Traces in symmetric monoidal categories

TRA CES IN SYMMETRIC MONOIDAL CA TEGORIES KA TE PONTO AND MICHAEL SHULMAN Contents Int ro duction 1 1. T races and fixed p oints 2 2. T races 5 3. Examples o f traces 8 4. Twisted traces and transfers 13 5. Prop erties of traces 17 6. F unctorialit y of traces 20 7. Vistas 23 References 23 Introduction The original notion of tra c e is, o f course, the trace of a s quare matrix with ent ries in a field. An important and far-reaching c a tegorical generalization of this notion applies to any endomorphism of a dualizable obje ct in a symmetric monoidal category ; see [ 11 , 26 ]. This generaliza tion has a num b er of applications which are o ften clos ely con- nected with the study of fixed p oin ts. One applica tion of particular imp ortance is the Le fschetz fixed p oin t theor e m and its v a rian ts and genera lizations, man y of which can be deduced direc tly from the natur alit y a nd functoriality of the canonical symmetric monoidal tra ce. The purp ose of this expo sitory note is to describ e this notion of trace in a sym- metric monoida l category , along with its imp ortant proper ties (including natur al- it y and functor ialit y), and to give as many examples a s p ossible. Among other things, this no te is intended as ba c k ground for [ 42 ] and [ 43 ], in which the s y m- metric monoidal trace is ge neralized to the context of bica tegories and indexed monoidal catego ries, and [ 38 – 41 ], whic h give applicatio ns o f the bicateg orical trace to fixe d p oin t theo r y . In § 1 we descr ibe o ne wa y to understand the connection b et ween tr aces and fixed p oint s. This provides motiv ation for the formal definitio ns in § 2 . In § 3 we give many examples o f the trace. These include top ological examples connected to the Le fschetz fixed p oin t theorem and its g eneralizations as well as examples arising in other con texts. In § 4 we define a gener a lization of the trac e from § 2 . This Date : V ersi on of October 25, 2013. Both authors were sup p orted b y National Science F oundation p ostd o cto ral fellowships during the writing of this pap er. 1 2 KA TE PONTO AND MICHAEL SHULMAN trace ar ises in many applications a nd is a generaliza tion of the cla ssical tra nsfer. Then in § 5 we describ e “co herence” prop erties of the tra ce, while in § 6 we descr ibe its functor ialit y a nd naturality , including the Lefschetz fixed po in t theo rem as an application. Finally , in § 7 we remark on some g eneralizations. 1. Trac es and fixed points A common feature of all the examples we will consider is that tr ac es give infor- mation ab out fixe d p oints . Th us, b efore embarking on formalities , in this section we will attempt to giv e some intuition for why this should b e so. Suppo se w e are working in a mono idal c a tegory , and consider a morphism whos e source and target ar e tensor pro ducts, s uch as f : A ⊗ B ⊗ C → D ⊗ B . W e think of s uc h an f as a “pro cess” which takes three inputs, of types A , B , and C , and pro duces t wo outputs, of types D and B . In keeping with this intuition, we draw f as follo ws: f B D C A B This is an exa mple of string diagr am notation for monoida l ca tegories, which is “Poincar´ e dual” to t he usual so rt of diagr ams: instea d of dra wing ob jects as vertic es and morphisms as arr ows co nnecting these vertices, we dra w ob jects as arr ows and morphisms as vertic es , often with boxes around them. See [ 22 , 24 , 3 7 , 45 ] for more ab out string diagram calculus. In pa rticular, we note that Joy al a nd Street [ 22 ] prov ed that the “v alue” of a string diag ram is inv aria n t under defor ma tions of dia- grams, so that we can prove th eor ems b y top ological r easoning; see P ropos itio n 2.4 for an example. If the sour ce a nd target of the morphism f ab ov e are the same, then a fix e d p oint of f is a mo rphism f † : ∗ → X (where ∗ denotes the unit for the monoida l structure) suc h that f X f † X ∗ = f † X ∗ W e will b e in terested in tra ces of mor e gener al morphisms. F or these we will need to b e able to duplicate inputs and outputs, which we dr a w as follows: A A A This is only possible if our monoidal categ ory is c artesian monoidal, in which case the abov e duplicatio n pro cess is the dia gonal ∆ : A → A × A . Now suppo se only one of the inputs of f matches its output: TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 3 f X A X An ( A -p ar ametrize d) fix e d p oint of f is a morphism f † : A → X such that f X A f † X A = f † X A i.e. “ f ( a, f † ( a )) = f † ( a ) for a n y a ∈ A ” or “ f † ( a ) ∈ X is a fixed p oin t of f ( a, − ) for an y a ∈ A .” A common wa y to lo ok for fixed points in concrete situations is by iter ation: we start with s ome x 0 ∈ X and compute x 1 = f ( a, x 0 ), x 2 = f ( a, x 1 ), and so on. If ever x n +1 = x n , we’v e found a fixed p oin t. But even if not, we c a n hope that the sequence ( x 0 , x 1 , x 2 , . . . ) will “ con verge” to a fixe d po in t. Two contexts where this works are the contraction mapping theor em in top ology and the least- fix ed-point combinator in domain semantics. In order to mimic this in abs tr act languag e, we need a notion of fe e db ack , i.e. a wa y to plug the o utput of a given morphism into its input. In diag rammatic terms, given a morphism one o f who se inputs matches one of its outputs: g B X A X we wan t to construct a new mo r phism in which the X input and output have b een “fed ba c k int o each o ther” somehow: g B A X This is called a tr ac e o f the morphism f . In fact, Hyland [ 4 ] a nd Hasegaw a [ 17 ] hav e indep enden tly observed the following (see § 5 ). Theorem 1.1. In a c artesian m o noidal c ate gory, to give a notion of tr ac e is pr e- cisely the same as to give a fixe d-p oint op er ator which assigns to every morphism A × X → X a fi xe d p oint A → X in a c oher ent way. 4 KA TE PONTO AND MICHAEL SHULMAN This relationship is especially imp ortant in computer science. How ever, in topol- ogy , we are int eres ted in maps which may hav e zero, one, or many fixed p oin ts. Thu s, we can’t exp ect to ha ve a fixed- p oint op erator acting o n the whole category , since there is no way to sp ecify a fixed p oin t for a map which has no fixed p oin ts. Instead, w e would like to kno w, given a map, do es it ha ve any fixed p oints, a nd if so, how many and what ar e they? Thus, we need an o peration which pro duces, in- stead of a fixed p oin t, some sort of “in v a r ian t” carr ying information a bout whatever fixed p oin ts a map may ha ve. A fruitful appro ac h to this is to map our cartes ian monoidal category C in to a la rger catego ry D in which mor phisms can be “sup er- impo sed” or “added.” Then we ma y hop e for a tra ce or a fixed-p oin t op erator in D which computes the “sum” of all the fixed points that a map ma y have (or “zero” if it has none). Often our functor Z : C → D will b e like the free ab elian group functor, and the “fixed po in t” of Z ( f ) will b e something like P f ( a )= a a . And just as the free ab elian group functor maps cartes ian pr o ducts not to car tesian pr oducts, but to tensor products, if we wan t Z to be a monoidal functor, w e usually cannot expect D to b e car tesian mo noidal, only symmetric mo no idal. Therefor e, a trace in D no longer implies a fixed po in t op erator. Ho wever, since C is car tesian, w e still hav e diagonal morphisms for ob jects in the image o f Z , a nd this is a ll we really need. In fact, if we c an find a catego r y D which is suitably “additive,” then it often comes with a canonical notion of tra ce for free. The idea is to split the “feedbac k” diagram in to a comp osition of three piec es: g B A M = g B A M The mor phism M M (from the unit o b ject to M ⊗ M ) is called a “ coev aluation” or “ unit.” If M is of the for m Z ( X ) for some X ∈ C , then the co ev alua tion is supp osed to pick out a formal sum such as P x ∈ X x ⊗ x . (T o b e precis e, the second s tring la beled M is actually its “dual” M ⋆ , and so the sum is actually P x ∈ X x ⊗ x ⋆ .) Similarly , the morphism M M TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 5 is called an “ev a luation” or “counit.” F or M = Z ( X ), the ev aluation is supp osed to b e suppo rted on pairs o f the for m x ⊗ x (o r, more precisely , x ⋆ ⊗ x ) , and to give zero when applied to a pair x ⊗ x ′ for x 6 = x ′ . If this is the case, then the comp osite f A X X will a c t as follows: a 7→ X x ∈ X a ⊗ x ⊗ x 7→ X x ∈ X f ( a, x ) ⊗ x 7→ X x ∈ X f ( a, x ) ⊗ f ( a, x ) ⊗ x 7→ X x ∈ X f ( a,x )= x f ( a, x ) . Thu s, as desir ed, it pic ks o ut the sum of all the fixed po in ts of f . T races constructed in this wa y from ev aluatio n and coev a luation maps are called c anonic al traces. In [ 24 ], Joy al a nd Street show ed that any traced monoidal cat- egory D can b e embedded in a larg er o ne Int( D ), in such a wa y that the given traces in D a re iden tified with canonical traces in In t( D ). Therefore, for the pur- po ses of finding fixed-p oin t inv ariants, there is no lo ss in r estricting our attention to ca nonical traces. How ever, the choice o f a par ticular D do es restrict the maps for which we can calculate o ur fixed-p oin t inv aria n t, since the resulting ob jects in D must a dmit ev al- uations a nd coev aluations. This prop erty is ca lled being dualizable . F or instance, in the free ab elian g roup on X , the sum P x ∈ X x ⊗ x is only defined when X is finite. A giv en functor C → D thus induces a notio n o f “finiteness” on ob jects of C . W e will see in § 3 tha t differ en t c hoices of D can drastically affect the notion o f finiteness, as well as the utility a nd computability of trace s. How ever, in most ap- plications the choice of D is s traigh tforward, and the res ultin g finiteness restrictio n not o nerous. 2. Trac es W e now mov e o n to the abstract study of canonical traces in symmetric monoidal categorie s; in the next section w e will sp ecialize to a num b er of examples a nd see how w e o btain information ab out fixed po in ts. W e b egin with the formal definition of dualiz abilit y . 6 KA TE PONTO AND MICHAEL SHULMAN Co ev aluation η : I → M ⊗ M ⋆ M M ⋆ Ev alua tion ε : M ⋆ ⊗ M → I M ⋆ M Figure 1. Co ev aluation and ev alua tion M M M ⋆ M ⋆ M M = M M ⋆ = M ⋆ M ⋆ M M M ⋆ M ⋆ Figure 2. The triangle identities Let C b e a sy mmetric monoidal ca tegory with pro duct ⊗ and unit ob ject I . W e will omit t he asso ciativit y and unit isomorphisms of C f rom the notation (effectively pretending that C is strict, a s is allo wable b y the coherence theorem), and we write s for any instance or co mposite of instanc e s of the symmetry isomorphism of C . Definition 2.1 . An ob ject M of C is dualizable if there e xists an ob ject M ⋆ , called its dual , and maps I η − → M ⊗ M ⋆ M ⋆ ⊗ M ε − → I satisfying the triangle iden tities (id M ⊗ ε )( η ⊗ id M ) = id M and ( ε ⊗ id M ⋆ )(id M ⋆ ⊗ η ) = id M ⋆ . W e call ε the ev aluation and η the coev aluation ; so me authors c all them the c ounit and the unit . W e s a y that C is compac t clo sed if every o b ject is dualizable. As sugges ted in § 1 , dual pa irs ar e repres e n ted gr aphically by tur ning around the direction of arr o ws; see Figure 1 . Note that the unit ob ject I is repr esen ted by the lack of any strings, such as in the input to η and the output of ε . Strictly sp eaking, there s ho uld b e b o xes at the ends of these “ c aps” a nd “cups” la b eled by η and ε resp ectively , but these lab els are almost universally omitted in string diagram notation (this is als o justified by a theorem o f Joyal and Street). The triangle iden tities for a dual pair translate graphica lly as “bent s trings can b e straightened;” see Figure 2 . An y tw o duals of an ob ject M are isomorphic; an isomorphism can b e constructed from η and ε . If M ⋆ is a dual of M , then M is a dual o f M ⋆ . And if M and N are dualiz able, any map f : Q ⊗ M → N ⊗ P has a dual or mate f ⋆ : N ⋆ ⊗ Q → P ⊗ M ⋆ , given by the comp osite N ⋆ ⊗ Q id ⊗ id ⊗ η − − − − − − → N ⋆ ⊗ Q ⊗ M ⊗ M ⋆ id ⊗ f ⊗ id − − − − − − → N ⋆ ⊗ N ⊗ P ⊗ M ⋆ ε ⊗ id ⊗ id − − − − − − → P ⊗ M ⋆ . TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 7 f M M ⋆ M = I η   M ⊗ M ⋆ f ⊗ id   M ⊗ M ⋆ ∼ =   M ⋆ ⊗ M ε   I or just f M ⋆ Figure 3. The tra ce In particular, if M is dualizable, any endomor phism f : M → M has a dual f ⋆ : M ⋆ → M ⋆ . There are v arious equiv a len t characteriza tions of dualizable ob jects. F or exam- ple, when C is clos ed, M is dualizable if and only if the canonical map M ⊗ Hom( M , I ) → Hom( M , M ) is a n isomorphism. How ever, for us the above definition is most appropr ia te. W e now mov e on to the simplest form of tra ce. Definition 2.2. Le t C be a symmetric monoidal category , M a dualiza ble ob ject of C and f : M → M an endomor phism of M . The trace of f , denoted tr( f ), is the follo wing co mp osite: (2.3) I η − → M ⊗ M ⋆ f ⊗ id − → M ⊗ M ⋆ s − → ∼ M ⋆ ⊗ M ε − → I . The Eul er c haracteristic of a dualiza ble M is the tr a ce o f its iden tity map. The trace of a morphism transla tes as gra phically as “feeding its output int o its input;” see Figure 3 . References for this notion of tr ace include [ 11 , 14 , 23 , 24 , 26 ]. It is an endomor- phism of the unit ob ject I , and do e s not depend on the choice of dual for M or o n the c hoice of the maps η and ε . It also has the following fundament al prop ert y . Prop osition 2.4 (Cyclicit y) . F or any f : M → N and g : N → M with M , N b oth dualizable, we have tr ( f g ) = tr( g f ) . Pr o of. The following pro of is r eally only render ed comprehensible by string diagr am notation (see Figure 4 ). tr( f g )= ε s ( f g ⊗ id) η = ε s ( f ⊗ id)( g ⊗ id) η = ε s ( f ⊗ id)(id ⊗ ε ⊗ id)( η ⊗ id ⊗ id)( g ⊗ id) η = ε s (id ⊗ ε ⊗ id)( f ⊗ id ⊗ id ⊗ i d)(id ⊗ id ⊗ g ⊗ id)( η ⊗ η )=( ε ⊗ ε ) s (id ⊗ id ⊗ g ⊗ id)( f ⊗ id ⊗ id ⊗ id )( η ⊗ η )= ( ε ⊗ ε ) s (id ⊗ id ⊗ g ⊗ id)(id ⊗ id ⊗ η )( f ⊗ id ) η = ε s (id ⊗ ε ⊗ id)( g ⊗ id ⊗ id ⊗ id)( η ⊗ id ⊗ id)( f ⊗ id) η = ε s ( g ⊗ id)(id ⊗ ε ⊗ id)( η ⊗ id ⊗ id)( f ⊗ id) η = ε s ( g ⊗ id)( f ⊗ id) η = ε s ( g f ⊗ id ) η =tr( g f )  W e will consider additional pr operties of the trace in § 5 . 8 KA TE PONTO AND MICHAEL SHULMAN f g N M ⋆ M = f g N N ⋆ N M M ⋆ M = f g N N ⋆ N M M ⋆ M = f g N N ⋆ N M M ⋆ M = f g N ⋆ N M Figure 4. Cyclicity of the tra ce 3. Examples of traces Example 3.1 . Let C = V ect k be the catego ry of vector spaces over a field k . A vector s pace is dualizable if and only if it is finite-dimensional, and its dual is the usual dual vector spa ce. W e hav e I = k a nd C ( I , I ) ∼ = k by multiplication. Using this identification, Definition 2.2 recov ers the usual trace of a matr ix. The Euler characteristic of a vector space is its dimension. Example 3.2 . Let C = M o d R be the catego ry of mo dules ov er a commutativ e ring R . The dualizable ob jects are the finitely genera ted pro jectiv es. As in V ect k , we hav e I = R and C ( R, R ) ∼ = R , so every endomorphism of a finitely g enerated pro jective mo dule has a tr a ce whic h is an element of R . The Euler c hara cteristic of such a mo dule is its rank, regarded a s an e le men t of R (so that, for instance, the Euler characteris tic of a rank- p free ( Z /p )-module is zero). Example 3.3 . Again, let R b e a commutativ e r ing and consider the category Ch R of chain complexes of R -mo dules, with its s ymmetric monoidal tensor pro duct. The “corr ect” sy mmetry isomo rphism M ⊗ N ∼ = N ⊗ M in tro duces a sign: a ⊗ b 7→ ( − 1) | a || b | ( b ⊗ a ). The dualizable ob jects are again the finitely generated pro jectiv es, the unit is ag ain R itself (in degr ee 0), and endomorphisms of the unit can a gain be ident ified with elements o f R . The trace of an endomorphism of a finitely generated pro jective chain co mplex, called its L efschetz numb er , is the alternating sum o f its degreewise traces. Likewise, the E uler characteristic of s uc h a chain co mplex is the alternating sum of its ranks. This genera lizes straightforwardly to mo dules o ver a DGA. Example 3.4 . There is also a symmetric monoidal ca tegory Ho( Ch R ), called the de- rive d c ate gory of R , obta ined from Ch R by forma lly inv e r ting the quasi-is omorphisms (morphisms which induce isomor phisms on a ll homology gro ups). The dualizable ob jects in Ho( Ch R ) are those that are quasi-is omorphic to an ob ject that is dual- izable in Ch R , and the t wo kinds of tra ces also agree. Example 3 .5 . Let n Cob be the categor y whose ob jects are closed ( n − 1)-dimensional manifolds, and whose morphisms ar e diffeomor phism cla sses of cobor dis ms. Com- po sition is b y gluing , cylinders M × [0 , 1] give iden tities, and disjoint union supplies a symmetric monoidal struc tur e. The unit ob ject is the empt y set ∅ , and an endo- morphism of ∅ is just a closed n -manifold. TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 9 Every ob ject of n C o b is dua lizable: the ev aluation a nd co ev aluatio n are b oth M × [0 , 1], r egarded either as a cobor dism from ∅ to M ⊔ M o r from M ⊔ M to ∅ . The trace of a cob ordism fr om M to M is the c lo sed n -manifold obtained by gluing the t wo comp onen ts of its b oundary to gether. In particular, the E ule r characteristic of a c losed ( n − 1)-manifold M is M × S 1 , regarded as a cobor dis m from ∅ to itself. Example 3 .6 . In a c artesian mono idal ca tegory , the only dualizable ob ject is the terminal ob ject. Th us, in this ca se there are no interesting traces. How e ver, a s suggested in § 1 , often we can obtain useful dualities and traces by applying a functor from such a categ ory to a non- cartesian monoida l categor y . Such a functor F induces a notion of “finiteness” on its domain in the evident w ay: X is “finite” if F ( X ) is dualizable. F or an endomorphism f of such an X , we can then compute the trace of F ( f ). Probably t he simplest such functor is the free ab elian group functor Z [ − ] : Set → Ab . Of c o urse, Z [ X ] is dualizable in Ab if a nd only if X is a finite set. If f : X → X is an e ndo morphism of a finite set, then the trace of Z [ f ] : Z [ X ] → Z [ X ] is easily seen to b e simply the nu mber of fixed po in ts of f . T his justifies the hop e ex pr essed in § 1 tha t by mapping a ca rtesian monoidal categor y into an “additive” one, we could extr act information ab out fixed p oin ts which may or may not b e pr esen t. The next few examples c a n also be viewed in this light. Example 3.7 . Supp ose tha t instead of a set we star t with a top o logical space. The category T op is, o f course, ca rtesian monoidal, so in o rder to obtain interesting du- alities we need to apply a functor landing in so me non-cartesian monoidal category . One obvious g uess, by analo gy with E xample 3.6 , would b e the catego ry of ab elian top ological gro ups and the fre e ab elian top ological group functor . It usually turns out to be b etter, how ever, to use a mor e refined notion: the category Sp of sp e ct r a . F or the reader unfamiliar with sp ectra some intuition can be g ained as follo ws. A connective sp ectrum can b e thought o f as analog ous to an a belian topo logical group, except that its group structure is o nly a s sociative, unital, and commutativ e up to homo to p y a nd all higher homotopies. The passa ge from co nnectiv e sp ectra to ar bitrary sp ectra is then analog ous to the passag e from b ounded-b elo w chain complexes to arbitrar y ones. There is a symmetric monoidal catego ry Sp of sp ec- tra and a “ free” functor Σ ∞ + : T op → Sp , usua lly ca lled the su sp ension sp e ct rum functor. (Actually , there are many such ca tegories Sp , a ll equiv alent up to ho- motopy , but ea c h having different technical adv antages a nd dis adv ant ages ; s ee for instance [ 13 , 32 , 33 ]. W e w ill gener a lly gloss ov er s uc h distinctio ns .) The monoidal structure of Sp is ca lled the “sma sh product” ∧ , and its unit o b ject is the spher e sp e ctrum S (which can be identified with Σ ∞ + of a po in t). Since Sp is not car tesian monoidal, we can hop e for it to have an interesting duality theory . How ever, it turns out that in Sp it is only rea sonable to a sk for duality up to homotopy . Thus, instea d of Sp we usually work with the categor y Ho( Sp ) o btained from it by inv erting the “ stable equiv alences ;” this is ca lled the stable homotopy c ate gory . W e still ha ve a functor Σ ∞ + : T op → Ho( Sp ) which factors through Ho( T op ) (in which we inv ert the weak ho mo top y equiv alences). The rea son fo r the use of “ stable” is that for compact spa ces M and N , the homset Ho( Sp )(Σ ∞ + ( M ) , Σ ∞ + ( N )) ca n be iden tified with the set o f stable homoto py classes of maps from M to N , i.e. the colimit ov er n of the sets of ho mo top y clas ses of maps Σ n ( M + ) → Σ n ( N + ). 10 KA TE PONTO AND MICHAEL SHULMAN One can now show that if M is a clo sed smo oth manifold, or more generally a compact ENR (E uclidean Neighborho o d Retr act), then Σ ∞ + ( M ) is dualizable in Ho( Sp ); its dual is the Thom spe ctrum T ν of its stable nor mal bundle. This is pr o ven in [ 1 , 11 , 28 ]. The s e t of endomorphisms of the sphere s p ectrum S is colim n π n ( S n ), whic h is isomorphic to Z ; thus tr aces in Ho( Sp ) can be ident ified with in tegers. Using this identification, the trace of an endomor phis m can b e iden tified with its fixe d p oint index . The fix e d point index is an integer which is defined classically , for a map with isolated fixed p oin ts, as the sum o ver all fix ed p oin ts x of the degr ee of the self-map induced by th e “difference” of the identit y map and the endomor phism on a sufficiently small sphere s urrounding x . This turns out to b e homotopy in v aria n t, and so for an a rbitrary map it can be defined by homotoping to a map with isolated fixed p oin ts. In particular, this is necessa ry for the identit y map, whose fixed p oint index is the Euler characteristic of the manifold (this is, of course, the origin of the term “E uler c haracter is tic” for trace s of identit y maps in general). See [ 5 , 7 ] for the classical approa c h to the index and [ 8 , 9 , 11 ] for the identification o f this trace with the classical fixed point index and Euler characteris tic. F or compact spaces the induced notions of duality and trace can also b e formu- lated witho ut using the sta ble homotopy c a tegory; we replace S with the n -s phere S n for some large enough finite n . In this g uis e it is called n -duality ; references include [ 11 , 28 ]. Sp a nd Ch R are tw o instance s of a g eneral phenomenon: a symmetr ic monoidal category that has a n ass ociated sy mmetric mono idal homo top y categor y . A gen- eral theor y of when and how s ymmetric monoidal structures descend to homotop y categorie s is given by the study of monoidal mo del c ate gories , as in [ 18 , Ch. 4]. Example 3.8 . F or a compact Lie gr oup G , there is an e quivariant stable homotopy c ate gory Ho( G - Sp ), which is related to the ca tegory G - T op of G - s paces in the same way that Ho( Sp ) is related to T op ; see for instance [ 32 ]. Now the susp ension and stabiliza tion take place not relative to ordina ry sphere s S n , but relative to representations of G . The categor y Ho( G - Sp ) is also symmetr ic monoidal and admits a susp ension G -spectrum functor fr om G -spaces. The dualizable ob jects in Ho( G - Sp ) include the equiv a r ian t suspensio n sp ectra of closed smooth G -manifolds and compact G -ENR’s. Such dual pairs can be a lso describ ed us ing V -dualit y for a representation V . A re ference for equiv ariant dua lity is [ 28 ]. T races in Ho( G - Sp ) are a gain called fixed p oin t indices; see [ 38 , 46 , 47 ]. Example 3.9 . Another v ariatio n is to consider p ar ametrize d duality , which instea d of spaces or G -spaces starts from spaces over a bas e space B . In [ 35 ], May and Sigurdsson co nstruct a categor y Sp B of p ar ametrize d sp e ct r a over B , whic h is sym- metric monoidal, has a symmetric monoida l homotopy categor y Ho( Sp B ), and admits a susp ension functor Σ ∞ B , + from T op /B that is simila r to Σ ∞ + . If M is a fibration ov er B , then Σ ∞ B , + ( M ) is dualizable in Ho( Sp B ) if and only if each of its fib e rs is dualizable in the usual stable homotopy categor y . In particula r , a fibra tion of clo s ed smo oth manifolds gives rise to a dualizable pa rametrized sp ec- trum. The trace of a fibe rwise endomorphism is once again called its fixed p oin t index; se e [ 9 ]. R emark 3.10 . F or parametrized spaces and sp ectra it is often more illuminating to consider a differen t t yp e of duality called Costenoble-Waner du ality , and its TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 11 asso ciated notion of tr ace. These notions o f duality and trace do not tak e place in a symmetric monoidal categor y , but rather in a bicategor y arising from an indexed symmetric monoidal categ ory; se e [ 35 , Ch. 18] and [ 39 , 42 , 43 ]. Here are some more “to y” ex amples. Example 3.11 . Let Re l b e the ca tegory w ho se ob jects are sets and whos e morphis ms from M to N are re la tions R ⊂ M × N ; we write R : M − 7 − → N to avoid confusion with functions M → N . If S : N − 7 − → P is another re lation, their comp osite is S R = n ( m, p )    ∃ n with ( m, n ) ∈ R and ( n, p ) ∈ S o . A symmetric mono ida l structure on Rel is induced by the car tesian product o f sets (whic h is not the cartesian pr oduct in R el ). There is a functor Set → Rel which is the iden tity on o b jects and takes a function f : X → Y to its graph Γ f = { ( x, f ( x )) | x ∈ X } . Moreov er, every set is dualizable in Rel , and moreover is its own dua l; the relatio ns η and ε ar e b oth the identit y relatio n on X , considered as a re la tion ∗ − 7 − → X × X or X × X − 7 − → ∗ , res pectively . Thus every set is “finite” relative to this functor. How ever, the trade off is that traces contain cor responding ly less infor mation, since the only endomorphisms of the unit ∗ in R el are the empt y relation a nd the full o ne. If w e regar d these as the tru th values “ false” and “true,” resp ectiv ely , then the trace of a relation R : M − 7 − → M is the truth v alue o f the statement “ ∃ m : ( m, m ) ∈ R .” In particular, for a function f : M → M , tr(Γ f ) is true if and o nly if f ha s a fixed po in t. This example can be ge ne r alized to internal relations in an y suitably w ell-b eha ved category . Example 3 .12 . Let Sup denote the category of suplattic es : that is, its ob jects are p osets with all suprema and its morphisms are s uprem um-preserving ma ps. (Of course, a suplattice a lso ha s all infima, but suplattice maps need not pres erv e infima.) There is a tensor pr oduct o f suplattices, concisely descr ibed by saying that supla tt ice ma ps M ⊗ N → P represent functions M × N → P which preser v e suprema in each v ar iable separ ately . The unit o b ject is the suplattice I = (0 ≤ 1). W e can see an a nalogy b etw een Sup a nd Ab by reg arding s upr ema in a s uplat- tice as similar to sums in an abelia n gr oup. F or instance, there is a “free suplattice” functor Set → Sup whic h simply takes a set A t o its pow er set P ( A ); the “free gen- erators ” are the singleto n sets, and a subset B ⊆ A is the “formal sum” P x ∈ B { x } . Every such p o wer set is dualizable, so every set is “ finite” relative to the functor P . Explicitly , w e have P ( A ) ⋆ ∼ = P ( A ) with co ev aluation η (1) = _ a ∈ A { a } ⊗ { a } and ev aluation ε ( X ⊗ Y ) = ( 1 if X ∩ Y 6 = ∅ 0 otherwise . Note that a suplattice map f : P ( A ) → P ( A ) is equiv alent to a function A → P ( A ), and thereby to a r elation R f ⊂ A × A . The trace of such a map in Sup is easily verified to be id I if there is an a ∈ A with ( a, a ) ∈ R f and 0 otherwise; thus we essentially recapture Example 3 .11 . 12 KA TE PONTO AND MICHAEL SHULMAN How ever, not all dualizable supla ttices ar e power sets. F or instance, if A is an Alexandrov top ological spa ce (o ne wher e ar bitrary in tersectio ns of op en s e ts a re op en), then its op en-set lattice O ( A ) is a dualizable supla tt ice. In this case, for a contin uo us map f : A → A , the trace of f − 1 : O ( A ) → O ( A ) is id I if there is an a ∈ A such that a ≤ f ( a ) in the sp ecialization order, a nd 0 otherwise. (Recall that the sp e cializatio n or der of a top ological space is defined so that x ≤ y if and only if ev ery op en set containing y also con tains x .) Example 3.1 3 . Contin uing the analogy betw een Sup and Ab , it is natural to con- sider co mm utative mo noid ob jects in Sup a s analogous to commutativ e r ings. A commutativ e monoid ob ject in a symmetric monoidal catego r y C is an o b ject R with morphisms R ⊗ R → R a nd I → R sa tisfying eviden t a xioms. In particular , for any top ological s pace B , the op en-set lattice O ( B ) is a co mm utative mono id ob ject in Sup whose “multiplication” map is in tersectio n ∩ : O ( B ) ⊗ O ( B ) → O ( B ). Likewise, for an y con tinuous f : A → B , we have a monoid homomorphism f − 1 : O ( B ) → O ( A ). In this w ay a category of suitably nice topolog ical space s ca n be identified with the opp osite of a sub category of commutativ e mono ids in Sup ; see [ 25 ] and [ 21 , Ch. C1]. This is analog ous to how the ca tegory o f affine s chemes can be iden tified with the opp osite of the category of comm utative rings. If C ha s co equalizers pres e rv ed on bo th sides by ⊗ , then for any commutativ e monoid o b ject R the ca teg ory of R -modules in C is itself symmetric mono idal under the tensor pro duct g iv en by the usual c o equalizer M ⊗ R ⊗ N ⇒ M ⊗ N → M ⊗ R N . In particular, this applies to O ( B )-modules in Sup for any s pa ce B . Given a n y contin uo us map p : X → B , O ( X ) b ecomes a O ( B )-algebra (that is, there is a monoid homomo r phism p − 1 : O ( B ) → O ( X )) and thus a O ( B )-module. If p is a lo cal homeomorphis m (that is, X is the “e s pace etale” of a sheaf ov er B ), then O ( X ) is a dualizable O ( B )-mo dule that is its own dual: the ev aluation is O ( X ) ⊗ O ( B ) O ( X ) ε − → O ( B ) ( W , W ′ ) 7→ p ( W ∩ W ′ ) and the co ev aluation is O ( B ) η − → O ( X ) ⊗ O ( B ) O ( X ) U 7→ _ p ( W ) ⊆ U p | W is a homeomorphism W ⊗ W . The unit O ( B )-module is O ( B ), and an O ( B )-module map O ( B ) → O ( B ) is deter- mined uniquely by wher e it sends B ∈ O ( B ) (the unit for ∩ ). Any map f : X → X ov er B gives a map f − 1 : O ( X ) → O ( X ) of O ( B )-mo dules, who se trac e is charac- terized b y B 7→  b ∈ B   ∃ x ∈ p − 1 ( b ) : f ( x ) = x  ; that is, the s e t of b ∈ B s uc h tha t f | p − 1 ( b ) has a fixed po in t. In particula r, the Euler characteristic of a sheaf X is its supp ort p ( X ) ⊆ B . When B is the one-p oin t space, X mu st b e discrete, and we recapture p ow e r sets in Sup . F or mo re o n this po in t of view, see [ 44 ]. TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 13 W e hav e so far considered only symmet ric monoidal categ ories, but the defi- nitions we hav e given make sens e with only a br a iding, and there a r e interesting examples whic h a r e not symmetric. Example 3 .14 . Let T ang be the catego ry o f tangles : its ob jects are natural n um- ber s 0 , 1 , 2 , . . . , and its mo r phisms from n to m ar e tangles fr o m n p oints to m po in ts . A tangle is like a bra id, ex cept that str ings ca n b e turned around, so that n need not equal m ; see [ 14 ]. T ang is braided but not symmetr ic monoidal; its pro duct is disjoin t union and its unit ob ject is 0. Every ob ject is its o wn dua l, the endomorphisms of the unit are links, and the tra ce o f a n endo-tangle is its “ tangle closure” in to a link. There are orien ted and framed v ariants. How ever, the trace as w e ha ve defined it is not quite correct in the non-symmetric case. F or instance , with our definitions, the trace of the identit y id 2 in T a ng is tw o linke d circles, while the trace of the braiding s 2 is tw o u nlinke d cir cles; clea rly it would make more sense for this to be the other wa y round. This c an b e remedied with the notion of a b alanc e d mono idal categor y , whic h is a braided monoidal category in which e a c h ob ject is equipp ed with a “do uble- t wis t” automor phism; see [ 23 , 24 ]. Symmetr ic mono idal categorie s can be identified with bala nced ones in which every double-twist is the identit y . In a balanced mo noidal category , we define the tra ce of a n endomorphism f by including a double-twist with f in b et ween η and ε ; this remedies the problem noted abov e with T ang . F or simplicity , how e ver, in this pap er we will consider only the symmetric case. 4. Twisted tra ces and transfers The exa mples in § 3 show that the ca nonical tra c e defined in § 2 do es give use - ful information ab out fixed p oin ts, but usually it only indica tes their presence or absence, or at best counts the num b er o f them (with m ultiplicit y). How ever, in § 1 we saw that in the presence of “diag o nals”, we could hop e to ex tract no t merely the num b er of fixed p oin ts, but the fixed po in ts themselves. The us e of diagona ls in this w ay turns out to b e a sp ecial ca se of the following more gener a l notion o f “twisted trace.” Definition 4.1. Le t C be a symmetric monoidal category , M a dualiza ble ob ject of C , and f : Q ⊗ M → M ⊗ P a morphism in C . The trace tr( f ) of f is the following c o mposite: (4.2) Q η − → Q ⊗ M ⊗ M ⋆ f − → M ⊗ P ⊗ M ⋆ s − → ∼ M ⋆ ⊗ M ⊗ P ε − → P This trace is display ed gr aphically in Figure 5 . R emark 4.3 . Since C is sy mmetric, it may s e em o dd to wr ite the domain of f as Q ⊗ M but its codomain as M ⊗ P . Indeed, in the literature on symmetric monoida l traces it is mor e common to a lign the M ’s on one side, as in the right-hand version of Fig ure 5 . Our nota tion is chosen instead to match that of the bicategorical generaliza tion presented in [ 4 2 ], in which case the order we hav e chosen here is the only possibility . Of cours e, when Q = P = I is the unit ob ject, this reduces to the previous notion of trace. It is also cyclic, in a suitable sense. 14 KA TE PONTO AND MICHAEL SHULMAN f Q P M M ⋆ = Q id ⊗ η   Q ⊗ M ⊗ M ⋆ f ⊗ id   M ⊗ P ⊗ M ⋆ ∼ =   M ⋆ ⊗ M ⊗ P ε ⊗ id   P or just f Q P M ⋆ Figure 5. The “ t wisted” trace f Q P g K L N M M ⋆ M ⋆ M = g K L f Q P M N N ⋆ N ⋆ N Figure 6. Cyclicity of the “ t wisted” trace Lemma 4 .4. If M and N ar e du alizable and f : Q ⊗ M → N ⊗ P and g : K ⊗ N → M ⊗ L ar e morphisms, then tr  ( g ⊗ id P )(id K ⊗ f )  = tr  s ( f ⊗ id L )(id Q ⊗ g ) s  . The string dia gram for this lemma is Figure 6 , which should b e co mpared to Figure 4 . As pr omised, the trace using a “dia gonal mo r phism” is a sp ecial case o f the general notion of t wisting. Definition 4.5. Let M ∈ C b e a dualizable ob ject equipp ed with a “diago nal” morphism ∆ : M → M ⊗ M , and let f : M → M b e an endomo rphism of M . Then the trace of f with resp ect to ∆ is the trace of ∆ ◦ f : M → M ⊗ M . The trace of id M with respect to ∆ is called the transfer of M . TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 15 The tr ace of f with r e spect to ∆ is a morphism I → M ; by cyclicity it is also equal to the trace o f ( f ⊗ id) ◦ ∆. Where do such diagona ls c o me from? Of c o urse, if C is c artesian monoidal, then any ob ject M has such a diago nal, but w e have seen ( Example 3.6 ) that in this case there are few dua lizable o b jects. How ever, we hav e also seen that for many o f the trac e s we ar e interested in, the dualiz able ob jects are in the image of a symmetric monoidal functor whose domain is cartesia n monoidal, and such a functor preserves the existence of diagonals. That is, if X is an ob ject of a car tesian monoidal catego r y C and Z a symmetric mono idal functor with domain C , then the diagonal X → X × X gives rise to a diag onal Z ( X ) → Z ( X ) ⊗ Z ( X ). This is usually the sour ce o f “diago nals” in examples. Example 4.6 . Let S = Set , C = Ab , and Z be the free ab elian g roup functor Z [ − ]. In this case the induced diago nal Z [ X ] → Z [ X ] ⊗ Z [ X ] sends a g enerator x to x ⊗ x . If X is finite, s o tha t Z [ X ] is dualizable, the trac e of Z [ f ] with r e spect to this diagonal is P f ( x )= x x ∈ Z [ X ]. In particular, the transfer of Z [ X ] is P x ∈ X x ∈ Z [ X ]. Note t hat while the o rdinary trace of Z [ f ] records only the numb er of fixed po ints of f , its trace with resp ect to ∆ reco r ds what those fixed p oin ts are (as e lemen ts of Z [ X ]). Example 4.7 . As a more so phisticated version of the previous example, let S = T op and le t C = Ho( Sp ) b e the stable homotopy catego ry . Since T op is ca rtesian monoidal and the suspensio n sp ectrum functor Σ ∞ + is strong monoidal, the diagonal M → M × M of a n y space induces a diagonal ∆ : Σ ∞ + ( M ) → Σ ∞ + ( M ) ∧ Σ ∞ + ( M ) . Thu s, when M is n -dualiza ble, w e ca n define traces a nd transfers with resp ect to ∆. This example is the orig inal use of the term tr ansfer . In this ca se, the tr a nsfer of an n - dualizable space M is a map S → Σ ∞ + ( M ), which is by definition an elemen t of π s 0 ( M + ), the 0 th stable homotopy gro up of M . If M is co nnected, there is an isomorphism π s 0 ( M + ) ∼ = H 0 ( M + ) under which the image of the tra ns fer is χ ( M ) times the generator o f H 0 ( M + ); see [ 28 , II I.8.4]. Mo re genera lly , if f is a n e ndo morphism of M , then the trace of f with resp ect to ∆ is the fixed p oin t transfer defined by Dold in [ 10 ]. W e hav e the same intuit ion a s for the previous e x ample; while the fixed p oin t index o nly c ounts the fixed p oin ts, the fixed point transfer r e c or ds them. There a re a lso equiv ariant and para metr ized tra nsfers. F or example, the Beck er- Gotleib transfer [ 3 ] is the parametrized transfer of a fibration with compact manifold fiber s. Example 4.8 . Recall fr o m Example 3.11 that we als o have a functor Set → Rel which is the identit y o n ob jects a nd sends a function f to its graph Γ f . In this case, for a n y endofunction f : M → M , the tra ce of Γ f with resp ect to Γ ∆ is the set of all fixed p oin ts o f f , regarded as a relation from ⋆ to M . In particular, the transfer o f M is M itself so reg arded. Example 4.9 . Likewise, if Σ is the free suplattice functor P : Set → Sup from Example 3 .12 , then for any endofunction f : M → M the tr ace of P ( f ) with resp ect to P (∆ ) is also the set o f fixed points of f , now regar de d as an elemen t of P ( M ). 16 KA TE PONTO AND MICHAEL SHULMAN Twisted traces not arising from dia gonals a re less common, but they do o ccur. Example 4.10 . Let f : Q × M → M b e a function b et w een sets, where M is finite. Then the tr a ce of Z [ f ] : Z [ Q ] ⊗ Z [ M ] → Z [ M ] in Ab is the homomor phism Z [ Q ] → Z which s ends each genera to r q ∈ Q to the num b er of fixed points of f ( q, − ). W e ca n a ls o combine this with a transfer, by consider ing ( f , pr 2 ) : Q × M → M × M . This induces a map Z [ Q ] ⊗ Z [ M ] → Z [ M ] ⊗ Z [ M ], whose trace Z [ Q ] → Z [ M ] sends eac h g enerator q ∈ Q to the sum of the fixed p oints of f ( q , − ). Similarly , for any set M and any function f : Q × M → M , the trace of Γ f in Rel is the relation fro m Q to ∗ defined by tr(Γ f ) = { q ∈ Q | f ( q, − ) ha s a fixed p oin t } , and the trace of the induced relation Q × M → M × M is { ( q , m ) ∈ Q × M | m is a fixed po int of f ( q, − ) } . Example 4 .1 1 . If f : Q × M → M is a contin uous map of top ological spaces and Σ ∞ + M is dualizable, the trace of f is the stable homo top y cla s s of a map Q → S 0 . Using explicit des criptions of the co ev aluation a nd ev aluatio n for Σ ∞ + M , it is not difficult to verify this stable map is homotopically trivial if the set { ( q , m ) ∈ Q × M | f ( q , m ) = m } is empty . Let π : Q × M → Q be the first co ordinate pro jectio n. Note that f determines a fiber wise ma p F : Q × M → Q × M by F ( q , m ) = ( q , f ( q , m )). The tra ce of F as descr ibed in E xample 3.9 coincides with the trace o f f under a corres p onding compariso n of fib e r wise stable endomor phism of the unit ob ject over Q with the stable maps from Q to S 0 . This trace is related to the high er Euler char acteristics in [ 15 ]. Example 4.12 . F or any top ological space A we hav e an “intersection” morphism ∩ : O ( A ) ⊗ O ( A ) → O ( A ) in Sup . If A is moreover Alexandrov, so that O ( A ) is dualizable in Sup , then for a n y f : A → A , the tr ace o f f − 1 ◦ ∩ is the function O ( A ) → I that takes U ⊂ A to 1 if U contains a p o in t x with x ≤ f ( x ) and to 0 otherwise. If w e iden tify a suplattice map g : O ( A ) → I with the clos ed subset \ K closed g ( A \ K )=0 K (whic h determines it), then the tra ce of f − 1 ◦ ∩ is iden tified with the closure of { a ∈ A | a ≤ f ( a ) } . On the other hand, if B is another Alexandrov space wit h a map m : A × B → A , then we hav e an induced suplattice map m − 1 : O ( A ) → O ( A × B ) ∼ = O ( A ) ⊗ O ( B ). Its tra ce is the s uplattice map I → O ( B ) which takes 1 to the op en set { b ∈ B | ( ∃ a ∈ A )( a ≤ m ( a, b )) } . Example 4.1 3 . If p : X → B is any lo cal homeo morphism, then regarding O ( X ) as a O ( B )-mo dule w e a gain hav e an “intersection” mor phism ∩ : O ( X ) ⊗ O ( B ) O ( X ) → O ( X ) (corr esponding to the dia gonal X → X × B X ). F or any f : X → X ov er B , TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 17 the tra ce of f − 1 ◦ ∩ is the O ( B )-mo dule map O ( X ) → O ( B ) that sends V ∈ O ( X ) to  b ∈ B   ∃ x ∈ p − 1 ( b ) ∩ V : f ( x ) = x  . In par ticular, the trace of ∩ itself sends each V ∈ O ( X ) to its supp ort. On the o ther hand, if q : Y → B is a nother lo cal homeo morphism and f : X × B Y → X is a map over B , then the trace of f − 1 : O ( X ) → O ( X ) ⊗ O ( B ) O ( Y ) is the map O ( B ) → O ( Y ) sending the unit B to the o pen set  y ∈ Y   ∃ x ∈ p − 1 ( q ( y )) : f ( x , y ) = x  . 5. Proper t ies of traces In addition to c y clicit y , the (twisted) symmetric monoidal tra ce satisfies many useful naturality prop erties whic h we summarize here. W e omit most pro ofs, which are stra ig h tforward diagr am c hases (and are esp ecially ea sy in string diag ram no- tation). References include [ 11 , 24 , 28 , 34 ]. W e b egin with in v ariance under dualization. Rec all that any f : Q ⊗ M → M ⊗ P has a mate f ⋆ : Q ⊗ M ⋆ → M ⋆ ⊗ P , and since M ⋆ is also dua lizable (with dua l M ), after comp osing with symmetry isomorphisms on either side we ca n tak e the trace o f f ⋆ as w ell. Prop osition 5.1 . If M is dualizable and f : Q ⊗ M → M ⊗ P is any morphism, then tr( f ) = tr( s f ⋆ s ) . In the simple ca se of square matrices in Ab or Mo d R , this says that the trace of a matrix is equa l to the trace of its tra ns pose. Next w e have a na turalit y pr operty . Prop osition 5.2. L et M b e dualizable, let f : Q ⊗ M → M ⊗ P b e a map, and let g : Q ′ → Q and h : P → P ′ b e t wo maps. Then h ◦ tr( f ) ◦ g = tr  (id M ⊗ h ) ◦ f ◦ ( g ⊗ id M )  . In other wor ds, the function tr : C ( Q ⊗ M , M ⊗ P ) − → C ( Q, P ) is natur al in Q and P . F or example, recall from E xample 4.10 that for a function f : Q × M → M , the trace o f the induced map in Ab is the map tr( f ) : Z [ Q ] → Z which sends each q ∈ Q to the num b er of fixed p oin ts of f ( q , − ). In this case, natura lit y in Q means tha t given g : Q ′ → Q , comp osing tr( f ) with Z [ g ] : Z [ Q ′ ] → Z [ Q ] counts the num b er of fixed p oints of f ( g ( q ′ ) , − ). Prop osition 5.2 a ls o implies that quite generally , trac e s “calcula te fixed p oin ts”, as described informally in § 1 . Corollary 5.3 (Fixed p oint prop ert y) . If M is dualizable, ∆ : M → M ⊗ P is a map, f : M → M is an endomorphism, and h : P → P is such that ( f ⊗ h )∆ = ∆ f , then h ◦ tr(∆ f ) = tr(∆ f ) . In the situation discussed after Definition 4.5 of a symmetr ic monoidal functor Z : C → S , with C a cartes ia n monoidal categor y , for a ny morphism f : M → M in C we hav e that ( f × f )∆ = ∆ f . Applying the above corolla ry in S then implies Z ( f ) ◦ tr( Z (∆ f )) = tr( Z (∆ f )). 18 KA TE PONTO AND MICHAEL SHULMAN Note that tr( Z (∆ f )) is a map I → Z ( M ) in S , and not in the original categor y C . How ever, w e can see the connections to fixed po in ts explicitly in a couple examples. If w e co nsider the example where C = Set , S = Ab , a nd Z is the free ab elian g roup functor, then tr(∆ f ) = P f ( x )= x x as computed in Example 4.6 , and Corollar y 5.3 implies that this element of Z [ M ] is fixed by Z [ f ]. The cor responding example Σ : T op → Sp in Example 4.7 behaves similarly . The next few proper ties are of more technical interest. An additiona l naturality prop ert y follows dir ectly from cyclicit y . Prop osition 5.4. L et M and N b e dualizable and let f : Q ⊗ M → N ⊗ P and h : N → M b e maps. Then tr(( h ⊗ id P ) f ) = tr( f (id Q ⊗ h )) . In fancier lang uage, this sa ys that the function tr : C ( Q ⊗ M , M ⊗ P ) − → C ( Q, P ) is “ extraordinary- na tural” (see [ 12 ]) in the dualizable ob ject M . W e now consider compatibility of traces with the monoidal structur e. No te that the unit I is alwa ys dualizable with I ⋆ = I . Prop osition 5.5. If f : Q ⊗ I → I ⊗ P is a morphism in C , then tr( f ) = f (mo dulo unit isomorphisms). If M and N are dualizable, then s o is M ⊗ N , with dua l M ⋆ ⊗ N ⋆ . In this cas e, if w e have a map f : Q ⊗ N ⊗ M − → M ⊗ N ⊗ P, we can either take the trac e of f s with r espect to M ⊗ N , or w e can first take the trace o f f with resp ect to M and then with res pect to N ; either wa y results in a map Q → P . Prop osition 5. 6. In the ab ove situation, we have tr( f s ) = tr(tr( f ) ) . Alternatively , we could hav e tw o maps f : Q ⊗ M → M ⊗ P and g : K ⊗ N → N ⊗ L . Prop osition 5. 7. In the ab ove situation, we have tr  s ( f ⊗ g ) s  = tr( f ) ⊗ tr( g ) . T aking N = I we o btain the following. Corollary 5.8. If M is dualizable and f : Q ⊗ M → M ⊗ P and g : K → L ar e maps, then tr( s ( f ⊗ g )) = tr( f ) ⊗ g . On the other hand, it is not hard to show that Prop osition 5.7 follows from Corollar y 5.8 together with Prop osition 5.6 . Finally , if M and N are dualizable and we hav e maps f : Q ⊗ M → M ⊗ P and g : P ⊗ N → N ⊗ K , then we hav e the comp osite (id M ⊗ g )( f ⊗ id N ) : Q ⊗ M ⊗ N − → M ⊗ N ⊗ K . The next result then follows fr om Prop osition 5.2 and P ropos ition 5.6 . Corollary 5. 9. In the ab ove situation, we have tr  (id M ⊗ g )( f ⊗ id N )  = tr( g ) ◦ tr( f ) . TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 19 In particula r , w e can apply these res ults to un twisted traces. No te that by the Eckmann-Hilton argument, the t wo op erations ⊗ a nd ◦ o n C ( I , I ) agree (up to unit isomor phisms) and make it a commutativ e monoid. W e thereby o bta in the following. Corollary 5. 10. If C is s ymm et ric monoi dal, M and N ar e dualizable, and f : M → M and g : N → N ar e endomorphisms, then tr( f ⊗ g ) = tr( f ) ⊗ tr( g ) = tr( f ) ◦ tr( g ) . One final pro perty o f traces that sho uld b e mentioned here is the following. Prop osition 5.11. If M is dualizable, then the tr ac e of id M ⊗ M : M ⊗ M → M ⊗ M is id M : M → M . In [ 24 ] the ab ov e prop erties were taken to constitute the fo llowing definition. Definition 5.12. A symmetric mono idal categor y C is traced if it is equipp ed with functions tr : C ( Q ⊗ M , M ⊗ P ) → C ( Q , P ) satisfying the conclusio ns of Pro positions 5.2 , 5.4 , 5.5 , 5 .6 , and 5.11 as well as Corollar y 5.8 . Actually , [ 24 ] dea ls with the more general case of a ba lanced monoidal c ategory; we have s implified things by tr e ating only the symmetric case. A simila r set of axioms is consider ed in [ 31 ]. Evidently if C is co mpact closed (every ob ject is dualizable), then it is traced in a canonical (and, in fa c t, unique) way . Conversely , it is shown in [ 24 ] that any traced symmetric monoidal category ca n b e embedded in a compact closed o ne , in a wa y that identifies the given trace with the ca no nical trace. On the other hand, muc h of the int eres t of the canonica l symmetric monoidal trace lies in its applica bilit y to particular interesting dualizable ob jects in ca tegories wher e not every ob ject is dualizable. W e end this section by making the connection to fixed-p oin t op erators mentioned in § 1 precise. If C is a cartes ian monoidal categ ory , a tr ace as in Definition 5.12 defines na tural functions F : C ( Q × M , M ) → C ( Q, M ) where for f : Q × M → M , F ( f ) is defined b y tr( Q × M id × ∆ − − − − → Q × M × M f × id − − − → M × M ) . Then the ab o ve prop erties b ecome the following. • F or f : Q × M → M , F ( f ) = f ◦ (id Q × F ( f )) ◦ (∆ Q × id M ) . • F or f : Q × N → M and g : Q × M → N , F ( f ◦ (id Q × g ) ◦ (∆ Q × id M )) = f ◦ (id Q × F ( f ◦ (id Q × f ) ◦ (∆ Q × id M ))) ◦ (∆ Q × id M ) . • F or f : Q × Q × M → M , F ( f ◦ (id Q × ∆ M )) = F ( F ( f )) . 20 KA TE PONTO AND MICHAEL SHULMAN These are the conditions that define a fixed-p oint op erator . Conversely , any fixed-p oin t op erator F o n a cartesian monoidal ca tegory defines a trace, where the trace o f f : Q × M → M × P is Q F ( f ◦ (i d A × π )) − − − − − − − − − → P × M pr 1 − − → P. Theorem 1 .1 says that these tw o constructions are inverses: thus the existence of a fixed-p oint op erator on a ca rtesian monoidal categ ory is eq uiv alent to that of a trace. 6. Functoriality of traces One of the ma in adv antages o f having a n abstr act formulation of trace is that disparate notions of trace whic h all fall into the gener a l framework can be compared functorially . In this section we summarize the relev ant results and their applicability in so me examples, including the Lefschetz fixed point theo rem. Recall that a l ax symme tri c monoidal functor F : C → D b et ween sym- metric mo noidal catego ries cons is ts of a functor F and natural tra nsformations c : F ( M ) ⊗ F ( N ) − → F ( M ⊗ N ) i : I D − → F ( I C ) satisfying appropria te coherence axioms. W e sa y F is normal if i is an isomo rphism, and s trong if c and i are both isomorphisms. Prop osition 6.1. L et F : C → D b e a normal lax symmetric monoidal functor, let M ∈ C b e dualizable with dual M ⋆ , and assume that c : F ( M ) ⊗ F ( M ⋆ ) → F ( M ⊗ M ⋆ ) is an isomorphism (as it is when F is str ong). Then F ( M ) is dualizable with dual F ( M ⋆ ) . Pr o of. Suppo se given M with dual M ⋆ exhibited b y η and ε . Then the maps I D i − → F ( I C ) F ( η ) − − − → F ( M ⊗ M ⋆ ) c − 1 − − → F ( M ) ⊗ F ( M ⋆ ) and F ( M ⋆ ) ⊗ F ( M ) c − → F ( M ⋆ ⊗ M ) F ( ε ) − − − → F ( I C ) i − 1 − − → I D show that F ( M ) is dualizable with dual F ( M ⋆ ).  In the abov e situation, we sa y that F preserv es the dual M ⋆ of M . Actually , a s ligh tly weaker condition on F suffices fo r the above conclus ion; see [ 6 ]. Prop osition 6.2. If F pr eserves the dual M ⋆ of M , and mor e over c : F ( P ) ⊗ F ( M ) → F ( P ⊗ M ) is an isomorphism (as it is whenever P = I and F is normal), then for any map f : Q ⊗ M → M ⊗ P , we have F (tr( f )) = tr  c ◦ F ( f ) ◦ c − 1  . In p articular, for an endomorphism f : M → M , we have F (tr( f )) = i ◦ tr ( F ( f )) ◦ i − 1 Pr o of. Use the dual F ( M ⋆ ) of F ( M ) to ev a luate the righ t hand side.  TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 21 Example 6.3 . If R and S are co mm utative rings and φ : R → S is a ring homo- morphism, then extension o f scala rs defines a strong symmetric monoida l func- tor ( − ⊗ R S ) fro m R -modules to S -mo dules. If M is a dualizable R -mo dule and f : Q ⊗ M → M ⊗ P is a map of R -modules, Prop osition 6.2 implies tr( f ⊗ R S ) = tr( f ) ⊗ R S . If Q and P are bo th the r ing R , then as usual, we ca n think of the traces of f and f ⊗ R S as elements o f R and S , resp ectiv ely; in this ca se, we ha ve tr( f ⊗ R S ) = tr( f ) ⊗ R S = φ ( tr ( f )). Example 6.4 . Homology is a normal la x symmetric monoidal functor from Ch R or Ho( Ch R ) to the categor y GrMo d R of gra ded R -mo dules. The K ¨ unneth theorem implies that the natural tr ansformation c : H ( M ∗ ) ⊗ H ( N ∗ ) → H ( M ∗ ⊗ N ∗ ) is an isomo r phism if M p and H p ( M ∗ ) are pro jective for each p . When these con- ditions are sa tisfied (such as when the g round r ing R is a field), Prop osition 6.2 implies tr( H ∗ ( f )) = H ∗ (tr( f )) for any map of chain complexes f : M ∗ → M ∗ . In other w ords, the Lefsc hetz num be r is the same whether it is calculated a t the lev el of chain complexes or homology . Example 6 .5 . By co mposing the rational cellular chain complex functor with a functorial CW approximation, we obtain a functor T op → Ch Q . In fa ct, this functor fa ctors throug h Sp via a similar construction of CW spectra , and we ha ve an induced functor Ho( Sp ) → Ho( Ch Q ) which is stro ng symmetric monoidal. It follows that the fix e d p oint index of a co ntin uous map is equal to the Lefschetz nu mber of the induced map on chain co mplexes. Combining this with the pr evious example, a nd using ratio na l co efficient s s o the K¨ unneth theorem holds, w e o btain the L efschetz fixe d p oint the or em : if f : M → M is a contin uous m ap, where Σ ∞ + ( M ) is dualiz able, and tr H ∗ ( f , Q ) 6 = 0, then tr( f ) 6 = 0, and thus f has a fixed p oin t. This example was one of the original motiv ations for the abstract study of traces in [ 11 ]. Example 6.6 . Genera lizing the previous exa mple, if Σ ∞ + ( M ) is dualizable in Ho( Sp ) and f : Q × M → M is a co n tinuous ma p, then the trace of Σ ∞ + ( f ) in Ho( Sp ) is a morphism Σ ∞ + ( Q ) → S in Ho( Sp ). W e can then take the ra tional homology of this map to obtain a map tr( H ∗ ( f + )) : H ∗ ( Q + ) → Z . On the other ha nd, we can a ls o apply rational homolog y b efore taking the trace; this wa y we obtain a map H ∗ ( f + ) : H ∗ ( Q + ) ⊗ H ∗ ( M + ) → H ∗ ( M + ) whose trace is a map H ∗ ( Q + ) → Z . Prop osition 6.2 then sho ws H ∗ (tr f + )) = tr( H ∗ ( f + )) . When Q is a p oint, the set o f morphisms Σ ∞ + ( Q ) → S in Ho( Sp ) and the set of morphisms H 0 ( Q + ) → Z in Ab can b oth b e identified with Z , so no informa tion ab out tra ces is lo s t by passa g e to rational homo logy . F or g eneral Q , informatio n is lost, but this is not necessa rily a bad thing: the s et of maps Σ ∞ + ( Q ) → S can be difficult to ca lculate, while the set of maps H 0 ( Q + ) → Z will usually b e muc h easier to describ e. Example 6.7 . An n - di mensional top olo gic al field the ory [ 2 ] with v alues in a sym- metric monoidal catego ry C (s uch as V ect k ) is a stro ng symmetric monoidal func- tor Z : n Cob → C . Since e very ob ject M of n Cob is dualiz able, so is each ob ject 22 KA TE PONTO AND MICHAEL SHULMAN Z ( M ). Th us, the tr ace of a cob ordism B from M to itself is mapp ed to an endo - morphism o f the unit o f C , which ca n b e regar ded as an algebra ic inv ariant of B computed by the field theory Z . If n = 1, then 1 Cob is the free s ymmetric mono ida l categ ory on a dualiza ble ob ject; thus a 1-dimensiona l TFT is just a dualizable ob ject. Likewise, if n = 2, then 2 Cob is the free symmetric mono ida l categ ory o n a F rob enius alg e br a; see , for instance, [ 27 ]. F or a higher-dimensiona l ge ne r alization, see [ 30 ]. Finally , since monoidal ca tegories form no t just a ca tegory but a 2-categ ory , it is natural to ask als o how traces interact with monoidal natural tra ns formations. Recall that if F, G : C → D are lax symmetric monoidal functors, a m o noidal nat- ural transformation is a na tural tra nsformation α : F → G which is compatible with the monoida l constr ain ts o f F and G in an ev iden t wa y . Prop osition 6.8. L et F, G : C → D b e normal lax symmetric monoidal functors, let α : F → G b e a monoidal n atu r al tr ansformation, let M b e dualizable in C , and assume that F and G pr eserve its dual M ⋆ . Then α M : F ( M ) → G ( M ) is an isomorphi sm, and for any f : Q ⊗ M → M ⊗ P , the squar e F ( Q ) tr ( c ◦ F ( f ) ◦ c − 1 ) / / α Q   F ( P ) α P   G ( Q ) tr ( c ◦ G ( f ) ◦ c − 1 ) / / G ( P ) c ommut es. In p articular, for an endomorphism f : M → M , we have tr( F ( f )) = tr( G ( f )) . Pr o of. Since F ( M ) and G ( M ) hav e duals F ( M ⋆ ) a nd G ( M ⋆ ) resp ectiv ely , the morphism α M ⋆ : F ( M ⋆ ) → G ( M ⋆ ) has a dual ( α M ⋆ ) ⋆ : G ( M ) → F ( M ). A diagram chase (see [ 6 , Pro p. 6]) shows tha t this is a n inv erse to α M . Then since G ( f ) = α M ◦ ( F ( f ) ) ◦ ( α M ) − 1 , cyclic it y implies that tr( F ( f )) = tr( G ( f )).  R emark 6.9 . In pa rticular, if C is compact clos e d and F , G : C → D a r e stro ng monoidal, then any monoidal transfor mation F → G is an isomor phis m. Thus, when we say 1 Cob is the free symmetric monoidal categor y o n a dualiz a ble ob ject, as in Example 6 .7 , we r eally mean that the categ ory of stro ng monoidal functors 1 Cob → D is equiv alent to the gr oup oid of dualizable o b jects in D and iso mor- phisms b et ween them. This also genera lizes to higher dimensions. Prop osition 6.8 implies some useful “co mparisons betw een co mparisons” of w ays to co mpute traces . Example 6.1 0 . As in Example 6.4 , since Q is a field, the K ¨ unneth theore m implies that the functor H ∗ ( − ; Q ) fro m Ho( Sp ) to the catego ry GrV e ct Q of graded Q - vector spaces is strong symmetric monoidal. While integral ho mology H ∗ ( − ; Z ) is not s trong symmetric monoidal, the K ¨ unneth theorem implies that it b ecomes so if we quotient b y tor sion; thus H ∗ ( − ; Z ) / T o rsion is a strong symmetric monoidal functor from Ho( Sp ) to GrMo d Z . Hence we can co mput e Lefschetz num be r s us ing int egra l homo logy as w ell, a nd it is natura l to want to compar e the tw o results. TRA CES IN S YMMETRIC MONOID AL CA TEGORIES 23 As in Example 6.3 , ex tension of s c a lars along the inclusion ι : Z → Q defines a strong symmetr ic monoidal functor fro m GrMo d Z to GrV ect Q . Thus we have t wo functors H ∗ ( − ; Q ) and  H ∗ ( − ; Z ) / T o rsion  ⊗ Q from Ho( Sp ) to GrV ect Q , and the same inclusion also defines a natural transfor - mation α :  H ∗ ( − ; Z ) / T o rsion  ⊗ Q − → H ∗ ( − ; Q ) . Therefore, we can combine Prop ositions 6.2 a nd 6.8 to compare the Lefschetz n um- ber s computed using H ∗ ( − ; Z ) and H ∗ ( − ; Q ). Explicitly , supp ose Σ ∞ + ( M ) is dualiz a ble and f : M → M is an endomo rphism in Ho( Sp ). Then Prop osition 6.8 implies that, first of a ll, α is an isomorphism  H ∗ ( M ; Z ) / T orsion  ⊗ Q ∼ = H ∗ ( M ; Q ) , and secondly , t he tra ce of  H ∗ ( f ; Z ) / T orsion) ⊗ Q is the sa me as the trace of H ∗ ( f ; Q ). Since this trac e is not twisted, the obser v ation at the end of Example 6 .3 implies tr   H ∗ ( f ; Z ) / T orsion  ⊗ Q  = ι  tr  H ∗ ( f ; Z ) / T orsion   ; th us the Lefschetz num b er of f computed using H ∗ ( − ; Z ) / T o rsion is the sa me as the Lefsc hetz num b er computed using H ∗ ( − ; Q ). 7. Vist as The symmetric monoidal tr ace describ ed in this pap er can b e generalized in v a ri- ous directions. W e hav e already mentioned its generalizations to b alanc e d m onoidal c ate gories (at the end o f § 3 ) and to tr ac e d monoidal c ate gories ( Definition 5.12 ). There are also str a igh tforward ge neralizations to symmetric monoidal 2-categories and s ymmetric monoidal n -categorie s (mo dulo a definition of the la tter). Categorifying in a different direction, in [ 39 ] the first author intro duced a gen- eral notion of tr a ce for bic ate gories , reg arded as “monoidal categ ories with many ob jects”. 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