Completeness of dagger-categories and the complex numbers

Completeness of † -categories and the complex n um b ers Jamie Vicary Oxford Universit y Computing Lab orat ory jami e.vic ary@co mlab.ox.ac.uk July 8, 2010 Abstract The complex n u m b ers are an imp ortant part of quan tum theory , but are difficult to motiv ate from a theoretic al p ersp ectiv e. W e describ e a simple formal framewo r k for theories of ph ysics, and sho w that if a theory of ph ysics presen ted in this manner satisfies c ertain completeness prop erties, then it necessarily includes the complex n u m b ers a s a mathematic al ingredien t. Cent r al to our approac h are the tec hniqu es of catego r y theory , and w e introdu ce a n ew category-theoretica l to ol, called the † -limit , whic h go v ern s the w ay in wh ic h systems can b e com b ined to form larger sys tems. These † -limits can b e used to charac terize the prop erties of the † -fu n ctor on the category of finite-dimensional Hilb ert spaces, a n d so can b e used as an equiv alent defin ition of the inner pro du ct. One of our mai n r esults is that in a non trivial monoidal † -category with finite † -limits and a simple tensor u nit, th e semiring of s calars em b eds in to an in volutiv e field of c haracteristic 0 and orderable fixed fi eld. 1 In tro ducti on The purp o se of this pap er is to describ e a set o f properties of a the ory of ph ysics , whic h together imply that the theory mak es us e of the complex num b ers. These pro p erties a r e phrased in terms of the w ay that ph ysical pro cesses in teract with eac h other, and as a result are in tuitive and phys ical. The approach as a whole is a robust one; w e are not concerned with man y fine details of the theory , suc h as t he na ture of dynamics, or the w a y that measuremen t is describ ed. There is a v ast literature of inv estigations into the mathematical foundations of quan tum theory , whic h v aries greatly in approach and persp ectiv e. Some of this w ork tackle s the problem of deriving the structure of quan tum theory from ph ysical or op eratio na l principles, a small sample of whic h is [4, 7, 10, 12, 14]. It is this t yp e of researc h that has the strongest connection to the ideas presen ted here. There is also a la r g e b o dy of w ork in v estigating the prop erties of generalized quan tum theories based on fields different to the complex n um b ers [6, 16, 18, 19, 2 5], a gainst whic h the results presen ted here serv e a s a foil. 1 T o apply our metho d to a pa r ticular theory o f phys ics, w e first need to obtain from the theory a fa mily of systems , equipped with a family of pr o c esses whic h g o fro m one system to another. W e will often denote processes as f : A ✲ B , whic h indicates a process f going from sys t em A to system B . It is sometimes useful to imag ine that systems are sets of states , and that pro cesses are function s taking states of one system in t o states of another, but w e will not rely on an y suc h in terpretation. F or an y t w o ‘head-to- tail’ pro cesses f : A ✲ B and g : B ✲ C we require that there exists a comp osite pro cess f ; g : A ✲ C , in terpreted as the pro cess f follo wed by the pro cess g . W e require that this comp osition is associative , and for any system A w e require the existence of a ‘trivial’ pro cess id A : A ✲ A whic h is the iden tit y for comp osition. These are exactly the axioms of a c ate gory , and w e will make essen tial use of the to ols of category theory to pro v e our r esults. W e call this category the c ate gory o f pr o c es ses asso ciated to a particular theory . It will not necessarily completely define t he theory; other imp ortant asp ects, suc h as observ ation or measuremen t, a re likely to b e outside of its remit. Also, v ery few real- w orld theories of ph ysics will b e naturally pres ented in terms of a catego r y of pro cesses, but for man y theories there will nev ertheless b e natural candidates for suc h a category . If an y of these candidates ha ve the pr o p erties w e will desc r ib e, that will indicate that the underlying theory someho w mak es use o f on the complex n um b ers. F or the case of the theory of quan t um me chanics , whic h certainly mak es use o f the complex num bers, w e migh t define the category of pro cesses to hav e separable Hilb ert spaces as systems and b ounded linear maps as pro cesses, and the category o btained in this w ay satisfies the prop erties w e will describ e. The first pro p ert y that w e require is that eac h pro cess has an adjoint , whic h can be considered as a formal ‘rev ersal’ of the pro cess. W e use the term ‘a dj o in t’ sinc e this is a generalization of a familiar op era t ion from quan tum theory , taking the adjoin t of a b ounded linear map b etw een Hilb ert space s. F or an y f : A ✲ B its adjoin t is a pro cess f † : B ✲ A ; w e require t ha t ( f † ) † = f for any pro cess f , and also that ( f ; g ) † = g † ; f † for an y comp osable pro cesses f and g . These properties define a functor from our category to itself, and w e call this the † -functor . A sec ond prop ert y that w e require is s up erp osition : for an y tw o parallel pro cesses f , g : A ✲ B there must exis t a third pro cess f + g : A ✲ B , where + is an asso ciativ e, unital, comm utativ e op eration with the prop ert y that ( f + g ); h = f ; h + g ; h for any h : B ✲ C and any system C . Finally , w e require a notion of c omp ound system : for an y t w o systems A a nd B there m ust exist a compound sys tem A ⊗ B , where ⊗ is a n asso ciative , unital op eration. 1 A useful in tuitio n for this is the systems A a nd B existing sim ultaneously , but indep enden tly and without necessarily in teracting. The unit for the comp ounding op eration ⊗ is a sys tem I , called the neutr al or unit system, such that I ⊗ A = A = A ⊗ I for all system s A . This comp ounding op eration m ust also b e defined on pro cesses, so for all f : A ✲ B and g : C ✲ D there exists a pro cess f ⊗ g : A ⊗ C ✲ B ⊗ D ; this must in teract we ll with comp osition, satisfying the compatibilit y equation ( f ⊗ g ); ( h ⊗ j ) = ( f ; h ) ⊗ ( g ; j ) for all appropriate pro cesses f , g , h and j . If we interpret the pro cess f ⊗ g as represen ting pro cesses f and g o ccurring sim ultaneously and indep enden tly , then this compatibilit y equation mak es in tuitiv e se nse: it sa ys that p erforming f and g sim ultaneously , and then p erfor ming h and j 1 Exp erts in categor y theory will note that w e ar e des cribing a s tr ict monoidal category here; a w eak one would do just as w ell. 2 sim ultaneously , is the same as p erforming f fo llow ed by h , while sim ultaneously p erforming g fo llo w ed b y j . Supp ose now that we ha v e a whole collection of systems and pro cesses, of the fo llowing general fo rm: J G H A B C D E F (1) In t his diagra m, letters represen t syste ms and arrow s represen t pro cesses. W e call this t yp e of diagram a finite for est-shap e d multigr aph : there are a finite n umber of connected comp onen ts, eac h of whic h is a finite tree with a ro ot at the top and lea ve s a t the b ottom, and w e a llo w the po ssibilit y of m ultiple parallel branc hes betw een no des. This giv es us a collection of allo wed pro cesses with whic h to turn leaf systems at the b ot t o m into no de systems at the top. W e could understand this ph ysically as de scribing a simple sort of nondeterministic dynamics, where w e evolv e f rom an initial system at the b ott om of the diagr a m t o a final system tow ards the top, making a c ho ice of pro cess whenev er more than one is av a ila ble. Assuming for a moment t ha t our systems are comp osed of sets o f states, and our pro cesses are functions, w e can ask the follo wing: are there an y states of initial sy stems whic h will alw ays transform in to the same final state, regardless o f the processes c hosen? W e could also take a more computatio nal p ersp ectiv e, and regard the pro cesses as c o nstr aints ; an analogous question w ould then b e to find the initial states whic h satisfy these constrain ts. The answ er to this is pro vided b y the notion of limit , an imp or t a n t and widely-used to ol in category theory . F or our purp oses, a limit is described b y a system L equipped with a family of pro cesse s l X : L ✲ X , where X ranges ov er eac h of the leaf system s in our diagram. The se processes must also satisfy a univ ersality prop erty . W e can interpret the limit syste m L as comprising all of t he leaf systems in our diagr a m combine d together, but with states iden tified when they ev olv e in the same w a y under the action of the pro cesse s in the diagr a m. W e can visualize this with the follo wing diagram: J G H A B C D E F L l A l B l C l D l E l F (2) The limit system and its asso ciated pro cesses a re dra wn in gra y here. Our final r equiremen t is that this limit is compatible with the † -functor on our category of pro cesses, which allo ws us to formally ‘rev erse’ pro cesses. Supp ose that w e comp ose the 3 pro cesses l A : L ✲ A and l † A : A ✲ L ; this comp osite ev olv es a state of L into a state o f A , and then ev olv es this bac k again into a state of L . W e can consider this as taking a state of L and retaining only that part o f it whic h arises from A . This make s sense, as w e describ ed the L as b eing constructed from the com bination of all of the leaf sys t ems. It is reasonable to require that a state of L is precisely sp ecified by the sum total of its restrictions to a ll of the leaf systems . Using our sup erp osition op eration, w e can express this principle with the follo wing equation: l A ; l † A + l B ; l † B + l C ; l † C + l D ; l † D + l E ; l † E + l F ; l † F = id L . (3) W e call this the normaliza tion c ondition . If w e can find a limit satisfying this conditio n then w e call it a † -limit , and if a category has † -limits for all finite forest-shap ed m ultigraphs then w e sa y that it has al l fini te † -limits . Categories with all finite † -limits ha v e man y in teresting prop erties, whic h w e ex plo re throughout this pap er. One useful prop ert y is that a category can hav e at most a single sup erp osition rule (the ‘+ ’ op eration) such that all finite † -limits exist! So the superp osition rule is more lik e a pr op erty of the † -limits than a structure o n t he underlying category , and w e do not need to sp ecify it explicitly . Ha ving † -limits also implies other useful features, including nonde gener acy (or p ositivity ) of the † -functor, and c anc el lability for the sup erp osition op eration, as w e will explore later. W e can no w state an intere sting result. Supp o se w e ha ve a category of pro cesses whic h has a † -functor, comp ound systems and all finite † - limits, suc h tha t the ‘neutral system’ I — the unit for constructing comp o und s ystems — is ‘simple’, meaning that t he only system smaller than it is the empt y system. Then we can sho w that the analogue of ‘quan tum amplitudes’ in this category tak e v alues in an inv olutiv e field with ch a racteristic 0, and with orderable fixed field. W e in terpret this field as analogous to C , the in v o lution as analogous to complex conjugation, and the orderable fixed field as analo g ous to R . F urthermore, supp ose that the results of measuremen ts in our t heory a re v alued in this orderable fixed field. Then if ev ery b ounded sequence o f measu remen t results has a le ast upp er b ound, and these least upp er b ounds a re preserv ed when w e add a constan t to our measuremen t results, it f o llo ws that our inv olutiv e field is C itself, the orderable fixed field is R , and the order is the familiar order on the real num b ers. An imp ortan t inspiration fo r the dev elopmen t of † -limits came from the category FdHilb , whic h has finite-dimensional Hilb ert spaces a s ob jects and linear maps as mor phisms. This can b e considered as a category of pro cesses which eme rg es from q uan tum theory . That category has a † -f unctor, giv en b y taking linear maps to their adjoints, a nd with this † -functor the cat ego ry FdHilb ha s all finite † -limits. In fact, a s we explore later with Theorem 5.2, this † -functor can actually b e completely defined by its completeness prop erties. Since kno wing the adjoin ts of the bounded linear maps to a Hilbert space is the same as knowing the inner pro duct on it, this giv es a new axiomatization o f inner pro ducts. In a n utshell, what w e do in this pap er is to observ e ho w close the abstract theory of monoidal † -categories comes to describing the structure o f real phys ical theories, a nd t hen to ‘tak e it seriously’. This is not a new idea. In particular, it has b een adv anced with m uc h success in the field o f quan t um computation, esp ecially b y Abramsky and Co ec ke [3, 8]. W e b eliev e that this is a f ruitful p ersp ectiv e whic h holds the promise of deliv ering significan t 4 further results in the future. W e note that in teresting work has already b een car r ied out whic h ta k es the extends the results described here, adding a xioms tha t imply that the resulting category em b eds in to a categor y of Hilb ert spaces [15]. Ac kno wledgemen ts I am grateful to Samson Abramsky , Chris Isham, Zurab Janelidze, Paul Levy and Paul T a ylor, and especially to the anon ymous referee, Kevin Buzzard, Chris Heunen and P eter Selinger, for useful commen ts a nd discussions. I ha v e used P aul T aylor’s diagra ms pac k age, and I am gra t eful for financial su pp ort from the EPSR C and the ONR. I am also grateful to the progra m committe e of Cate gory The ory 2008 for the opp o rtunit y to presen t some early ve rsions of these results. 2 † -F unctors , † -categ ories and † - l imits The † -functor Of all the categorical structures that w e will mak e use of, the most fundamen tal is the † -functor, first made explicit in the con text of categorical quan tum mec hanics b y Abramsky and Co ec k e [1, 3]. As described in the in tro duction, it is motiv ated b y the pro cess of taking the adjoint of a line a r map b etw een tw o Hilb ert spaces: for any bounded linear map of Hilb ert spaces f : H ✲ J , the adjoint f † : J ✲ H is t he unique map satisfying h f ( φ ) , ψ i J = h φ, f † ( ψ ) i H (4) for all φ ∈ H and ψ ∈ J , where the angle brac k ets represen t the inner pro ducts for eac h space. Abstractly , we define a † -functor as a con t r a v arian t functor from a categor y to itself, whic h is the iden tity on o b j ects, and whic h satisfies † ◦ † = id C . A † -c ate gory is a category equipped with a particular c hoice of † -functor. These are sometimes known instead as Hermitian c ate gories or ∗ - c ate gories , but w e prefer t he ‘ † ’ notation, since it is snappier and more flex ible than ‘Hermitian’, a nd the sym b ol ‘ ∗ ’ is also used to denote duals for o b jects in a monoidal category . Although it is often uninformative to name something after the sym b ol that denotes it, in our view this is out we ig hed by the con venie nce of having a straigh tforward naming con v en tio n [23] for ‘ † -v ersions’ of man y familiar constructions, suc h as † -bipro ducts, † -equalizers, † -k ernels, † -limits, † -sub ob jects and so on, all of whic h w e will encoun ter b elo w. The inner pro duct on a Hilb ert spac e is used to calculate the a dj o in t of a linear map, and in fact the pro cess has a con v erse [2]: kno wledge of the adjoin ts can b e used to reconstruct the inner pro duct. T o sho w this, w e use the fact t ha t v ectors φ ∈ H correspond to linear maps C ✲ H b y considering the image of the n um b er 1 under any suc h map. F or any t wo v ectors φ, ψ ∈ H w e can calculate the inner pro duct a s h φ, ψ i H = h φ (1) , ψ (1) i H = h 1 , φ † ( ψ (1)) i C = φ † ( ψ (1)) , 5 where the last step fo llows fro m the fact that the inner pro duct o n the complex n umbers is determined b y m ultiplication. F or this reason, the † -functor can b e thought of not only as an abstraction of the construction of adjoin t linear maps, but also a s an a bstraction of the inner pro duct. Ho w ev er, w e note that an arbitr ary † -functor might giv e rise to ‘inner pro ducts’ whic h are quite badly-b eha ve d: for example, in a category with zero morphisms, w e might ha ve h φ, φ i = 0 for φ 6 = 0. The † -functors whic h arise from inner pro ducts are c haracterized in the last section of the pap er, in Theorem 5.2. W e write the action of a † -functor on a morphism f : A ✲ B as f † : B ✲ A , and w e refer to the morphism f † as the a d joint of f . W e also mak e the following straigh tfo rw ard definitions, ta ken from the v o cabulary of functional analysis: a morphism is unitary if its adjoin t is its in v erse ( f ; f † = id A and f † ; f = id B ), an isometry if its adjoin t is its retraction ( f ; f † = id A ), a nd is self-adjoint if it equals its adj o in t ( f = f † ). If a morphism f : A ✲ B is an isometry , w e also sa y that A is a † -sub obje ct o f B . If tw o ob jects in a † -category ha ve a unitary morphism going b etw een them, w e say that they are unitarily is omorphic ; if ev ery pair o f isomorphic ob jects are unitarily isomorphic, then the † -categor y is a unitary † -c ate gory . Man y imp ort a n t † -categories are unitary; fo r example, the † -category of Hilb ert spaces with † -functor give n b y adjo in t, the † -category of manifo lds a nd cobordisms w ith † -functor giv en by taking the opp osite cob ordism, or an y 2–Hilb ert space [5]. There is a natural notion of equiv alence b etw een † -categories, whic h w e call unitary † -e quivalenc e . Let C and D b e † - categories, with † -f unctors † : C ✲ C and ‡ : D ✲ D . These † -categories are unitarily † -equiv alent if there exists a functor F : C ✲ D b etw een them whic h is part of an adjoin t equiv alence of categories, suc h that the unit a nd counit natural transformations are unitary at ev ery stage, and if it c ommutes with the † -functors, satisfying F ◦ † = ‡ ◦ F . As we later sho w in Lemma 5.1, a functor can b e made a part of a unitary † -equiv alence iff it comm utes with the † -functors, and is full, faithf ul, and unitarily essen tially surjectiv e. Merely equipping a category with a † -f unctor is certainly not trivial, but is p erhaps not itself particularly p o werful. Ho w ev er, in teresting phenomena start to arise when w e relate the † -functor to other constructions that w e can mak e with t he category . It often pays off to do this enth usiastically , a p olicy whic h des erv es to be v ery clearly stated. The Philosoph y of † -Categories. When working with a † -c ate gory, al l imp ortant structur es should b e chose n so that they ar e c omp atible with the † -functor. Of course, this is a rule of thum b rather than a tec hnical statemen t; what coun ts as a n ‘imp ortan t structure ’, and w ha t ‘compatible’ should mean, will depend up on the setting. Ho we ver, there are many situations in whic h applying this philosoph y b ears interes ting results: • The constructions made in this pap er are a prime example, where w e require limits to b e compatible with the † -functor. • In the study of top olo gical quan tum field theories, it is ph ysically w ell-motiv ated to require that the functor defining the field theory should b e compatible with the † -functor o n the category of cob ordisms and the † -functor on the catego r y of Hilb ert spaces. This g iv es a unitary topolog ical quan tum field theory . 6 • If a F rob enius algebra in Hilb has its m ultiplication relat ed to its comultiplication b y the † -functor , then it is a C*-algebra [9, 24]. • When w or king with a monoidal † -category , it is often useful to require that the left unit, righ t unit, asso ciativity and braiding isomorphisms a re unitary at ev ery stage [2, 5, 24]. Constructing † -limits As mentioned earlier, † -limits are the central categorical construction whic h w e will use to pro ve our results. W e use dia g rams in the shap e of fo rest-shap ed m ultig r a phs with a finite n um b er of lea ve s, suc h as example (1) in the in tro duction. These are defined a s diagrams with a finite num b er of connected componen ts, eac h of w hic h is a directed tree orien ted from a finite n umber of lea v es at the b ot t om to a ro o t at the to p, and for which m ultiple parallel branc hes b etw een no des are allow ed. Note that there is no a m biguit y a b out whic h ob jects are the lea v es; they are exactly the sys t ems in the dia gram whic h are not t he target of any pro cess in the dia g ram (except for an iden tit y pro cess.) In the rest of the paper, in the con t ext of † -limits, w e will often simply refer to these forest-shap ed multigraphs as diagr ams . W e sa y that suc h a diagram is finite when it has a finite num b er of arro ws. On pa ge 16 w e giv e a generalized definition of † -limit whic h applies to a muc h larger class of diagrams, but † -limits of finite forest-shap ed m ultigraphs are sufficien t to obtain our results. Let F : J ✲ C define suc h a diagram in the category C . A c one f or t his diagram is an ob ject X in C , equipped with c one m a p s x S : X ✲ F ( S ) for all o b jects S of J , suc h that for an y map f : A ✲ B in J the equation x A ; F ( f ) = x B holds. A l i m it f o r the diagram is a sp ecial cone L , equipp ed with cone maps l S : L ✲ F ( S ), suc h that for any cone ( X , x S ) for the diagram, there is a unique map m : X ✲ L suc h that x S = m ; l S for all ob jects S in J . F or mor e information a b out limits in category theory ha v e a lo ok a t an y in tro ductory category theory textb o ok, suc h as [22]. If C is a † -category with a sup erp osition rule ‘+’ on the ho m-sets — or tec hnically , whic h is e n riche d in c ommutative monoids — then a † -limit for a diagram F : J ✲ C is a limit for the diagram in the usual sens e, suc h that the normalization condition X S l S ; l † S = id L , S is a leaf in J (5) holds, where the maps l S : L ✲ F ( S ) are the pro jection maps from the limit ob ject to the lea v es. Since w e require all diagrams to only ha v e finite nu mber of lea ves , this is a finite sum, and will alw ay s b e w ell-defined. It seems that this definition is sensitiv e to the definition of the sup erp osition rule ‘+’ used to define the summation, but in fact it is not, a s explained b y Lemma 2.4. If a † -category has a † -limit for ev ery diagram then w e say it has al l † -limits . If it only has a † -limit for all finite diagrams, then w e sa y that it has all finite † -limits . These † -limits are, in part icular, ordinar y limits, and so will b e isomorphic to any other ordinary limit. Ho w ev er, b etw een themselv es, † -limits satisfy a stro ng er univ ersal prop erty — they are unique up to unique unitary isomorphism. 7 Lemma 2.1. In a † -c ate gory, any † -limit is unique up to unique unitary isomo rp hism. Pr o of. Let F : J ✲ C b e a diagram, and let ( L, l S ) and ( M , m S ) b e † -limits for this diagr a m, where l S : L ✲ F ( S ) and m S : M ✲ F ( S ) are the respective limit maps, and S is a v ariable that ranges o v er the leaf ob j ects of J . Then b y the properties of limits, there mus t b e a unique comparison isomorphism c : L ✲ M w ith the prop erty that c ; m S = l S for all S . By the normalization condition w e ha ve the equations P S l S ; l † S = id L and P S m S ; m † S = id M , and we emplo y these in the fo llowing w ay to sho w that c ; c † = id L : c ; c † = c ; id M ; c † = c ;  X S m S ; m † S  ; c † = X S c ; m S ; m † S ; c † = X S l S ; l † S = id L . It can be sho wn in a similar w a y that c † ; c = id M , a nd so c is unitary . † -Pro ducts and † -equalizers W e will mak e substan tial use of t w o particularly imp ortan t t yp es of † -limit. The first t yp e of † - limit is a finite † -pr o d uct , which is the † -limit of a finite discrete diagram, for whic h ev ery ob ject is a leaf. The second t yp e is a finite † -e qualizer , whic h is the limit of a diagram consisting of a finite n um b er o f arrow s, all of whic h ha v e the same source ob ject a nd the same target ob ject; this has exactly one leaf v ertex. W e can dra w these † -limits as follo ws, with the diagram in blac k and the † -limit a nd its asso ciated maps in grey: • • · · · • L B L E l ✲ • • . . . l 1 l 2 l N l 1 ; l † 1 + l 2 ; l † 2 + · · · + l N ; l † N = id L B l ; l † = id L E (6) The r elev an t form of the normalization condition ( 5 ) is giv en underneath eac h diagra m. W e emphasize that a † -equalizer is exactly a con v en t ional catego ry-theoretical equalizer, suc h that the equalizing map is an isometry . This extra isometry condition is a na tural one to consider in a † -category , since equalizers are alw ays monic, and the isometry condition can b e considered as a strengthening of t he monic prop ert y . W e also define a † -kernel to b e a † -equalizer of a parallel pair consisting of an arr ow and the zero arro w. This researc h programme w as b orn out of a study of the prop erties of † -categories with † -equalizers, a nd I am grateful to P eter Selinger for suggesting them as a construction. A first useful result is t ha t † -pro ducts are ex a ctly † -bipr o ducts , whic h are w ell-kno wn generalizations of the concept of ‘orthogonal direct sum’: fo r an y t w o ob jects A and B , their † -bipro duct is an ob ject A ⊕ B equipped with inje ction m orphisms i A : A ✲ A ⊕ B and i B : B ✲ A ⊕ B satisfying the follo wing equations: i † A ; i A + i † B ; i B = id A ⊕ B i A ; i † A = id A i B ; i † B = id B (7) i A ; i † B = 0 A,B i B ; i † A = 0 B ,A The adjoin ts t o the injec t io n morphisms are called the pr oje ction mo rp h isms . This definition of † -bipro duct generalizes in an ob vious w a y to an y finite list of ob jects. 8 Lemma 2.2. The † -limit of a discr ete diagr am (that is, a † -pr o duct) is the † -bi p r o d uct of the obje cts of the diagr am, and the c one maps ar e the † -bipr o duct pr oje ctions. Pr o of. W e prov e our lemma for the case of a discrete diagram with tw o ob jects; the extension to an y finite discrete diagram of ob jects is straightforw ard. Consider the † -limit of the diagram consisting of tw o ob jects, A and B . The † -limit is a limit ob ject L , equip p ed with morphisms l A and l B whic h satisfy l A ; l † A + l B ; l † B = id L . (8) Since L is the limit, there is a unique map h 0 B ,A , id B i : B ✲ L with h 0 B ,A , id B i ; l A = 0 B ,A and h 0 B ,A , id B i ; l B = id B , where 0 B ,A : B ✲ A is the unit for the enric hment in comm utative monoids. Precomp osing (8) with t his map w e obtain l † B = h 0 B ,A , id B i , and so we ha v e l † B ; l A = 0 B ,A , l † B ; l B = id B . (9) Similarly w e can sho w that l † A = h id A , 0 A,B i : A ✲ L , whic h leads to the equations l † A ; l B = 0 A,B , l † A ; l A = id A . (10) Altogether, these equations witnes s the fact that L is the † -bipro duct of A and B , with pro jections l A , l B and injections l † A , l † B . In a category with biproducts there is a unique enric hmen t in commutativ e monoids, whic h can b e defined in the following w ay : A f + g ✲ B A ⊕ A ∆ A ❄ f ⊕ g ✲ B ⊕ B ∇ B ✻ (11) Here, t he diago n al map ∆ A : A ✲ A ⊕ A is the unique map ha ving the property t ha t ∆ A ; i 1 † = ∆ A ; i 2 † = id A , where i 1 † , i † 2 : A ⊕ A ✲ A a re the pro jections on to the first and second comp onen t of the bipro duct resp ectiv ely . The c o diagonal ∇ B : B ⊕ B ✲ B is defined in a similar w a y as the unique map satisfying i 1 ; ∇ B = i 2 ; ∇ B = id B . It is straigh tforward to sho w that the bipro duct op eratio n on morphisms satisfies ( f ⊕ g ) † = f † ⊕ g † for ev ery pair of morphisms f and g . Also, w e ha ve f ; ( g + h ) = ( f ; g ) + ( f ; h ) and ( g + h ); j = ( g ; j ) + ( h ; j ) for all morphisms f , g , h, j of the correct types, as can b e directly c hec ked b y applying equation (11 ). The diagonal ∆ A and the co diagonal ∇ A are adjoin t to each other, as demonstrated b y the following lemma. 9 Lemma 2.3. F or an y † -bipr o duct A ⊕ A , the diagonal ∆ A : A ✲ A ⊕ A and c o diagona l ∇ A : A ⊕ A ✲ A satisfy ∆ A † = ∇ A . Pr o of. W e see that id A = (id A ) † = (∆ A ; p i ) † = p i † ; ∆ A † , where i ∈ { 1 , 2 } and p i is a pro jector on to one of the factors of the bipro duct. But id A = p i † ; ∇ A for all i is the defining equation for the co diagonal, a nd so ∆ A † = ∇ A . F rom this lemma, and from the definition of f + g giv en b y equation (11), it follo ws that the comm utativ e monoid structure is compatible with the action o f the † -functor, satisfying ( f + g ) † = f † + g † (12) for all parallel morphisms f and g . In a categor y with bipro ducts w e ha v e a matrix calculus a v ailable to us: a morphis m f : L i A i ✲ L i B i corresp onds to a matrix o f morphisms f i,j : A i ✲ B j , and comp osition of morphisms is giv en b y matr ix m ultiplication. In an y † -category with † -bipro ducts, it can b e shown that the adjoint of a matrix has the followin g form:       f g · · · x h j . . . . . . y z       † =       f † h † · · · y † g † j † . . . . . . x † z †       (13) This is just the familiar matrix conjugate-transp ose op eration, with the ‘conjugate’ of each en try in the matrix b eing its adjoin t. The category H ilb has all finite † -limits, a nd so in particular has both † -bipro ducts and † -equalizers: the † - bipro duct of a finite list o f Hilb ert spaces is given by their direct sum, and for some parallel set o f linear maps A ✲ B , their † - equalizer is given b y an isometry with image equal to the lar g est subspace of A o n whic h all the linear maps agree. Uniqueness of t he sup erp osit ion rule Because of the normalization condition (5) it seems that the definition o f † - limits depends on the c hoice of the superp osition rule ‘+’, whic h w e refer to as the enrichment in c o m mutative monoids . This is true, but can b e easily o verc o me, thanks to the follow ing fa ct: if by some enric hmen t in commutativ e monoids a † - cat ego ry at least has † -limits of discrete diagrams and of the empty diagram, then the category in fact admits a unique enric hment in comm utativ e monoids. So in particular, if a † -category admits an enric hmen t in commu t ativ e monoids suc h that it has all finite † -limits, then that enric hment is determined uniquely . This can b e sho wn b y considering Lemma 2.2 along with the follo wing w ell-kno wn result. Lemma 2.4. Supp ose that a c ate gory has a zer o obje ct and al l finite bipr o ducts. Then it has a unique enrichm ent in c ommutative monoids. Pr o of. F or any hom-set Hom( A, B ), write 0 A,B : A ✲ B for the unique morphis m w hich factors through the zero ob ject, and e 0 A,B : A ✲ B for the unit morphism enco ded b y the 10 enric hmen t in commutativ e monoids. Clearly 0 A, 0 = e 0 A, 0 and 0 0 ,A = e 0 0 ,A , since those hom- sets only con tain a single elemen t. Us ing the axiom that e 0 A,B ; f = e 0 A,C for all ob jects C and all morphisms f : B ✲ C , we obtain 0 A,B = 0 A, 0 ; 0 0 ,B = e 0 A, 0 ; 0 0 ,B = e 0 A,B , and so t he zero morphisms and the unit morphisms for the enric hmen t coincide. As a result, for the rest of this pro of, we will use 0 A,B to represen t b o th the zero and unit morphisms. In a categor y with bipro ducts, for an y f , g : A ✲ B , w e can define a mo r phism f ⊞ g : A ✲ B as A ∆ A ✲ A × A α − 1 ✲ A + A ( f g ) ✲ B , (14) where ( f g ) is the unique map with i 1 ; ( f g ) = f and i 2 ; ( f g ) = g . The map α − 1 is t he in v erse of the map α : A + A ✲ A × A , whic h is the unique map suc h that: i 1 ; α ; p 1 = id A i 1 ; α ; p 2 = 0 A,A i 2 ; α ; p 1 = 0 A,A i 2 ; α ; p 2 = id A (15) Here i 1 and i 2 are the copro duct injections in to A + A , and p 1 and p 2 are the pro duct pro jections out of A × A . W e will demonstrate that α − 1 = p 1 ; i 1 + p 2 ; i 2 . Consider α ; α − 1 = α ; p 1 ; i 1 + α ; p 2 ; i 2 . Then i 1 ; α ; α − 1 = i 1 and i 2 ; α ; α − 1 = i 2 , and b y the unive r sal prop ert y satisfied b y copro duct injections, w e m ust hav e α ; α − 1 = id A . W e can sho w similarly that α − 1 ; α = id A , and so α − 1 and α are in vers e. Substituting our express ion for α − 1 in to equation (14 ), w e obtain f ⊞ g = ∆ A ; ( p 1 ; i 1 + p 2 ; i 2 ); ( f g ) = ∆ A ; p 1 ; i 1 ; ( f g ) + ∆ A ; p 2 ; i 2 ; ( f g ) = id A ; f + id A ; g = f + g . (16) But f ⊞ g w as defined without reference to the CMon - enric hmen t op era t ion ‘+’, and so it follo ws that this is the only enric hment t ha t can exis t . Prop erties of † -categories with † -limits The existence of all finite † -limits in a † -category guaran tees some in teresting prop erties. As a general rule of t hum b, these prop erties are those which are f a miliar from the category of complex Hilb ert spaces. Nondegeneracy The first prop erty we will ex a mine is n o nde gener acy , also called p ositivity b y some authors [13, Definition 8.9]. In a † -category with a zero ob ject, w e define the † -functor to be nondegenerate if f ; f † = 0 implies f = 0 for all morphisms f . W e sho w now that this prop ert y is closely link ed to the existenc e of † -equalizers. 11 Lemma 2.5 (No ndegeneracy) . In a † -c ate g ory with a zer o o b j e ct and finite † -e qualizers, the † -functor is nonde gener ate. Pr o of. Let f : A ✲ B b e an ar bitrary morphism satisfying f ; f † = 0 A,A . Then f mus t factor through the † -k ernel of f † as indicated by the following comm uting diagram, where the facto r ising morphism is denoted ˜ f , and ( K, k ) for ms the † -ke rnel of f † : A K ⊂ k ✲ B f † ✲ 0 B ,A ✲ A ˜ f f (17) By definition w e hav e k ; f † = 0 K,A , and w e apply the † -functor to obt a in f ; k † = 0 A,K . Also, since ( K , k ) is a † - k ernel, k is an isometry , whic h means k ; k † = id K . W e can now demonstrate that f is zero: f = ˜ f ; k = ˜ f ; k ; k † ; k = f ; k † ; k = 0 A,K ; k = 0 A,B . An imp ortant feature of this pro of, whic h will recur in other pro ofs throughout this pap er, is that altho ugh the † -functor is use d sparingly , it is used crucially: in this cas e, to translate k ; f † = 0 K,A in to f ; k † = 0 A,K . The category of complex Hilb ert spaces has finite † -equalizers, and so this lemma can b e seen as ‘explaining’ why that catego ry has a nondegenerate † -functor. Conv en tionally , the nondegeneracy prop ert y in Hilb w ould instead be pro ve d using the fact that inne r pro ducts on Hilb ert spaces ar e necessarily p ositive definite . In this wa y , it is clear that there is some connection b et we en p ositive - definiteness of inner pro ducts and the existence of † -equalizers; w e formalize this lat er with Theorem 5.2, whic h demonstrates that in a † - category with † -equalizers, eac h ob ject is endo w ed with a canonical notion of inner pro duct. Cancellabilit y W e no w study v ar io us cancellabilit y properties satisfie d b y the a dditiv e structure on the hom-sets. Sa y that a comm uta t iv e monoid is c anc el lable if, for an y three elemen ts a, b, c in the monoid, a + c = b + c ⇒ a = b . W e are mot iv ated to study this condition s ince, in particular, it is satisfied by the a ddition of linear maps b etw een Hilbert spaces . W e no w sho w that it follo ws as a conse quence of havin g † -limits. Lemma 2.6 (Cancellable addition) . In a † -c ate gory with al l finite † -lim its, hom-set addition is c anc el lable; that is, for arbitr ary f , g , h in the same hom-set, f + h = g + h ⇒ f = g . (18) Pr o of. Let f , g , h : A ✲ B b e morphisms satisfying the equation f + h = g + h . Then w e can form the follo wing comm uting diagram, consisting of a † -equalizer ( E , e ) for the parallel 12 pair ( f h ) and ( g h ) along with tw o cones ( A, i 2 ) and ( A, ∆ A ): ˜ i 2 A i 2 =  0 A,A id A  E ⊂ e =  e 1 e 2  ✲ A ⊕ A ( f h ) ✲ ( g h ) ✲ B e ∆ A A ∆ A =  id A id A  (19) The mo r phism i 2 is the injection of the second factor in to the † -bipro duct, and the morphism ∆ A is the diago na l for the † -bipro duct. Since i 2 and ∆ A are cones they m ust fa ctor ize uniquely through e , and w e denote these factorizations b y ˜ i 2 and e ∆ A resp ectiv ely . The condition tha t e is an isometry give s the equation e 1 ; e † 1 + e 2 ; e † 2 = id E . (20) Precomp o sing with ˜ i 2 giv es e † 2 = ˜ i 2 , and p o stcomp o sing this with e 1 and e 2 resp ectiv ely giv es e † 2 ; e 1 = 0 A,A , (21) e † 2 ; e 2 = id A . (22) Similarly , precomposing (20) with e ∆ A giv es us e † 1 + e † 2 = e ∆ A , a nd p ostcomp osing with with e 1 and e 2 resp ectiv ely giv es e † 1 ; e 1 + e † 2 ; e 1 = id A , ( 2 3) e † 1 ; e 2 + e † 2 ; e 2 = id A . ( 2 4) W e will sho w that i 1 = ( id A 0 A,A ) : A ✲ A ⊕ A is a cone for the parallel pair , whic h directly leads to the required conclusion f = g . W e m ust find a factorising morphism c : A ✲ E whic h giv es i 1 up on comp o sition with e : E ✲ A ⊕ A . W e c ho ose c = e † 1 , and so w e mus t sho w that e † 1 ; e 1 = id A and e † 1 ; e 2 = 0 A,A . The first of these is obtained b y applying equation (21) to equation (23), and the second by applying the † -functor t o equation (21). An imp ortan t observ ation is that it seems to b e imp ossible to av oid the use of the † -functor for the final stage of this proof . Without it, the strongest equation that we can easily deriv e for the endomorphism e † 1 ; e 2 is e † 1 ; e 2 + id A = id A , (25) obtained b y com bining equations (22 ) and (24). Of course, without the cancellabilit y prop ert y that w e are trying to pro v e, this is not enough to establis h that e † 1 ; e 2 = 0 A,A . One use for t his lemma is to demonstrate that a particular category do es not ha v e all finite † -limits, whic h is usually more difficult tha n c hec king whether ho m-set addition is cancellable. F or example, the category Rel of sets and relations is a † -category with † -functor giv en by relational con v erse, and it has finite † -bipro ducts. Since id 1 + id 1 = id 1 in this category , it do es no t hav e cancellable addition, and so by the t heorem do es not ha ve all † -limits. (Of course, since Rel do es not eve n hav e equalizers, this is not surprising.) 13 W e no w in v estigate another for m of cancellabilit y . In a catego ry enric hed in comm utativ e monoids, for an y natural n umber n and a ny morphism f , we define the n -fold sum of f to b e n · f := f + f + · · · + f , where we sum o v er a total of n copies of f . W e can then pro ve the following lemma. Lemma 2.7. I n a † -c ate gory with al l finite † -limits, for any f , g in the same hom-set, if ther e exists a nonzer o n with n · f = n · g , then f = g . Pr o of. Consider the follo wing comm utativ e diagr am, where f , g : A ✲ B are morphis ms satisfying n · f = n · g : E A ⊕ n ( f f · · · f ) ✲ ( g g · · · g ) ✲ B A e ∆ ✻ e ∆ (26) The diagonal morphism ∆ : A ✲ A ⊕ n is a cone for the para llel pair, and so it f actors uniquely thro ugh the † -equalizer e : E ✲ A ⊕ n as e ∆ : A ✲ E . Let p i : A ⊕ n ✲ A b e the pro jection on to the i th factor of the † -bipro duct, and define e i := e ; p i : E ✲ A as the i th elemen t o f the † -equalizer morphism e : E ✲ A ⊕ n . W e hav e ∆ = e ∆; e = e ∆; e ; e † ; e = ∆; e † ; e , and b y p ostcomp osing with p 1 w e o bt a in id A = P i ∈ N e † i ; e 1 where N is a set with n elemen ts. T aking the a djoin t of this giv es id A = P i ∈ N e † 1 ; e i . Sinc e e is a cone w e ha ve P i ∈ N ( e i ; f ) = P i ∈ N ( e i ; g ), and b y precomposing with e † 1 and reorganising w e obtain ( P i ∈ N e † 1 ; e i ); f = ( P i ∈ N e † 1 ; e i ); g . W e ha v e already sho wn that P i ∈ N e † 1 ; e i = id A , and so w e obtain f = g . Finally we sho w that the n -fo ld sum op eration has an in v erse fo r an y p ositiv e n . It follows from this that we can construct fr actions of mor phisms. Lemma 2.8. In a † -c ate gory with al l finite † -limits, for e ach obje ct A and e ach nonzer o natur al numb er n , ther e exists a unique morphism id A n : A ✲ A with n · id A n = id A . Pr o of. Consider the equalize r dia gram consisting of the pro jection maps p i : A ⊕ n ✲ A . Let e : E ✲ A ⊕ n b e their † -equalizer, and let ∆ : A ✲ A ⊕ n b e the n -fold diago nal map, whic h is also an equalizer. Then there is a unique map e ∆ : A ✲ E mediating betw een thes e equalizers. E A ⊕ n p 1 ✲ · · · p n ✲ A A e ∆ e ∆ (27) Let e i : E ✲ A b e the i th comp onen t o f the † - equalizer e , defined by e i = e ; p i . Since e is an equalizer for the morphisms p i , eac h of these comp onen ts e i are equal. Then id A = ∆; p 1 = e ∆; e 1 = e ∆; e ; e † ; e 1 = ∆; e † ; e 1 = P i e † i ; e 1 = P i e † 1 ; e 1 = n · e † 1 ; e 1 , and w e can define id A n := e † 1 ; e 1 . It follows from Lemma 2.7 that this morphism is the unique one with the necess a r y prop ert y . 14 Exc hange lemma The final prop erty that we pro ve is a n ‘exc hange lemma’, whic h iden tifies a r estriction on the algebra of morphism comp osition in t he presence of † -limits. It can b e seen a s a stronger form of the nondegeneracy prop erty demonstrated in Lemma 2.5. W e will use this exc hange lemma in a n esse ntial w a y in the next section, to pro ve that our generalized real nu mbers admit a total order. Lemma 2.9 (Exc hange) . In a † -c ate gory with al l finite † -limits, for any p ar al lel morphisms f and g , f † ; f + g † ; g = f † ; g + g † ; f ⇒ f = g . (28) Pr o of. Let f , g : A ✲ B b e morphisms satisfying f † ; f + g † ; g = f † ; g + g † ; f . As migh t b e exp ected from the earlier lemmas, our pro of strategy is to construct a † -equalizer diagram, whic h in this case consists of the parallel pair ( f g ) a nd ( g f ). W e next deduce the existence of certain cones, ( B , p ) and ( B , q ), whic h factorize throug h the † -equalizer ( E , e ) via ˜ p and ˜ q resp ectiv ely: ˜ p B p =  f † g †  E ⊂ e =  e 1 e 2  ✲ A ⊕ A ( f g ) ✲ ( g f ) ✲ B ˜ q B q =  g † f †  (29) Since e : E ✲ A ⊕ A is a † -equalizer w e ha v e p † = e † ; ˜ p † = e † ; e ; e † ; ˜ p † = e † ; e ; p † , and similarly q † = e † ; e ; q † . The equalising morphism e is a cone, and given that ( f g ) = p † and ( g f ) = q † , w e obtain e ; p † = e ; q † . It is then straightforw ard to see that ( f g ) = p † = e † ; e ; p † = e † ; e ; q † = q † = ( g f ), and so f = g as required. W e call this the ‘exc hange lemma’ since, passing from one side of the main equation to the other, the morphism s f and g exc hange p ositions. Man y in teresting relations arise as sp ecial cases of this lemma. Cho osing g = 0 A,B w e obtain the nondegeneracy result of Lemma 2.5, f † ; f = 0 B ,B ⇒ f = 0 A,B , so the exc hange lemma can b e seen as a g eneralization of t his. Another in t eresting sp ecial case is g = id A,A , whic h gives , for all f : A ✲ A , id A + f † ; f = f + f † ⇒ f = id A . (30) Finally , choo sing f and g to be endomorphisms and f = g † , we obtain f ; f † + f † ; f = f ; f + f † ; f † ⇒ f = f † , (31) whic h giv es a new w a y to identify self-adjoint endomorphisms. Of course, since Hilb is our primordial example o f a † -category with all finite † -limits, the exc hange lemma and its corollaries holds there. Ho w ev er, in this category — or in an y 15 † -category fo r whic h hom-set addition is inv ertible — the exc ha nge lemma is equiv alen t to the nondegeneracy condition, b y moving terms across the equalit y and factorizing: f † ; f + g † ; g = f † ; g + g † ; f ⇔ ( f † − g † ); f = ( f † − g † ); g ⇔ ( f − g ) † ; ( f − g ) = 0 In a general † -category with † -limits, how ev er, the exc hange lemma is more g eneral, since although hom-set addition will b e cancellable b y L emma 2 .6 , it will not necessarily b e in v ertible. It seems like ly that without the † -functor there w ould b e no analog ue to the results in this section. F o r this reason, w e ar gue that the † -functor is an imp o r t a n t mathematical structure whic h deserv es to b e studied in its o wn righ t. More general † -limits The definition of † -limits can b e substan tially generalized, allo wing us to compute † -limits of (a lmost) arbitrary diagrams rather than just those in the shap e of finite forest-shaped m ultigraphs. In t he case that our † -category is unitary , meaning that any pair o f isomorphic ob jects hav e a unitary isomorphism going b et we en them, this more general t yp e of † - limit can alw ays b e constructed fro m the simpler t yp e, and in fact me rely ha ving zero ob j ects, finite † -pro ducts and finite † -equalizers giv es enough p o we r to construct them. The rest of the pap er do es not depend on this subsection, so it can b e safely skipped. T o describ e this bigger class of † -limits, w e b egin b y considering arbitrary finite diagrams. These are finite sets of systems and pro cesse s, closed under composition, suc h that for ev ery pro cess its initial and final systems are included, and for ev ery system its identit y pro cess included. Here is a drawing of a simple diagr a m, where for clarity w e lea v e out the iden tity pro cesses: D A C E B h j l k g f m (32) Supp ose that these processes compose in the followin g w ay: g ; j = h f ; h = j g ; f = id A f ; g = id B m ; m = m (33) Then our pro cesses a re closed under compo sition, and the diagram is w ell-defined. Note that w e allo w cycles in t hese more general diagrams, as long as we mak e sure to retain closure under comp osition. W e now choose a privileged subset Ω of the sy stems in the diagra m, called the supp orting subset , and w e refer to its elemen ts as the supp orting obje cts . The only constrain t w e impo se on Ω is that, b y starting a t systems in Ω and fo llo wing pro cesses in t he diag r a m, we m ust b e able to reac h ev ery system. So { A, C , E } w o uld b e a n allow ed c hoice fo r Ω, as we can get to B b y follo wing g : A ✲ B , and to D by follo wing h : A ✲ D (or alternatively k 16 or l .) An illegal c hoice for Ω w ould b e { C, D , E } , as neither A nor B can b e reac hed starting from those ob jects. It is alw ay s v alid to tak e Ω to con tain all the o b jects in the dia g ram. Ho we ver, it is vital that w e hav e the freedom to tak e Ω as an y supp orting subset, not only the maximal one: otherwise we w o uld no t b e able to construct † - equalizers, whic h w e rely on for man y of our results. Giv en a particular diagram F : J ✲ C , and a v alid c hoice o f supporting subset Ω of the ob jects of J , a † -limit for this diagram is a limit system L in the usual sense, equipp ed with limit maps l S : L ✲ F ( S ) satisfying the following normalization condition: X S ∈ Ω l S ; l † S = id L . (34) This is v ery similar t o the previous definition of † -limits, but our normalization condition do es not in v olv e the limit maps to the leaf ob jects ( a s our diagrams will not in general be forest- shap ed), but rather to the ob j ects in the supporting subset. W e dra w an ex a mple of this for the example diagram giv en earlier, with the supp orting subset c hosen to b e Ω = { A, C , D , E } : D A C E B L l A l C l D l E l A ; l † A + l C ; l † C + l D ; l † D + l E ; l † E = id L (35) An y † - limit obtained from a forest-shap ed m ultigra ph, a s describ ed in previous sections, is clearly also a † -limit in this more general sense, where the supp ort ing subset Ω is tak en to b e the set of leav es of the diagram. W e also men tion tha t it is straigh tfor ward to pro ve an extension of Lemma 2.1 sho wing that these more general t yp es of † -limit are unique up to unique unitary isomorphism. This more general t yp e of † -limit can b e computed for an y diagram that admits a finite set of supp orting ob jects. One approac h to the standard theory of categorical limits [20] states that a category has limits exactly when the diagonal functor ∆ : C ✲ C J has a left adjo in t. It would b e desirable to find a generalization of this a ppro ac h that w orks for the case of † - limits, p erhaps b y replacing categories by † - categories throughout. Ho w eve r, the author has b een unable to dev elop a t heory along these lines. One problem that is encoun tered is that, in the t heory of † -limits presen ted here, non - † -catego ries a r e still imp orta n t — for example, as the diagram category f o r a † - equalizer. Imp ortance of the c hoice of support ing subset The maps from the limit ob j ect for a g eneral † -limit dep end significan tly on the choice of supp orting subset Ω ⊆ Ob( J ). As an example, consider the f ollo wing simple diagram in 17 Hilb , the category o f Hilb ert spaces: C C 2 C 2 ( 2 0 0 1 )  1 2 0 0 1  ( 1 0 ) ( 2 0 ) (36) Eac h of the ob jects has a canonical basis, and w e represen t t he morphisms of the diag ram as matrices with r esp ect to those bases. The limit ob ject for this diagram can be tak en to b e C , regardless of the c hoice of supp orting subset. If w e tak e the supp orting subset to o nly con tain the o b j ect C in the middle of the diagram, then the † -limit morphism is the linear map 1 : C ✲ C , whic h clearly satisfies the norma lizat io n condition. Instead, supp ose we take the suppo rting subset to contain a ll the ob jects of the diagram; then the limit maps are, in order of ob jects from left to right, ( 1 / √ 6 0 ) : C ✲ C 2 , 1 / √ 6 : C ✲ C and ( 2 / √ 6 0 ) : C ✲ C 2 . It is easy to c hec k that these a lso satisfy the correct normalizatio n condition. The p o wer of the † -limit construction is that these are essen tially unique, up to unique unitar y isomorphism. F or any ob ject J ∈ Ob( J ), w e can associate a canonical self-adjoin t morphis m l † J ; l J : F ( J ) ✲ F ( J ). This is uniquely define d for a giv en supp orting subset, a prop erty that follo ws straigh tfo r w ardly from the fact that the † -limit is unique up to unique unitary isomorphism. Note that the ob ject J do es not itself hav e to b e in Ω. F or the example just described, for the case that ev ery ob ject is in the supporting subs et, these self-adjoint morphisms are, from left to righ t, ( 1 6 0 0 0 ) , 1 6 and ( 2 3 0 0 0 ) . No w supp ose that our † -categor y has a w ell-defined no t ion of tr ac e for endomorphisms, v alued in some semiring, suc h that T r( f ; g ) = T r( g ; f ) for all opp ositely-directed f and g , and T r( h + j ) = T r( h ) + T r( j ) for all h and j whic h are b oth endomorphisms o f the same ob ject. Restricting to ob jects in the supp orting subset and summing ov er these traces, w e see that X S ∈ Ω T r ( l † S ; l S ) = X S ∈ Ω T r ( l S ; l † S ) = T r X S ∈ Ω l S ; l † S ! = T r(id L ) . (37) In many con texts the s calar T r(id L ) represen ts the size of the ob ject L , and so it is apparen t that eac h scalar T r ( l † S ; l S ) — whic h in many categories will b e ‘p ositive ’ in a suitable sense — indicates ‘ho w muc h’ of L arises from the ob ject S . Note that although T r (id L ) will, in man y commonly-encoun tered categor ies, necess a r ily b e an ‘in teger’, there is no suc h restriction on the v alues T r ( l † S ; l S ) . Also, since ev ery diagra m has a canonical choice of supp orting subset giv en b y all the ob jects, this giv es rise to a canonical w eigh ting, or ‘measure’, on the ob jects of the diag r a m. F or the example described ab o v e, in order of ob jects from left to rig ht, these w eigh ting s are 1 6 , 1 6 and 2 3 , which sum to T r(id C ) = 1 a s required. An existence theorem for † -limits W e no w examine the possibility of constructing arbitra ry † - limits from sp ecial ones, the † -equalizers a nd † -bipro ducts. W e will find that t his is p ossible as long as our category is 18 unitary , meaning that ev ery pair of isomorphic ob jects has a unitary isomorphism going b et we en them. This can b e seen as a n extension of the conv en tional existenc e theorem for limits, althoug h the pro of do es not transfer straightforw ardly since † -limits are significantly differen t from ordinar y limits. W e b egin by examining how to obta in arbitrary finite † -equalizers fro m simpler t yp es of † -limit. Lemma 2.10. If a † -c ate gory has binary † -e qualizers and b inary † -bipr o ducts, then it h as al l finite † -e qualizers. Pr o of. Let f i : A ✲ B be a set of par allel a rro ws indexed b y i ∈ I , a finite set. Then we can construct the I -fold † -bipro duct B ⊕ I , and define a column v ector F : A ✲ B ⊕ I as the unique morphism with t he prop erty that F ; p i = f i , where p i : B ⊕ I ✲ B is the pro j ection on to the i th fa cto r . Let ∆ : B ✲ B ⊕ I b e the diagonal map, and construct the fo llowing † -equalizer: E e ✲ A F ✲ f 1 ; ∆ ✲ B ⊕ I (38) P ostcomp osing with p i : B ⊕ I ✲ B w e obtain e ; F ; p i = e ; f 1 ; ∆; p i , whic h simplifies to e ; f i = e ; f 1 . It follo ws that e ; f i = e ; f j for a ll i, j ∈ I , and so e : E ✲ A is a cone f o r the morphisms f i : A ✲ B . Now let x : X ✲ A b e any map such that x ; f i = x ; f j for all i, j ∈ I . Then x is also a cone for the morphisms F and f 1 ; ∆, a nd so fa ctorizes uniquely through e : E ✲ A . It f o llo ws that the morphism e is t he † -equalizer of the morphisms f i : A ✲ B . W e will also require the t w o follo wing tec hnical lemma, whic h say s that w e can ta k e the ‘square r o ot’ o f an y ‘natura l n um b er’. Lemma 2.11. In a unitary † -c ate gory with binary † -e qualizers and bin a ry † -bipr o ducts, for e ach obje ct A and e ach natur al numb er n , ther e is an isomorphism r : A ✲ A with r ; r † = n · id A . Pr o of. W rite p i : A ⊕ n ✲ A for the pro jection of the † -biproduct on to its i th factor, and consider a ll these maps together as forming an equalizer diagram: E A ⊕ n p 1 ✲ · · · p n ✲ A A m u e ∆ (39) The n -f o ld diagonal map ∆ : A ✲ A ⊕ n is a n equalizer for these maps, since giv en an y x : X ✲ A ⊕ n with x ; p i = x ; p j for all v alid i a nd j , x factors unique ly through ∆ as x ; p 1 ; ∆. By Lemma 2.10 w e can construct the † -equalizer of the maps p i , whic h w e denote by e : E ✲ A ⊕ n . Since e and ∆ are b oth equalizers, there is a unique is o morphism m : A ✲ E with m ; e = ∆; and since A and E a r e isomorphic, b y unitarit y of the † -category , t here exists 19 some unitary morphism u : E ✲ A . Defining an endomorphism r := m ; u : A ✲ A , w e see that r ; r † = m ; u ; u † ; m † = m ; m † = m ; e ; e † ; m † = ∆; ∇ = n · id A , (40) where in the fo urt h expression we ha ve inserted the iden tity in the form id E = e ; e † . Since b oth m and u are isomorphisms it fo llo ws that r : A ✲ A is also an isomorphism. W e now describ e a new fundamen tal construction called the † -interse c tion . In a † -category , giv en a finite family of is o metries x i : X i ✲ A , their † -interse ction is defined to b e a pullbac k ( P , π i ) suc h that eac h of the maps π i : P ✲ X i is an isometry . The no tion of † -in tersection is a geometrical one: giv en a family of isometries represen ting s ub ob jects of a giv en ob ject, the † -in tersection is an isometry represen ting the inte r section of all these sub ob jects. Of course, this in tersection could b e zero. W e no t e that the † -in tersection of a family o f isometries is not give n b y their † -pullbac k, apart fro m the trivial case where w e are ta king the † - in tersection of a single isometry . W e no w giv e an existence theorem for † -in tersections. Lemma 2.12. If a unitary † -c ate gory has al l binary † -e qualizers and binary † -bipr o ducts then it has al l fi nite † -interse ctions. Pr o of. Let x i : X i ⊂ ✲ A b e our family of isometries in a unitary † -category C , indexed b y a finite set J . W e cons t r uct the † -bipro duct L i ∈ J X i , with canonical pro jections p i : L i ∈ J X i ✲ X i . Considering our family of isometries as a diagram in C , w e can construct its † -pullback b y forming the † -equalizer e : E ✲ L i ∈ J X i of the morphisms p i ; x i : L i ∈ J X i ✲ A , makin g use of Lem ma 2.10. T he cone maps of the † -limit are then giv en b y e ; p i : E ✲ X i . It is straightforw ard to ch ec k that they form a limit, and the normalization condition is satisfied since P i ∈ J e ; p i ; p † i ; e † = e ; ( P i ∈ J p i ; p † i ); e † = e ; e † = id E . An y of the comp osites e ; p i ; x i : E ✲ A , all of whic h are equal, in tuitiv ely represen ts the in tersection of the isometries x i : X i ✲ A . How ev er, these comp osites are not isometries in general; w e m ust add a normalization factor . W e construct the † -in t ersection of the morphisms x i as s := r E , | J | ; e ; p i ; x i : E ✲ A , fo r any ch o ice of i ∈ J , whe r e r E , | J | : E ✲ E is an isomorphism satisfying r E , | J | ; r † E , | J | = | J | · id E as describ ed in Lemma 2.1 1, and | J | is the n umber of elemen ts of J . Our morphism s do es indeed factor throug h the pro jections of a pullback in the necessary w ay , since we hav e already sho wn that the morphisms e ; p i form the pro jections of a † -pullbac k, and since limits are preserv ed b y isomorphisms, so do the morphisms r E , | J | ; e ; p i . T o show that s is an isometry is to sho w that s ; s † = id E , and b y 20 Lemma 2.7 , it suffices to show that | J | · ( s ; s † ) = | J | · id E : | J | · ( s ; s † ) = X i ∈ J s ; s † = X i ∈ J  ( r E , | J | ; e ; p i ; x i ); ( r E , | J | ; e ; p i ; x i ) †  = X i ∈ J  r E , | J | ; e ; p i ; x i ; x † i ; p † i ; e † ; r † E , | J |  = r E , | J | ; e ;  X i ∈ J p i ; p † i  ; e † ; r † E , | J | = r E , | J | ; e ; e † ; r † E , | J | = r E , | J | ; r † E , | J | = | J | · id E . (41) This completes the pro of. Finally , we w eav e these lemmas together to obtain an exis t ence theorem for † - limits. Theorem 2.13 (Existenc e theorem for † -limits) . A unitary † -c ate gory has al l finite † -limits iff it has a zer o obje ct, binary † -e qualizers and binary † -bipr o ducts. Pr o of. If a † -category has all finite † -limits then it has these three constructions; the zero ob ject is the † -limit of the empty diagram, bina r y † -equalizers are manifestly † -limits, and binary † -bipro ducts are † -limits by Lemma 2.2. Con v ersely , consider a unitary † -category C with a zero ob ject, binary † -equalizers and binary † -bipro ducts. By Lemma 2.10 suc h a category actually has all finite † - equalizers, and it is straightforw ard to obtain all finite † -bipro ducts from binary † -bipro ducts. Since finite bipro ducts exist the category is enric hed in comm utative monoids, and so the notion o f a † -limit is w ell-defined. Consider a diagra m F : J ✲ C , with a c hosen supp orting subs et Ω ⊆ Ob( J ). W e will sho w that this has a † -limit. If Ω is empt y then J mus t also b e empt y , and the † -limit of F is g iv en by the zero ob j ect in C . Otherwise, form the † -bipro duct in C of t he images F ( S ) of the ob jects in the supporting subset, for all S ∈ Ω. W e denote this † -bipro duct b y L F (Ω) , and write the pro jections on to the facto r s as p S : L F (Ω) ✲ F ( S ) for all S ∈ Ω. F or eac h T ∈ Ob( J ), denote b y A T the set of arrows in J whic h go fr o m an o b ject in Ω to T , and for eac h arrow f ∈ A T denote it s domain supp orting ob ject b y σ ( f ) ∈ Ω, so we ha ve f : σ ( f ) ✲ T . F or eac h f ∈ A T , w e can cons t ruct a morphism [ f ] : L F (Ω) ✲ F ( T ) as the following comp osite: [ f ] := L F (Ω) p σ ( f ) ✲ F ( σ ( f )) F ( f ) ✲ F ( T ) (42) Let e T : E T ✲ L F ( S ) b e the † -equalizer in C of the arrows [ f ] for all f ∈ A T . Our candidate for the † -limit is the † -inte r section of the isometries e T , o v er all ob jects T ∈ Ob ( J ). W e denote this † -in tersection b y π T ; e T : P ✲ L F (Ω) , whic h has the same v alue for an y T ∈ Ω; the morphisms π T : P ✲ E T are a family of isometric pullbac k pro jections, whic h are guaranteed to exist b y Lemma 2.12. The † -limit maps to the ob jects in the supp orting subs et are l S := π T ; e T ; p S : P ✲ F ( S ) for an y T ∈ Ob( J ) , and for all S ∈ Ω. 21 W e m ust sho w that these maps form a unive r sal, norma lized cone for the diagram. First, w e sho w that the maps l S : P ✲ F ( s ) satisfy the normalizatio n condition (5): X S ∈ Ω l S ; l † S = X S ∈ Ω π T ; e T ; p S ; ( π T ; e T ; p S ) † = X S ∈ Ω π T ; e T ; p S ; p † S ; e † T ; π † T = π T ; e T ;  X S ∈ Ω p S ; p † S  ; e † T ; π † T = π T ; e T ; e † T ; π † T = π T ; π † T = id E . (43) T o establis h that the morphisms l S define a cone, w e m ust sho w that the equ ation l σ ( f ) ; F ( f ) = l σ ( g ) ; F ( g ) is satisfied for all T ∈ Ob( J ) and all f , g ∈ A T . By the definition of the cone m aps l σ ( f ) ; F ( f ) = π T ; e T ; [ f ], and s ince e T ; [ f ] = e T ; [ g ] w e se e that the cone pro p ert y holds. T o establish the univ ersal prop erty , consider a cone of morphisms x S : X ✲ F ( S ) for all S ∈ Ω; the cone pro p ert y is that for all T ∈ Ob( J ) and all f , g ∈ A T , w e ha v e x σ ( f ) ; F ( f ) = x σ ( g ) ; F ( g ). Le t e x : X ✲ L F ( S ) b e the unique morphism suc h that e x ; p S = x S for all S ∈ Ω. Then by the cone prop erty , for all T ∈ Ob( J ) a nd a ll f , g ∈ A T w e hav e e x ; [ f ] = e x ; [ g ], and so for all T ∈ Ob( J ) there is a unique morphism χ T : X ✲ E T with e x = χ T ; e T . Since ( P , π T ) for m a pullbac k of the morphisms e T , there m ust in turn b e a unique morphism e χ : X ✲ P suc h that e χ ; π T = χ T . Since each e T has a retraction, e χ is also the unique morphism with the prop ert y that e χ ; π T ; e T = χ T ; e T = e x . It f o llo ws that e χ is the unique morphism with e χ ; π T ; e T ; p S = e x ; p S for all S ∈ Ω, and so it is also the unique morphism with e χ ; l S = x S . So ( P ; l S , S ∈ Ω) indeed giv es a † - limit for the diagram F : J ✲ C , with Ω the suppo r t ing subset. 3 Em b e d ding the s calars into a field Our main theorem of this section is stated most naturally in a monoi d al † -c ate gory . Con v entionally , this means a monoidal category whic h is a lso a † -category , such that the unit and asso ciato r natural isomorphisms are unitary . While this giv es the category nicer prop erties as a whole, w e will not need to use them. So, for our purp oses, a monoidal † -category can b e simply tak en to mean a monoidal category whic h is also a † - category . In an y monoidal category , w e define the sc alars to b e the hom-set Hom( I , I ). This will hav e a certain a moun t of extra structure, dep ending on the prop erties of the a m bien t category . At t he very least, as is well-kno wn, it is a commutativ e monoid, where monoid m ultiplication is giv en b y morphism comp osition. Our main result concerns the scalars in a monoidal † -category with all finite † - limits, whic h ha v e the structure of a se miring with inv olution. W e will pro v e the follo wing theorem: Theorem 3.1. In a nontrivial monoid a l † -c ate gory with simp le tensor unit, and with al l finite † -limits, the involutive semiring of sc alars has an involution-pr eserving emb e dding into an involutive field with cha r acteristic 0 and or der able fixe d field. The pro of of this theorem will b e give n piece-b y-piece throughout this section. Just to b e clear, b y ‘field’ w e mean a classical a lg ebraic field: a commu t ativ e ring with m ultiplicativ e 22 in v erses for ev ery no nzero elemen t. By ‘c haracteristic 0’ w e mean tha t no finite sum o f the form 1 + 1 + · · · + 1 giv es zero. By ‘inv olutiv e semiring’ and ‘in volutiv e field’ w e mean a structure equipp ed with an o rder-2 automorphism that resp ects addition and multiplic a tion, and by ‘fixed field’ w e mean the subfield on whic h the automorphism acts trivially . By ‘simple tensor unit’, w e mean that ev ery monic map in to the tensor unit is either zero or an isomorphism; in other w ords, it has no prop er sub ob j ects. The connection b et we en this theorem and the complex n um b ers is give n b y the following w ell-kno wn characterization of the subfields of the complex n umbers. 2 Theorem 3.2. The subfields of the c omplex numb ers ar e pr e cisely the fields of c h ar acteris- tic 0 which ar e at most of c ontinuum c ar dinality. It follow s immediately that, if w e hav e a monoidal † -category satisfying the conditions of Theorem 3.1 for whic h the scalars are at most contin uum cardinalit y , they m ust embed as a semiring into the complex n um b ers. Ho w ev er, we cannot guaran tee that there will b e an involution-pr eserving embedding into the complex n um b ers, whic h translates the action o f the † -functor on the scalars into complex conjugation on the complex n um b ers. W e deal with this in the next section. In addition to the † -limits whic h w e studied in the previous section, The orem 3.1 requires t wo extra conditions: nontrivialit y , and t hat the monoidal unit ob ject is simple. Both a re natural, in the sense tha t they prev ent the theorem fro m b eing ‘ob viously’ false. A field is required to ha ve 0 6 = 1, and this translates to the condition that o ur category is no n trivial. Also, if we had a monoidal † -categor y satisfying the conditions of the theorem, w e could tak e the cartesian pro duct of this categor y with itself; this has an ob vious monoidal structure for whic h the monoidal unit do es ha v e prop er † -subo b j ects, the scalars being pairs of scalars in the original category . Suc h a semiring can nev er em b ed in to a field, since it contains zer o divisors , nonzero elemen ts a and b whic h satisfy ab = 0. Requiring the monoidal unit to lac k prop er subob jects blo c ks this o b vious source of coun terexamples. The scalars as a semiring W e b egin by showing that the scalars in a monoidal category form a comm utative monoid. W e establish this with the classic argumen t due to Kelly a nd La pla za [17], r elated to the Ec kmann-Hilton argumen t. W e note that this commutativit y property is the o nly reason that we pro v e Theorem 3.1 for the scalars in a monoidal category; it w ould hold for an y comm utativ e endomorphism mono id on an ob ject without prop er † -sub o b jects. 2 This theor em is often considere d s urprising, given that there seem to b e ‘obvious’ count e r ex- amples: for example, a field of r ational funct ions , whic h has elements given by equiv alenc e cla sses of ra tios of complex polynomia ls P ( x ) /Q ( x ), where Q ( x ) is not the zero polynomial, and where P ( x ) /Q ( x ) ∼ P ′ ( x ) /Q ′ ( x ) when P ( x ) Q ′ ( x ) = P ′ ( x ) Q ( x ). An e mbedding of such a field in to the co mplex nu mber s is difficult to visualize, since it will b e highly noncontin uous with re s pect to the natural top o logies inv olved. T o prove the theo rem, take any field o f characteristic 0 and at most co n tinuum cardinality , and add to it a contin uum of tr a nscendentals, obtaining a field o f precisely contin uum ca rdinality . Then ta ke the algebraic completion. The res ult is iso mo rphic to the c o mplex num b ers, since it is an a lgebraically - closed field of characteristic 0 and contin uum cardina lit y . 23 Lemma 3.3. In a monoidal c ate gory, the sc alars ar e c ommutative. Pr o of. W e presen t the standard commutativ e diagram in the form of a cub e, whic h holds for any t wo scalars a, b : I ✲ I . The coherence equation ρ I = λ I is essen tial. I a ✲ I I a ✲ b ✲ I b ✲ I ⊗ I λ I ❄ a ⊗ id I ✲ I ⊗ I λ I ρ I ❄ I ⊗ I λ − 1 I ρ − 1 I ✻ a ⊗ id I ✲ id I ⊗ b ✲ I ⊗ I ρ − 1 I ✻ id I ⊗ b ✲ (44) W e next sho w t ha t, if the monoidal category also has bipro ducts, the scalars form a comm utativ e semiring. A semiring, sometimes called a rig , is a structure similar to a ring but whic h is not req uired to ha ve hav e additiv e in v erses for all elemen ts. In this pap er a ring alw ays has a m ultiplicativ e unit, and the zero elemen t satisfies 0 x = x 0 = 0 for all elemen ts x in the ring. In a category with bipro ducts, the hom-sets hav e a comm utativ e monoid s tructure as described b y equation (11) . In terpreting this monoid structure as addition, and compo sition of scalars as m ultiplication, these structures combine to give the scalars in a monoidal category with bipro ducts the structure of a comm utativ e semiring. T o prov e this, w e need to sho w that a ( b + c ) = ( ab ) + ( ac ) for a ll scalars a, b, c ; t his follows from naturalit y of the diagonal and codiag onal maps, as discussed earlier on page 9. W e also require 0 a = a 0 = 0 for all scalars a , whic h follo ws from the definition of the zero morphisms . This comm utative semiring of scalars acts in a natural w a y on the hom- sets of the category . F o r an y morphism f : A ✲ B and any sc a la r a , w e define a · f : A ✲ B as follo ws: A a · f ✲ B I ⊗ A λ A ❄ a ⊗ f ✲ I ⊗ B λ − 1 B ✻ (45) In fact, this gives the eac h hom-set the structure of a semimo dule ov er the scalars, whic h is the natura l notion of mo dule extende d from a ring to a semiring. W e no w consider the extra structure giv en by t he † -functor and † - bipro ducts. The † -functor giv es us an in volution on the scalars, sending a : I ✲ I to a † : I ✲ I . This in v o lution is o rder-rev ersing for m ultiplication, due to the con t r av a riance of the † -functor, 24 and distributes ov er addition as explained in the discussion around equation (12). This giv es the scalars t he structure of an involutive semirin g . In the case that the unit isomorphisms asso ciated to the monoidal structure are unitary , the hom-sets then become involutive semimo dules for this semiring, but w e will not need this extra structure. One aim o f this researc h is to understand the categorical structure of the complex n um b ers, whic h is certainly a n in v olutiv e semiring, so the category theory is generating the correct kind of structure. Of course, the complex n um b ers are f a r more than just a semiring, and w e will now see how some of the necessary extra prop erties arise. Em b edding in to a field T o a chiev e our goal o f em b edding the scalars into a field, it is clear that additive cancellabilit y is a necessary prop ert y . W e demonstrated this for all ho m- sets in † -catego r ies with finite † -bipro ducts and finite † -equalizers in Lemma 2 .6. Another prop ert y whic h is clearly necessary is cancellable multiplication. Definition 3.4. A comm utativ e semiring has c anc el lable multiplic ation when, f o r an y three elemen ts a, b, c in t he semiring, ac = bc, c 6 = 0 ⇒ a = b . W e no w sho w tha t the sc alars ha v e this prop ert y in any category of the ty p e whic h w e are considering. The condition that the monoidal unit has no prop er † -sub ob jects is clearly crucial here, but this is far from the only ro le play ed b y this condition in proving the theorem. Lemma 3.5. I n a mo noidal † -c ate gory with simple tensor unit, a zer o obje ct and finite † -e qualizers, the sc alars have c anc el lable multiplic ation. Pr o of. Supp ose that the scalars did not hav e cancellable m ultiplication. Then there would exist scalars a, b , c with c 6 = 0, suc h that a 6 = b but ac = bc . W e consider the follo wing comm uting diagram: I E ⊂ e ✲ I a ✲ b ✲ I ˜ c c (46) The † - equalizer morphism e : E ✲ I giv es a † -subob ject of I . It is not zero, since c factors through it and c 6 = 0; also, since a 6 = b , it cannot b e a n isomorphism. It follo ws that I has a prop er † -sub ob ject, but this contradicts our h yp othesis. It follow s that the scalars ha v e cancellable multiplication. As a first step to wards em b edding t he scalars in to a field, w e first em b ed them into a ring. Give n our semiring S o f scalars, w e can construct it s d iffer enc e ring D ( S ). Elemen ts of D ( S ) are equiv alence classes of ordered pairs ( a, b ) of elemen ts of S , whic h w e write using the suggestiv e not a tion a − b . The equiv alence relation is give n by a − b ∼ c − d iff a + d = c + b. (47) 25 It is a standard exercise to show that this is symmetric, transitiv e and reflexiv e, for which w e rely on the fact that the scalars hav e cancellable addition. Addition and m ultiplication are defined on represen tatives of the equiv alence classes in the familiar algebraic wa y: ( a − b ) + ( c − d ) = ( a + c ) − ( b + d ) (48) ( a − b )( c − d ) = ( ac + bd ) − ( ad + bc ) (49) These are w ell-defined on equiv alence class es. W e see that the scalars in our category em b ed in to their difference semiring, under the ob vious mapping a ✲ a − 0. F or tw o eleme nts to b e sen t to the same elemen t of the difference ring would mean that a − 0 ∼ b − 0, but applying the definition of the equiv a lence relation then giv es a = b , so the mapping is fa ithful. As we will see, the difference ring em b eds in to a field if and o nly if it has cancellable m ultiplication. F rom Definition 3.4, this condition is ( a − b )( c − d ) ∼ ( a − b )( e − f ) , a − b ≁ 0 ⇒ c − d ∼ e − f (50) for all c hoices of elemen ts a, b, c, d, e, f ∈ S . Using the definition of the equiv alence relation to write this directly in t erms of the elemen ts of the underly ing semiring, w e obtain a ( c + f ) + b ( d + e ) = a ( d + e ) + b ( c + f ) , a 6 = b ⇒ c + f = d + e. (51) Defining A := c + f and B := d + e , this reduces to the condition aA + bB = aB + bA, a 6 = b ⇒ A = B . (52) W e no w sho w that this holds in an y category of the ty p e w e a r e w orking with. In some w ays , this condition resem bles that of the exc hange lemma 2.9, but it is log ically indep enden t from it. Lemma 3.6. In a monoidal † -c ate gory with simple tensor unit and al l finite † -limits, any choic e of sc alars A, B , a, b : I ✲ I satisfies the implic ation aA + bB = aB + bA, a 6 = b ⇒ A = B . Pr o of. W e hav e already sho wn that the scalars in suc h a categor y are comm utativ e and ha v e cancellable addition and multiplication, and we will use thes e prop erties throughout. Let A, B , a, b b e scalars satisfying aA + bB = aB + bA and a 6 = b . If a = 0 then bB = bA , and cancelling the nonzero b w e obtain B = A ; the case b = 0 is similar. Con ve rsely , if A = 0 then bB = aB , and B = A = 0 is the only p ossibility , or B would cance l contradicting our assumption that a 6 = b ; the case B = 0 is similar. In each o f these cases, therefore, t he implication holds. W e now consider the case in which none of the four scalars are zero. W e construct the follo wing commutativ e diagram where ( E , e ) is a † - equalizer for the parallel pair ( A B ) and 26 ( B A ), and ( I , p ) and ( I , q ) are cones: ˜ p I p =  a b  E ⊂ e =  e 1 e 2  ✲ I ⊕ I ( A B ) ✲ ( B A ) ✲ I ˜ q I q =  b a  (53) F or eac h cone, w e denote the unique factorization through the equalizer with a tilde. Using the matrix calculus a nd t he † -equalizer equation e ; e † = id E w e see that p = p ; e † ; e and q = q ; e † ; e , and writing these out in comp onen ts, w e obtain the follo wing: a = a ; e † 1 ; e 1 + b ; e † 2 ; e 1 (54) b = a ; e † 1 ; e 2 + b ; e † 2 ; e 2 (55) b = b ; e † 1 ; e 1 + a ; e † 2 ; e 1 (56) a = b ; e † 1 ; e 2 + a ; e † 2 ; e 2 (57) The first tw o equations come fr o m the comp onen ts of p , and the second tw o fro m the comp onen ts of q . Multiplying equation (54) b y b and (56) b y a and eq uat ing the righ t-ha nd sides , this giv es ba ; e † 1 ; e 1 + b 2 ; e † 2 ; e 1 = ab ; e † 1 ; e 1 + a 2 ; e † 2 ; e 1 . (58) W e apply commutativit y and additive cancellabilit y to obtain b 2 ; e † 2 ; e 1 = a 2 ; e † 2 ; e 1 . (59) W e note that the quantit y e † 2 ; e 1 is a scalar. Either it is zero, or it is nonzero and it can b e cancelled to giv e a 2 = b 2 . W e will consider these cases separately . First we a ssume that e † 2 ; e 1 6 = 0 I ,I and a 2 = b 2 . Defining c := a + b , w e see that ca = a 2 + ba = b 2 + ab = cb. (60) So ca = cb , and if c 6 = 0 it will cancel from b oth sid es to giv e a = b . Ho w ev er, by assumption a 6 = b , a nd so w e must ha ve c = 0 and a + b = 0. R eturning to our equation aA + bB = aB + bA and a dding b ( A + B ) to b oth sides, w e obtain aA + bB + b ( A + B ) = aB + bA + b ( A + B ) ⇒ ( a + b ) A + 2 bB = ( a + b ) B + 2 bA ⇒ 2 bB = 2 bA. (61) Since 2 : I ✲ I is giv en by ∆ I ; ∇ I = ∆ I ; (∆ I ) † where ∆ I : I ✲ I ⊕ I is the diagonal for the bipro duct, b y Lemma 2.5 it m ust be nonzero, a nd so it can b e cancelled from b oth sides. 27 By assumption b 6 = 0, and so it can b e cancelled as w ell. This giv es B = A a s required. The only unresolv ed case is e † 2 ; e 1 = 0. Alternativ ely , we could ha v e m ultiplied equation (5 5) b y a and equation (57 ) b y b and equated the righ t-hand sides. This leads to a similar conclusion: either e † 1 ; e 2 6 = 0 and A = B , or e † 1 ; e 2 = 0 and the theorem is not immediately resolve d. Since this line of argumen t is indep enden t fr om the previous one, the only remaining case to consider is that e † 2 ; e 1 = e † 1 ; e 2 = 0. W e ha ve not yet used the fact tha t the equalizer e : E ✲ I ⊕ I is a cone, whic h is a sserted b y the following equation: e 1 ; A + e 2 ; B = e 1 ; B + e 2 ; A. (62) Comp osing o n the left with e † 2 , we obtain e † 2 ; e 1 ; A + e † 2 ; e 2 ; B = e † 2 ; e 1 ; B + e † 2 ; e 2 ; A. (63) Applying e † 2 ; e 1 = 0, t his g iv es e † 2 ; e 2 ; B = e † 2 ; e 2 ; A. (64) T o deal with this w e ne ed to kno w the v alue of the scalar e † 2 ; e 2 . W e observ e that ∆ I = ( id I id I ) : I ✲ I ⊕ I is a cone, a nd so there exists some e ∆ I : I ✲ E satisfying e ∆ I ; e = ∆ I . Using the † -equalizer equation e ; e † = id E w e obtain e ∆ I = ∆ I ; e † = e † 1 + e † 2 . P ostcomp osing with e 2 giv es the equation e ∆ I ; e 2 = 1 = e † 1 ; e 2 + e † 2 ; e 2 . (65) Applying the assumption that e † 1 ; e 2 = 0, this giv es e † 2 ; e 2 = 1. Equation (6 4) then giv es B = A as needed, whic h completes the pro of. A t the cost o f a more long-winded pro of w e ha v e av oided using the † -functor explicitly here. W e are certainly relying on it indirectly , how ev er, as w e require that additio n in the semiring is cancellable; this w as pro ve d in Lemma 2.6, and it do es not seem that the use of the † -functor in that pr o of can b e av oided. F or an y non trivial comm utativ e ring R with cancellable m ultiplication, w e can obtain its quotient fiel d Q ( R ) in to whic h R em b eds. Elemen ts of Q ( R ) are equiv alence classe s of pairs ( s, t ) o f elemen ts o f R with t 6 = 0. W e write these pairs in the form s t , to resem ble a fraction. The equiv alence r elat io n is giv en b y s t ∼ u v iff sv = u t. (66) This is symmetric, transitiv e and reflexiv e, as required. W e rely on the cancellable m ultiplication to demonstrate transitivit y . Multiplication and addition are defined on represen tativ es o f the equiv alence class es a s if they w ere con ven tional fractions: s t · u v = su tv (67) s t + u v = sv + ut tv (68) 28 These op erations are we ll- defined on the equiv alence classes. F urthermore, the ring R em b eds in to Q ( R ) under the mapping r ✲ r 1 , and this is faithful since r 1 ∼ s 1 ⇒ r = s . It is straigh tfo r w ard to see that this em b edding preserv es m ultiplication and addition. W e require the comm utativ e ring R to b e nontrivial, satisfying 0 6 = 1 , since a field m ust satisfy this b y definition. This leads to the requiremen t tha t the monoidal categor y from whic h w e obtain our scalars mus t b e nontrivial, ha ving more than one morphism. W e m ust require this explicitly , since the one-morphism category otherwise satisfies all of our conditions: it is a monoidal † -category with all finite † -limits, for whic h the monoidal unit ob ject ha s no prop er † -sub ob jects. Altogether, for a non t r ivial monoidal † -category with all finite † -limits, in whic h the monoidal unit is simple, w e hav e sho wn that the comm utative semiring S of scalars em b eds in to the comm utative difference ring D ( S ); that this ring has cancellable m ultiplicatio n; and that an y ring R with cancellable m ultiplication em b eds into its quotient field Q ( R ). It follo ws that the semiring S em b eds into Q ( D ( S )), and so the scalars in our monoidal category embed in to a field. Establishing the c haracteristic W e next sho w t ha t the semiring S of scalars ha s c hara cteristic 0. Since we ha v e sho wn that this semiring em b eds in to the field Q ( D ( S )), it fo llows that this field must also ha ve c haracteristic 0. Lemma 3.7. In a non trivial monoi d a l † -c ate gory w i th finite † -bipr o ducts and † -e qualizers, for which the mon o idal unit obje ct has no † -sub obj e cts, the sc alars have char acteristic 0. Pr o of. Supp ose that scalar additio n is not of c hara cteristic 0 . Then there exists some nonzero scalar a : I ✲ I , and p o sitiv e nat ura l num b er n , suc h that a + · · · + a = 0 (69) where the sum con tains n copies of a . This sum is equal to n · a , whe r e n : I ✲ I is a scalar giv en by ∆ n I ; ∇ n I , for ∆ n I the n -fold co diagona l of I and ∇ n I the n -fold diag onal. F ro m t he † -bipro duct prop erty it follows that ∇ n I = (∆ n I ) † b y Lem ma 2.3, and from the † -equalizer prop ert y it follows in turn that n = ∆ n I ; (∆ n I ) † 6 = 0 by Lemma 2.5. How ev er, b y Lemma 3.4, the pro duct of t w o nonzero scalars cannot b e zero. W e conclude tha t our o r iginal a ssumption w as wrong, and that scalar addition is of c har a cteristic 0. In v olution and ordering The action of the † -functor giv es the s calar s the structure of an involutive semiring, equipping it with an in v o lution tha t respects semiring addition a nd m ultiplication: we ha ve ( a + b ) † = a † + b † b y Lemma 2.3 , and ( ab ) † = a † b † b y functorialit y . An in v o lution is usually required to b e order-rev ersing for m ultiplication, whic h is satisfied in a natura l w ay since the † -functor is con trav aria n t, but w e can neglect this here as the scalars are comm utativ e. 29 The self-adjoin t scalars a r e those scalars satisfying a = a † . These self-adjoint sc a lars are closed under m ultiplication and addition, and so fo rm a subsemiring. It is easy to see that the field Q ( D ( S )) in to whic h the scalars S em b ed inherits the in volution, and so is an in v o lutiv e field. The self-adjoint elemen ts of Q ( D ( S )) also form a field, and the self-adjoint scalars em b ed in to this field. W e no w demonstrate that the self-adjoint scalars admit an or der . An order on a semiring is a reflexiv e total order on the underlying set, suc h that the follo wing conditions hold: a ≤ b ⇒ a + c ≤ b + c (70) 0 ≤ a, 0 ≤ b ⇒ 0 ≤ ab (71) W e will not work directly with these conditions. Instead, w e will tak e adv an tage of the fact that o ur s cala r s em b ed in to a field, and use the follo wing classical theorem on orders for fields (f or a pr o of, see [21, Theorem 3.3.3].) Theorem 3.8. A field admits an or der if and only if a finite sum of squar es of nonzer o elements is never zer o. W e will use this theorem to show that the self-adjoin t elemen ts of the field Q ( D ( S )) admit an order. It then follo ws straightforw ardly that the semiring of self-adjoint elemen ts of S admits an order, through its in v olutio n-preserving em b edding in to Q ( D ( S )). Ho w ev er, w e emphasize that there is no guaran tee that this order will b e unique, o r that there will b e a canonical choice of order. W e a ctually pro ve a more gene ral theorem, on sums of s q uar e d norms of elemen ts o f Q ( D ( S )) . Definition 3.9. F or a field with in volution a ✲ a † , the squar e d no rm of a is aa † . Lemma 3.10. L et S b e the semiring of sc alars in a nontrivial monoidal † -c ate gory with simple tensor unit, and with al l fi nite † -limits. Then a finite sum o f squar e d norms of nonzer o elements of the field Q ( D ( S )) is nev e r zer o. Pr o of. W e m ust sho w that, giv en an y finite sum satisfying a 1 a 1 † + a 2 a 2 † + · · · + a N a N † = 0 (72) where each a i is a n elemen t of Q ( D ( S )) , eac h a i is actually zero. By construction, each a i is a formal quotien t b i /c i of some pair of elemen ts b i , c i in D ( S ). W r it ing the sum in terms of thes e quotien ts, and m ultiplying through b y eac h denominator, w e obtain another sum in the form of ( 7 2) in whic h eac h term is a squared no r m of an eleme nt of Q ( D ( S )) with trivial denominator; in other w ords, an elemen t of D ( S ). W riting these elemen ts as formal ordered pairs d i − e i , where d i , e i are elemen ts of S , w e obtain the sum ( d 1 − e 1 )( d 1 − e 1 ) † + ( d 2 − e 2 )( d 2 − e 2 ) † + · · · + ( d N − e N )( d N − e N ) † = 0 . (73) W e define the morphism d : I ✲ I ⊕ N to be the column v ector with comp onents ( d 1 , d 2 , . . . , d N ), and the mo r phism e : I ✲ I ⊕ N to b e the column v ector with comp onen t s 30 ( e 1 , e 2 , . . . , e N ). By matrix m ultiplicatio n, w e see that equation (73) is precisely equiv a len t to the equation d ; d † + e ; e † = d ; e † + e ; d † . (74) W e can now a pply the exc hange lemma 2.9 to conclude that d = e , and so e i = d i for all i . It fo llo ws that eac h of the orig ina l a i = d i − e i denom w as zero, and that the sum of squared norms w as in fact a sum o f zeros, which prov es the lemma. F rom this lemma w e see that a finite sum o f squares of nonzero self-adjo int elemen ts of the field Q ( D ( S )) is nonzero. So b y Theorem 3.8 the self-adjoint elemen ts of Q ( D ( S )) admit an ordering, and in general they will admit man y differen t orderings. By extension, the self-adjoin t elemen ts of the scalar semiring S also admit an ordering, since they em b ed in to the self-adjo int elemen ts of Q ( D ( S )). This concludes the pro of of the main theorem. 4 Completin g the scalars W e hav e sho wn that, in a monoidal † -category with all finite † -limits that satisfies the conditions of the previous section, the scalars share many prop erties with the complex n um b ers. In particular, the self-adjoin t scalars will admit an o rder, just as the real n um b ers do. The order on the real n umbers is a sp ecial one: in pa rticular, it is De dekind-c omplete , whic h fo r a t o tal order means that ev ery subset with an upper b ound has a least upper b ound, and ev ery subset with a low er b o und has an greatest lo w er b o und. The real num b ers are also a field, and the field structure in teracts w ell with the Dedekind- completeness prop ert y of t he underlying total order: if X is a set of elemen ts of R with upper b ound W ( X ), then w e hav e W ( X + r ) = W ( X ) + r , w here r is a real n um b er a nd X + r denotes the set { x + r | x ∈ X } , and similarly V ( X + r ) = V ( X ) + r . If a totally-ordered semiring has a Dedekind-complete underlying tota lly-ordered set, and has an addition op erat io n satisfying these extra compatibilit y conditions, then w e call it a De dekin d -c omplete semiring . In this section, w e will show tha t this Dedekind-completeness prop ert y is the extra abstract property required to c haracterize the complex n um b ers. T o w ork tow ards this, w e first prov e a useful lemma. Lemma 4.1. Supp os e a c omm utative semiring c ontains the p ositive r ational numb ers and is ad d i tively c anc el lab le, multiplic atively c anc el lable, total ly-or der e d and De dekind-c omplete. Then it has the fol low i n g pr op e rties: 1. ( Means.) F or any p air of eleme nts a < b we c an c onstruct their ‘me an ’ as 1 2 ( a + b ) , which satisfies a < 1 2 ( a + b ) < b . 2. ( Partial subtraction.) F or any p air of p ositive elements a an d b with a < b , ther e exists an element c with c + a = b . 3. ( No p ositiv e infinitesim a ls.) F or any p ositive element a , ther e exi s ts a n a tur a l numb er n such that an > 1 . 4. ( No p ositiv e infinite elemen ts.) F or any p ositive elem e nt a , ther e exists a natur al numb er n such that a < n . 31 5. ( D ense p ositive rationals.) F or a ny two une qual p ositive elements, ther e is a r ational numb er b etwe en them. 6. ( R eal n um b ers.) The semiring is isom orphic to either the semiring R ≥ 0 of nonne gative r e al numb ers, or the field R of al l r e al numb ers. Pr o of. W e prov e these prop erties sequen tially , at times using low er-n um b ered prop erties to aid the pro of of higher-num b ered ones. Throughout, let L b e a comm utat iv e se miring satisfying the h yp o theses of the lemma, and let a, b ∈ L b e v ariables v alued in the semiring. 1. (Means.) Since a < b w e ha ve a + a = 2 a < a + b , and multiply ing b y the fraction 1 2 , w e obtain a < 1 2 ( a + b ). Similarly , w e can also sho w that 1 2 ( a + b ) < b . 2. (P artial subtraction.) F or an y pair of elemen ts a, b satisfying 0 < a < b , consider the follo wing sets: J = { x ∈ L , x + a > b } (75) K = { x ∈ L, x + a < b } (76) The set J has a lo we r b ound 0 and the set K has a n upp er b ound b , so the gr eat est lo we r b ound V ( J ) and greatest upp er b ound W ( K ) b oth exis t b y Dedekind-completeness . If W ( J ) + a = b or V ( K ) + a = b then w e hav e disco v ered c and w e ar e done, so supp ose that neither hold. Supp ose that V ( J ) + a < b : then V ( J + a ) < b b y the preserv ation of infima b y addition, but this is not possible, since b w ould then serv e a s a greater low er b ound. Similarly , w e can rule out W ( K ) + a > b . The only remaining situation is that in whic h W ( K ) + a < b < V ( J ) + a , from whic h it follo ws b y additiv e cancellabilit y that W ( K ) < V ( J ). Construct the mean o f W ( K ) and V ( J ) as m := 1 2 ( W ( K ) + V ( J )); then b y prop erty 1, W ( K ) < m < V ( J ) . (7 7) Consider the v alue of m + a . Supp ose tha t m + a < b ; then m ∈ K a nd s o m ≤ W ( K ), but this contradicts equation (77). Similarly , supp ose that m + a > b ; then m ∈ J and so m ≥ V ( K ), and this again leads to a con tradiction. The only remaining p o ssibilit y is tha t m + a = b , and so w e are done. 3. (No infinitesimals.) Cons ider the set I = { x ∈ L, x > 0 , ∀ n ∈ N nx < 1 } , (78) the elemen ts of whic h w e call the infinitesimals. Supp ose the set I is not empt y; since the elemen t 1 serv es as an upp er b ound, the suprem um W ( I ) mus t therefore exist, and will satisfy W ( I ) > 0 since it is certainly greater than eac h p ositiv e infinitesimal. Su pp ose W ( I ) is not itself an infinitesimal; then there exists some m ∈ N with m W ( I ) > 1, and multiplying b y the rational n umber 1 m it follo ws that W ( I ) > 1 m . But then 1 m serv es as a lo w er upper b ound to the infinitesimals than W ( I ); this giv es a con tradiction, and so W ( I ) m ust b e an infinitesimal. Since W ( I ) > 0 it follo ws that 2 W ( I ) > W ( I ); t he quan tit y 2 W ( I ) is therefore not an infinitesimal, and there m ust exist some p ∈ N with 2 p W ( I ) > 1. But since 2 p is a natural n um b er, W ( I ) is not infinitesimal, and so w e hav e a contradiction. It follows that the set I is empt y . 32 4. (No p ositiv e infinite elemen ts.) This prop erty is pro v ed in a similar w a y to prop ert y 3. Define the set H = { x ∈ L, ∀ n ∈ N x > n } , (79) con taining the infinite e lements , and assume that it is not empt y . Clearly this s et has a p ositiv e lo w er b ound given by an y natural n umber, so b y Dedek ind- completeness it m ust ha ve a p ositive gr eatest upp er b ound V ( H ). Since 1 2 V ( H ) < V ( H ) it follo ws that 1 2 V ( H ) is not an infinite elemen t, and so there exists some n ∈ N with 1 2 V ( H ) < n ; from this we see that V ( H ) < 2 n , and so V ( H ) itself is not an infinite elemen t. But then 2 n is a greater lo wer b ound for the eleme nts of H , whic h con tradicts t he definition of V ( H ). The only remaining p ossibilit y is that the set H is empty . 5. (Dense p ositive rationa ls.) Let a, b ∈ L b e tw o unequal p o sitiv e elemen ts without a rational n umber b etw een them. Without loss of generalit y , assume a < b . By pro p ert y 2 there exists a p ositiv e elemen t c ∈ L with a + c = b , and by prop erty 3 there exists some natural n umber n ∈ L with nc > 1. It follows that nb = na + nc > na + 1. W rite p ∈ L for the smallest natura l n umber greater than na , whic h exists b y property 4; it satisfie s na + 1 > p > na . Then nb > na + 1 > p > na . Multiplying by the rational 1 n w e obtain b > p n > a , and w e ha ve pro ved the prop ert y . 6. (Real n umbers.) F or an y p ositive elemen t a , define the set Q + 0, and iden tif ying b 2 with a real num ber, w e can find a p ositiv e elemen t c ∈ L with c 2 = b 2 , and a p ositiv e elemen t 1 c ∈ L whic h is the recipro cal o f c . Then defining x = b c + 1, w e see that x 2 =  b c + 1  2 =  b 2 c 2 + 1 + 2 b c  = 2 + 2 b c = 2  1 + b c  = 2 x. (80) Supp ose that x 6 = 0 ; from the m ultiplicative cancellabilit y prop erty this implies that x = b c + 1 = 2, and therefore that b = c . But this is not p ossible, since b < 0 and c > 0. W e conclude that x = 0, and therefore that b c + 1 = 0 and b c = − 1. It follo ws that the semiring is in fact a ring, and that the negativ e elemen ts are in bij ection with the p ositiv e elemen ts under m ultiplication b y − 1. W e therefore obta in a n isomorphism b et w een the entire ring and the real n umbers R by the metho d describ ed in the previous para graph, a nd it is clear that our semiring is not only a ring, but a field. W e now com bine this lemma with Theorem 3.1 to prov e out main result, whic h demonstrates the existence of complex n umbers in a category based only its completenes s prop erties. Note that the statemen t of this theorem only mak es sense in the ligh t of Theorem 3.1, whic h guaran tees that the self-adjoint sc a la rs will admit a total order compatible with the sem ir ing structure. 33 Theorem 4.2. In a monoi d al † -c ate gory with simple tensor unit, which h a s al l finite † -limits, and for which the sel f - a djoint sc alars ar e De dekind -c omplete, the sc alars have an involution- pr eserving emb e dding in to the c omplex numb ers. Pr o of. W riting S for the se miring of sc a la rs, w e write L ⊆ S for the subsemiring of self- adjoin t scalars. This semiring is comm utative b y Lemma 3 .3 , contains the p ositive rational n um b ers by Lemma 2 .8, is additiv ely cancellable b y Lemma 2.6, is multiplicativ ely cancellable b y Lemma 3.5, admits a total ordering b y Th eorem 3.1, and in fact admits an addition- compatible Dedekind-complete ordering by h yp othesis. Lemma 4.1 therefore applies and L is either R ≥ 0 or R , the latter being the smallest field in to w hich L em b eds. It follo ws that Q ( D ( L )) = D ( L ) = R , where Q ( − ) and D ( − ) construct the smallest field con taining a particular ring and and smallest ring containing a particular semiring resp ectiv ely , in the manner describ ed in section 3. By Theorem 3.1 w e kno w that S h a s an in v o lution-preserving em b edding into the in v o lutiv e field Q ( D ( S )), and it fo llows immediately that the subsemiring L has an em b edding into F ⊆ Q ( D ( S )), the subfield consisting of t he self-adjoint elemen ts. In f a ct, this em b edding is surjectiv e, as we now sho w. Consider some elemen t r = a − b ∈ D ( S ) where a, b ∈ S ; if r is self-adjoint, then this implies a + b † = b + a † . But since r = ( a + a † ) − ( a † + b ) w e see that r can be expre ssed as the difference of elemen ts of L , and so the self-adjoin t subring of D ( S ) is precisely D ( L ). Now consider an elemen t s ∈ F ⊆ Q ( D ( S )), so s = c/d as a formal ratio of elemen ts c, d ∈ D ( S ). If s is self-adjoin t then c † /d † = c/d as formal ratios, whic h means that c † d = cd † in D ( S ). But then w e can write c/d = cd † /dd † , demonstrating that c/d is in fact a ratio of self-a djoin t elemen ts of D ( S ), whic h are precisely elemen ts of D ( L ). W e therefore se e that, as subsets, Q ( D ( L )) = F ⊆ Q ( D ( S )). In particular, since Q ( D ( L )) = R w e hav e F = R , a nd w e will use this identific a t io n freely in the rest of the pro of. W e will demonstrate an inv olution-preserving embedding of Q ( D ( S )) into t he complex n um b ers. Since L is either R ≥ 0 or R , then Q ( D ( L )) = D ( L ) = R . Supp o se that the in v o lution on t he scalars is trivial; t hen L = S , a nd Q ( D ( S )) = Q ( D ( L )) = R ⊂ C , so the theorem holds. Otherwis e, let x ∈ Q ( D ( S )) b e an eleme nt of our field suc h that x † 6 = x ; then y := x − x † is a nonzero elemen t satisfying y † = − y , and y † y ∈ F is a nonzero real n umber. Supp ose t ha t y † y < 0; then − y † y is a po sitiv e real n umber with a p ositive ro ot r ∈ F satisfying r † r + y † y = 0. But b y Lemma 3.10 this cannot b e the case, and w e conclude that y † y > 0. Let s ∈ F b e the positive ro o t of y † y satisfying s 2 = y † y , and define j = y /s . Then j † = y † /s † = − y /s = − j and j 2 = y 2 /s 2 = − y † y / s 2 = − 1, and j satisfies the prop erties that w e exp ect of i ∈ C . With this in mind, for all eleme nts z ∈ Q ( D ( S )) w e define Re( z ), Im( z ) ∈ F by Re( z ) = 1 2 ( z + z † ) , (81) Im( z ) = 1 2 j ( z − z † ) . (82) These a re the unique elemen ts of F suc h that z = Re( z ) + j Im( z ). F rom t his decomp osition w e obtain an obv ious field homomorphism σ : Q ( D ( S )) ✲ C giv en b y σ ( z ) = R e( z ) + i Im( z ), where i ∈ C is a square ro ot of − 1, and where w e are using the iden tification of F with the real n um b ers. Th is homomorphism is clearly injectiv e, and it is surjec tiv e since an y elemen t 34 k ∈ C is equal to σ ( ℜ ( k ) + j ℑ ( k )), so it is a field isomorphism. Since the semiring S has an in v o lution-preserving em b edding in to Q ( D ( S )), it therefore also has an in volution-preserv ing em b edding in to C , with in v o lution giv en by complex conjugation. Finally w e will show tha t if the inv olution on the scalars is non trivial, then the scalar semiring is actually isomorphic to C , with inv olution giv en b y complex conjugation. W e ha v e demonstrated the existence of a n inv olution-preserving em b edding of S into C , and in what follo ws w e will use this em b edding freely . Since we know that S at least con tains R ≥ 0 , w e only nee d to sho w that it also con ta ins i , since it will then con tain the entire comple x plane. Supp ose some nonzero elemen t a ∈ S has ℜ ( a ) = 0 ; then a = ir for some r ∈ R ⊂ C . W rite r + ∈ R ≥ 0 for the p ositiv e ro ot of r 2 ; then since R ≥ 0 ⊂ S , w e hav e 1 /r + ∈ R ≥ 0 ⊂ S . It f ollo ws that a (1 /r + ) = ± i ∈ S , and so either this quan tit y or its adjoin t is i ∈ S . So, if w e can sho w the existence of a nonzero elemen t of S with zero r eal part, our result will follo w. W e kno w that there exists some b ∈ S with b 6 = b † . Supp ose ℜ ( b ) = 0; then w e are done. Supp ose instead that ℜ ( b ) < 0; then defining c := −ℜ ( b ) ∈ R ≥ 0 ⊂ S w e see that ℜ ( b + c ) = 0, so w e are do ne. F inally , supp ose that ℜ ( b ) > 0; then f r om a simple consideration of the geometry of t he complex plane, it is straightforw ard to see that there exists some natura l n um b er n with b n ∈ S suc h that ℜ ( b n ) < 0 , but we just demonstrated t ha t the existence of suc h an elemen t implies i ∈ S . W e conclude that whatev er the v alue of ℜ ( b ) w e hav e i ∈ S , and so the in v olutive semiring S can b e iden tified with the field C , with in volution giv en b y complex conjugation. In particular, the scalars can b e iden tified with either R ≥ 0 or R with trivial in v olution, or C with complex conjugation as in v olutio n. 5 Categoric al descriptio n of inner pro duc ts In this section, w e will see ho w † -limits can b e used to define the † - functor o n the category FdHilb of finite-dimensional Hilb ert spaces. Since kno wing the † -functor on t his categor y is equiv alen t to kno wing the inner pro ducts on all the ob jects, w e also obta in a new w ay to describe inner pro ducts. W e b egin with a useful tec hnical lemma. If C and D are † - categories and F : C ✲ D is a functor, then we sa y that F c ommutes with the † -functors if † ◦ F = F ◦ † , where the first † is on the category D and the second is on the catego ry C . Also, w e recall the definition of a unitarily essential ly surje ctive functor as a functor with ev ery o b ject in the co domain unitarily isomorphic to some ob j ect in the functor’s image, and unitary † -e quivalenc e as an equiv alence betw een t wo † -categories whic h comm utes with the † -functors, and for whic h the natural isomorphisms are unitary at ev ery stage. Lemma 5.1. Supp ose that that ther e is a functor b etwe en two † -c ate gories which is ful l, faithful, unitarily essential ly surje ctive, and c ommutes with the † -functors. Then it forms p art of a unitary † -e quivalenc e. Pr o of. W e pro v e this by extending the con v en tional ar gumen t [20, Theorem IV.4.1] that a full, faithful and essen tially surjectiv e functor forms part o f an equiv a lence. Suppo se that 35 a functor F : C ✲ D has the prop erties describ ed in the hy p othesis. Then for any ob j ect d ∈ D , w e can find a ob ject G 0 ( d ) ∈ C and an unitary morphism η d : d ✲ F ( G 0 ( d )). W e w ant t o promo t e the function G 0 : Ob( D ) ✲ Ob( C ) in to a functor G : D ✲ C , suc h that η b ecomes a nat ural transformation. The naturalit y square f or η lo oks like this: d f ✲ d ′ F G ( d ) η d ❄ F G ( f ) ✲ F G ( d ′ ) η d ′ ❄ (83) It follows that F G ( f ) = η † d ; f ; η d ′ , and since F is full and fa ithful, this uniquely defines G . Constructing this equation for the adjoin t o f f w e ha v e F G ( f † ) = η † d ′ ; f † ; η d , and taking the adjoin t of this e quatio n giv es F ( G ( f † ) † ) = η † d ; f ; η d ′ . It follows that F G ( f ) = F ( G ( f † ) † ), and since F is f ull and faithful G ( f † ) = G ( f ) † , so G comm utes with the † -functors. T o fully demonstrate the unitary † -equiv alence w e still need to construct a unitary natural transformation ǫ : GF ⇒ id C . W e define this by F ( ǫ c ) = η † F c ; since F is full and faithf ul, this defi nition is v alid. It is easy to sho w that these morphis ms are unitary and natura l, and in fa ct, the equiv alence is an adjoin t equiv alence. W e no w pro ve the main theorem of this section. Theorem 5.2. L et † : FdV ect ✲ FdV ect b e a † -functor on the monoidal c ate g ory of finite-dimensiona l c omplex ve ctor sp ac es. Then the fol lo wing pr o p erties ar e e quivalent: 1. e quipp e d with † , FdV ect has al l finite † -lim its and De dekind-c omplete self-adjo int sc alars; 2. ther e is a choic e of inner pr o duct on e ach obje ct of FdV ect such that the † -functor acts by taking adjoints with r esp e ct to these inner pr o ducts; 3. ther e is a unitary † -e quivalenc e b etwe en FdV ect with its sp e cifie d † -functor, an d FdHilb with its c anonic al † -functor. Pr o of. W e b egin with the implication 1 ⇒ 2. The complex num b ers are presen t in V ect as endomorphisms of the one-dimensional v ector space, and the † -functor gives it an in v olution; w e denote this inv olutiv e field b y ( C , † ). This could b e differen t to ( C , ∗ ), the comple x n um b ers equipp ed with complex conjugatio n as inv olution. Ho w eve r , b y Theorem 4.2, there m ust b e an inv olution-preserving field isomorphism χ : ( C , † ) ✲ ( C , ∗ ). Since χ preserv es the in v olution w e ha v e χ † = ∗ χ , and since χ is in v ertible, w e see that the in volution induced b y the † -functor is c onjugate to complex conjugation. F or ev ery ob ject A in V ect w e define a putativ e inner pro duct for all φ, ψ : C ✲ A as h φ, ψ i := χ ( ψ ; φ † ) : C ✲ C . W e m ust sho w tha t this satisfies the a xioms of an inner pro duct. W e first establish t hat ∗ ( h φ , ψ i ) = h ψ , φ i , b y observing that ∗ ( h φ, ψ i ) = ∗ χ ( ψ ; φ † ) = χ † ( ψ ; φ † ) = χ ( φ ; ψ † ) = h ψ , φ i . No w, supp ose that some v ector φ : C ✲ A has negativ e norm under this inner pro duct; without loss of g enerality w e assume that it is normalized, 36 so that φ satisfies h φ, φ i = − 1. No w consider the column v ector ( 1 φ ) : C ✲ C ⊕ A ; this will hav e a norm o f zero, which is ruled out b y the † -equalizer prop ert y as established b y Lemma 2.5. W e conclude that h φ, φ i > 0 for a ll nonzero φ . Linearity of the inner pro duct follo ws straightforw ardly from the pr o p erties of † -bipro ducts. Altogether, the construction h φ, ψ i := χ ( ψ ; φ † ) is linear in the second argument, conjugate-symmetric and p ositive - definite, and hence is a g enuine inner pro duct. It is then trivial that for all f : A ✲ B , φ A : C ✲ A and φ B : C ✲ B , w e hav e h φ B , ( φ A ; f ) i = h ( φ B ; f † ) , φ A i , and so the † -functor tak es linear maps to their adjoin ts and w e hav e prov ed the implication. F or the implication 2 ⇒ 3, the c hoice of functor is o b vious: ev ery o b ject of FdV ect has an assigned inner pro duct, and since a finite-dimensional complex v ector space with inner pro duct is nece ssarily a Hilb ert space, w e ha v e a functor in to FdHilb . This functor is full, faithful and essen tially surjectiv e, since ev ery Hilb ert space is define d up to isomorphism b y its cardinalit y and there will b e Hilb ert spaces of ev ery finite cardinality in the image of the functor. Finally , it is clear that the inclusion is compatible with the action of the † -functor, and since tw o isomorphic Hilb ert spaces alwa ys ha v e a unitary isomorphism b et w een them, w e ha v e a unitary † - equiv alence by Lemma 5.1. Finally , w e consider the implication 3 ⇒ 1. Let F : FdHilb ✲ FdV ect b e a functor forming part of the unitary † -equiv alenc e; then it giv es rise to an in volution-preserving field homomorphism F : ( C , ∗ ) ✲ ( C , † ), where ∗ is the complex conjugation op eration a nd † represen ts the action of t he † -functor on the scalars of FdV ect . Since F also giv es rise to a field isomorphism betw een the self-adjoint elemen ts o f b oth fields, and since the self-adjoint elemen ts of ( C , ∗ ) a re Dedekind-complete under the unique order on the re al num b ers, it follo ws that the self-adjoin t elemen ts of ( C , † ) also admit a unique order, whic h is Dedekind- complete. The implication is completed with the s t raigh tfo r ward fa ct that, j ust as limits are preserv ed b y equiv a lences, † -limits are preserv ed b y unitary † -equiv alences. A similar theorem would hold for the category of all complex v ector spaces, but we would then b e dealing with inner-pro duct space s rather than Hilb ert spaces. 6 T ec hnical dis c ussion Our results giv e a n abstract c haracterization of the pro p erties endo w ed b y the complex n um b ers on a phys ical theory . More impo r tan tly , this abstract c haracterization — f ormalized b y T heorem 4.2 — admits a relatively clear phy sical in terpretation. The most imp ortan t structure is the requiremen t of ha ving all finite † -limits, a t yp e of completen ess property whic h can be in terpreted as the ability to tak e the direct sum of separate ph ysical systems, mo dulo the action of pro cesses, in a w ay whic h preserv es norms. Another crucial structure is D edekind completeness, which is the requiremen t that, in the totally-o r dered set of self- adjoin t scalars, ev ery b ounded set has a least upp er b ound and a greatest lo wer bo und, and that thes e b ounds g et along with addition of scalars. In con ven tional quan tum ph ysics these self-adjoin t scalars represen t the results of measuremen ts, a nd Dedekind completenes s is a prop ert y that w e observ e experimen tally . The final prop erty is that the theory has a simple tensor unit; ph ysically , this means t hat there exists a ‘trivial system’ that b eha v es in sensible w ay , suc h that the o nly smaller sys tem is the empt y system. 37 The most arguable ph ysical property is p erhaps that of Dedekind completeness. Ev en without this Theorem 3 .1 still a pplies, telling us tha t the theory is built o n an in v olutive field of c haracteristic 0, with an orderable fixed field whic h is not the real num b ers. It is in teresting to consider the ro le play ed b y the † -functor in these results, whic h represen ts our abilit y to turn an y pro cess f : A ✲ B in to a pro cess f † : B ✲ A . Tw o imp ortan t lemmas, the cancellable addition lemma 2.6 and the exc hange lemma 2.9, seem to rely crucially on the † -functor. It seems that the p o we r of the † - functor lies in its abilit y to add a n extra degree of symmetry to a sy stem of equations. F o r example, in the pro of of Lemma 2.6, the role of the † -functor is t o pro v e e † 1 ; e 2 = 0 from the known equation e † 2 ; e 1 = 0. The underlying † -equalizer diagram do es not ha ve a symmetry ex changing e 1 and e 2 , but the existence of the † -functor forces the existence of suc h a sym metry , prov ing the theorem. This contrasts with the pro of of Lemma 3.6, for whic h the diagr a m do es hav e a symmetry exc hanging e 1 and e 2 , and the † -functor is not directly required for the pro of. It is possible to consider v arian ts of † -bipro ducts and † -equalizers that do not rely on the † -functor — such as bipro ducts, and equalizers that ha v e retractions — but it do es not seem that these would b e p o werful enough to prov e analog ous results . W e hop e that these results, and others that rely crucially on properties of the † -functor (suc h as [9, 11, 24]), will stim ulate in terest in t he † -functor as a fundamental construction, b oth in category theory and in the foundations of quantum theory . References [1] Sa mson Abra msky and Bob Co ec ke. A categorical seman tics of quan tum proto cols. Pr o c e e dings of the 19th A nnual IEEE Symp osium on L o gic in Computer Scienc e , pag es 415–425, 2 0 04. IEE E Computer Science Press . [2] Sa mson Abramsky and Bob Co ec k e. Abstract ph ysical traces. The ory and Applic ations of Cate gories , 14(6):1 11–124, 2005. [3] Sa mson Abramsk y and Bo b Co ec ke . Handb o ok of Quantum L o gic and Q uan tum Structur e s , v olume 2, chapter Categorical Quan tum Mec hanics. Elsevier, 2008. [4] D iederik Aerts. Comp endium of Quantum Physics , c hapter Op erational Quan tum Mec hanics, Quan tum Axiomatics a nd Qua ntum Str uctures, pages 434– 440. Springer, 2008. [5] Jo hn C. Baez. Higher-dimensional algebra I I: 2- Hilb ert spaces. A dvanc es in Mathematics , 127 :1 25–189, 1997. [6] Stephen P . Brum b y and Girish C. Joshi. Exp erimen tal status of quaternionic quan tum mec hanics. Chaos, Sol i tons & F r actals , 7(5):7 4 7–752, 1996 . [7] R ob Clifton, Jeffrey Bub, and Hans Halv orson. Chara cterizing quantum theory in terms of informat io n-theoretic constrain ts. F oundations of Physics , 33 :1561–1591 , 2003. 38 [8] Bo b Co eck e. In tro ducing categories to the practicing ph ysicist. In What is Cate go ry The ory? , pages 45–74. P olimetrica Publishing, 2006. [9] Bo b Co ec ke , Dusk o P a vlovic , and Ja mie Vicary . Comm utativ e dagger-Frob enius algebras in FdHilb are orthogonal bases. (RR - 08-03), 2008. T ec hnical Rep ort. [10] Giacomo D ’Ariano. Probabilistic t heories: What is special ab out quan tum mec hanics? In A. Bokulic h and G. Jaeger, editors, Philoso phy o f Quantum Inf o rmation and Entanglement . Cam bridge Univ ersity Press, 2010. T o app ear. [11] Sergio Doplic her and John E. Rob erts. A new dualit y theory for compact groups. Inventiones Mathematic ae , 98:157 –218, 1989. [12] Andreas D¨ oring and Chris Isham. New Structur es in Physics , c hapter ‘What is a Thing?’: T o p os Theory in the F o undations of Ph ysics. 2008. [13] Hans Halvorson and Mic hael M ¨ uger. Handb o ok of the Philosophy of Physic s , c hapter Algebraic quantum field theory . North Holland, 2006 . [14] Lucien Hardy . Quantum theory f rom five reasonable axioms. Unpublis hed, 2001. [15] Chris Heunen. An em b edding theorem for Hilbert categories. The ory and Applic ations of Cate gories , 22(13) :321–344, 2009. [16] La wrence P . Horo witz. Hypercomplex quan tum mec hanics. F oundations of Physic s , 26(6):851– 862, 1996. [17] Gregory M. Kelly and Miguel L. Laplaza. Coherence for compact closed categories. Journal of Pur e and Applie d A l g ebr a , 19:193–2 1 3, 1980. [18] Stefano De Leo and Khaled Ab del-Khalek. Octonionic quan tum mec hanics and complex geometry . Pr o gr ess of The or etic al Physics , 96(4):823 –831, 1996. [19] F elix Lev. Wh y is quantum ph ysics based on complex num b ers? Finite Fields and their Applic ations , 12:336–35 6, 2006. [20] Saunders Mac Lane. Cate gories for the Working Mathematician . Springer, 19 9 7. 2nd edition. [21] Dav id Mark er. Mo del The ory: An Intr o duction . Springer, 2002. [22] Colin McLarty . Elementary Cate gories, Elem entary T op oses . Oxfo r d Univ ersity Press, 1995. [23] P eter Selinger. Idemp oten ts in dag g er categories. In Pr o c e e dings of the 4th International Workshop on Quantum Pr o gr amming L a nguages , July 2006. [24] Jamie Vicary . Categorical form ulation of finite-dimensional quan tum algebras. Communic ations in Mathematic al Physics , 2008. T o app ear. [25] Ap ostol V ourdas. G alois quan tum systems. Journal of Physics A , 38 :8453–8471 , 2005. 39