TRACES IN MONOIDAL CATEGORIES

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIET Volume 364, Number 8, Augus 2012, Pages S (2012) Aricle elecronically published on March 29, 2012 TRACES IN MONOIDAL CATEGORIES STEPHAN STOL AND PETER TEICHNER Absrac. This paper conains he consrucion, examples and properies of a race and a race pairing for cerain morphisms in a monoidal caegory wih swiching isomorphisms. Our consrucion of he caegorical race is a common generalizaion of he race for endomorphisms of dualizable objecs in a balanced monoidal caegory and he race of nuclear operaors on a opological vecor space wih he approximaion propery. In a forhcoming paper, applicaions o he pariion funcion of super-symmeric field heories will be given. Conens 1. Inroducion Moivaion via field heories Thickened morphisms and heir races Thickened morphisms The race of a hickened morphism Traces in various caegories The caegory of vecor spaces Thick morphisms wih semi-dualizable domain Thick morphisms wih dualizable domain The caegory Ban of Banach spaces A caegory of opological vecor spaces The Riemannian bordism caegory Properies of he race pairing The symmery propery of he race pairing Addiiviy of he race pairing Braided and balanced monoidal caegories Muliplicaiviy of he race pairing 4461 References Inroducion The resuls of his paper provide an essenial sep in our proof ha he pariion funcion of a Euclidean field heory of dimension 2 1 is an inegral modular funcion [ST1]. While moivaed by field heory, he wo main resuls are he consrucion Received by he ediors Ocober 21, 2010 and, in revised form, April 29, Mahemaics Subjec Classificaion. Primary 18D10; Secondary 46A32, 81T99. Boh auhors were parially suppored by NSF grans. They would like o hank he referee for many valuable suggesions. The firs auhor visied he second auhor a he Max-Planck-Insiu in Bonn during he Fall of 2009 and in July He would like o hank he insiue for is suppor and for is simulaing amosphere. c 2012 American Mahemaical Sociey Revers o public domain 28 years from publicaion 4425

2 4426 STEPHAN STOL AND PETER TEICHNER of races and race pairings for cerain morphisms in a monoidal caegory. Le C be a monoidal caegory wih monoidal uni I C [McL]. Quesion. Wha condiions on an endomorphism f C(, ) allow us o consruc a well-defined race r(f) C(I,I) wih he usual properies expeced of a race? Theorem 1.7 below provides an answer o his quesion. Our consrucion is a common generalizaion of he following wo well-known classical cases: (1) If C is a dualizable objec (see Definiion 4.17), hen every endomorphism f has a well-defined race [JSV, Proposiion 3.1]. (2) If f : is a nuclear operaor (see Definiions 4.25 and 4.28) on a opological vecor space, henf has a well-defined race provided has he approximaion propery, i.e., he ideniy operaor on can be approximaed by finie rank operaors in he compac open opology [Li]. Le TV be he caegory of opological vecor spaces (more precisely, hese are assumed o be locally convex, complee, Hausdorff), equipped wih he monoidal srucure given by he projecive ensor produc (see Secion 4.5). Then an objec TV is dualizable if and only if is finie dimensional, whereas every Hilber space has he approximaion propery. Hence exending he race from endomorphisms of dualizable objecs of C o more general objecs is analogous o exending he noion of race from endomorphisms of finie-dimensional vecor spaces o cerain infinie-dimensional opological vecor spaces. In fac, our answer will involve analogues of he noions nuclear and approximaion propery for general monoidal caegories which we now describe. The following noion is our analogue of a nuclear morphism. Definiion 1.1. A morphism f : in a monoidal caegory C is hick if i can be facored in he form (1.2) I id id b I for morphisms : I, b: I. As explained in he nex secion, he erminology is moivaed by considering he bordism caegory. In he caegory Vec of vecor spaces, wih monoidal srucure given by he ensor produc, a morphism f : is hick if and only if i has finie rank (see Theorem 4.1). In he caegory TV a morphism is hick if and only if i is nuclear (see Theorem 4.27). If f : is a hick endomorphism wih a facorizaion as above, we aemp o define is caegorical race r(f) C(I,I) o be he composiion (1.3) I s, b I. This caegorical race depends on he choice of a naural family of isomorphisms s {s, : } for, C. We do no assume ha s saisfies he relaions (5.8) required for he braiding isomorphism of a braided monoidal caegory. Apparenly lacking an esablished name, we will refer o s as swiching isomorphisms. We would like o hank Mike Shulman for his suggesion. For he monoidal caegory Vec, equipped wih he sandard swiching isomorphism s, :, x y y x, he caegorical race of a finie rank

3 TRACES IN MONOIDAL CATEGORIES 4427 (i.e., hick) endomorphism f : agrees wih is classical race (see Theorem 4.1). More generally, if is a dualizable objec of a monoidal caegory C, heabove definiion agrees wih he classical definiion of he race in ha siuaion (Theorem 4.22). In general, he above race is no well defined, since i migh depend on he facorizaion of f given by he riple (, b, ) raher han jus he morphism f. As we will see in Secion 4.4, his happens, for example, in he caegory of Banach spaces. To undersand he problem wih defining r(f),leuswrie r(,, b) C(I,I) for he composiion (1.3) and Ψ(,, b) C(, ) for he composiion (1.2). There is an equivalence relaion on hese riples (see Definiion 3.3) such ha r(,, b) and Ψ(,, b) depend only on he equivalence class [,, b]. In oher words, here are well-defined maps r: Ĉ(, ) C(I,I), Ψ: Ĉ(, ) C(, ), where Ĉ(, ) denoes he equivalence classes of riples (,, b) forfixed, C. We noe ha by consrucion he image of Ψ consiss of he hick morphisms from o. We will call elemens of Ĉ(, ) hickened morphisms. If f Ĉ(, ) wih Ψ( f) f C(, ), we say ha f is a hickener of f. Using he noaion C k (, ) for he se of hick morphisms from o,i is clear ha here is a well-defined race map r: C k (, ) C(I,I)makinghe diagram (1.4) C k r (, ) C(I,I) Ψ r Ĉ(, ) commuaive if and only if has he following propery: Definiion 1.5. An objec in a monoidal caegory C wih swiching isomorphisms has he race propery if he map r is consan on he fibers of Ψ. For he caegory Ban of Banach spaces and coninuous maps, we will show in Secion 4.4 ha he map Ψ can be idenified wih he homomorphism (1.6) Φ: Ban(, ), w f (v wf(v)), where is he Banach space of coninuous linear maps f : C equipped wih he operaor norm and is he projecive ensor produc. Operaors in he image of Φ are referred o as nuclear operaors, and hence a morphism in Ban is hick if and only if i is nuclear. I is a classical resul ha he race propery for a Banach space is equivalen o he injeciviy of he map Φ which in urn is equivalen o he approximaion propery for : he ideniy operaor of can be approximaed by finie rank operaors in he compac-open opology, see e.g., [Ko, 43.2(7)]. Every Hilber space has he approximaion propery, bu deciding wheher a Banach space has his propery is surprisingly difficul. Grohendieck asked his quesion in he 1950s, bu he firs example of a Banach space wihou he approximaion propery was found by Enflo only in 1973 [En]. Building on Enflo s work, Szankowski showed in 1981 ha he Banach space of bounded operaors on an (infinie-dimensional) Hilber space does no have he approximaion propery [Sz].

4 4428 STEPHAN STOL AND PETER TEICHNER Theorem 1.7. Le C be a monoidal caegory wih swiching isomorphisms, i.e., C comes equipped wih a family of naural isomorphisms s, :. If C is an objec wih he race propery, hen he above caegorical race r(f) C(I,I) is well defined for any hick endomorphism f :. This compares o he wo classical siuaions menioned above as follows: (i) If is a dualizable objec, hen has he race propery, and any endomorphism f of is hick. Moreover, he caegorical race of f agrees wih is classical race. (ii) In he caegory TV of opological vecor spaces (locally convex, complee, Hausdorff), a morphism is hick if and only if i is nuclear, and he approximaion propery of an objec TV implies he race propery. Moreover, if f : is a nuclear endomorphism of an objec wih he approximaion propery, hen he caegorical race of f agrees wih is classical race. The firs par sums up our discussion above. Saemens (i) and (ii) appear below as Theorems 4.22 and 4.27, respecively. I would be ineresing o find an objec in TV which has he race propery bu no he approximaion propery. To moivae our second main resul, Theorem 1.10, we noe ha a monoidal funcor F : C D preserves hick and hickened morphisms and gives commuaive diagrams for he map Ψ from (1.4). If F is compaible wih he swiching isomorphisms, hen i also commues wih r. However, he race propery is no funcorial in he sense ha if some objec C has he race propery, hen i is no necessarily inheried by F () (unless F is essenially surjecive and full or has some oher special propery). In paricular, when he funcor F is a field heory, hen, as explained in he nex secion, his non-funcorialiy causes a problem for calculaing he pariion funcion of F. We circumven his problem by replacing he race by a closely relaed race pairing (1.8) r: C k (, ) C k (,) C(I,I) for objecs, of a monoidal caegory C wih swiching isomorphisms. Unlike he race map r: C k (, ) C(I,I) discussed above, which is only defined if has he race propery, no condiion on or is needed o define his race pairing r(f,g) as follows. Le f Ĉ(, ), ĝ Ĉ(,) be hickeners of f and g, respecively (i.e., Ψ( f) f and Ψ(ĝ) g). We will show ha elemens of Ĉ(, ) can be pre-composed or pos-composed wih ordinary morphisms in C (see Lemma 3.9). This composiion gives elemens f g and f ĝ in Ĉ(, ) which we will show o be equal in Lemma Hence he race pairing defined by r(f,g) : r( f g) r(f ĝ) C(I,I) is independen ofhechoiceof f and ĝ. WenoehaΨ( f g) Ψ(f ĝ) f g C k (, ) and hence if has he race propery, hen (1.9) r(f,g) r(f g) for f C k (, ), g C k (,). In oher words, he race pairing r(f,g) is a generalizaion of he caegorical race of f g, defined in siuaions where his race migh no be well defined. The race pairing has he following properies ha are analogous o properies one expecs o hold for a race. We noe ha he relaionship (1.9) immediaely

5 TRACES IN MONOIDAL CATEGORIES 4429 implies hese properies for our race defined for a hick endomorphism of objecs saisfying he race propery. Theorem Le C be a monoidal caegory wih swiching isomorphisms. Then he race pairing (1.8) is funcorial and has he following properies: (1) r(f,g) r(g, f) for hick morphisms f C k (, ), g C k (,). If has he race propery, hen r(f,g) r(f g) and symmerically for. (2) If C is an addiive caegory wih disribuive monoidal srucure (see Definiion 5.3), hen he race pairing is a bilinear map. (3) r(f 1 f 2,g 1 g 2 )r(f 1,g 1 )r(f 2,g 2 ) for f i C k ( i, i ), g i C k ( i, i ), provided s gives C he srucure of a symmeric monoidal caegory. More generally, his propery holds if C is a balanced monoidal caegory. We recall ha a balanced monoidal caegory is a braided monoidal caegory equipped wih a naural family of isomorphisms θ {θ : } called wiss saisfying a compaibiliy condiion (see Definiion 5.12). Symmeric monoidal caegories are balanced monoidal caegories wih θ id. For a balanced monoidal caegory C wih braiding isomorphism c, : and wis θ :, one defines he swiching isomorphism s, : by s, : (id θ ) c,. There are ineresing examples of balanced monoidal caegories ha are no symmeric monoidal, e.g., caegories of bimodules over a fixed von Neumann algebra (monoidal srucure given by Connes fusion) or caegories of modules over quanum groups. Traces in he laer are used o produce polynomial invarians for knos. Originally, we only proved he muliplicaive propery of our race pairing for symmeric monoidal caegories. We are graeful o Gregor Masbaum for poining ou o us he classical definiion of he race of an endomorphism of a dualizable objec in a balanced monoidal caegory which involves using he wis (see [JSV]). The res of his paper is organized as follows. In Secion 2 we explain he moivaing example: we consider he d-dimensional Riemannian bordism caegory, explain wha a hick morphism in ha caegory is, and show ha he pariion funcion of a 2-dimensional Riemannian field heory can be expressed as he relaive race of he hick operaors ha a field heory associaes o annuli. Secion 2 is moivaional and can be skipped by a reader who wans o see he precise definiion of Ĉ(, ), he consrucion of r, and a saemen of he properies of r which are presened in Secion 3. In Secion 4 we discuss hick morphisms and heir races in various caegories. In Secion 5 we prove he properies of r and deduce he corresponding properies of he race pairing saed as Theorem 1.10 above. 2. Moivaion via field heories A well-known axiomaizaion of field heory is due o Graeme Segal [Se] who defines a field heory as a monoidal funcor from a bordism caegory o he caegory TV of opological vecor spaces. The precise definiion of he bordism caegory depends on he ype of field heory considered: for a d-dimensional opological field heory, he objecs are closed (d 1)-dimensional manifolds and morphisms are d-dimensional bordisms (more precisely, equivalence classes of bordisms where we idenify bordisms if hey are diffeomorphic relaive boundary). Composiion is given by gluing of bordisms, and he monoidal srucure is given by disjoin union.

6 4430 STEPHAN STOL AND PETER TEICHNER For oher ypes of field heories, he manifolds consiuing he objecs and morphisms in he bordism caegory come equipped wih an appropriae geomeric srucure; e.g., a conformal srucure for conformal field heories, a Riemannian meric for Riemannian field heories, or a Euclidean srucure ( Riemannian meric wih vanishing curvaure ensor) for a Euclidean field heory. In hese cases more care is needed in he definiion of he bordism caegory o ensure he exisence of a well-defined composiion and he exisence of ideniy morphisms. Le us consider he Riemannian bordism caegory d-rbord. The objecs of d-rbord are closed Riemannian (d 1)-manifolds. A morphism from o is a d-dimensional Riemannian bordism Σ from o, ha is, a Riemannian d-manifold Σ wih boundary and an isomery Σ. More precisely, a morphism is an equivalence class of Riemannian bordisms, where wo bordisms Σ, Σ are considered equivalen if here is an isomery Σ Σ compaible wih he boundary idenificaions. In order o have a well-defined composion by gluing Riemannian bordisms, we require ha all merics are produc merics near he boundary. To ensure he exisence of ideniy morphisms, we enlarge he se of morphisms from o by also including all isomeries. Pre- or pos-composiion of a bordism wih an isomery is he given bordism wih boundary idenificaion modified by he isomery. In paricular, he ideniy isomery provides he ideniy morphism for as objec of he Riemannian bordism caegory d-rbord. A more sophisicaed way o deal wih he issues addressed above was developed in our paper [ST2]. There we do no require he merics on he bordisms o be a produc meric near he boundary; raher, we have more sophisicaed objecs consising of a closed (d 1)-manifold equipped wih a Riemannian collar. Also, i is echnically advanageous no o mix Riemannian bordisms and isomeries. This is achieved in ha paper by consrucing a suiable double caegory (or equivalenly, a caegory inernal o caegories), whose verical morphisms are isomeries and whose horizonal morphisms are bordisms beween closed (d 1)-manifolds equipped wih Riemannian collars. The 2-morphisms are isomeries of such bordisms, relaive boundary. When using he resuls of he curren paper in [ST1], we ranslae beween he approach here using caegories versus he approach via inernal caegories used in [ST2]. Le E be d-dimensional Riemannian field heory, ha is, a symmeric monoidal funcor E : d-rbord TV. For he bordism caegory d-rbord he symmeric monoidal srucure is given by disjoin union; for he caegory TV i is given by he projecive ensor produc. Le be a closed Riemannian (d 1)-manifold and Σ be a Riemannian bordism from o iself. Le Σ gl be he closed Riemannian manifold obained by gluing he wo boundary pieces (via he ideniy on ). Boh Σ and Σ gl are morphisms in d-rbord: Σ:, Σ gl :. We noe ha is he monoidal uni in d-rbord, and hence he vecor space E( ) can be idenified wih C, he monoidal uni in TV. In paricular, E(Σ gl ) Hom(E( ), E( )) Hom(C, C) C is a complex number. Quesion. How can we calculae E(Σ gl ) C in erms of he operaor E(Σ): E() E()?

7 TRACES IN MONOIDAL CATEGORIES 4431 We would like o say ha E(Σ gl )isherace of he operaor E(Σ), bu o do so we need o check ha he condiions guaraneeing a well-defined race are me. For a opological field heory E his is easy: in he opological bordism caegory, every objec is dualizable (see Definiion 4.17), hence E() is dualizable in TV which is equivalen o dim E() <. By conras, for a Euclidean field heory he vecor space E() is ypically infinie dimensional, and hence o make sense of he race of he operaor E(Σ) associaed o a bordism Σ from o iself, we need o check ha he operaor E(Σ) is hick and ha he vecor space E() hashe race propery. I is easy o prove (see Theorem 4.38) ha every objec of he bordism caegory d-rbord has he race propery and ha among he morphisms of d-rbord (consising of Riemannian bordisms and isomeries), exacly he bordisms are hick. The laer characerizaion moivaed he adjecive hick, since we hink of isomeries as infiniely hin Riemannian bordisms. I is sraighforward o check ha being hick is a funcorial propery in he sense ha he hickness of Σ implies ha E(Σ) is hick. Unforunaely, as already menioned in he inroducion, he race propery is no funcorial, and we canno conclude ha E() hasherace propery. Replacing he problemaical race by he well-behaved race pairing leads o he following resul. I is applied in [ST1] o prove he modulariy and inegraliy of he pariion funcion of a super-symmeric Euclidean field heory of dimension 2. Theorem 2.1. Suppose Σ 1 is a Riemannian bordism of dimension d from o,andσ 2 is a Riemannian bordism from o. LeΣΣ 1 Σ 2 be he bordism from o iself obained by composing he bordisms Σ 1 and Σ 2,andleΣ gl be he closed Riemannian d-manifold obained from Σ by idenifying he wo copies of ha make up is boundary. If E is d-dimensional Riemannian field heory, hen E(Σ gl )r(e(σ 2 ),E(Σ 1 )). Proof. By Theorem 4.38 he bordisms Σ 1 : and Σ 2 : are hick morphisms in d-rbord, and hence he morphism r(σ 1, Σ 2 ): is defined. Moreover, every objec d-rbord has he race propery (see Theorem 4.38) and hence r(σ 1, Σ 2 ) r(σ 2 Σ 1 ) r(σ). In par (3) of Theorem 4.38 we will show ha r(σ) Σ gl. Then funcorialiy of he consrucion of he race pairing implies E(Σ gl )E(r(Σ 1, Σ 2 )) r(e(σ 1 ),E(Σ 2 )). 3. Thickened morphisms and heir races In his secion we will define he hickened morphisms Ĉ(, )andherace r( f) C(I,I) of hickened endomorphisms f Ĉ(, ) for a monoidal caegory C equipped wih a naural family of isomorphisms s, :. We recall ha a monoidal caegory is a caegory C equipped wih a funcor : C C C

8 4432 STEPHAN STOL AND PETER TEICHNER called he ensor produc, a disinguished elemen I C, and naural isomorphisms α,, :( ) ( ) l : I r : I (associaor), (lef uni consrain), (righ uni consrain) for objecs,, C. These naural isomorphisms are required o make wo diagrams (known as he associaiviy penagon and he riangle for uni) commuaive; see [McL]. I is common o use diagrams o represen morphisms in C (see for example [JS1]). The picures U V W f g g g g represen a morphism f : U V W and he ensor produc of morphisms g : and g :, respecively. The composiion h g of morphisms g : and h: is represened by he following picure. h g g h Wih ensor producs being represened by he juxaposiion of picures, he isomorphisms I I sugges deleing edges labeled by he monoidal uni I from our picure; e.g., he picures (3.1) b represen morphisms : I and b: I, respecively. Rephrasing Definiion 1.1 of he inroducion in our picorial noaion, a morphism f :

9 TRACES IN MONOIDAL CATEGORIES 4433 in C is hick if i can be facored in he following form. f (3.2) b Here sands for op and b for boom. We will use he noaion C k (, ) C(, ) for he subse of hick morphisms Thickened morphisms. I will be convenien for us o characerize he hick morphisms as he image of a map Ψ: Ĉ(, ) C(, ), he domain of which we refer o as hickened morphisms. Definiion 3.3. Given objecs, C, ahickened morphism from o is an equivalence class of riples (,, b) consising of an objec C, and morphisms : I, b: I. To describe he equivalence relaion, i is useful o hink of hese riples as objecs of a caegory and o define a morphism from (,, b) o(,,b ) o be a morphism g C(, ) such ha he following hold. g and b g (3.4) b Two riples (,, b) and(,,b )areequivalen if here are riples ( i, i,b i )for i 1,...,n wih (,, b) ( 1, 1,b 1 )and(,,b )( n, n,b n ) and morphisms g i beween ( i, i,b i )and( i+1, i+1,b i+1 ) (his means ha g i is eiher a morphism from ( i, i,b i )o( i+1, i+1,b i+1 )orfrom( i+1, i+1,b i+1 )o( i, i,b i )). In oher words, a hickened morphism is a pah componen of he caegory defined above. We wrie Ĉ(, ) for he hickened morphisms from o. As suggesed by he referee, i will be useful o regard Ĉ(, )asacoend; his will sreamline he proofs of some resuls. Le us consider he funcor S : C op C Se given by (,) C(I, ) C(, I). Then he elemens of C S(, ) are riples (,, b) wih C(I, ) and b C(, I). Any morphism g : induces maps C(,) S(g,id ) C(, ) and C(,) S(id,g) C(, ).

10 4434 STEPHAN STOL AND PETER TEICHNER We noe ha for any (, b ) S(,), he wo riples (, S(g, id )(, b )) (,, b g) and (,S(id,g)(, b )) (,g, b ) represen he same elemen in Ĉ(, ). In fac, by consrucion Ĉ(, )ishe coequalizer S(g,id ) coequalizer S(,) S(id S(, ),g), C g C(, ) i.e., he quoien space of C S(, ) obained by idenifying all image poins of hese wo maps. This coequalizer can be formed for any funcor S : C op C Se; i is called he coend of S, and following [McL, Ch. I, 6], we will use he inegral noaion C S(, ) for he coend. Summarizing our discussion, we have he following way of expressing Ĉ(, )asacoend: C (3.5) Ĉ(, ) C(I, ) C(, I). Lemma 3.6. Given a riple (,, b) as above, le Ψ(,, b) C(, ) be he composiion on he righ-hand side of equaion (3.2). Then Ψ only depends on he equivalence class [,, b] of (,, b), i.e., he following map is well defined. (3.7) Ψ: Ĉ(, ) C(, ) [,, b] Ψ(,, b) ev We noe ha by consrucion he image of Ψ is equal o C k (, ), he se of hick morphisms from o. As menioned in he inroducion, for f C(, ) we call any f [,, b] Ĉ(, ) wih Ψ( f) f a hickener of f. Remark 3.8. The difference beween an orienable versus an oriened manifold is ha he former is a propery, whereas he laer is an addiional srucure on he manifold. In a similar vein, being hick is a propery of a morphism f C(, ), whereas a hickener f is an addiional srucure. To make he analogy beween hese siuaions perfec, we were emped o inroduce he words hickenable or hickable ino mahemaical English. However, we finally decided agains i, paricularly because he hick-hin disincion for morphisms in he bordism caegory is jus perfecly suied for he purpose.

11 TRACES IN MONOIDAL CATEGORIES 4435 Proof. Suppose ha g : is an equivalence from (,, b) o(,,b )in he sense of Definiion 3.3. Then he following diagram shows ha Ψ(,,b ) Ψ(,, b). b g b b Thickened morphisms can be pre-composed or pos-composed wih ordinary morphisms o obain again hickened morphisms as follows: Ĉ(,W) C(, ) C(,W) Ĉ(, ) Ĉ(, W), ([,, b],f) [,, b (id f)], Ĉ(, W), (f,[,, b]) [, (f id ), b]. The proof of he following resul is sraighforward and we leave i o he reader. Lemma 3.9. The composiion of morphisms wih hickened morphisms is well defined and compaible wih he usual composiion via he map Ψ in he sense ha he following diagrams commue: Ĉ(,W) C(, ) Ĉ(, W) C(,W) Ĉ(, ) Ĉ(, W) Ψ id C(,W) C(, ) Ψ C(, W), id Ψ C(,W) C(, ) Ψ C(, W). Corollary If f : is a hick morphism in a monoidal caegory C, hen he composiions f g and h f are hick for any morphisms g : W, h:. Lemma Le f 1 Ĉ(, ), f 2 Ĉ(U, ), andf i Ψ( f i ). Then f 1 f 2 f 1 f 2 Ĉ(U, ). Proof. Le f i [ i, i,b i ]. Then f 1 f 2 [ 1, 1,b 1 (id 1 f 2 )] and 1 U 1 U 1 U b 1 (id 1 f 2 ) f g 2 b 1 b 1 b 2 b 2

12 4436 STEPHAN STOL AND PETER TEICHNER where 1 2 g b 1 2. Similarly, f 1 f 2 [ 2, (f 1 id 2 ) 2,b 2 ], and f (f 1 id 2 ) 2 b g This shows ha g is an equivalence from ( 1, 1,b 1 (id 1 f 2 )) o ( 2, (f 1 id 2 ) 2,b 2 ) and hence hese riples represen he same elemen of Ĉ(U, ) as claimed The race of a hickened morphism. Our nex goal is o show ha for a monoidal caegory C wih swiching isomorphism s, :, hemap (3.12) r: Ĉ(, ) C(I,I) [,, b] r(,, b) is well defined. We recall from equaion (1.3) of he inroducion ha r(,, b) is defined by r(,, b) b s,. In our picorial noaion, we wrie i as (3.13) r(,, b) b where is shorhand for s,. We noe ha he nauraliy of s, is expressed picorially as g h h g (3.14) for morphisms g : 1 2, h: 1 2.

13 TRACES IN MONOIDAL CATEGORIES 4437 Lemma r(,, b) depends only on he equivalence class of (,, b). In paricular, he map (3.12) is well defined. Proof. Suppose ha g : is an equivalence from (,,b )o(,, b) (see Definiion 3.3). Then g r(,,b ) r(,, b). g b b b b In erms of he coend descripion, his lemma is acually obvious because r is he composiion Ĉ(, ) C C(I, ) C(, I) C C(I, ) C(, I) C(I,I), where he firs map is given by swiching and he second by composiion. 4. Traces in various caegories The goal in his secion is o describe hick morphisms in various caegories in classical erms and relae our caegorical race o classical noions of race. A a more echnical level, we describe he map Ψ: Ĉ(, ) C(, ) and is image C k (, ) in hese caegories. In he firs subsecion his is done for he caegory of vecor spaces (no necessarily finie dimensional). In he second subsecion we show ha Ψ is a bijecion if is dualizable and hence any endomorphism of a dualizable objec has a well-defined race. Traces in caegories for which all objecs are dualizable are well sudied [JS2]. In he hird subsecion we inroduce semi-dualizable objecs and describe he map Ψ in more explici erms. In a closed monoidal caegory, every objec is semi-dualizable. In he following subsecion we apply hese consideraions o he caegory of Banach spaces. Then we discuss he caegory of opological vecor spaces, and finally he Riemannian bordism caegory The caegory of vecor spaces. Theorem 4.1. Le C be he monoidal caegory Vec F of vecor spaces (no necessarily finie dimensional) over a field F equipped wih he usual ensor produc and he usual swiching isomorphism s, :, x y y x. (1) A morphism f : is hick if and only if i has finie rank.

14 4438 STEPHAN STOL AND PETER TEICHNER (2) The map Ψ: Ĉ(, ) C(, ) is injecive; in paricular, every finie rank endomorphism f : has a well-defined caegorical race r(f) C(I,I) F. (3) For a finie rank endomorphism f, is caegorical race r(f) agrees wih is usual (classical) race which we denoe by cl-r(f). For a vecor space, le be he dual vecor space, and define he morphism (4.2) Φ: C(F, ) C(, ) by sending : F o he composiion F id id ev F. We noe ha he se C(F, ) can be idenified wih via he evaluaion map (1). Using his idenificaion, Φ maps an elemenary ensor y g o he linear map x yg(x) which is a rank 1 operaor. Since finie rank operaors are finie sums of rank one operaors, we see ha he image Φ consiss of he finie rank operaors. The classical race of he rank one operaor Φ(y g) is defined o be g(y); by lineariy his exends o a well-defined race for all finie rank operaors. The proof of Theorem 4.1 is based on he following lemma which shows ha Ψ is equivalen o he map Φ. Lemma 4.3. For any vecor spaces, he map (4.4) α: C(F, ) Ĉ(, ), [,,ev] is a bijecion and Ψ α Φ. Here ev : F is he evaluaion map g x g(x). In subsecion 4.2, we will consruc he map Φ and prove his lemma in he more general conex where is a semi-dualizable objec of a monoidal caegory. Proof of Theorem 4.1. Saemen (1) follows immediaely from he lemma. To prove par (2), i suffices o show ha Φ is injecive which is well known and elemenary. For he proof of par (3) le f : be a hick morphism. We recall ha is caegorical race r(f) is defined by r(f) r( f) for a hickener f Ĉ(, ). So Ψ( f) f and his is well defined hanks o he injeciviy of Ψ. Using ha α is a bijecion, here is a unique C(F, ) wih α() [,,ev] f. Wenoe ha r(α()) r([,,ev]) C(F, F) F is he composiion F s, ev F. In paricular, if (1) y g,hen 1 y g g y g(y) and hence r(f) g(y) cl-r(f) is he classical race of he rank one operaor Φ() f given by x yg(x). Since boh he caegorical race of α() andhe classical race of Φ() depend linearly on, his finishes he proof. Remark 4.5. We can replace he monoidal caegory of vecor spaces by he monoidal caegory SVec of super vecor spaces. The objecs of his caegory are jus /2- graded vecor spaces wih he usual ensor produc. The grading involuion on is he ensor produc ɛ ɛ of he grading involuions on and,

15 TRACES IN MONOIDAL CATEGORIES 4439 respecively. The swiching isomorphism is given by x y ( 1) x y y x for homogeneous elemens x, y of degree x, y /2. Then he saemens above and heir proof work for SVec as well, excep ha he caegorical race of a finie rank endomorphism T : is is super race sr(t ) : cl-r(ɛ T ) Thick morphisms wih semi-dualizable domain. The goal of his subsecion is o generalize Lemma 4.11 from he caegory of vecor spaces o general monoidal caegories C, provided he objec C saisfies he following condiion. Definiion 4.6. An objec of a monoidal caegory C is semi-dualizable if he funcor C op Se, C(, I) is represenable, i.e., if here is an objec C and naural bijecions (4.7) C(, ) C(, I). By oneda s Lemma, he objec is unique up o isomorphism. I is usually referred o as he (lef) inernal hom and denoed by C(, I). To pu his definiion in conex, we recall ha a monoidal caegory C is closed if for any C he funcor C C, has a righ adjoin; i.e., if here is a funcor C(, ): C C and naural bijecions (4.8) C(, C(, )) C(, ),,, C. In paricular, in a closed monoidal caegory, every objec is semi-dualizable wih C(, I). The caegory C Vec F is an example of a closed monoidal caegory; for vecor spaces, he inernal hom C(, )ishevecorspace of linear maps from o. Hence every vecor space is semi-dualizable wih semi-dual C(, I). Oher examples of closed monoidal caegories are he caegory of Banach spaces (see subsecion 4.4) and he caegory of bornological vecor spaces [Me]. As also discussed in [Me, p. 9], he symmeric monoidal caegory TV of opological vecor spaces wih he projecive ensor produc is no an example of a closed monoidal caegory (no maer which opology on he space of coninuous linear maps is used). Sill, some opological vecor spaces are semi-dualizable (e.g., Banach spaces), and so i seems preferable o sae our resuls for semi-dualizable objecs raher han objecs in closed monoidal caegories. In Lemma 4.11 we will generalize a saemen abou he caegory Vec F o a saemen abou a general monoidal caegory C. To do so, we need o consruc he maps Φ and α in he conex of a general monoidal caegory. This is sraighforward by using he same definiions as above, jus making he following replacemens: (1) Replace F, he monoidal uni in Vec F, by he monoidal uni I C. (2) Replace, he vecor space dual o, by, he semi-dual of C. Here we need o assume ha C is semi-dualizable which is auomaic for any objec of a closed caegory such as Vec F. (3) For a semi-dualizable objec C, heevaluaion map (4.9) ev: I

16 4440 STEPHAN STOL AND PETER TEICHNER is by definiion he morphism ha corresponds o he ideniy morphism under he bijecion (4.7) for. I is easy o check ha his agrees wih he usual evaluaion map F for C Vec F. The map Φ: C(I, ) C(, ) is given by he following picure. ev (4.10) The following lemma is a generalizaion of Lemma 4.3. Lemma Le be a semi-dualizable objec of a monoidal caegory C. Then for any objec C he map (4.12) α: C(I, ) Ĉ(, ), [,,ev] is a naural bijecion which makes he diagram C(I, ) Ĉ(, α ) Φ Ψ C(, ) commuaive. In paricular, a morphism f : is hick if and only if i is in he image of he map Φ (see equaion (4.10)). Proof. The commuaiviy of he diagram is clear by comparing he definiions of he maps Φ (see Equaion (4.10)), α (Equaion (4.12)), and Ψ (Equaion (3.7)). To see ha α is a bijecion, we facor i in he form (4.13) C(I, ) C C(I, ) C(, ) C C(I, ) C(, I). Here he firs map sends C(I, )o[,,id ]. This map is a bijecion by he coend form of oneda s Lemma, according o which for any funcor F : C Se he map (4.14) F (W ) C F () C(, W) [W,, id W ] is a bijecion (see [Ke, Equaion 3.72]. The second map of equaion (4.13) is induced by he bijecion C(, ) C(, I). By consrucion he ideniy map id corresponds o he evaluaion map via his bijecion, and hence he composiion of hese wo maps is α.

17 TRACES IN MONOIDAL CATEGORIES 4441 Remark The referee observed ha here is a nea inerpreaion of Ĉ(, ) in erms of he oneda embedding C F : Fun(C op, Se), C(,). The monoidal srucure on C induces a monoidal srucure on F, heconvoluion ensor produc [Day], defined by (M N)(S) : V,W C(S, V W ) M(V ) N(W ) for an objec S C. Equipped wih he convoluion produc, he funcor caegory F is a closed monoidal caegory (see equaion (4.8)) wih inernal lef hom F(N,L) F given by F(N,L)(S) F(N( ),L(S )). We can regard C as a full monoidal subcaegory of F via he oneda embedding. In paricular, every objec C has a lef semi-dual F(, ) F and hence by he previous lemma we have a bijecion F(, ) F(I, ). Explicily, he semi-dual is given by (S) F(C(,), C(S,I)) C(S, I). The referee observed ha he oneda embedding induces a bijecion (4.16) Ĉ(, ) F(, ) F(I, ). In paricular, morphism f C(, ) is hick if and only if is image under he oneda embedding is hick. To see ha he above map is a bijecion, we evaluae he righ-hand side F(I, )( )(I) V,W C(I,V W ) C(V, ) (W ) W W C(I, W ) (W ) C(I, W ) C(W, I), which we recognize as he coend descripion of Ĉ(, ) (equaion (3.5)). The second equaliy is a consequence of (he coend form of) he oneda lemma (equaion (4.14)) Thick morphisms wih dualizable domain. Asmenionedinheinroducion, here is a well-known race for endomophisms of dualizable objecs in a monoidal caegory (see, e.g., [JSV, 3]). Afer recalling he definiion of dualizable and he consrucion of ha classical race, we show in Theorem 4.22 ha our caegorical race is a generalizaion. Definiion 4.17 ([JS2, Def. 7.1]). An objec of a monoidal caegory C is (lef) dualizable if here is an objec C (called he (lef) dual of ) and morphisms ev: I (called evaluaion map) and coev: I (coevaluaion

18 4442 STEPHAN STOL AND PETER TEICHNER map) such ha he following equaions hold. (4.18) coev ev id ev coev id If is dualizable wih dual, hen here is a family of bijecions (4.19) C(, ) C(, ), naural in, C, givenby f f ev wih inverse coev g g In paricular, a dualizable objec is semi-dualizable in he sense of Definiion 4.6. Example A finie-dimensional vecor space is a dualizable objec in he caegory Vec F.Weake o be he vecor space dual o, ev o be he usual evaluaion map, and define coev: F by 1 i e i e i, where {e i } is a basis of and {e i } is he dual basis of.iisnohardoshow ha a vecor space is dualizable if and only if i is finie dimensional. Definiion Le C be a monoidal caegory wih swiching isomorphisms. Le f : be an endomorphism of a dualizable objec C. Then he classical race cl-r(f) C(I,I) is defined by he following. coev cl-r(f) : f ev

19 TRACES IN MONOIDAL CATEGORIES 4443 This definiion can be found for example in secion 3 of [JSV] for balanced monoidal caegories (see Definiion 5.12 for he definiion of a balanced monoidal caegory and how he swiching isomorphism is deermined by he braiding and he wis of he balanced monoidal caegory). In fac, he consrucion in [JSV] is more general: hey associae o a morphism f : A B araceinc(a, B) if is dualizable; specializing o A B I gives he classical race described above. Theorem Le C be a monoidal caegory wih swiching isomorphisms, and le be a dualizable objec of C. Then (1) The map Ψ: Ĉ(, ) C(, ) is a bijecion. In paricular, any morphism wih domain is hick, and any endomorphism f : has a well-defined caegorical race r(f) C(I,I). (2) The caegorical race of f is equal o is classical race cl-r(f) defined above. Remark Par (1) of he heorem implies in paricular ha if is (lef) dualizable, hen he ideniy id is hick. The referee observed ha he converse holds as well. To see his, assume ha id is hick. Then id is in he image of Ψ: Ĉ(, ) C(, ) and hence by Lemma 4.11 in he image of Φ: C(I, ) C(, ). If coev C(I, ) belongs o he pre-image Φ 1 (id ), hen i is sraighforward o check ha equaions (4.10) hold and hence is dualizable. The firs equaion holds by consrucion of Φ; o check he second equaion, we apply he bijecion (4.7) (for ) o boh sides and obain ev for boh. Proof. To prove par (1), i suffices by Lemma 4.11 o show ha he map Φ: C(I, ) C(, ) (see equaion (4.10)) is a bijecion. Comparing Φ wih he naural bijecion (4.19) for dualizable objecs C, we see ha his bijecion is equal o Φ in he special case I. To prove par (2), we recall ha by definiion r(f) r( f) C(I,I) for any f Ψ 1 (f) Ĉ(, ) (see equaion (1.4)). In he siuaion a hand, Ψ is inverible by par (1), and using he facorizaion Ψ Φ α 1 provided by Lemma 4.11, we have f Ψ 1 (f) αφ 1 (f) α((f id ) coev) [, (f id ) coev, ev]. Here he second equaliy follows from he explici form of he inverse of Φ (which agrees wih he map (4.19) for I). Comparing r( f) (equaion (3.13)) and cl-r(f) (Definiion 4.21), we see r(f) r(, (f id ) coev, ev) cl-r(f) The caegory Ban of Banach spaces. Le be a Banach space and f : a coninuous linear map. There are classical condiions (f is nuclear and has he approximaion propery, see Definiion 4.25) which guaranee ha f has a welldefined (classical) race, which we again denoe by cl-r(f) C. For example, any Hilber space H has he approximaion propery and a coninuous linear map f : H H is nuclear if and only if i is race class. The main resul of his subsecion is Theorem 4.26 which shows ha hese classical condiions imply ha f has a well-defined caegorical race and ha he caegorical race of f agrees wih is classical race. Before saing his resul, we review he (projecive) ensor produc of Banach spaces and define he noions nuclear and approximaion propery.

20 4444 STEPHAN STOL AND PETER TEICHNER The caegory Ban of Banach spaces is a monoidal caegory whose monoidal srucure is given by he projecive ensor produc, defined as follows. For Banach spaces,,heprojecive norm on he algebraic ensor produc alg is given by z : inf{ x i y i z x i y i }, where he infimum is aken over all ways of expressing z alg as a finie sum of elemenary ensors. Then he projecive ensor produc is defined o be he compleion of alg wih respec o he projecive norm. I is well known ha Ban is a closed monoidal caegory (see equaion (4.8)). For Banach spaces, he inernal hom space Ban(, ) is he Banach space of coninuous linear maps T : equipped wih he operaor norm T : sup x, x 1 T (x). In paricular, every Banach space has a lef semi-dual Ban(, I) in he sense of Definiion 4.17 which is jus he Banach space of coninuous linear maps C. The caegorically defined evaluaion map ev: C (see equaion (4.9)) agrees wih he usual evaluaion map defined by f x f(x). The map Φ Ban(C, ) Ban(, ) (see equaion (4.10)) is deermined by sending y f o he map x yf(x) (see he discussion afer Theorem 4.1). We noe ha he morphism se Ban(,) is in bijecive correspondence o he coninuous bilinear maps. This bijecion is given by sending g : o he composiion g χ, whereχ: is given by (x, y) x y. In paricular, if is he Banach space dual o, wehavea morphism (4.24) ev: C deermined by g x g(x) called he evaluaion map. Definiion 4.25 ([Sch, Ch. III, 7]). A coninuous linear map beween Banach spaces is nuclear if f is in he image of he map Φ: Ban(, ) deermined by y g (x yg(x)). A Banach space has he approximaion propery if he ideniy of can be approximaed by finie rank operaors wih respec o he compac-open opology. Theorem Le, Ban. (1) A morphism f : is hick if and only if i is nuclear. (2) If has he approximaion propery, i has he race propery. (3) If has he approximaion propery and f : is nuclear, hen he caegorical race of f agrees wih is classical race. Proof. This resul holds more generally in he caegory TV which we will prove in he following subsecion (see Theorem 4.27). For he proofs of saemens (2) and (3) we refer o ha secion. For he proof of saemen (1) we recall ha a morphism f : in he caegory C Ban is hick if and only if i is in he image of he map Ψ: Ĉ(, ) C(, ) and ha i is nuclear if i is in he image of Φ: C(I, ) C(, ). Hence par (1) is a corollary of Lemma 4.11, according o which hese wo maps are equivalen for any semi-dualizable

21 TRACES IN MONOIDAL CATEGORIES 4445 objec of a monoidal caegory C. In paricular, since Ban is a closed monoidal caegory, every Banach space is semi-dualizable A caegory of opological vecor spaces. In his subsecion we exend Theorem 4.26 from Banach spaces o he caegory TV whose objecs are locally convex opological vecor spaces which are Hausdorff and complee. We recall ha he opology on a vecor space is required o be invarian under ranslaions and dilaions. In paricular, i deermines a uniform srucure on which in urn allows us o speak of Cauchy nes and hence compleeness; see [Sch, secion I.1] for deails. The morphisms of TV are coninuous linear maps, and he projecive ensor produc described below gives TV he srucure of a symmeric monoidal caegory. I conains he caegory Ban of Banach spaces as a full subcaegory. Theorem Le, be objecs in he caegory TV. (1) A morphism f : is hick if and only if i is nuclear. (2) If has he approximaion propery, hen i has he race propery. (3) If has he approximaion propery, and f : is nuclear, hen he caegorical race of f agrees wih is classical race. Before proving his heorem, we define nuclear morphisms in TV and he approximaion propery. Then we will recall he classical race of a nuclear endomorphism of a opological vecor space wih he approximaion propery as well as he projecive ensor produc. Definiion A coninuous linear map f TV(, )isnuclear if i facors in he form (4.29) p f 0 j 0 0, where f 0 is a nuclear map beween Banach spaces (see Definiion 4.25) and p, j are coninuous linear maps. The definiion of nucleariy in Schaefer s book ([Sch, p. 98]) is phrased differenly. We give his more echnical definiion a he end of his secion and show ha a coninuous linear map is nuclear in his sense if and only if i is nuclear in he sense of he above definiion. Approximaion propery. An objec TV has he approximaion propery if he ideniy of is in he closure of he subspace of finie rank operaors wih respec o he compac-open opology [Sch, Chaper III, secion 9]. (Our compleeness assumpion for opological vecor spaces implies ha uniform convergence on compac subses is he same as uniform convergence on pre-compac subses.) The classical race for nuclear endomorphisms. Le f be a nuclear endomorphism of TV, and le I ν : be a ne of finie rank morphisms which converges o he ideniy on in he compac-open opology. Then f I ν is a finie rank operaor which has a classical race cl-r(f I ν ). I can be proved ha he limi lim ν cl-r(f I ν ) exiss [Li, Proof of Theorem 1] and is independen of he choice of he ne I ν (his also follows from our proof of Theorem 4.27). The classical race of f is defined by cl-r(f) : lim cl-r(f I ν ). ν

22 4446 STEPHAN STOL AND PETER TEICHNER Projecive ensor produc. The projecive ensor produc of Banach spaces defined in he previous secion exends o opological vecor spaces as follows. For, TV he projecive opology on he algebraic ensor produc alg is he fines locally convex opology such ha he canonical bilinear map χ: alg, (x, y) x y, is coninuous; see [Sch, p. 93]. The projecive ensor produc ishe opological vecor space obained as he compleion of alg wih respec o he projecive opology. I can be shown ha i is locally convex and Hausdorff and ha he morphisms TV(,) are in bijecive correspondence o coninuous bilinear maps ; his bijecion is given by sending f : o f χ; see [Sch, Chaper III, secion 6.2]. Semi-norms. For checking he convergence of a sequence or coninuiy of a map beween locally convex opological vecor spaces, i is convenien o work wih seminorms. For TV and U a convex circled 0-neighborhood (U is circled if λu U for every λ C wih λ 1), one ges a semi-norm (4.30) x U : inf{λ R >0 x λu} on. Conversely, a collecion of semi-norms deermines a opology, namely he coarses locally convex opology such ha he given semi-norms are coninuous maps. For example, if is a Banach space wih norm, we obain he usual opology on. As anoher example, he projecive opology on he algebraic ensor produc alg is he opology deermined by he family of semi-norms U,V paramerized by convex circled 0-neighborhoods U, V defined by z U,V : inf{ n x i U y i V z i1 n x i y i }, where he infimum is aken over all ways of wriing z alg as a finie sum of elemenary ensors. I follows from his descripion ha he projecive ensor produc defined above is compaible wih he projecive ensor produc of Banach spaces defined earlier (see [Sch, Chaper III, secion 6.3]). Our nex goal is he proof of Theorem 4.27, for which we will use he following lemma. i1 Lemma (1) Any morphism : C in TV facors in he form 0 0 j where 0 is a Banach space.

23 TRACES IN MONOIDAL CATEGORIES 4447 (2) Any morphism b: C facors in he form p 0 b b 0 where 0 is a Banach space. For he proof of his lemma, we will need he following wo ways of consrucing Banach spaces from a opological vecor space : (1) Le U be a convex, circled neighborhood of 0. Le U be he Banach space obained from by quoiening ou he null space of he semi-norm U and by compleing he resuling normed vecor space. Le p U : U be he eviden map. (2) Le B be a convex, circled bounded subse of. We recall ha B is bounded if for each neighborhood U of 0 here is some λ C such ha B λu. Le B be he vecor space B : n1 nb equipped wih he norm x B : inf{λ R >0 x λb}. If B is closed in, hen B is complee (by our assumpion ha is complee), and hence B is a Banach space (see [Sch, Ch. III, 7; p. 97]). The inclusion map j B : B is coninuous hanks o he assumpion ha B is bounded. Proof of Lemma To prove par (1) we use he fac (see, e.g., Theorem 6.4 in Chaper III of [Sch]) ha any elemen of he compleed projecive ensor produc, in paricular he elemen (1), can be wrien in he form (4.32) (1) λ i y i z i wih y i 0, z i 0, λ i <. i1 Le B : {y i i 1, 2...} {0}, and le B be he closure of he convex, circled hull of B (he convex circled hull of B is he inersecion of all convex circled subses of W conaining B ). We noe ha B is bounded, hence is convex, circled hull is bounded, and hence B is bounded. We define j : 0 o be he map j B : B. To finish he proof of par (1), i suffices o show ha (1) is in he image of he inclusion map j B id : B. Iisclearhaeachparialsum n i1 λ iy i z i belongs o he algebraic ensor produc B alg, and hence we need o show ha he sequence of parial sums is a Cauchy sequence wih respec o he semi-norms B,V on B alg ha define he projecive opology (here V runs hrough all convex circled 0-neighborhoods V ). Since y i B, i follows y i B 1, and hence we have he esimae n n n λ i y i z i B,V λ i y i B z i V λ i z i V. i1 i1 Since z i 0, we have z i V 1 for all bu finiely many i, and his implies ha he parial sums form a Cauchy sequence. To prove par (2) we recall ha he morphism b: C corresponds o a coninuous bilinear map b : C. The coninuiy of b implies ha here are i1

24 4448 STEPHAN STOL AND PETER TEICHNER convex, circled 0-neighborhoods V, U such ha b (z, x) < 1forz V, x U. I follows ha b (z, x) z V x U for all z, x. Hence b exends o a coninuous bilinear map b : U C for he compleion U of. This corresponds o he desired morphism b 0 : U C; he propery b b 0 (id p U ) is clear by consrucion. Defining he map p: 0 o be p U : U, we obained he desired facorizaion of b. Remark The proofs above and below imply ha, for fixed objecs, TV, he hickened morphisms TV(, ) acually form a se. Any riple (,, b) canbe facored ino Banach spaces 0, 0, 0 as explained in he hree picures below. By heargumenabove,wemayacuallychoose 0 U,whereU runs over cerain subses of. Since is fixed, i follows ha he arising Banach spaces U range over a cerain se. Finally, by Lemma 4.11, he Banach space 0 may be replaced by U wihou changing he equivalence class of he riple. Therefore, he given riple (,, b) isequivalenoaripleofheform( U,,b ). Since he collecion of objecs U TV forms a se, we see ha TV(, )isaseaswell. In his paper, we have no addressed he issue of wheher Ĉ(, )isasebecause his problem does no arise in he examples we discuss: he argumen above for C TV is he hardes one; in all oher examples we acually idenify Ĉ(, ) wih some very concree se. This problem is similar o he fac ha pre-sheaves on a given caegory do no always form a caegory (because naural ransformaions do no always form a se). So we are following he radiion of reaing his problem only if forced o. Proof of Theorem The facorizaion (4.29) shows ha a nuclear morphism f : facors hrough a nuclear map f 0 : 0 0 of Banach spaces. Then f 0 is hick by Theorem 4.26 and hence f is hick, since pre- or pos-composiion of a hick morphism wih an any morphism is hick. To prove he converse, assume ha f is hick, i.e., ha i can be facored in he form f (id b)( id ) : I b: I. Then using Lemma 4.31 o facorize and b, weseehaf can be furher facored in he following form. 0 p f 0 0 j b 0

25 TRACES IN MONOIDAL CATEGORIES 4449 This implies ha f has he desired facorizaion he following holds. 0 p 0 f j,where f 0 b 0 0 I remains o show ha f 0 is a nuclear map beween Banach spaces. In he caegory of Banach spaces a morphisms is nuclear if and only if i is hick by Theorem A firs glance, i seems ha he facorizaion of f 0 above shows ha f 0 is hick. However, on second hough one realizes ha we need o replace by a Banach space o make ha argumen. This can be done by using again our Lemma 4.31 o facorize 0 and hence f 0 furher in he following form. 0 0 f 0 0 j b 0 0 This shows ha f 0 is a hick morphism in he caegory Ban and hence nuclear by Theorem The key for he proof of pars (2) and (3) of Theorem 4.27 will be he following lemma. Lemma For any f TV(,), hemap (4.35) TV(, ) TV(I,I)C, g r( f g), is coninuous wih respec o he compac-open opology on TV(, ). To prove par (2) of Theorem 4.27, assume ha has he approximaion propery, and le I ν : be a ne of finie rank operaors converging o he ideniy of in he compac open opology. Then by he lemma, for any f TV(, ), he ne r(f Îν) r( f I ν ) converges o r( f id ) r( f). Here Îν TV(, ) are hickeners of I ν. They exis since every finie rank morphism is nuclear and hence hick by par (1). This implies he race condiion for, since r( f) lim ν r(f Îν) depends only on f.