker(coker ( m )). F or (ii): if m ≤ n , then coker ( m ) ◦ ker(cok er( m )) = 0, M m * * T T T T T T T T T T T T T T T   X coker( n ) 5 5 k k k k k k k k k k k k k k k coker( m ) ) ) S S S S S S S S S S S S S S S N n 4 4 j j j j j j j j j j j j j j j O O 0 4 4 i i i i i i i i i i i i i i i i ker(cok e r ( n )) O O so k e r( c o k er( m )) ≤ ker(cok er( n )). F or (iii): since ker(coker ( m )) ∈ ClSub( X ), we hav e ker(cok er(ker(cok er( m )))) = ker(coker ( m )) by Lemma 3.  Lemma 6 Ther e is a r efle ction Sub( X ) ker (c ok er( − )) / / ⊥ ClSub( X ) ? _ o o . Pro of W e ha ve to pro ve that ker(coker( m )) ≤ n iff m ≤ n for a mono m and a † -mono n . By (i) of Lemma 5 we have m ≤ ker(coker ( m )), proving one directio n. The conv erse direction is just (ii) o f L emma 5.  The previo us lemma could b e in ter preted a s a moral justification for studying the (replete) semilattice of clo sed s ubob jects instea d of tha t of sub ob jects. 3.3 Pro jections Instead o f closed sub ob jects, it turns o ut we can also co nsider pro jections. A pr oje ction on X is a mor phism p : X → X s atisfying p ◦ p = p = p † . W e define Pro j( X ) as the set of all pro jectio ns on X . It is par tially or dered by defining p ⊑ q iff p ◦ q = p . Prop osition 1 Ther e is an or der isomorphism C lSub ( X ) ∼ = Pro j( X ) . Pro of Any closed sub ob ject m yields a pro jection mm † . Conv er sely , any pro- jection p gives a closed sub ob ject ℑ ( p ). Let us verify that these maps are each o thers inv er ses. Starting with a close d sub o b ject repr esen ted by m , we end up with ℑ ( mm † ). Since m is † -mono a nd 6 m † is † -epi, this is already a facto risation in the sense o f Lemma 3, and hence ℑ ( mm † ) = m as c losed subob jects. Co n versely , a pro jection p maps to ii † , where p factors as p = ie fo r an epi e : X → I and † -mono i = ℑ ( p ). By functoriality o f the facto r isation it fo llows from pp = p that p i = i . Now i = pi = p † i = ( ie ) † i = e † i † i = e † , so indeed ii † = i e = p . Finally let us consider the o rder. If m ≤ n as sub ob jects, say m = nϕ for a † -mono ϕ , then mm † nn † = nϕϕ † n † nn † = nn † nn † = nn † , so indeed mm † ⊑ nn † . Co n versely , if p ⊑ q , then pq = p , whence ℑ ( pq ) = ℑ ( p ), so tha t indeed ℑ ( p ) ≤ ℑ ( q ) by functoria lit y of the factor isation.  Consequently , ev e r y result w e der iv e ab out the pa rtial o rder of clo sed sub- ob jects holds for the pr o jections and vice versa. 4 Existen tial quant ifier This sec tio n establis hes an existential quantifier, i.e. a left adjoint to the inverse image functor that satisfies the B ec k-Chev alley condition. Prop osition 2 ClSub( X ) is a lattic e. Pro of Since we a lready know that ClSub( X ) is a meet-s e milattice, it suffices to show that it has joins and a lea st element. J oins follow from e.g. [Bor, 4.2.6]. Explicitly , M ∨ N  , 2 m ∨ n / / X is giv en by ℑ ( s ), wher e s = [ m, n ] : M ⊕ N → X . The c lo sed smallest sub ob ject, the bottom element o f ClSub( X ), is given b y 0  , 2 0 / / X .  The † -mono m : M ֌ Y aris ing in the factorisa tion of a morphism f : X → Y of H is called the (dir e ct) image of f , denoted Im( f ). Notice that Im( f ) defines a unique † -subo b ject, although the representing † -mo no is only unique up to a † -iso. This † -sub ob ject is denoted ∃ f . More gener ally , for n : N  , 2 / / X in ClSub( X ), we define ∃ f ( N ) = Im( f n ) , which gives a well-defined ma p ∃ f : ClSub( X ) → ClSub( Y ) for any mor phism f : X → Y of H . Theorem 1 L et f : X → Y b e a morphism of H . The map ∃ f : ClSub( X ) → ClSub( Y ) is monotone and left-adjo int to f − 1 : ClSub( Y ) → ClSub( X ) . If g : Y → Z is another morph ism then ∃ g ◦ ∃ f = ∃ g ◦ f : ClSub( X ) → ClSub( Z ) . Also ∃ id = id . 7 Pro of W e follow the pro of of [But, Lemma 2.5]. F or monotonicity of ∃ f let M ≤ N in ClSub( X ). Fir s t facto rise n and then M → ∃ f N to get the follo wing diagram. X f / / Y N / / / / _ L R n O O ∃ f N _ L R O O M / / / / _ L R O O  d l m 5 5 I _ L R O O Now M / / / / I  , 2 / / Y is a n epi- † -mono factoris ation of f m , so I represents ∃ f M , and ∃ f M ≤ ∃ f N . T o show the adjunction, let M ∈ ClSub( X ) and N ∈ ClSub( Y ), a nd c onsider the solid arr ows in the following diag ram. X f / / Y f − 1 N  _ _ L R O O / / N _ L R n O O M / / / / K @ I m E E                < < y y y y ∃ f M = = { { { { M A J F F               If ∃ f M ≤ N then the right das hed map ∃ f M → N exists and the outer square commutes. Hence, s ince f − 1 N is a pullbac k , the left dashed map M → f − 1 N exists, and M ≤ f − 1 N . Conv er s ely , if M ≤ f − 1 N , factorise the map M → N to g et the image of M under f . In par ticular, this image then factors through N , whence ∃ f M ≤ N . Finally , the identit y ∃ g ◦ ∃ f = ∃ g ◦ f just states how left adjoin ts comp ose.  4.1 The B ec k-Chev alley condition Recall the Be ck-Cheval ley c ondition : if the left squar e b elo w is a pullback, then the right o ne must commute. P _  q / / p   Y g   X f / / Z ⇒ ClSub( P ) ∃ p   ClSub( Y ) ∃ g   q − 1 o o ClSub( X ) ClSub( Z ) f − 1 o o (BC) It ensures that the semantics of the existential quant ifie r is sound with r espect to substitution. T o show that our ∃ f satisfies (BC), we will assume that the monoidal unit C of our pre-Hilb ert catego r y H is a simple genera tor. Recall 8 that an ob ject C is calle d a gener ator when f x = g x for all x : C → X implies f = g : X → Y . It is called simple when Sub( C ) = { 0 , C } . In this case, [Heu, Theorem 4.6] shows that H is enr ic hed ov er Ab elian groups, so that we can talk of adding and subtracting mor phisms. Lemma 7 In a pr e-Hilb ert c ate gory whose monoidal unit is a simple gener ator, epi’s ar e stable u nder pul lb ack. Pro of The pr o of of [Bor, Pro position 1.7.6] works verbatim.  The previous lemma entails that H is a r e gular c ate gory , and hence that all results of [But] a pply . Thus, in s uc h a catego ry H one can soundly interpret regular logic, in particular the exis tential quantifier. Theorem 2 In a pr e-Hilb ert c ate gory whose m onoidal unit is a simple gener a- tor, (BC) holds. Pro of The pr o of of [But, Lemma 2.9 ] works v erba tim.  Also the F r ob enius identity holds. Let f : X → Y b e a mor phism of Hilb . Let M ∈ ClSub( X ) and N ∈ ClSub( Y ). Then ∃ f ( M ∧ f − 1 N ) = ∃ f M ∧ N as † -sub ob jects of Y . F or a pro of, w e refer to [But, Lemma 2.6]. 5 Orthogonalit y W e will now recover the orthogo nal s ubspace constructio n from the † - functor in any pre - Hilbert categ ory . The idea is to mimick the fact that ker( f ) ⊥ = ℑ ( f † ) in Hilb . Prop osition 3 Ther e is an involutive fu n ctor ( − ) ⊥ : ClSub ( X ) op → ClSub( X ) determine d by m ⊥ = ker( m † ) for m ∈ ClSub( X ) . Pro of T o show that the ab ov e definition extends functorially , let m, n ∈ ClSub( X ) be such that m ≤ n . Say that m factor s through n by m = ni for i : M → N . Then m † ◦ ker( n † ) = i † ◦ n † ◦ ker( n † ) = i † ◦ 0 = 0 . Hence k e r( n † ) factors throug h ker( m † ), that is, n ⊥ ≤ m ⊥ . W e finish the pro of b y showing that ⊥ is inv olutive: m ⊥⊥ = (ker( m † )) ⊥ = ker(k er( m † ) † ) = ker(cok er( m )) = m. Here, the last equation follows from Lemma 3.  The functor ( − ) ⊥ co opera tes with ∧ and ∨ as exp ected. Lemma 8 ClSub( X ) is an ortho c omplemente d lattic e, t ha t is, m ∧ m ⊥ = 0 and m ∨ m ⊥ = 1 for al l m ∈ ClSub ( X ) . (A forteriori, the c otu pl e [ m, m ⊥ ] is a † -iso.) 9 Pro of Recall that m ∧ m ⊥ is defined as the † -pullback M ∧ M ⊥ _   , 2 p / / _ _ _ _ _   q      M ⊥ _   ker( m † )   M  , 2 m / / X Because m is a † -mono, we hav e q = m † ◦ m ◦ q = m † ◦ ker( m † ) ◦ p = 0 ◦ p = 0 . Hence m ∧ m ⊥ = m ◦ q = m ◦ 0 = 0. T o prove the second claim, let f satisfy f ◦ [ m, m ⊥ ] = 0. Then f ◦ m = 0, so f factors thro ug h coker( m ) as f = g ◦ coker ( m ). Also f ◦ m ⊥ = 0 , so g = g ◦ ker ( m † ) † ◦ ker( m † ) = f ◦ ker ( m † ) = 0 , whence f = 0 . So, by Lemma 2, [ m, m ⊥ ] is epi. Hence [ m, m ⊥ ] factors as id ◦ [ m, m ⊥ ], but a lso as ( m ∨ m ⊥ ) ◦ p . So m ∨ m ⊥ m us t b e a † -iso. That is, m ∨ m ⊥ = 1 . Let us prove tha t [ m, m ⊥ ] is also a † -mono , and hence even a † -iso : [ m, m ⊥ ] † ◦ [ m, m ⊥ ] = h m † , ker( m † ) † i ◦ [ m, ker( m † )] =  m † ◦ m m † ◦ ker( m † ) ker( m † ) † ◦ m ker( m † ) † ◦ ker( m † )  = id M ⊕ M ⊥ .  How ever, ( − ) ⊥ has po or ‘substitution prop erties’, as it do es not comm ute with pullbacks. F or a coun ter example in Hilb , let X = C 2 , Y = C , f = π : X → Y : ( x, y ) 7→ x and m = 0 : 0 → Y . Then f − 1 ( m ⊥ ) = C 2 , but ( f − 1 ( m )) ⊥ = { ( x, 0) | x ∈ C } . In spite of this, a sp ecial case of “( − ) ⊥ is stable under pullbacks” still holds : we now recov er orthomo dularit y of ClSub( X ) using the previous lemma. Theorem 3 ClSub( X ) is an orthomo dular lattic e, that is, m ∨ ( m ⊥ ∧ n ) = n whenever m ≤ n ∈ ClSub( X ) . Pro of Say that m fac tors through n as m = nd . Let ( p, q ) b e the pullback o f ker( m † ) alo ng n . Then n ◦ [ q , d ] = [ nq , nd ] = [ p ◦ ker( m † ) , m ]. Hence, if we can show that [ q , d ] is epi, this would b e the fac torisation of [ p ◦ ker( m † ) , m ], and m ∨ ( m ⊥ ∧ n ) = n . First w e show that M ⊕ ( M ⊥ ∧ N ) is a pullback o f n a nd [ m, m ⊥ ]. Let f and g s a tisfy n ◦ g = [ m, ker( m † )] ◦ f . Then there are unique h 1 , h 2 making the 10 following diagr am commute. P g   h 2 A A A A A h 1 { { w w w w w w ED BC π 2 ◦ f o o GF @A π 1 ◦ f / / M ⊥ ∧ N _   , 2 q / / _   p   N _   n   M  l r d o o id  _ M ⊥  , 2 ker( m † ) / / X M  l r m o o This makes the fo llo wing int o a pullback diagram. P g   h h 1 ,h 2 i & & M M M M M M f ' ' M ⊕ ( M ⊥ ∧ N ) _  [ d,q ] / / id ⊕ p   N _   n   M ⊕ M ⊥ [ m, ker( m † )] / / X Since isomorphisms are stable under pullbac k, [ d, q ] is iso. In particula r , it is epi, and the theorem is e s tablished.  Corollary 1 Ther e c annot b e right adjoi n ts f − 1 ⊣ ∀ f for al l morphisms f of H , that satisfy the Be ck-Cheval ley c ondition. Pro of If there would be, then ∧ would have a right adjoint in every ClSub( X ) [AB, 3.4.16]. That is, there w ould be an implication. But the prime example Hilb shows that ClSub( X ) is in general not a Heyting alge bra.  Lemma 9 The funct or ⊥ : ClSub( X ) op → ClSub( X ) is an e quivalenc e of c at- e gories. In p articular, it is b oth left and right adjoint t o its opp osite ⊥ op : ClSub( X ) → ClSub( X ) op . Pro of This means precis e ly that m ⊥ ≤ n iff n ⊥ ≤ m , which holds since ⊥ is inv olutive.  The fo llowing theorem, inspired by [Pal], provides a connection b et ween a d- joint morphis ms in a pre-Hilb ert catego ry and a djoin t functors betw ee n lattices of † -sub ob jects. It explic a tes the rela tionship b et ween ∃ f and ∃ f † . Theorem 4 F or a morph ism f : X → Y , define f ⊥ = ⊥ Y ◦∃ op f : ClSub ( X ) op → ClSub( Y ) . Then ( f ⊥ ) op ⊣ ( f † ) ⊥ . 11 Pro of In general, for g : Y → X , the a djunction ( f ⊥ ) op ⊣ g ⊥ means tha t for M ∈ ClSub( X ) and N ∈ ClSub( Y ), ClSub( Y ) op ( ⊥ op Y ◦∃ f ( M ) , N ) ∼ = ClSub( X )( M , ⊥ X ◦∃ op f † ( N )) . That is, n ≤ ker( ℑ ( f m ) † ) iff m ≤ ker( ℑ ( g n ) † ). That means that in L 0 / /   ( ker( l † ) $ $ I I I I I I I I I J $ $ j † $ $ H H H H H H H H H M q O O     , 2 m / / X l † 6 6 @ v v v v v v v v v g † / / Y n †  , 2 N M  , 2 m / / i $ $ $ $ I I I I I I I I I I X f / / Y n †  , 2 coker( k )   ( H H H H H H H H H N I 6 6 @ k : : v v v v v v v v v 0 / / K p O O    (1) we must show that there is a p making the low er dia g ram comm ute iff ther e is a q making the upp er o ne comm ute, for the sp ecial case g = f † . So, let such a q b e giv e n. Then n † ◦ k ◦ i = n † ◦ f ◦ m = j ◦ l † ◦ m = j ◦ l † ◦ ker( l † ) ◦ q = j ◦ 0 ◦ q = 0 = 0 ◦ i, and since i is epi, n † k = 0. Hence n † factors thro ugh co k er( k ) via some p . Conv ersely , given p , we hav e j † ◦ l † ◦ m = n † ◦ f ◦ m = n † ◦ k ◦ i = p ◦ coker ( k ) ◦ k ◦ i = p ◦ 0 ◦ i = 0 = j † ◦ 0 , so since j † is mono, l † m = 0. Hence m facto r s thr ough k er( l † ) via some q .  In a diagra m, the adjunction of the previous theorem is the fo llowing. ClSub( X ) ⊢ ∃ f / / ClSub( Y ) ⊥ op Y   ClSub( X ) op ⊥ X O O ClSub( Y ) op ∃ op f † o o A conv erse to this theorem needs so me prepar ation, a nd the as sumption that the monoidal unit is a simple g enerator. Lemma 10 L et C b e a simple obje ct in a pr e-Hilb ert c ate gory. If f , g : X → C satisfy ker( f ) ≤ ker( g ) , then g = sf for some s : C → C . Unless f = 0 , this s is unique. Pro of Consider ∃ f X ∈ ClSub( C ). Either ∃ f X = 0, or ∃ f X is an iso a nd hence a † -iso since it is a † -mono . 12 If ∃ f X = 0, then f = 0 . So ker ( f ) is a † -iso, and since ker( f ) ≤ ker( g ), a lso ker( g ) is † - iso, whence g = 0 . Th us g = 0 f . If ∃ f X is a † - iso, in particular it is epi, and so is f . It ca n be factorised as a † -epi f ′ follow ed b y a mono s f . I ( ( s f ( ( Q Q Q Q Q Q Q X f ′ - 3 : m m m m m m m f / / / / C Now either s f = 0 or s f is iso. If s f = 0 then ∃ f X = 0 and hence f = 0, so that we ar e done by g = 0 f . Hence we ma y ass ume s f iso. Since ker( f ′ ) ≤ ker( f ) ≤ ker( g ) w e a r e thu s left with the following situatio n. L  # + ker( g ) ( ( P P P P P P P P C X f ′ . 3 ; n n n n n n n n g ( ( P P P P P P P P P K _ L R p O O . 3 ; ker( f ′ ) 6 6 n n n n n n n n C Now f ′ = coker(k er( f ′ )), and g ◦ ker( f ′ ) = g ◦ k e r( g ) ◦ p = 0 ◦ p = 0 . Hence ther e is a unique s ′ such that g = s ′ ◦ f ′ . Finally , putting s = s ′ s − 1 f satisfies g = s ′ f ′ = s ′ s − 1 f f = sf .  In a monoidal categ ory , morphisms s : C → C play the role of sca lars, and multiplication with them is natural. As ment io ned before , if C is a simple generator , then the s c alars comprise an inv olutive field [Heu , Theorem 4 .6]. The following lemma s umma r ises so me well-known (and easily pr oved) results. Lemma 11 L et H b e a monoida l c ate gory. Then H ( C, C ) is an involutive semiring t ha t acts on H by sca lar multiplication as fol lows: for s : C → C and f : X → Y , s • f is define d by X s • f / / ∼ =   Y C ⊗ X s ⊗ f / / C ⊗ Y ∼ = O O Mor e over, sc alar multiplic ation is natur al, t ha t is, ( s • g ) ◦ f = g ◦ ( s • f ) . Final ly, s • f = s ◦ f for s : C → C and f : X → C .  Now we can state and prove a conv erse to Theorem 4. Theorem 5 In a pr e-Hilb ert c ate gory whose monoida l unit is a simple gener- ator, if ( f ⊥ ) op ⊣ g ⊥ , t hen g = s • f † for a sc alar s . Unless f = 0 , this s is unique. 13 Pro of The adjunction of the hypothesis means that there is a q making the upper dia g ram in (1) c o mm ute iff ther e is a p making the lower o ne commute. So, if n † f m = 0 , then n † k i = 0, and be c a use i is epi hence n † k = 0. So p exists, whence q exists, so that n † g † m = j † 0 q = 0. T aking m = ker( n † f ) th us gives tha t k e r( n † f ) ≤ ker( n † g † ) fo r a ll n . Applying Lemma 10 yields that for a ll n : C → Y , there exists s n : C → C suc h that n † g † = s n n † f . Using Lemma 1 1 and dualising , this beco mes: for all n : C → Y , there is s n : C → C with g n = ( s † n • f † ) n . W e will show that all s n are in fact equal to each other (or zero). If all y : C → Y would hav e y = 0, then Y ∼ = 0, in whic h case g = 0 • f † . Otherwise, pic k an y : C → Y with y 6 = 0. There is an s : C → C with g y = ( s † • f † ) y . Put n ′ = y y † n : C → Y and n ′′ = ker( y † ) ◦ ker( y † ) † ◦ n : C → Y . Then n ′ + n ′′ = [id , id ] ◦ (( y ◦ y † ◦ n ) ⊕ (ker ( y † ) ◦ ker( y † ) † ◦ n )) ◦ h id , id i = [ y , y ⊥ ] ◦ [ y , y ⊥ ] † ◦ n = n. Moreov e r, ( s † n ′ • f † ) n ′ = g n ′ = g y y † n = ( s † • f † ) y y † n = ( s † • f † ) n ′ , so s n ′ = s . Finally ( s † n ′ • f † ) n ′ + ( s † n ′′ • f † ) n ′′ = g n ′ + g n ′′ = g n = ( s † n • f † ) n = ( s † n • f † ) n ′ + ( s † n • f † ) n ′′ . Hence s n = s n ′ = s for a ll n : C → Y , and w e hav e g n = ( s † • f † ) n . B ut since C is a generato r , g = s † • f † . Reviewing our choice of s in the ab ov e pro of, w e see that it is unique unless f = 0.  As a conseq uence, we find that, mo dulo s calars, the passa ge from morphisms f to functors ⊥ ◦∃ op f is one-to-one. A Fibred accoun t W e can summarise our results in ter ms of fibre d categ ory theor y [Jac]. There ar e fibrations Sub( H ) → H and ClSub( H ) → H . The latter is in fact a fibra tion o f meet-semilattices by Lemma 4. The reflection of Lemma 6 is a fibre d reflection. Our functor ∃ of Theorem 1 is a fibred copr oduct, and hence truely provides a existential qua n tifier. The assig nmen ts H → Sub( H ), X 7→ id X assemble into a fibred terminal ob ject 1 : H → Sub( H ), also for H → ClSub( H ). The fibrations Sub( H ) → H and ClSub( H ) → H a dmit comprehension. This means that 1 : H → Sub ( H ) 14 has a right adjoint, usually denoted by {−} : Sub ( H ) → H . Indeed, if we take { m : M ֌ X } = M , then Sub( H )(id X , m ) ∼ = H ( X , { m } ). In fact, the fibration ClSub( H ) → H is a bifibration by Theorem 1 a nd [Jac, 9.1.2] – no tice that the Beck-Chev alle y condition is no t needed for this. Thus, ClSub( H ) op → H op , ( m : M  , 2 / / X ) 7→ X is also a fibr ation. The following prop osition shows that ortho g onalit y can be extended to a functor b etw een fibrations, but it is not a fibred functor, basically bec a use it do es not commu te with pullback. Prop osition 4 ( − ) ⊥ extends to a funct or ClSub ( H ) op → ClSub ( H ) satisfying ClSub( H ) op   ( − ) ⊥ / / ClSub( H )   H op ( − ) † / / H (2) However, it is not a fibr e d functor. Pro of W e can under stand ( − ) ⊥ as a functor ClSub( H ) op → ClSub( H ) b y extending its actio n on morphisms a s follows. Let ( f , g ) b e a morphism m → n , that is, let f : X → Y a nd g : M → N satisfy f m = ng . W e are to define a morphism ( f , g ) ⊥ : n ⊥ → m ⊥ , that is , a pair f ⊥ : Y → X and g ⊥ : N ⊥ → M ⊥ satisfying f ⊥ ◦ n ⊥ = m ⊥ ◦ g ⊥ . Put f ⊥ = f † . Then m † ◦ f † ◦ n ⊥ = g † ◦ n † ◦ ker( n † ) = g † ◦ 0 = 0 , so there is a g ⊥ such that f † ◦ n ⊥ = ker( m † ) ◦ g ⊥ = m ⊥ ◦ g ⊥ . It must be g ⊥ = ( m ⊥ ) † ◦ m ⊥ ◦ g ⊥ = ker( m † ) † ◦ f † ◦ n ⊥ = coker( m ) ◦ f † ◦ ker( n † ) . This explicitly defines the functor ( − ) ⊥ : ClSub( H ) op → ClSub ( H ). It makes the s quare (2) c omm ute. Now, a morphism ( f , g ) : m → n of ClSub( H ) is Cartesian (ov er f ) iff f = ng m † = nn † f mm † . Consequen tly , the morphism ( f , g ) ⊥ : n ⊥ → m ⊥ in ClSub( H ) op is Cartesian iff f † = ker( m † ) ◦ ker( m † ) † ◦ f † ◦ ker( n † ) ◦ ker( n † ) † . Thu s, ( − ) ⊥ is a fibred functor iff f † = k e r( m † ) ker( m † ) † f † ker( n † ) ker( n † ) † whenever f = n n † f mm † for any morphism f and † -mono’s m and n . Finally , we come to our co unterexample. T ake m = κ M for M 6 = 0 , f = mm † and n = id M ⊕ M . Then f = mm † = mm † mm † = nn † f mm † . But ker( m † ) = κ ′ so ker( m † ) † = π ′ and ker( n † ) = 0, so ker( m † ) ◦ ker( m † ) † ◦ f † ◦ ker( n † ) ◦ ker( n † ) † . = κ ′ ◦ π ′ ◦ f † ◦ 0 = 0 6 = f † . Hence ( − ) ⊥ is not a fibred functor.  15 References [AB] Steve Awodey and Andrej Bauer . Intro duction to categor ic al logic. [Abr] Samson Abramsky . T emp erley-lieb alg ebra: F rom knot theory to lo gic and computation via quantum mec hanics. In Go ong Chen, Louis Ka uffm a n, and Sam Lomo naco, edito r s, Mathematics of Quantum Computing and T e ch- nolo gy , pages 4 15–458. T aylor and F ra ncis, 2 007. [Bir] Garrett Birkhoff. L attic e The ory . American Mathematical So ciet y , 19 48. [Bor] F ra ncis Bor c eux. 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