An embedding theorem for Hilbert categories

An em b edding theorem fo r H il b ert categories Chris Heunen Octob er 23, 2018 Abstract W e axiomatically define ( pre-)Hilbert categories. The axioms resem- ble those for monoidal A b elian catego ries with the addition of an inv o- lutive functor. W e then prove em b edding theorems: any locally small pre-Hilb ert category whose monoidal unit is a simple generator embed s (w eakly) monoidally in to the category of pre-Hilb ert spaces and adjointa b le maps, preserving adjoin t morphisms and all fin ite (co)limits. A n interme- diate result that is imp ortan t in its own righ t is that t he s calars in suc h a category necessarily form an invo lut iv e field. In case of a H il b ert category , the embedding exten d s to the category of Hilb ert spaces and continuous linear maps. The axioms for ( pre-)Hilbert categories are weak er than the axioms found in other approac hes to axiomatizing 2-Hilb ert spaces. Nei- ther enric hment nor a complex base field is presupp osed. A comparison to other approaches will b e made in th e introd uction. 1 In tro duction Mo dules over a ring are fundamen tal to algebr a . Distilling their c ategorical prop erties results in the definition o f Abelia n categ ories, which play a pro minen t part in algebraic geometry , cohomology and pure categor y theory . The pro to- t ypic a l Ab elian c ategory is that o f mo dules ov er a fixed r ing. Indeed, Mitchell’s famous embedding theo rem states that a n y small Abelian category embeds in to the catego ry of mo dules over some ring [Mitc hell, 1965, F reyd, 19 64 ]. Likewise, the ca tegory Hi lb o f (co mplex ) Hilb ert spaces a nd contin uous lin- ear tra nsformations is of par amoun t imp ortance in quantum theory and func- tional a nalysis. So is the category preHil b of (complex) pre-Hilb ert spac e s and adjointable maps . Although they closely resemble the category of mo dules (ov er the complex field), neither Hilb nor preHil b is Ab elian. A t the heart of the failure of Hilb and preHilb to b e Abelia n is the existence of a functor providing adjoint mor phis ms , called a dagg er, that witnesses self-duality . Hence the pro of metho d of Mitc hell’s embedding theor em does not apply . This article evens the situation, b y combining ideas from Ab elian categories and dagger categories . The latter ha ve been used fruitfully in mo deling as- pec ts of quantum physics recently [Abramsky & Co eck e, 2004, Seling er, 2007, Selinger, 20 08 ]. W e a xiomatically define (pr e-)Hilb ert c ate gories . The axioms 1 closely resemble those of a monoidal Abe lia n c a tegory , with the a dditio n o f a dagger. Their names a re justified by proving appropr iate embedding theo r ems: roughly sp eaking, pre-Hilbe r t c a tegories e mbed into preHilb , and Hilb ert cat- egories em b ed in to Hil b . These embeddings are in general not full, and only weakly monoidal. But otherwise they pr eserv e all the s tr ucture o f pre-Hilb ert categorie s, including all finite (co)limits, and adjoint morphisms (up to an iso- morphism of the induced base field). T o sketc h the historical context of these embedding theorems, let us start by r e calling that a catego ry is called Ab elian when: 1. it has finite bipro ducts; 2. it has (finite) equalis e rs a nd co equalisers; 3. ev ery mono morphism is a kernel, and every e pimo rphism is a cokernel. W e can p oint o ut alrea dy that Definition 1 b elow, o f pr e -Hilbert categ ory , is re- mark ably similar, except for the o ccurence of a da gger. F rom the a bov e a xioms, enrichmen t ov er Ab elian groups follows. F or the Abelian embedding theorem, there are (at lea s t) tw o ‘different’ pro ofs, one by Mitc hell [Mitchell, 196 5 ], and one b y Lubkin [Lubkin, 196 0 ]. Both op erate b y firs t embedding into the cat- egory Ab of Ab elian groups, and then adding a scalar multiplication. This approach can b e extended to also ta k e tensor pro ducts into account [H` ai, 20 02 ]. How ever, a s Ab is no t a self-dual category , this str ategy does not extend straightforwardly to the setting o f Hilbert spaces. Several author s hav e used an involution on the given ca tegory in this c o n text befo re. Specific a lly , by a dagger on a categor y C we mea n a functor † : C op → C that satisfies X † = X on ob jects and f †† = f on mor phisms. F or example, [Ghez, Lima & Rober ts, 1985, Prop osition 1.1 4 ] pr o ves that an y C*-c ate gory embeds into Hilb . Here, a C*-catego r y is a categ ory suc h that: 1. it is enriched over co mplex B anac h spaces a nd linea r con tra ctions; 2. it has a n an tilinea r dag ger; 3. every f : X → Y satisfies f † f = 0 ⇒ f = 0 , and there is a g : X → X with f † f = g † g ; 4. k f k 2 = k f † f k for every morphism f . The embedding of a C*-catego ry into Hi l b uses p ow erful ana lytical metho ds, a s it is basically a n extension of the Gelfand-Naimark theorem [Gelfand & Naimark, 19 43 ] showing that every C*-alge br a ( i.e. one-ob ject C* -category) can be realized concretely as an a lgebra of op erators on a Hilber t s pa ce. Compare the previ- ous definition to Definition 1 be low: the axio ms of (pre -)Hilbert categorie s a re m uch weak er. F or example, nothing ab o ut the base fie ld is built into the defi- nition. In fact, one of o ur main r esults der iv es the fact that the base semiring is a field. F or the same reaso n, our situation also differ s fro m T annakian c ate- gories [Deligne, 1990], that a re o therwise somewha t similar to our (pre-)Hilb ert 2 categorie s. Moreov er, (pre- )Hilbert ca tegories do no t presupp ose any enr ic h- men t, but derive it from prio r pr inc iples . A related em b edding theorem is [Doplicher & Rob erts, 198 9 ] (see also [Halv or son & M¨ uger, 2007] for a ca tegorical acco un t). It characterizes categ ories that are equiv a len t to the category of finite-dimensional unitary representations of a uniquely determined compact sup ergroup. Without ex plaining the p ostulates, let us mention that the catego ries C considered: 1. are enriched ov er complex vector spaces; 2. hav e an antilinear dagg er; 3. hav e finite bipr oducts; 4. hav e tensor pro ducts ( I , ⊗ ); 5. satisfy C ( I , I ) ∼ = C ; 6. every pro jection dagg er splits; 7. every ob ject is compa c t. Our definition o f (pre-)Hilb ert categ o ry also requir e s 2,3, a nd 4 a bov e. F urther- more, we will also use an analogue of 5, namely that I is a simple gener ator . But notice, again, that 1 a bov e presupp oses a base field C , a nd enrichmen t over complex vector spaces , wherea s (pre-)Hilb ert ca tegories do not. As will b ecome clear, our definition and theor e ms function re g ardless of dimension; we will come back to dimensionality and the compact ob jects in 7 a bov e in Subsectio n 7 .1. This is ta k en a step further b y [Baez, 199 7 ], which follows the “categor ifica- tion” programme originating in homotopy theory [Ka pranov & V o ev o dsky , 1994]. A 2-Hilb ert sp ac e is a category that: 1. is enr iched ov er Hilb ; 2. has an ant ilinea r dag ger; 3. is Ab elian; The categor y 2Hil b of 2-Hilb ert spac es turns out to b e monoidal. Hence it makes sense to define a symmetric 2-H*-algebr a a s a comm utative monoid in 2Hilb , in which furthermore every o b ject is compact. Then, [Baez, 19 97 ] proves that every symmetric 2-H* - algebra is equiv alent to a catego ry of contin uous uni- tary finite-dimensional representations of some compact superg roupoid. Again, the pro of is basically a ca tegorification o f the Gelfand-Naimark theor em. Al- though the motiv a tion for 2-Hilber t spaces is a categ orification o f a s ingle Hilb ert space, they res e m ble our (pre-)Hilb ert categories, that could b e seen as a ch a r- acterisatio n of the category o f a ll Hilb ert spaces. How ever, there are impor ta n t differences. Firs t o f all, ax iom 1 ab ov e again presupp oses bo th the complex n um- ber s as a ba se field, and a nontrivial enrichmen t. F or example, as (pre-)Hilb ert categorie s a ssume no enrichmen t, we do not hav e to consider co herence with 3 conjugation. Moreov er , [Ba ez, 1997] considers o nly finite dimensions, whereas the categ ory of a ll Hilb ert space s , rega rdless of dimension, is a prime e x am- ple o f a (pre- )Hilb ert c a tegory (see also Subsection 7.1). Finally , a 2-Hilb ert space is an Abelia n catego ry , wher eas a (pre-)Hilber t catego r y need not be (see Appendix A). Having sketc hed ho w the present work differs fr om existing work, let us end this in tro duction b y making our approach a bit more pr ecise while describing the structure o f this paper. Section 2 in tro duces our a xiomatisation. W e then emb a rk on proving em b edding theor e ms fo r such categories H , under the as- sumption that the monoidal unit I is a generator. First, we establish a functor H → sHMo d S , em b edding H in to the category of strict Hilb ert semimo dules ov er the involutiv e semiring S = H ( I , I ). Section 3 deals with this rigor ously . This extends pr evious work, that s ho ws tha t a category H with just bipr o d- ucts and tens o r pro ducts is enriched ov er S -semimo dules [Heunen, 2008]. If moreov er I is simple, Section 4 pr o ves that S is an in volutive field of charac- teristic zero. This is an improvemen t ov er [Vicary , 200 8 ], on which Section 4 draws for inspiration. Hence sHMo d S = p reH ilb S , and S embeds in to a field isomorphic to the complex num b ers. Ext ension of sc alars giv es an em b edding preHilb S → preHi l b C , discussed in Section 5. Finally , when H is a Hilb ert category , Section 6 shows that Cauc hy completion induces an embedding in to Hilb o f the image of H in preHi lb . Compo sing these functors then gives a n embedding H → Hilb . Along the wa y , we a lso discuss ho w a great dea l of the structure o f H is preser ved under this em b edding: in addition to b eing (weakly) monoidal, the embedding preserves a ll finite limits a nd colimits, and preser v es adjoint morphisms up to an isomor phism of the co mplex field. Section 7 con- cludes the main b o dy of the pap er, and Appendix A cons ide r s re le v ant asp ects of the categ o ry H ilb itse lf. 2 (Pre-)Hilb ert categorie s This section intro duces the ob ject of study . Let H be a ca tegory . A functor † : H op → H with X † = X on ob jects and f †† = f on morphis ms is called a dagger ; the pair ( H , † ) is then called a dagger c ate gory . Such ca tegories ar e automatically is omorphic to their oppo site. W e can conside r coherence of the dagger with resp ect to a ll sor ts of structur es. F or example, a morphism m in such a ca tegory that satisfies m † m = id is called a dagger mono(morphism) and is denoted  , 2 / / . Likewise, e is a dagger epi(morphism) , denoted  , 2 , when ee † = id . A morphis m is called a dagge r isomorphism when it is both dagger epic and dagger monic. S imila rly , a bipro duct on such a ca tegory is called a dagger bipr o du ct when π † = κ , where π is a pro jection a nd κ an injection. This is equiv alent to demanding ( f ⊕ g ) † = f † ⊕ g † . Also, an eq ua liser is called a dagger e qualiser when it can be r epresen ted by a da gger mono , and a kernel is ca lled a dagger kernel when it can be repr esen ted by a dagg er mono . Finally , a dag ger ca tegory H is called dagger monoidal when it is e q uipped with monoidal structure ( ⊗ , I ) that is compatible with the dagger, in the sens e that 4 ( f ⊗ g ) † = f † ⊗ g † , and the coherence isomo rphisms a re dag ger isomorphisms. Definition 1 A c ate gory is c al le d a pre-Hilber t c a tegory when: • it has a dagger; • it has finite dagger bipr o ducts; • it has (finite) dagger e qualisers; • every dagger mono is a dagger kernel; • it is symmetric dagger monoidal. Notice that no enrichmen t of any kind is as sumed. Instea d, it will follow. Also, no men tion is made of the complex num ber s or any other base field. This is a notable difference with other approaches men tioned in the Introduction. Our main theore m will a ssume that the monoidal unit I is a gener ator , i.e. that f = g : X → Y when f x = g x for all x : I → X . A final condition we will use is the following: the monoida l unit I is called simple when Sub( I ) = { 0 , I } and H ( I , I ) is at most of cont inuum cardina lit y . Intuitiv ely , a simple ob ject I ca n be thought o f as being “ o ne-dimensional”. The definition of a simple ob ject in abs tract algebr a is usually given without the size requirement, which we require to ens ure that the induced base field is not to o la rge. With an ey e tow ard future g eneralisation, this pap er postp ones assuming I s imple a s long as p ossible. The category Hilb itse lf is a lo cally s mall pre- Hilb ert category whose mono idal unit is a simple genera tor, and so is its s ub categor y fdHil b o f finite-dimensional Hilber t spaces (see App endix A). Finally , a pre -Hilbert categor y who s e morphisms are b ounded is c alled a Hilb ert c ate gory . It is eas ier to define this las t axiom rigo rously after a disc us sion of scalar s , and so w e defer this to Sectio n 6. 3 Hilb ert semimo dules In this section, we study Hilb ert semimo dules, to be defined in Definition 2 below. It turns out tha t the str ucture of a pre-Hilb ert catego ry H gives rise to an em b edding o f H int o a categ ory of Hilbert se mimodules. Let us first recall the notions of semir ing and semimodule in some detail, as they migh t b e unfamiliar to the reader. A s emiri ng is ro ug hly a “ring tha t does not neces sarily hav e subtra ction”. All the semirings we use will b e commutativ e. Explicitly , a co mm utative semir- ing consists of a set S , t wo elemen ts 0 , 1 ∈ S , and tw o binary op erations + and 5 · on S , suc h tha t the following equations hold fo r all r, s, t ∈ S : 0 + s = s, 1 · s = s, r + s = s + r , r · s = s · r, r + ( s + t ) = ( r + s ) + t , r · ( s · t ) = ( r · s ) · t, s · 0 = 0 , r · ( s + t ) = r · s + r · t. Semirings are also kno wn as rigs . F or more information w e refer to [Golan, 199 9]. A semimo dule o ver a comm utative semiring is a generalis ation of a mo dule ov er a co mm utative r ing , which in turn is a gener alisation o f a vector space ov er a field. Explicitly , a semimo dule over a commutativ e semir ing S is a s et M with a sp ecified elemen t 0 ∈ M , equipp ed with functions + : M × M → M a nd · : S × M → M satisfying the following equatio ns for a ll r, s ∈ S and l , m, n ∈ M : s · ( m + n ) = s · m + s · n, 0 + m = m, ( r + s ) · m = r · m + s · m, m + n = n + m, ( r · s ) · m = r · ( s · m ) , l + ( m + n ) = ( l + m ) + n, 0 · m = 0 , 1 · m = m, s · 0 = 0 . A function b et ween S -semimo dules is called S -semilinear when it preserves + and · . Semimo dules ov er a commutativ e s e miring S and S -semilinear tra ns for- mations form a catego ry SMo d S that lar gely b ehav es like that of mo dules over a commutativ e ring. F or example, it is s ymmetric monoidal closed. The tenso r pro duct of S -semimodules M and N is genera ted by elements o f the form m ⊗ n for m ∈ M and n ∈ N , sub ject to the following r elations: ( m + m ′ ) ⊗ n = m ⊗ n + m ′ ⊗ n, m ⊗ ( n + n ′ ) = m ⊗ n + m ⊗ n ′ , ( s · m ) ⊗ n = m ⊗ ( s · n ) , k · ( m ⊗ n ) = ( k · m ) ⊗ n = m ⊗ ( k · n ) , 0 ⊗ n = 0 = m ⊗ 0 , for m, m ′ ∈ M , n, n ′ ∈ N , s ∈ S a nd k ∈ N . It satisfies a universal prop ert y that differs s ligh tly from that of mo dules over a ring: every function from M × N to a comm utative mono id T that is semilinear in b oth v aria bles separa tely factors uniquely through a semilinear function from M ⊗ N to T / ∼ , where t ∼ t ′ iff there is a t ′′ ∈ T with t + t ′′ = t ′ + t ′′ . F or mo r e infor mation a bout semimo dules, we r efer to [Golan, 1999], or [Heunen, 2008] for a categoric a l p ersp e c tiv e. A comm uta tive involutive semiring is a co mm utative semiring S equipp ed with a s emilinear in volution ‡ : S → S . An element s of an inv o lutive semir- ing is called p ositive , denoted s ≥ 0, when it is of the form s = t ‡ t . The set of all p ositive elements of a n involutiv e semiring S is deno ted S + . F or every semimo dule M o ver a commutative in volutive semiring, there is also a semi- mo dule M ‡ , whose ca rrier set and addition are the same as b efore, but who se 6 scalar m ultiplicatio n sm is defined in terms of the scala r multiplication of M by s ‡ m . An S -semilinear map f : M → N also induces a map f ‡ : M ‡ → N ‡ by f ‡ ( m ) = f ( m ). Thus, an inv olution ‡ o n a commutativ e semiring S induces an inv olutive functor ‡ : SMo d S → SMo d S . Now, just as pre-Hilb ert s pa ces are vector spa ces equipp ed with an inner pro duct, w e can consider semimo dules with an inner pro duct. Definition 2 L et S b e a c ommutative involutive semiring. An S - s emi m o dule M is c al le d a Hilbert semimo dule when it is e quipp e d with a morphism h− | −i : M ‡ ⊗ M → S of SMo d S , satisfying • h m | n i = h n | m i ‡ , • h m | m i ≥ 0 , and • h m | −i = h n | − i ⇒ m = n . The H ilb ert semimo dule is c al le d stric t if mor e over • h m | m i = 0 ⇒ m = 0 . F or e xample, S itself is a Hilber t S -semimodule by h s | t i S = s ‡ t . Recall that a semiring S is multiplic atively c anc el lative when sr = st and s 6 = 0 imply r = t [Gola n, 1 999]. Thus S is a strict Hilb ert S -semimo dule iff S is multiplicativ ely cancellative. The follo wing choice of morphisms is a lso the standard choice of morphisms betw ee n Hilb ert C*-mo dules [Lance, 1 9 95 ]. 1 Definition 3 A semimo dule homomorphism f : M → N b etwe en Hilb ert S - semimo du les is c al le d adjoint a ble when ther e is a semimo dule homomorphism f † : N → M such that h f ‡ ( m ) | n i N = h m | f † ( n ) i M for al l m ∈ M ‡ and n ∈ N . The adjo int f † is unique since the power tr a nspose of the inner pro duct is a mono morphism. Howev er, it do es not necessarily exist, ex cept in sp ecial situations like (complete) Hilber t spaces ( S = C or S = R ) and bo unded semi- lattices ( S is the Bo olean semir ing B = ( { 0 , 1 } , max , min), see [P as ek a, 1999]). Hilber t S - semimodules and adjointable maps organise themselv es in a category HMo d S . W e denote by s HMo d S the full sub category of strict Hilb ert S - semimo dules. The choice of morphisms e nsures that HMo d S and sHM od S are dagg er categ ories. Let us study some o f their pro perties. The follow- ing lemma could b e rega rded as an analo gue of the Riesz-Fisc her theorem [Reed & Simon, 1 972, Theorem I II.1]. Lemma 1 HMo d S is enriche d over SMo d S , and HMo d S ( S, X ) = SMo d S ( S, X ) ∼ = X , wher e we s uppr esse d the for getful functor H M od S → SMo d S . 1 There is another analogy for th i s choice of morphisms. W ri ting M ∗ = M ⊸ S f or the dual S -semimo dule of M , Definition 2 resembles that of a ‘diagonal’ ob ject of the Chu con str uction on SMo d S . The Ch u construction provides a ‘generalised topology’, l ik e an inner pro duct prov i de s a ve ctor space pr o vides with a m et r i c and hence a topol ogy [Barr, 1999]. 7 Pro of F or X , Y ∈ HMo d S , the zero map X → Y in SM od S is self-adjoint , and hence a morphism in HMo d S . If f , g : X → Y a re adjointable, then so is f + g , as its adjoint is f † + g † . If s ∈ S a nd f : X → Y is adjointable, then so is sf , as its adjoint is s ‡ f † : h sf ( x ) | y i Y = s ‡ h f ( x ) | y i Y = s ‡ h x | f † ( y ) i X = h x | s ‡ f † ( y ) i X . Since comp osition is bilinear , HMo d S is thus enriched over SMo d S . Suppo se X ∈ HMo d S , and f : S → X is a morphis m of SMo d S . Define a morphism f † : X → S of SMo d S by f † = h f (1) | −i X . Then h f ( s ) | x i X = h sf (1) | x i X = s ‡ h f (1 ) | x i X = s ‡ f † ( x ) = h s | f † ( x ) i S . Hence f ∈ HM od S ( S, X ). Obviously HMo d S ( S, X ) ⊆ SMo d S ( S, X ). The fact that S is a gener ator for HMo d S prov es the last cla im SMo d S ( S, X ) ∼ = X . Notice from the pr o of of the a bov e lemma that the inner pro duct o f X can b e reconstructed from HMo d S ( S, X ). Indeed, if we temp orarily define x : S → X by 1 7→ x for x ∈ X , then we can use the adjoint b y h x | y i X = h x (1) | y i X = h 1 | x † ( y ) i S = x † ( y ) = x † ◦ y (1) . W e can go further b y providing HMo d S ( S, X ) itself with the structure o f a Hilber t S -s emimodule: for f , g ∈ HMo d S ( S, X ), put h f | g i HMo d S ( S,X ) = f † ◦ g (1). Then the a bov e lemma ca n b e stre ng thened a s follo ws. Lemma 2 Ther e is a dagger isomorphism X ∼ = HMo d S ( S, X ) in HMo d S . Pro of Define f : X → HMo d S ( S, X ) by f ( x ) = x · ( − ), a nd g : HMo d S ( S, X ) → X by g ( ϕ ) = ϕ (1). Then f ◦ g = id and g ◦ f = id , and moreover f † = g : h x | g ( ϕ ) i X = h x | ϕ (1) i X = ( x · ( − )) † ◦ ϕ (1) = h x · ( − ) | ϕ i HMo d S ( S,X ) = h f ( x ) | ϕ i HMo d S ( S,X ) Recall that (a s ubset of ) a semiring is c alled zer osumfr e e when s + t = 0 implies s = t = 0 for all elements s and t in it [Golan, 1999]. Prop osition 1 HMo d S has finite dagger bipr o ducts. When S + is zer osumfr e e, sHMo d S has finite dagger bipr o ducts . Pro of Let H 1 , H 2 ∈ HMo d S be given. Cons ider the S -semimo dule H = H 1 ⊕ H 2 . Eq uip it with the inner pro duct h h | h ′ i H = h π 1 ( h ) | π 1 ( h ′ ) i H 1 + h π 2 ( h ) | π 2 ( h ′ ) i H 2 . (1) Suppo se that h h | −i H = h h ′ | −i H . F or every i ∈ { 1 , 2 } and h ′′ ∈ H i then h π i ( h ) | h ′′ i H i = h h | κ i ( h ′′ ) i H = h h ′ | κ i ( h ′′ ) i H = h π i ( h ′ ) | h ′′ i H i , 8 whence π i ( h ) = π i ( h ′ ), and so h = h ′ . Thus H is a Hilbert semimo dule. The maps κ i are morphisms o f HMo d S , as their adjoints a re g iven by π i : H → H i : h h | κ i ( h ′ ) i H = h π 1 ( h ) | π 1 κ i ( h ′ ) i H 1 + h π 2 ( h ) | π 2 κ i ( h ′ ) i H 2 = h π i ( h ) | h ′ i H i . F or sHMo d S we need to verify that H is str ict when H 1 and H 2 are. Supp ose h h | h i H = 0. Then h π 1 ( h ) | π 1 ( h ) i H 1 + h π 2 ( h ) | π 2 ( h ) i H 2 = 0 . Since S + is zero- sumfree, we hav e h π i ( h ) | π i ( h ) i H i = 0 for i = 1 , 2. Hence π i ( h ) = 0, beca use H i is strict. Th us h = 0, and H is indeed strict. Prop osition 2 HMo d S is symmetric dagg er monoidal. W hen S is mult ip lic a- tively c anc el lative, sHMo d S is symmetric dagger monoidal. Pro of Let H , K b e Hilbert S -semimodules; then H ⊗ K is a gain an S -semimo dule. Define an equiv alence r elation ∼ on H ⊗ K b y setting h ⊗ k ∼ h ′ ⊗ k ′ iff h h | −i H · h k | −i K = h h ′ | −i H · h k ′ | −i K : H ⊕ K → S. This is a congruence r elation (see [Golan, 1999]), so H ⊗ H K = H ⊗ K/ ∼ is again an S -semimodule. Defining an inner pro duct on it by h [ h ⊗ k ] ∼ | [ h ′ ⊗ k ′ ] ∼ i H ⊗ H K = h h | h ′ i H · h k | k ′ i K . makes H ⊗ H K into a Hilber t semimo dule. Now let f : H → H ′ and g : K → K ′ be morphisms of HMo d S . Define f ⊗ H g : H ⊗ H K → H ′ ⊗ H K ′ by ( f ⊗ H g )([ h ⊗ k ] ∼ ) = [ f ( h ) ⊗ g ( k )] ∼ . This is a well-defined function, for if h ⊗ k ∼ h ′ ⊗ k ′ , then h f ( h ) | − i H ′ · h g ( k ) | − i K ′ = h h | f † ( − ) i H · h k | g † ( − ) i K = h h ′ | f † ( − ) i H · h k ′ | g † ( − ) i K = h f ( h ′ ) | −i H ′ · h g ( k ′ ) | −i K ′ , and hence ( f ⊗ H g )( h ⊗ k ) ∼ ( f ⊗ H g )( h ′ ⊗ k ′ ). Mor e over, it is a djo intable, a nd hence a mo rphism of HMo d S : h ( f ⊗ H g )( h ⊗ k ) | ( h ′ ⊗ k ′ ) i H ′ ⊗ H K ′ = h f ( h ) ⊗ g ( k ) | h ′ ⊗ k ′ i H ′ ⊗ H K ′ = h f ( h ) | h ′ i H ′ · h g ( k ) | k ′ i K ′ = h h | f † ( h ′ ) i H · h k | g ( k ′ ) i K = h h ⊗ k | f † ( h ′ ) ⊗ g † ( k ′ ) i H ⊗ H K = h h ⊗ k | ( f † ⊗ g † )( h ′ ⊗ k ′ ) i H ⊗ H K In the same way , one shows that the co herence iso morphisms α , λ , ρ and γ of the tenso r pro duct in SMo d S resp ect ∼ , and descend to dagger isomor phisms in HMo d S . F or ex ample: h λ ( s ⊗ h ) | h ′ i H = h sh | h ′ i H = s ‡ h h | h ′ i H = h s | 1 i S · h h | h ′ i H = h s ⊗ h | 1 ⊗ h ′ i S ⊗ H H = h s ⊗ h | λ − 1 ( h ′ ) i S ⊗ H H , 9 so λ † = λ − 1 . A r outine chec k shows that ( ⊗ H , S ) makes HMo d S int o a sym- metric monoidal ca tegory . Finally , let us verify tha t these tensor pro ducts desce nd to sHMo d S when S is multiplicatively cance lla tiv e. Supp ose 0 = h [ h ⊗ k ] ∼ | [ h ⊗ k ] ∼ i H ⊗ H K = h h | h i H · h k | k i K . Then since S is m ultiplicatively cance lla tiv e, either h h | h i H = 0 or h k | k i H = 0. Since H and K a re assumed strict, this means that either h = 0 or k = 0. In bo th cases we conclude [ h ⊗ k ] ∼ = 0, so that H ⊗ H K is indeed strict. Now supp ose H is a non trivia l 2 lo cally small pre- Hilbert categ ory with monoidal unit I . Then S = H ( I , I ) is a commutativ e inv olutive semiring, a nd H is enriched ov er SMo d S . Explicitly , the zero morphism is the unique one tha t factors thro ugh the zero ob ject, the sum f + g of t wo morphisms f , g : X → Y is given b y X ∆ / / X ⊕ X f ⊕ g / / Y ⊕ Y ∇ / / Y , and the m ultiplicatio n of a morphism f : X → Y with a scala r s : I → I is determined by X ∼ = / / I ⊗ X s ⊗ f / / I ⊗ Y ∼ = / / Y . The scalar m ultiplication works more generally for symmetric monoidal cat- egory [Abramsky , 2005]. The fact that the above provides an enrichmen t in SMo d S (and that this enric hment is furthermore functorial) is prov ed in [Heunen, 2008]. Hence there is a functor H ( I , − ) : H → SMo d S . If I is a g enerator, this functor is faithful. W e will now show that this functor in fact facto r s through sHMo d S . Lemma 3 L et H b e a n o n trivi al lo c al ly smal l pr e-Hilb ert c ate gory. D enote by I its monoidal un it. Then S = H ( I , I ) is a c ommutative involutive semiring, and S + is zer osumfr e e. When mor e over I is simple, S is multiplic atively c anc el lative. Pro of F or the pro of that S is a semiring we refer to [Heunen, 20 08 ]. If I is sim- ple, [Vicary , 2 008 , 3.5 ] shows that S is multiplicativ ely canc e lla tiv e, and [Vicary , 200 8, 3.10] shows that S + is zerosumfree in any ca se. Theorem 1 L et H b e a nontrivial lo c al ly smal l pr e-Hilb ert c ate gory. Denote its m ono idal u nit by I . Ther e is a functor H ( I , − ) : H → sHMo d S for S = H ( I , I ) . It pr eserves † , ⊕ , and kernels. It is monoidal when I is simple. It is faithful when I is a gener ator. Pro of W e hav e to put an S -v alued inner pro duct on H ( I , X ). Inspired b y Lemma 2, we define h− | −i : H ( I , X ) ‡ ⊗ H ( I , X ) → H ( I , I ) by (linea r extension of ) h x | y i = x † ◦ y for x, y ∈ H ( I , X ). The Y oneda lemma shows that its p o wer 2 The unique trivial semiri ng S with 0 = 1 i s sometimes excluded from consideration by con ven tion. F or example, fields usually require 0 6 = 1 by definition. In our case, the semiring S is trivi al iff the category H i s trivial, i.e . when H i s the one-morphism (and hence one-ob ject) category . F or this r eason many results in this pap er assume H to b e nontrivial, but the main result, T heorem 4, holds regardlessly . 10 transp ose x 7→ x † ◦ ( − ) is a mo no morphism. Thus H ( I , X ) is a Hilb ert S - semimo dule. A forteriori, [Vicary , 20 08 , 2.11] sho ws that it is a strict Hilber t semimo dule. Moreov er , the image of a mo rphism f : X → Y of H under H ( I , − ) is indeed a morphism of sHMo d S , that is, it is a djoin table, since h f ◦ x | y i H ( I ,Y ) = ( f ◦ x ) † ◦ y = x † ◦ f † ◦ y = h x | f † ◦ y i H ( I ,X ) for x ∈ H ( I , X ) and y ∈ H ( I , Y ). This also shows that H ( I , − ) preserves † . Also, by definition of pro duct w e hav e H ( I , X ⊕ Y ) ∼ = H ( I , X ) ⊕ H ( I , Y ), s o the functor H ( I , − ) preser v es ⊕ . T o sho w that H ( I , − ) preser ves kernels, supp ose that k = ker( f ) : K  , 2 / / X is a kernel of f : X → Y in H . W e hav e to sho w that H ( I , k ) = k ◦ ( − ) : H ( I , K )  , 2 / / H ( I , X ) is a kernel of H ( I , f ) = f ◦ ( − ) : H ( I , X ) → H ( I , Y ) in sH Mo d S . First of all, one indeed ha s H ( I , f ) ◦ H ( I , k ) = H ( I , f ◦ k ) = 0. Now supp ose tha t l : Z → H ( I , X ) also satisfies H ( I , f ) ◦ l = 0. That is, for all z ∈ Z , we hav e f ◦ ( l ( z )) = 0 . Since k is a kernel, for each z ∈ Z there is a unique m z : I → K with l ( z ) = k ◦ m z . Define a function m : Z → H ( I , K ) b y m ( z ) = m z . This is a well-defined module morphism, since l is ; for example, k ◦ m z + z ′ = l ( z + z ′ ) = l ( z ) + l ( z ′ ) = ( k ◦ m z ) + ( k ◦ m z ′ ) = k ◦ ( m z + m z ′ ) , so that m ( z + z ′ ) = m ( z ) + m ( z ′ ) be cause k is mono . In fa ct, m is the unique mo dule morphism satisfying l = H ( I , k ) ◦ m . Since k is a dagger mono, we hav e m = H ( I , k † ) ◦ l . So as a compo sition of adjointable module morphisms m is a well-defined morphism of s HMo d S . Thus H ( I , k ) is indeed a k er nel of H ( I , f ), and H ( I , − ) preserves kernels. If I is simple then s HMo d S is monoida l. T o show that H ( I , − ) is a monoidal functor we must give a natural transfor mation ϕ X,Y : H ( I , X ) ⊗ H ( I , Y ) → H ( I , X ⊗ Y ) and a morphis m ψ : S → H ( I , I ). Since S = H ( I , I ), w e can simply take ψ = id . Define ϕ by mapping x ⊗ y for x : I → X a nd y : I → Y to I ∼ = / / I ⊗ I x ⊗ y / / X ⊗ Y . It is eas ily seen that ϕ and ψ ma k e the r equired coherence diagrams commute, and hence H ( I , − ) is a mo noidal functor. 4 The s c alar field This section shows that the sc a lars in a pre-Hilber t ca teg ory whose mo noidal unit is a simple generator necessa rily form an inv o lutive field. First, we need a factorisatio n result, which is interesting in its own right. Lemma 4 In any dagger c ate gory: (a) A dagger mono which is epic is a dagger isomorphism. 11 (b) If b oth g f and f ar e dagger epic, so is g . (c) If m and n ar e dagger monic, and f is an isomorphism with nf = m , then f is a dagger isomorphism. Pro of F or (a), notice that f f † = id implies f f † f = f , from whic h f † f = id follows from the a ssumption that f is epi. F or (b): g g † = g f f † g = g f ( g f ) † = id . Finally , consider (c). I f f is isomorphism, in pa rticular it is epi. If both nf and n a re dagge r mono , then so is f , by (b). Hence by (a), f is dagg er is omorphism. Lemma 5 In any pr e-H ilb ert c ate gory, a morphism m is mono iff ker( m ) = 0 . Pro of Supp ose ker( m ) = 0. Let u , v satisfy mu = mv . Put q to be the da g ger co equaliser of u and v . Since q is dag ger e pic , q = c o k er( 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 dagger iso morphism, and in particular mo no . Hence, fro m 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 mo no, ker ( m ) = 0 follows from m ◦ k er( m ) = 0 = m ◦ 0. Lemma 6 Any morphi sm in a pr e-Hilb ert c ate gory c an b e factor e d as a dagger epi fol lowe d by a mono. This factorisation is unique u p to a unique dagger isomorphi sm . Pro of Let a mor phism f b e giv en. Put k = ker ( f ) and e = coker ( k ). Since f k = 0 (as k = ker( f )), f factors thr ough e (= coker( k )) as f = me . h   l               , 2 k / / e _   f / / g / / m ? ?            q  , 2 r O O s o o W e have to sho w that m is mono. Let g b e suc h that mg = 0. By Lemma 5 it suffices to sho w that g = 0. Since mg = 0, m factors throug h q = coker ( g ) 12 as m = rq . N ow q e is a dagger epi, b eing the co mp osite of tw o dagger epis. So q e = coker( h ) for some h . Since f h = r q eh = r 0 = 0, h facto rs through k (= ker( f )) as h = kl . Finally eh = e kl = 0 l = 0, s o e factors through q e = coker( h ) as q = sq e . But since e is (dagger) epic, this means sq = id , whence q is mono . It follows from qg = 0 that g = 0, and the factor isation is established. Finally , b y Le mma 4(c), the factoris a tion is unique up to a da gger isomorphism. W e just show ed that any Hilb ert catego ry has a facto risation system con- sisting of mo nos and dagge r epis. E quiv alently , it has a factoris a tion sys tem of epis and dagger mo no s. Indeed, if we ca n factor f † as an da gger epi follow ed by a mono, then taking the dagger s of tho s e, we find tha t f †† = f factors as a n e pi follow ed by a dagger mono. The combination of b oth factorisa tions yields that every mor phism c a n b e written as a dagge r epi, follow ed by a monic epimor- phism, follo wed by a dagger mono; this ca n b e thoug h t of a s a g eneralisation of p olar de c omp osition . Recall that a semifield is a co mm utative se mir ing in which every nonzero element has a multiplicativ e inv erse. Notice that the s c alars in the embedding theorem for Ab elian categ o ries do not necessarily have multiplicativ e inverses. Lemma 7 If H is a nontrivial pr e-Hilb ert c at e gory with simple monoidal unit I , then S = H ( I , I ) is a semifield. Pro of W e will show that S is a semifield b y pro v ing that any s ∈ S is either zero or iso morphism. F actorise s as s = me fo r a dagg er mono m : Im( s )  , 2 / / I and an epi e : I ։ Im( s ). Since I is simple, e ither m is zer o or m is isomorphism. If m = 0 then s = 0 . If m is iso morphism, then s is epi, so s † is mono . Again, either s † = 0, in which case s = 0, or s † is isomor phism. In this last case s is also isomorphis m. The following lemma shows that ev er y scala r also has an additive inv er se. This is always the case for the scalar s in the embedding theorem for Ab elian categorie s, but the usual pro of of this fact is denied to us bec ause epic monomor - phisms are not necessarily isomor phisms in a pre-Hilb ert ca teg ory (see Ap- pendix A). Lemma 8 If H is a nontrivial pr e-Hilb ert c ate gory whose monoidal unit I is a simple gener ator, then S = H ( I , I ) is a field. Pro of Applying [Golan, 1 999, 4.34] to the previous lemma yie lds that S is either zer osumfree, or a field. Assume, tow ards a cont r adiction, that S is zer o- sumfree. W e will show that the k ernel of the co diagonal ∇ = [id , id ] : I ⊕ I → I is zero. Supp ose ∇ ◦ h x, y i = x + y = 0 for x, y : X → I . Then for all z : I → X we ha ve ∇ ◦ h x, y i ◦ z = 0 ◦ z = 0 , i.e. xz + y z = 0 . Since S is as s umed zerosumfree hence xz = y z = 0, so h x, y i ◦ z = 0. Because I is a ge ne r ator then h x, y i = 0. Thu s k er( ∇ ) = 0. B ut then, by Le mma 5, ∇ is mono , whence κ 1 = κ 2 , which is a contradiction. 13 Collecting the pre vious r esults about the sc alars in a pre-Hilb ert category yields Theorem 2 b elo w. It uses a well-known characterisatio n of subfields of the complex num b ers, that we recall in the following tw o lemmas. Lemma 9 [Gril let, 2007, The or em 4.4] Any field of cha r acteristic zer o and at most c ontinuu m c ar dinality c an b e emb e dde d in an algebr aic al ly close d fi eld of char acteristic zer o and c ont inuum c ar dinality.  Lemma 10 [Chang & Keisler, 19 90 , Pr op osition 1.4.10] Al l algebr aic al ly close d fields of char acteristic zer o and c ontinu u m c ar dinality ar e isomorphic .  Theorem 2 If H is a nontrivial pr e-Hilb ert c ate gory whose monoidal unit I is a simple gener ator, then S = H ( I , I ) is an involutive field of char acteristic z er o of at most c ontinuum c ar dinality, with S + zer osumfr e e. Conse qu ently, ther e is a monomorphism H ( I , I ) ֌ C of fields. However, it do es n ot ne c essarily pr eserve the involution. Pro of T o establish c ha racteristic zero, we ha ve to prov e that for all scalars s : I → I the prop erty s + · · · + s = 0 implies s = 0, whe r e the s um contains n copies of s , for all n ∈ { 1 , 2 , 3 , . . . } .. So supp ose that s + · · · + s = 0 . By definition, s + · · · + s = ∇ n ◦ ( s ⊕ · · · ⊕ s ) ◦ ∆ n = ∇ n ◦ ∆ n ◦ s , where ∇ n = [id ] n i =1 : L n i =1 I → I and ∆ n = h id i n i =1 : I → L n i =1 I are the n -fold (co)diago nals. But 0 6 = ∇ n ◦ ∆ n = (∆ n ) † ◦ ∆ n by Lemma 2.11 of [Vicary , 2008], which states tha t x † x = 0 implies x = 0 for every x : I → X . Since S is a field by Le mma 8, this means that s = 0. This theorem is of interest to reconstructio n programmes, that try to derive ma jor results of quantum theor y fr om simpler mathema tica l assumptions , for among the things to b e reconstr ucted ar e the s calars. F or example, [Sol` er, 199 5 ] shows that if a n orthomo dular pre-Hilb ert s pace is infinite dimensional, then the base field is either R or C , and the s pace is a Hilber t space. With a scalar field, w e can sharp en the preserv ation of finite bipro ducts a nd kernels of Theor em 1 to preser v ation of all finite limits. Since H ( I , − ) preser v es the da gger, it hence also preser ves all finite colimits. (In other terms: H ( I , − ) is exact.) Corollary 1 The functor H ( I , − ) : H → s H Mo d H ( I ,I ) pr eserves al l finite lim- its and al l finite c olimits, for any pr e-Hilb ert c ate gory H whose monoidal u nit I is a simple gener ator. Pro of One easily chec ks that F = H ( I , − ) is an Ab -functor , i.e. that ( f + g ) ◦ ( − ) = ( f ◦ − ) + ( g ◦ − ) [Heunen, 2008]. Hence, F preser v es equalisers: F (eq( f , g )) = F (ker( f − g )) = ker( F ( f − g )) = ker( F f − F g ) = eq( F f , F g ) . Since we a lready know fro m Theor em 1 that F preser v es finite pro ducts, we can conclude that it pr eserv es all finite limits. And b ecause F preser v es the self-duality † , it also pres e rv es all finite c o limits. 14 5 Extension of scalars The main ide a underlying this section is to ex ploit Theorem 2. W e will construct a functor HMo d R → HMo d S given a mor phism R → S of commutative inv olu- tive semir ings, and apply it to the ab o ve H ( I , I ) → C . This is ca lled extension of sc alars , and is well known in the setting o f mo dules (see e.g. [Ash, 2000, 10.8.8]). Let us firs t co nsider in some mor e detail the constructio n o n s emimodules. Let R and S b e comm utative semirings, and f : R → S a homomorphism of semirings. Then any S -semimo dule M ca n be considered an R - semimodule M R by defining sca la r multiplication r · m in M R in terms of scalar m ultiplica tio n of M by f ( r ) · m . In particular, we can regar d S a s an R -semimo dule. Hence it makes sense to lo ok at S ⊗ R M . Somewhat more precis ely , we can v iew S as a left- S - r igh t- R -bisemimodule, and M a s a left- R -se mimo dule. Hence S ⊗ R M bec omes a left- S -semimo dule (see [Golan, 1999]). This construction induces a functor f ∗ : SMo d R → SMo d S , acting on morphisms g as id ⊗ R g . It is easily seen to b e stro ng monoida l a nd to preser ve bipr oducts a nd kernels. Now let us change to the setting where R and S are inv olutive semir ings, f : R → S is a morphism of inv olutive semirings, and we c o nsider Hilb ert semi- mo dules instead of s emimodules. The next theor em shows that this cons tr uction lifts to a functor f ∗ : sHM o d R → sH M od S (under some conditio ns on S and f ). Moreover, the fact that an y S -semimo dule ca n be seen a s an R -semimo dule via f immediately induces another functor f ∗ : S M od S → SMo d R . This one is calle d re st rictio n of s c alars . In fact, f ∗ is rig h t adjoint to f ∗ [Borceux, 199 4 , vol 1, 3.1.6 e]. How ever, since w e do no t know how to fashion an sesquilinea r R - v a lued form out of an S -v alued one in general, it seems impossible to constr uct an adjoint functor f ∗ : sHM o d S → sH Mod R . Theorem 3 L et R b e a c ommutative involutive semiring, S b e a mult iplic atively c anc el lative c ommutative involutive ring, and f : R ֌ S b e a monomorphism of involutive semirings. Ther e is a faithful functor f ∗ : sHM o d R → sHMo d S that pr eserves † . If R is multiplic atively c anc el lative, t he n f ∗ is str ong m ono idal. If b oth R + and S + ar e zer osumfr e e, then f ∗ also pr eserves ⊕ . Pro of Let M b e a strict Hilbert R -semimodule. Defining the car rier o f f ∗ M to b e S ⊗ R M turns it into a n S -semimo dule as befo re. F urnish it with h s ⊗ m | s ′ ⊗ m ′ i f ∗ M = s ‡ · s ′ · f ( h m | m ′ i M ) . Assume 0 = h s ⊗ m | s ⊗ m i f ∗ M = s ‡ sf ( h m | m i M ). Since S is m ultiplicatively cancellative, either s = 0 or f ( h m | m i M ) = 0. In the former case immediately s ⊗ m = 0. In the latter case h m | m i M = 0 since f is injective, and b ecause M is strict m = 0, whence s ⊗ m = 0. Since S is a ring, this implies tha t f ∗ M is a strict Hilber t S -semimo dule. F or if h x | −i f ∗ M = h y | −i f ∗ M then h x − y | −i f ∗ M = 0 , so in particular h x − y | x − y i f ∗ M = 0 . Hence x − y = 0 and x = y . Moreov er , the image of a mo r phism g : M → M ′ of sHMo d R under f ∗ is a 15 morphism of sHM od S , as its a djoin t is id ⊗ g † : h (id ⊗ g )( s ⊗ m ) | s ′ ⊗ m ′ i f ∗ M ′ = h s ⊗ g ( m ) | s ′ ⊗ m ′ i f ∗ M ′ = s ‡ s ′ f ( h g ( m ) | m ′ i ′ M ) = s ‡ s ′ f ( h m | g † ( m ′ ) i M ) = h s ⊗ m | s ′ ⊗ g † ( m ′ ) i f ∗ M = h s ⊗ m | (id ⊗ g † )( s ′ ⊗ m ′ ) i f ∗ M . Obviously , f ∗ is faithful, a nd preserves † . If dagge r bipro ducts are av aila ble , then f ∗ preserves them, s ince bipro ducts dis tr ibute ov er tensor pr oducts. If dagger tensor pro ducts ar e av ailable, showing that f ∗ preserves them comes down to giving an isomorphism S → S ⊗ R R and a natura l isomo rphism ( S ⊗ R X ) ⊗ S ( S ⊗ R Y ) → S ⊗ R ( X ⊗ R Y ). The ob vio us candidates fo r thes e satisfy the coher e nce diagr ams, mak ing f ∗ strong mono idal. Corollary 2 L et S b e an involutive field of char acteristic zer o and at most c on- tinuum c ar dinality. Then ther e is a str ong monoidal, faithful functor sHMo d S → sHMo d C that pr eserves al l fin ite limits and finite c olimits, and pr eserves † u p to an isomo rphism of the b ase field. Pro of The only claim that do es not follow from previous res ults is the s ta te- men t ab out pr eserv atio n o f finite (co)limits. This comes down to a ca lc ulation in the well-studied situation of module theory [Ash, 2000, Exercise 10.8 .5]. Note that the extension of s calars functor f ∗ of the previous theorem is full if and only if f is a reg ular epimorphism, i.e. iff f is sur jectiv e. T o see this, consider the inclusion f : N ֒ → Z . This is obviously not sur jective. Now, SMo d N can be ident ified with the catego ry cMon of commutativ e monoids, and SMo d Z can be identified with the category Ab of Ab elian g r oups. Under this identification, f ∗ : cMo n → Ab sends an ob ject X ∈ cMon to X ` X , with in verses b eing provided b y swapping the tw o terms X . F or a morphism g , f ∗ ( g ) sends ( x, x ′ ) to ( g x, g x ′ ). Consider h : X ` X → X ` X , deter mined b y h ( x, x ′ ) = ( x ′ , x ). If h = f ∗ ( g ) for s ome g , then ( x ′ , x ) = h ( x, x ′ ) = f ∗ g ( x, x ′ ) = ( g x, g x ′ ), so g x = x ′ and g x ′ = x for all x, x ′ ∈ X . Hence g must b e constant, contradicting h = f ∗ g . Hence f ∗ is not full. 6 Completion Up to now we hav e concerned ourselves with a lgebraic str ucture only . T o a rriv e at the catego ry of Hilbert spaces and contin uous linear maps, some a nalysis comes int o play . Lo oking back a t Definition 2 , we see tha t a strict Hilbert C -semimo dule is just a pr e-Hilbert space, i.e. a co mplex vector space with a po sitiv e definite sesq uilinear form o n it. An y pre-Hilb ert spa ce X c an b e com- pleted to a Hilbert space b X into which it de ns ely embeds [Reed & Simon, 19 7 2 , I.3]. 16 A morphism g : X → Y of sHMo d C amounts to a linear tra ns formation betw ee n pre-Hilb ert spaces tha t has an adjoint. So sHMo d C = p reH ilb . How- ever, these morphisms need not necess arily be b o unded, and only b ounded ad- joint a ble morphisms can be extended to the completion b X o f their domain [Reed & Simon, 1972, I.7]. Therefore , we imp ose another axiom o n the morphisms of H to ensure this. Basica lly , we rephrase the usual definition of b oundedness of a function betw ee n Banach s paces for morphism betw een Hilbe r t semimo dules. Recall from Lemma 2 that the s calars S = H ( I , I ) in a pre -Hilbert categor y H are alwa ys an in volutiv e field, with S + zerosumfree. Hence S is ordered by r ≤ s iff r + p = s for s ome p ∈ S + . W e use this or dering to define boundedness of a morphism in H , together with the norm induced b y the cano nical bilinea r form h f | g i = f † ◦ g . Definition 4 L et H b e a symmetric dagger monoid al dagger c ate gory with dag- ger bipr o ducts. A sc alar M : I → I is s aid to b o und a morphism g : X → Y when x † g † g x ≤ M † x † xM for al l x : I → X . A morphism is c al le d bo unded when it has a b ound. A Hilb ert ca tegory is a pr e-Hilb ert c ate gory who se mor- phisms ar e b ounde d. In particular, a morphism g : X → Y in sHMo d S is bounded when there is an M ∈ S satisfying h g ( x ) | g ( x ) i ≤ M † M h x | x i for all x ∈ X . Almost by definition, the functor H ( I , − ) of Theorem 1 pr eserves b ounded- ness o f morphisms when H is a Hilber t catego ry . The following lemma sho ws that a lso the extension of scalars of Theorem 3 pr eserv es b oundedness. It is noteworth y that a combinatorial condition (b oundedness) on the category H ensures a n analytic prop ert y (co n tinuit y) of its image in sHM od C , as we never even a ssumed a topo logy o n the scalar field, let alone assuming completeness. Lemma 11 L et f : R → S b e a morphism of involut ive semirings. If g : X → Y is b ounde d in sHMo d R , then f ∗ ( g ) is b ou n de d in sHMo d S . Pro of First, notice that f : R → S preserves the canonical o rder: if r ≤ r ′ , say r + t ‡ t = r ′ for r, r ′ , t ∈ R , then f ( r ) + f ( t ) † f ( t ) = f ( r + t † t ) = f ( r ′ ), so f ( r ) ≤ f ( r ′ ). Suppo se h g ( x ) | g ( x ) i Y ≤ M ‡ M h x | x i X for a ll x ∈ X a nd some M ∈ R . Then f ( h g ( x ) | g ( x ) i Y ) ≤ f ( M ‡ M h x | x i X ) = f ( M ) ‡ f ( M ) f ( h x | x i X ) for x ∈ X . Hence for s ∈ S : h f ∗ g ( s ⊗ x ) | f ∗ g ( s ⊗ x ) i f ∗ Y = h (id ⊗ g )( s ⊗ x ) | (id ⊗ g )( s ⊗ x ) i f ∗ Y = h s ⊗ g ( x ) | s ⊗ g ( x ) i f ∗ Y = s ‡ sf ( h g ( x ) | g ( x ) i Y ) ≤ s ‡ sf ( M ) ‡ f ( M ) f ( h x | x i X ) = f ( M ) ‡ f ( M ) h s ⊗ x | s ⊗ x i f ∗ X . Because elements of the form s ⊗ x form a basis for f ∗ X = S ⊗ R X , w e th us hav e h f ∗ g ( z ) | f ∗ g ( z ) i f ∗ Y ≤ f ( M ) ‡ f ( M ) h z | z i f ∗ X 17 for all z ∈ f ∗ X . In other w o rds: f ∗ g is bounded (namely , by f ( M )). Combining this section with Theorems 1 and 2, Cor ollary 2 and Lemma 11 now results in our main theorem. Notice that the co mpletion preser v es bipro d- ucts and kernels and thus equalisers, and s o pr eserves all finite limits and col- imits. Theorem 4 Any lo c al ly smal l H il b ert c ate gory H whose monoidal unit is a simple gener ator has a monoidal emb e dding into the c ate gory Hi lb of Hilb ert sp ac es and c ontinuous line ar m aps that pr eserves † (u p to an isomorphism of the b ase field) and al l finite limits and finite c olimits. Pro of The only thing left to prove is the cas e that H is trivial. But if H is a o ne-morphism Hilb ert ca tegory , its one ob ject m ust be the zero o b ject, and its one mo rphism mu s t b e the zer o morphism. Hence sending this to the zero - dimensional Hilbe r t space yields a fa ithful monoidal functor that prese rv es † and ⊕ , trivially preserving all (co)limits. T o finish, no tice that the embedding of the Hilbert categor y Hilb into itself th us c o nstructed is (isomorphic to ) the iden tity functor. 7 Conclusion Let us conclude by discuss ing several further is sues. 7.1 Dimension The embedding of Theorem 4 is s trong monoidal ( i.e. preser v es ⊗ ) if the ca non- ical (coherent) morphism is a n is omorphism H ( I , X ) ⊗ H ( I , Y ) ∼ = H ( I , X ⊗ Y ) , where the tensor pro duct in the left-hand side is that of (strict) Hilb ert semi- mo dules. This is a quite natural restriction, as it preven ts deg enerate cases like ⊗ = ⊕ . Under this condition, the em b edding prese rv es compact ob- jects [Heunen, 2008]. This means tha t c ompact o b jects corres p ond to finite- dimensional Hilbert spaces under the embedding in q ues tion. Our em b edding theorem also sho ws that ev er y Hilbe rt category embeds into a C*- category [Ghez, Lima & Rob erts, 1985]. This relates to r epresen ta tio n theo ry . Compare e.g. [Doplicher & Rob erts, 198 9 ], who esta blish a corresp ondence b et ween a compact gr oup and its catego ries of finite-dimensional, contin uous, unitary repr esen tations; the latter category is characterised by axio ms comparable to thos e of pre-Hilb ert categor ies, with moreov er every ob ject b e ing compact. Corollar y 1 o p ens the wa y to diagr am chasing (see e.g. [Borc e ux, 1 994, vol 2, Section 1 .9]): to prove that a diagra m co mm utes in a pre-Hilb ert cate- gory , it suffices to prove this in pr e-Hilbert spaces, wher e one has access to 18 actual e lemen ts. As discusse d ab o ve, when H is c o mpact, and the embed- ding H → preHilb is s trong monoidal, then the em b edding takes v alues in the category of finite-dimensio nal pre-Hilber t s paces. The latter coincides with the category o f finite-dimensional Hilber t spaces (since ev er y finite-dimensio na l pre-Hilb ert space is Cauch y co mplete). This partly explains the main cla im in [Selinger, 2008], namely that an equation holds in a ll dagg er traced sym- metric mo no idal categor ies if and only if it holds in finite-dimensional Hilbert spaces. 7.2 F unctor categor ies W e hav e us e d the assumption that the monoidal unit is simple in an essential wa y . But if H is a pre-Hilber t catego ry whose mono ida l unit is s imple, a nd C is any nontrivial small categor y , then the functor category [ C , H ] is a pre-Hilb ert category , alb e it one whose monoidal unit is not simple anymore. Perhaps the embedding theorem can b e extended to this exa mple. The conjecture would b e that an y pre-Hilb ert categ ory whose monoidal unit is a genera tor (but not nec- essarily simple), em b eds into a functor categ ory [ C , preHilb ] for some categor y C . This r equires reconstructing C from Sub( I ). Likewise, it w ould b e preferable to b e a ble to dr op the condition that the monoidal unit b e a g enerator. T o acco mplish this, one would need to find a dagger preserving embedding of a given pr e-Hilbert catego ry into a pre-Hilbert category with a finite set o f generators . In the Abelian case, this c a n b e done by moving fro m C to [ C , Ab ], in which ` X ∈ C C ( X , − ) is a g enerator. But in the setting of Hilbert catego ries there is no analogon of Ab . Also, Hilb ert categorie s tend not to hav e infinite copr oducts. 7.3 T op ology Our axiomatisa tion allow ed inner pro duct spaces ov er Q as a (pre-) Hilbert category . Additional axio ms, enforcing the bas e field to b e (Ca uc hy) complete and hence (isomorphic to) the real or complex n umbers, could p erhaps play a role in topo logising the ab o ve to yield an embedding in to sheav es of Hilb ert spaces. A forthco ming pap er studies sub ob jects in a (pre-)Hilb ert category , showing that quantum log ic is just an incarnation of categorica l logic. But this is a ls o interesting in relation to [Amemiya & Araki, 1 966], which shows that a pre-Hilbe r t space is complete if and only if its lattice of clo s ed subspaces is orthomo dular. 7.4 F ullness A natur al question is under what conditions the embedding is full. Imitating the answer for the embedding of Ab elian catego ries, w e can only obtain the following partial re s ult, since Hilber t catego r ies need not hav e infinite co products, as opp osed to Ab . An ob ject X in a pr e-Hilbert category H with mono idal unit I 19 is said to be finitely gener ate d when there is a dagger epi L i ∈ I I  , 2 X for some finite set I . Theorem 5 The emb e dding of The or em 1 is ful l when every obje ct in H is finitely gener ate d. Pro of W e hav e to prove that H ( I , − )’s action on morphisms, which we tem- po rarily deno te T : H ( X , Y ) → sHMo d S ( H ( I , X ) , H ( I , Y )), is surjective when X is finitely gener a ted. Let Φ : H ( I , X ) → H ( I , Y ) in sHMo d S . W e m ust find ϕ : X → Y in H such that Φ( x ) = ϕ ◦ x for all x : I → X in H . Suppos e firs t that X = I . Then Φ( x ) = Φ(id I ◦ x ) = Φ(id I ) ◦ x since Φ is a morphism o f S -semimo dules. So ϕ = Φ(id I ) satisfies Φ( x ) = ϕ ◦ x for all x : I → X in H . In g eneral, if X is finitely generated, ther e is a finite set I and a dagge r epi p : L i ∈ I I  , 2 X . Denote by Φ i the comp osite morphism H ( I , I ) T ( κ i ) / / H ( I , L i ∈ I I ) T ( p ) / / H ( I , X ) Φ / / H ( I , Y ) in sHMo d S . By the previo us case ( X = I ), for each i ∈ I there is ϕ i ∈ H ( I , Y ) such that Φ i ( x ) = ϕ i ◦ x for a ll x ∈ S . Define ¯ ϕ = [ ϕ i ] i ∈ I : L i ∈ I I → Y , and ¯ Φ = Φ ◦ T ( p ) : H ( I , L i ∈ I I ) → H ( I , Y ). Then, for x ∈ H ( I , L i ∈ I I ): ¯ Φ( x ) = Φ( p ◦ x ) = Φ( p ◦ ( X i ∈ I κ i ◦ π i ) ◦ x ) = X i ∈ I Φ( p ◦ κ i ◦ π i ◦ x ) = X i ∈ I Φ i ( π i ◦ x ) = X i ∈ I ϕ i ◦ π i ◦ x = ¯ ϕ ◦ x. Since p is a dagge r epi, it is a cokernel, say p = coker( f ). Now ¯ ϕ ◦ f = ¯ Φ( f ) = Φ( p ◦ f ) = Φ(0) = 0 , so there is a (unique) ϕ : X → Y with ¯ ϕ = ϕ ◦ p . Finally , for x : G → X , Φ( x ) = Φ( p ◦ p † ◦ x ) = ¯ Φ( p † ◦ x ) = ¯ ϕ ◦ p † ◦ x = ϕ ◦ p ◦ p † ◦ x = ϕ ◦ x. A The category of Hilb ert spaces W e deno te the categ ory of Hilber t s paces and contin uous linear tra nsformations by H ilb . First, w e show that Hilb is actually a Hilb ert category . Subsequently , we pr ove that it is not an Abelia n categor y . First, ther e is a dag g er in Hi lb , b y the Ries z representation theorem. The dagger of a mo rphism f : X → Y is its adjoint, i.e. the unique f † satisfying h f ( x ) | y i Y = h x | f † ( y ) i X . It is also well-kno wn that Hilb has finite dagg er biproducts: X ⊕ Y is ca rried by the direct sum of the underlying vector spaces, with inner pro duct h ( x, y ) | ( x ′ , y ′ ) i X ⊕ Y = h x | x ′ i X + h y | y ′ i Y . 20 F urthermore, Hi lb ha s kernels: the k er nel of f : X → Y is (the inclusion of ) { x ∈ X | f ( x ) = 0 } . Since ker ( f ) is in fact a closed subspa c e, its inclusion is isometric. That is , Hilb in fac t has dagg er kernels. Conseq uen tly k er( g − f ) is a dagger equa liser o f f and g in Hilb . W e now turn to the re quiremen t that every dagge r mono be a da gger kernel. Lemma 12 The monomorphisms in Hilb ar e t he inje ctive c ontinu ous line ar tr ansformations. Pro of If m is injectiv e, then it is obviously mono. Conv ers ely , suppo s e that m : X ֌ Y is mono. Let x, x ′ ∈ X satis fy m ( x ) = m ( x ′ ). Define f : C → X b y (contin uous linear extension o f ) f (1) = x , and g : C → X by (con tinuous linear extension of ) g (1) = x ′ . Then mf = mg , w henc e f = g and x = x ′ . Hence m is injectiv e. Recall that Hilb ert spaces have orthogo nal pro jectio ns , that is: if X is a Hilber t space, and U ⊆ X a closed subspace, then every x ∈ X c a n be written as x = u + u ′ for unique u ∈ U and u ′ ∈ U ⊥ , where U ⊥ = { x ∈ X | ∀ u ∈ U . h u | x i = 0 } . (2) The function that assigns to x the ab ov e unique u is a mor phis m X → U , the ortho gonal pr oje ction of X onto its close d subspace U . Prop osition 3 In H i lb , every da gger mono is a dagger kernel. Pro of Let m : M ֌ X b e a dagg er mono. In par ticular, m is a split mono, and hence its image is closed [Aubin, 2000, 4.5.2]. So, without loss of gener alit y , we can assume that m is the inclusio n of a closed s ubspace M ⊆ X . But then m is the dagger kernel of the ortho gonal pro jection of X onto M . All in all, Hilb is a Hilbert categ ory . So is its full subcateg ory fdHil b of finite-dimensional Hilb ert categories . A lso , if C is a small category a nd H a Hilber t category , then [ C , H ] is ag ain a Hilb ert categ ory . Since Hilb has bipro ducts, k erne ls and cokernels, it is a pre-Ab elian ca te- gory . But the behaviour of epis preven ts it from b eing a n Ab elian ca teg ory . Lemma 13 The epimorphisms in Hi lb ar e the c ontinu ous line ar tr ansforma- tions with dense image. Pro of Let e : X → Y sa tis fy e ( X ) = Y , and f , g : Y → Z satisfy f e = g e . Let y ∈ Y , say y = lim n e ( x n ). Then f ( y ) = f (lim n e ( x n )) = lim n f ( e ( x n )) = lim n g ( e ( x n )) = g (lim n e ( x n )) = g ( y ) . So f = g , whence e is epi. Conv ersely , supp ose that e : X ։ Y is epi. Then e ( X ) is a closed subspa ce of Y , so that Y / e ( X ) is again a Hilb ert space, and the pro jection p : Y → Y /e ( X ) is contin uous and linear. Consider a lso q : Y → Y /e ( X ) defined by q ( y ) = 0. 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