Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces

DERIVED K OSZUL DUAL ITY AND INV OLUTIONS IN THE ALGEBRAIC K -THEOR Y OF SP A CES ANDREW J. BLUMBERG AND MICHAEL A. MANDELL Abstract. W e int erpret differen t c onstructions of the algebraic K -theo ry of spaces as an instance of deriv ed Koszul (or bar) dualit y and also as an instance of Morita equiv alence. W e relate the in terpla y betw een these t w o descriptions to the homotopy in v olution. W e define a geometric analog of the Sw an th eory G Z ( Z [ π ]) i n terms of Σ ∞ + Ω X and show that it is the algebraic K -theory of the E ∞ ring sp ectrum D X = S X + . 1. In tr oduction Asso ciated to a spac e X are t w o ring sp ectra , Σ ∞ + Ω X , the fre e susp ension sp e c- trum on the bas ed lo o p space of X , a nd D X = S X + , the Spanier-Whitehead dual of X . W aldhause n [31] defined the algebraic K -theor y o f X , A ( X ), a s the K -theor y of the r ing sp ectrum Σ ∞ + Ω X . This theory has deep g eometric co nten t: when X is a manifold, A ( X ) contains the stable pseudo - isotopy theory of X , and when X is a finite complex, A ( X ) is a receptacle for “hig her tors io n inv ariants” [7 ] a nd close ly related to transfers [18]. On the other hand, the unsta ble ho motopy theory of X is enco ded in the E ∞ ring spec trum D X [20]. Recent work of Mora v a [24] conjectures the str ucture and pr op erties for a categ o ry of homotopy theor etic motives in ter ms of the sta biliz a tion of a categor y o f cor resp ondences; one candidate construc tio n put forward is built from algebraic K - theory of ring sp ectra o f the for m D X . Based o n computations in T H H motiv ated by string top olo gy , Ralph Cohen con- jectured a dua lit y b etw een K ( D X ) and A ( X ) as mo dules over K ( S ) [6]. Although the non-c onnectivity of D X means that tr ace metho ds fail to apply to K ( D X ), in this pap er we construct such a dua lit y in terms of derived Ko szul duality when X is a simply connected finite CW co mplex. In differential graded algebra , derived Kosz ul duality (o r bar duality) concerns the con trav aria nt adjunction be tw een the category of augmented differential graded algebras and itself [23, 16] (named for the sp ecial cas e of K oszul algebras [25, 1, 2]). The dual of an augmented differen tial graded k -algebr a A is an augmented differential g raded k -algebra E that mo dels the A - mo dule endomor phisms of k , End A ( k , k ). Under mild hypotheses, A ≃ End E ( k , k ); the contrav aria nt functors Hom A ( − , k ) and Hom E ( − , k ) form a n adjunction on the mo dule c a tegories and an equiv alence b etw een v arious thic k sub categor ies of the der ived categ o ries. In our co nt ext, Σ ∞ + Ω X forms a n augmented S -a lgebra, and we can iden tify the augmented S -algebra of Σ ∞ + Ω X -endomorphisms of S as the augment ed S -algebra Date : Septem ber 14, 2018. 2000 Mathematics Subje ct Classific ation. Pr imary 19D10; Seconda ry 18F25. The first author was supported in part by NSF gran t DMS-0906105. The second author was supported in part by NSF gran t DMS-0804272. 1 2 ANDREW J. BL UMBERG AND MICHAEL A. MANDELL D X [9, § 4.22 ]. In fact, the coherent Σ ∞ + Ω X -mo dule equiv alence S ∧ S ≃ S makes the endomorphism r ing sp ectrum natura lly comm utative, compatibly with the nat- ural commutativ e S -alg ebra structure o n D X . Int erpreting Ext Σ ∞ + Ω X ( − , S ) and Ext DX ( − , S ) as contra v arian t adjoints on der ived categor ies D Σ ∞ + Ω X / / D DX , o o we get equiv a lences upon restricting to certain s ub ca tegories. F or example, the s ub- category D c Σ ∞ + Ω X of compact ob jects o f D Σ ∞ + Ω X is the thick s ubca tegory ge ne r ated by Σ ∞ + Ω X and is eq uiv a lent under this adjunction to the thick sub categor y T DX ( S ) of D DX generated by S . This is reminiscent of W aldhausen’s co mparison of the stable category o f Ω X -spa ces with the sta ble ca teg ory o f re tr active spaces over X . W riting M DX ( S ) fo r the subcateg ory o f the mo del category of D X -mo dules that are iso morphic in D DX to ob jects in T DX ( S ), we pr ove the following theore m. In the following theorem and a ll theorems in this section, w e under stand X to b e a simply co nnected finite CW-complex. Theorem 1.1. K ( M DX ( S )) is we akly e quivalent to A ( X ) = K (Σ ∞ + Ω X ) . On the o ther hand, the s ubca tegory D c DX of compact ob jects of D DX is the thic k sub c ategory genera ted by D X and is equiv alen t under the adjunction ab ove to the thic k sub catego r y T Σ ∞ + Ω X ( S ) of D Σ ∞ + Ω X generated by S . The ca tegory T Σ ∞ + Ω X ( S ) is a geometric a nalogue of the categor y of finite r ank pro jective π 1 X -mo dules, whose K -theo r y G Z i ( Z [ π ]) was studied b y Swan [28]. W e call the K -theo ry of the categor y M Σ ∞ + Ω X ( S ) the geometr ic Swan theory of the spa ce X and denote it as G ( X ). W e prov e the following theorem. Theorem 1.2. G ( X ) = K ( M Σ ∞ + Ω X ( S )) is we akly e quivalent to K ( D X ) . In fact b oth G ( X ) and K ( D X ) are commutativ e ring spe ctra, and the equiv a- lence is a weak equiv a lence of r ing spe c tra. Likewise A ( X ) is a module spectrum ov er G ( X ) and K ( M DX ( S )) is a mo dule sp ectrum ov er K ( D X ); the equiv alence in Theorem 1.1 is a weak equiv alence o f module sp ectra. In fact, we have the following more pre c ise r esult. Theorem 1.3. The we ak e qu ivalenc e G ( X ) → K ( D X ) is a map of E ∞ ring sp e c- tr a. The we a k e quiva lenc e K ( M DX ( S )) → A ( X ) is a map of G ( X ) -mo dules. Because X is a finite CW complex, the Σ ∞ + Ω X -mo dule S is compact, and so we can in terpret the map on K -theory induced by inclusion of the thick s ubca tegory generated by S into the thick sub c a tegory of compa c t Σ ∞ + Ω X -mo dules in terms of W aldhausen’s fibratio n theorem. W e obtain a lo caliza tion sequence of K -theor y sp ectra (1.4) G ( X ) − → A ( X ) − → K ( C Σ ∞ + Ω X /ǫ ) where C Σ ∞ + Ω X /ǫ is the W aldhausen categ ory of compa ct Σ ∞ + Ω X -mo dules but with weak equiv a lences the maps whose cofiber is in T Σ ∞ + Ω X ( S ). (W e typically do no t hav e a corr esp onding transfer A ( X ) → G ( X ) b ecaus e S is not usua lly a compact D X -mo dule when X is a finite complex.) W e intend to study this seq uence further in a future pa per . Derived Koszul duality be t ween categor ie s o f mo dules ov er differen tial g raded algebras is a con trav ar iant pheno menon, but there is als o a n asso cia ted cov ariant DERIVED KOSZUL DUALITY AND INVOLUTIONS 3 Morita adjunction switc hing chirality fr o m left modules to rig h t mo dules [9]. (F or a s urvey on Morita theory in sta ble homotopy theory , see [27].) In the pr esence of an a nt i-inv olution (for e x ample, c o mmu tativity), we can us e the a nt i-inv olution to obtain a Morita adjunction b etw een categ o ries of left mo dules. In the context of D X and Σ ∞ + Ω X , we get tw o cov ar iant adjunctions D Σ ∞ + Ω X / / D DX o o given by the adjoint pairs Ext Σ ∞ + Ω X ( S, − ) , T o r DX ( − , S ) and T or Σ ∞ + Ω X ( − , S ) , E xt DX ( S, − ) . The fir s t restricts to an equiv alence T Σ ∞ + Ω X ( S ) ≃ D c DX (in fact, since S is compa c t as a n Σ ∞ + Ω X mo dule, the adjunction restricts to em bed D DX as the lo c alizing sub categ ory of S in D Σ ∞ + Ω X ). The second re s tricts to an equiv alence D c Σ ∞ + Ω X ≃ T DX ( S ) . These equiv a lences give r ise to equiv ale nce s on algebraic K -theory , akin to the equiv alences of Theorems 1.1 and 1.2. The comp osites ar e self-maps on A ( X ) and G ( X ). In Section 5, w e identif y these se lf-ma ps as the standard homo topy inv olutions. Exp erts will reco gnize that Theorems 1.1 and 1.2 fit in to the framework of [9] and [29]; the b enefit of the approa ch here is the description in terms of concrete mo dels, which allow more direct compariso ns than in the abstract appr oach, and more pre c ise r esults such as The o rem 1.3. Readers ma y also wonder ab out the connection to the work of Go r esky , Kott witz, and MacPherson on Koszul duality [14]. F rom our p ersp ective, they study the “dual” se tting in which G = Ω B G is compact and B G is infinite. Our techniques apply to re c over (integral) liftings of their equiv alences of der ived ca tegories; in fact, this cas e was studied in [16]. Because this ex ample is not connected as closely to A -theo ry , we have chosen to omit a deta iled discus sion. The pap er is organized as follows. In Section 2 w e review and sligh tly extend the passage from algebr aic structures o n W aldha us en catego ries to a lgebraic str uctures on K -theory s pec tr a, us ing the techn olog y dev elop ed in [12]. In Section 3 we in- tro duce the c oncrete mo dels for Σ ∞ + Ω X and the endomo rphism ring sp ectra, which allow a go o d point-set mo del fo r the adjunctions in the remaining s e c tions. Sec- tion 4 studies the contra v ariant adjunction and prov es Theor e ms 1.1, 1 .2, and 1 .3. Finally , Section 5 studies the p oint-set mo del o f the cov ar iant adjunctions of [9] and identifies the comp osite homotopy endomorphisms on A ( X ) and G ( X ) as the standard ho motopy inv olutions. The authors would like to thank Ra lph Cohen and Bruce Williams for a sking motiv ating questions, as well as Haynes Miller, Jo hn K le in, Ja ck Mora v a, and John Rognes for helpful conv ersations. 2. A lgebraic structures on Wald hausen K -theor y Since even b efore the a dvent of the theo ry of symmetric spec tr a [1 5], exp erts hav e under sto o d that any algebraic structure o n a W aldhausen categ o ry induces an analogo us structure on W aldhausen K -theor y . Sources for results o f this type in 4 ANDREW J. BL UMBERG AND MICHAEL A. MANDELL the litera ture include [31, p. 342], [1 3, App. A], [12]. W e briefly review the curr e n t state o f the theory here. W e re fer the r eader to [31, § 1.2] for the definition of a W aldhausen catego ry (called there a “ca tegory with cofibrations and weak equiv alences”). Reca ll that W aldhausen’s S • construction [31, § 1 .3] pro duces a simplicial W aldhausen ca tegory S • C from a W aldhausen categor y C and is defined as follows. Let Ar [ n ] deno te the category with ob jects ( i, j ) for 0 ≤ i ≤ j ≤ n and a unique map ( i, j ) → ( i ′ , j ′ ) for i ≤ i ′ and j ≤ j ′ . S n C is defined to b e the full s ub ca tegory of the ca tegory of functors A : Ar[ n ] → C such that: • A i,i = ∗ for all i , • The map A i,j → A i,k is a cofibratio n for all i ≤ j ≤ k , and • The diagram A i,j / /   A i,k   A j,j / / A j,k is a pushout squar e for all i ≤ j ≤ k , where w e write A i,j for A ( i, j ). T he las t tw o co nditions can be simplified to the hypothesis that e ach map A 0 ,j → A 0 ,j +1 is a co fibration and the induced maps A 0 ,j / A 0 ,i → A i,j are isomor phisms. This b ecomes a W aldhaus en category b y defin- ing a map A → B to b e a weak e q uiv alence when ea ch A i,j → B i,j is a w eak equiv alence in C , a nd to b e a c o fibration when each A i,j → B i,j and ea ch induced map A i,k ∪ A i,j B i,j → B i,k is a cofibration in C . Since S • C forms a simplicial W a ldha usen ca tegory , the construction can be it- erated to form S • S • . . . S • C . F or our purp oses , it is con venien t to ha ve an “a ll at once” construction of the q - th iterate S ( q ) • ,..., • C . F or this cons truction, we need the following ter minology (see also [26, § 2]). Definition 2 . 1. Le t [ n ] denote the order ed set 0 ≤ 1 ≤ · · · ≤ n . F or a W aldhausen category C , a functor C : [ n 1 ] × · · · × [ n q ] → C is cu bic al ly c ofibr ant mea ns that: (i) E very ma p C ( i 1 , . . . , i q ) → C ( j 1 , . . . , j q ) is co fibration, (ii) in e very sub- square (1 ≤ r < s ≤ q ) C ( i 1 , . . . , i q ) / /   C ( i 1 , . . . , i r + 1 , . . . , i q )   C ( i 1 , . . . , i s + 1 , . . . , i q ) / / C ( i 1 , . . . , i r + 1 , . . . , i s + 1 , . . . , i q ) , the induced map from the pushout to the lower-right en try C ( i 1 , . . . , i r + 1 , . . . , i q ) ∪ C ( i 1 ,...,i q ) C ( i 1 , . . . , i s + 1 , . . . , i q ) − → C ( i 1 , . . . , i r + 1 , . . . , i s + 1 , . . . , i q ) is a cofibratio n, (iii) a nd in genera l, in every m -dimensional sub-cub e s pe c ifie d by choos ing m distinct co o r dinates 1 ≤ r 1 < r 2 < . . . < r m ≤ n , the induced ma p from the colimit over the diagram o btained b y deleting Q = C ( i 1 , . . . , i r 1 + 1 , . . . , i r 2 + 1 , . . . , i r m + 1 , . . . , i n ) to Q is a cofibr ation. DERIVED KOSZUL DUALITY AND INVOLUTIONS 5 Construction 2.2 (Iterated S • Construction) . Let Ar[ n 1 , . . . , n q ] denote the cat- egory Ar[ n 1 ] × · · · × Ar[ n q ]. F or a functor A : Ar[ n 1 , . . . , n q ] = Ar[ n 1 ] × · · · × Ar[ n q ] − → C , we wr ite A i 1 ,j 1 ; ... ; i q ,j q for the v alue of A on the ob ject (( i 1 , j 1 ) , . . . , ( i q , j q )). F or a W aldhausen catego ry C , let S ( q ) n 1 ,...,n q C b e the full sub catego ry of functors A (as ab ov e) such that: • A i 1 ,j 1 ; ... ; i q ,j q = ∗ whenever i k = j k for some k . • The subfunctor C ( j 1 , . . . , j q ) = A 0 ,j 1 ; ··· ;0 ,j q : [ n 1 ] × · · · × [ n q ] − → C is cubically cofibra nt . • F or every ob ject ( i 1 , j 1 ; . . . ; i q , j q ) in Ar[ n 1 ] × · · · × Ar[ n q ], every 1 ≤ r ≤ q , and every j r ≤ k ≤ n r , the square A i 1 ,j 1 ; ... ; i q ,j q / /   A i 1 ,j 1 ; ... ; i r ,k ; ... ; i q ,j q   A i 1 ,j 1 ; ... ; j r ,j r ; ... ; i q ,i q / / A i 1 ,j 1 ; ... ; j r ,k ; ... ; i q ,i q is a pushout squar e. The sub categor y w S ( q ) n 1 ,...,n q C consists of the maps in S n 1 ,...,n q C that ar e o b jectwise weak equiv alences. W e understand S (0) C to b e C and we see that S (1) n C is S n C . F ollowing W aldhausen [3 1, p. 330], we define the K -theo r y sp ectrum of a W ald- hausen categor y C to b e the sp ectrum w ith q - th space K C ( q ) = N (w S ( q ) • ,..., • C ) = | N • (w S ( q ) • ,..., • C ) | , the g eometric re a lization of the nerve of the multi-simplicial ca tegory w S ( q ) • ,..., • C . The susp ension maps Σ K C ( q ) → K ( q + 1) are induced on diagrams by the pro jectio n map Ar[ n 1 ] × · · · × Ar[ n q ] × Ar[ n q +1 ] − → Ar[ n 1 ] × · · · × Ar[ n q ] . Defining an action of Σ q on K C ( q ) b y permuting the simplicial dir ections, we se e from the explicit description of S ( q ) • ,..., • C ab ove, that K C forms a symmetric spe c- trum. W e can enco de an alge braic s tr ucture on a set of symmetric sp ectra using a symmetric multic a te gory (also called c olor e d op er ad ). A s y mmetric m ulticategor y M enr iched in (small) catego ries consists of: • A se t of obje cts Ob M . • A (small) category of k -morphisms M k ( x 1 , . . . , x k ; y ) for all k = 0 , 1 , 2 , . . . and all x 1 , . . . , x k , y ∈ Ob M . • A unit ob ject 1 x in M 1 ( x ; x ) for each x ∈ Ob M • F or every p ermutation σ ∈ Σ k , an isomorphism σ ∗ : M k ( x 1 , . . . , x k ; y ) − → M k ( x σ 1 , . . . , x σk ) , compatibly ass emblin g to an a ction of Σ k on ` M k ( x 1 , . . . , x k ; y ). • Co mpo sition ma ps M n ( y 1 , . . . , y n ; z ) × ( M j 1 ( x 1 , 1 , . . . , x 1 ,j 1 ; y 1 ) × · · · × M j n ( x n, 1 , . . . , x n,j n ; y n )) − → M j ( x 1 , 1 , . . . , x n,j n ; z ) 6 ANDREW J. BL UMBERG AND MICHAEL A. MANDELL satisfying the ana logue of the usual conditions for a n o pe r ad [21, pp.1 –2]; these a r e written o ut in [12, § 2]. The following definition is standard: Definition 2.3. Let M be a symmetric m ulticategor y enriched in small catego ries. An M -a lgebra A in symmetric spectr a consis ts of a symmetric spectrum A ( x ) for each x ∈ Ob M and maps of symmetric spectr a N ( M k ( x 1 , . . . , x k ; y )) ∧ A ( x 1 ) ∧ · · · ∧ A ( x k ) − → A ( y ) for all k , x 1 , . . . , x k , y , which are compatible with the compositio n maps and identit y ob jects of M . Here (as ab ov e) N ( − ) deno tes the geo metric realization o f the nerve of the ca tegory . When k = 0 , we understand the map pictured abov e a s N ( M (; y )) ∧ S → A ( y ). T o define an M -a lgebra in W aldhaus en ca teg ories, we fir st need to descr ibe the kinds o f functors ob jects of M k should ma p to. Definition 2.4. Let C 1 , . . . , C n and D be W aldhausen ca tegories. A functor F : C 1 × · · · × C n − → D is mu ltiexact if it sa tisfies the following conditions: • F ( X 1 , . . . , X n ) = ∗ if a ny o f X 1 , . . . , X n is ∗ . • F is exact in ea ch v ariable (pres erves weak equiv alences, cofibra tions, and pushouts o ver cofibrations in each v ariable, keeping the o ther v aria bles fixed). • Given cofibrations X k, 0 → X k, 1 in C k for all k , the diagra m A ( i 1 , . . . , i n ) = F ( X 1 ,i 1 , . . . , X n,i n ) : [1 ] × · · · × [1] − → D is cubically cofibra nt . W e define the ca tegory of multiexact functors Mult n ( C 1 , . . . , C n ; D ) to have o b jects the m ultiexact functor s a nd maps the natural weak equiv alences. F or n = 0 , we define Mult 0 (; D ) to be w D , the s ubca tegory o f weak equiv alences in D . Because multiexact functor s comp o s e in to m ultiexact functors, the definition ab ov e makes the categ o ry of small W aldhausen categ ories into a symmetric multi- category enriched in categor ies. F ollowing [12], w e define an M -algebr a in W ald- hausen categor ie s a s a ma p of symmetric m ulticategories enriched in catego ries. Definition 2.5. Let M be a symmetric m ulticategor y enriched in small catego ries. An M -algebra C in W aldhausen categor ies consists of a W aldhause n catego ry C ( x ) for each x ∈ Ob M and functor s M k ( x 1 , . . . , x k ; y ) − → Mult k ( C ( x 1 ) , . . . , C ( x k ); D ) for all k , x 1 , . . . , x k , y , which a re compatible with the permutations, comp osition maps, a nd identit y ob jects of M . Recalling the universal prop erty of the smash pro duct of symmetr ic s pec tr a [15, 2.1.4], the following theor e m is immediate fro m ins p ectio n of the definitions ab ove. Theorem 2 . 6. Waldhausen ’s algebr a ic K -the ory functor natur al ly takes M -alge- br as in W aldhausen c ate gori es to M -algebr as in symmetric sp e ctr a. DERIVED KOSZUL DUALITY AND INVOLUTIONS 7 In par ticular, as ex plained in [12, § 9 ], the preceding theor em applies to descr ib e the alg e br aic structures on K -theo r y s pe c tra induced b y pa irings on the level of W aldhausen ca tegories . Supp ose that C is W aldhausen categor y which is als o a per mutativ e category , where the pro duct ⊗ : C × C → C is a biexact functor ; we will refer to C as a p ermutative Wald hausen c ate gory . Rec a ll that a p ermutative ca tegory is a rigidified for m of a sy mmetr ic mono idal category: a per mut ative catego ry is a symmetric monoida l catego r y where the pro duct satisfies strict a sso ciativity and unit r elations (the a sso ciativity and unit iso morphisms are the identit y). If C is a per mutativ e W aldhausen catego ry , a strict Waldhausen mo dule over C consis ts of a W aldha usen ca tegory Q and a biexact functor C × Q → Q sa tisfying the evident strict a sso ciativity and unit rela tions. The structure of a p ermutativ e W aldhausen categor y on C is equiv alen t to an algebra in W aldhausen c a tegories for the symmetric multicategory E Σ ∗ [12, § 3], where the unique o b ject of E Σ ∗ is taken to C . Then K C b eco mes an E Σ ∗ -algebra in symmetric sp ectr a; this is a par ticular type of E ∞ -algebra symmetric sp ectrum, which is a n asso cia tive ring symmetric sp ectra b y neglect of structure (the sym- metric multicategory of ob jects o f E Σ ∗ is the o p er ad Σ ∗ of sets). Similarly , the structure of a strict W aldha usen mo dule over C on Q is equiv alent to spec ifying an alg e bra in W aldhause n categor ies for the symmetr ic multicategory asso cia ted to E Σ ∗ parametrizing m o dules, called E ℓ M Σ ∗ in [1 2, § 9.1], such that the “ring ob ject” is taken to C and the “module ob ject” to Q . Then K Q b ecomes a K C -mo dule in symmetric sp ectra. Corollary 2. 7. L et C b e a p ermutative Waldhausen c ate gory. Then K C is nat- ur al ly an E Σ ∗ -algebr a symmetric sp e c trum, and in p articula r an asso ci ative ring symmetric sp e ctrum. Mor e over, if D is a strict Waldhausen C -mo dule, then K D is natur al ly a K C -mo dule. W orking with a p er m utative pro duct has the a ppea ling consequence that the m ulticategor y that arises is a familiar one, namely , the categor ic a l Barr att-Eccles op erad E Σ ∗ . How ev er, the catego ries that we work with in this pap er (and that tend to ar ise in pr actice) a re s ymmetric monoidal categor ies rather than per mut ative categorie s. This is no rea l limitation, s inc e a standar d cons tr uction [17] rectifies any s ymmetric monoidal ca tegory in to a n equiv alent p ermutativ e categ o ry: The rectification of C is a catego ry C ′ with ob jects the “words” in the ob jects of C , where a word ( X 1 , X 2 , . . . , X r ) corres po nds to the pro duct λ ( X 1 , . . . , X r ) = ( · · · ( X 1 ⊗ X 2 ) ⊗ · · · ) ⊗ X r in C ; we as so ciate the empty word in C ′ to the unit of the monoidal pro duct. The morphisms in C ′ are precisely the morphisms in C betw een the ass o ciated pro ducts C ′ (( X 1 , . . . , X r ) , ( Y 1 , . . . , Y s )) = C ( λ ( X 1 , . . . , X r ) , λ ( Y 1 , . . . , Y s )) Concatenation provides the permutativ e structure. Sending a word to the asso ci- ated pro duct λ defines a stro ng symmetric mo noidal functor C ′ → C . The inclusio n of C in C ′ as the singleton words is also a strong symmetr ic mono idal functor; the comp osite functor C → C is the identit y , while the comp osite functor C ′ → C ′ is naturally isomorphic to the identit y via the map corres po nding to the identit y map on the asso ciated product. When C is a W aldhausen ca tegory and ⊗ is biex a ct, we use the v ariant where we lo ok at words in o b jects that are not ∗ together with a distinguished zero ob ject ∗ , and for ce a concatenation in C ′ with ∗ to result 8 ANDREW J. BL UMBERG AND MICHAEL A. MANDELL in ∗ . The resulting category C ′′ bec omes a W aldhausen catego ry when we define the cofibrations and weak equiv alences to be thos e maps that corresp o nd to weak equiv alences and cofibratio ns in C . The functor s ab ove remain strong symmetric monoidal equiv alences, but now ar e exact functors as well. Alternatively , at the cost of complicating the multicategory in Coro llary 2.7, we c a n work dire c tly with symmetric monoidal W aldhausen categ ories (i.e., W ald- hausen categories that are sy mmetric mono idal under a biexact pro duct). Speci- fying such a structure on C is equiv alen t to s pec ifying the structure of an algebra ov er a certain symmetric multicategory B enric hed in small categories, defined as follows: Ob B is a single element. F or k = 1 , B 1 is the ca tegory with one ob ject and the identit y mo rphism. F or k > 1, B is the ca tegory with ob jects the lab elled planar bina ry trees with k leaves, having a unique morphism b etw een a ny t wo ob- jects. The p ermutation action p ermutes the la b els . As ab ove, there is a symmetric m ulticategor y pa r ametrizing mo dules in this s etting; an action of C on a W ald- hausen categor y D through a biexact functor endows ( C , D ) with the str ucture of an a lgebra ov er this mo dule multicategory . W e hav e the following conse q uence: Corollary 2.8 . L et C b e a symmetric monoidal Waldhausen c ate gory. Then K C is natur al ly an B -algebr a symmetric sp e ctrum. Mor e over, if D is a s ymm et ric monoidal Waldhausen C -mo dule, then K D is natur al ly a K C - mo dule (p ar ametrize d by t he mu ltic ate gory of E ∞ -mo dules asso ciate d to B ). 3. Mod els for endomorphism S -algebras and the d ouble centralizer conditio n Classically , for a k -algebra R and a R -mo dule M , the double centralizer condition for M is the requirement that the natura l ma p R − → End End R ( M ,M ) ( M , M ) be an isomo rphism. Dwyer, Greenlees, and Iyengar [9] studied the derived form of this conditio n. They study the example of R = Σ ∞ + Ω X and D X ≃ End R ( S, S ) in [9, § 4.2 2]. W e review this example in this section in terms of sp ecific mo dels we use in the r e mainder of the pa p er. In o ur context, we are interested in the case when X is a finite CW complex. As we will see be low, Dwyer’s results on con vergence of the Eilenberg-Mo o re sp ectra l sequence [8] imply that the double cent ralizer ma p cannot b e a weak eq uiv alence unless X is simply connected (as this is the only case in which π 1 X acts nilpo tent ly on H 0 (Ω X )). Once w e r estrict to this cont ext, we can assume without lo ss of generality that X is the geo metric realiza tion of a r educed finite s implicia l set. Then we ha ve a top ologic a l group mo del G for Ω X (given by the g eometric rea lization of the Kan lo op gr oup), a nd a free G -CW complex P whose quotient by G is X (the t wisted cartesian pro duct G • × τ X • for the universal t wisting function τ ; see for example Chapter VI of [22]). Notation 3 . 1. Let X , P , and G b e as ab ove. Let R = Σ ∞ + G , regarded as a n EKMM S -algebr a [11, IV.7.8]. Let S P = Σ ∞ + P , and let E = F R ( S P, S P ). W e reg ard S as a R -algebr a via the augmentation R → S (induced by the map G → ∗ ). The map S P → S (induced by the map P → ∗ ) is a weak equiv alence of R -mo dules. Although S P is no t cofibrant, it is semi-co fibrant [19, 1.2], meaning DERIVED KOSZUL DUALITY AND INVOLUTIONS 9 that the functor S P ∧ S ( − ) = P + ∧ ( − ) fro m S -mo dules to R -modules preserves cofi- brations and a cyclic cofibrations [1 9, 1.3(a )]. Since in EKMM S -mo dule categ ories all ob jects ar e fibr ant, E represents the correct endomorphism algebra E xt R ( S, S ) [19, 6.3 ]. These particula r mo dels show the strong parallel b etw een the do uble centralizer condition for Σ ∞ + Ω X and the bar dualit y theor y of [16]. The diago nal map P → P × P → X × P induces an X -como dule str ucture o n S P S P = Σ ∞ + P − → Σ ∞ + ( X × P ) ∼ = X + ∧ Σ ∞ + P = X + ∧ S P . This in tur n endows S P with a left D X - mo dule structure D X ∧ S S P − → D X ∧ S ( X + ∧ S P ) ∼ = ( D X ∧ X + ) ∧ S S P − → S ∧ S S P ∼ = S P. This left D X -mo dule structure commutes with the left R -mo dule structur e, and s o defines a map of S -alg ebras D X − → F R ( S P, S P ) = E . T o s ee that this map is a weak equiv alence, consider the following diag ram D X ∼ = % % K K K K K K K K K K / / F R ( S P, S P )   F R ( S P, S ) , where the righthand map is induced b y the map S P → S (induced by the map P → ∗ ), a nd the sla nted map is the isomorphism induced by the iso morphism P / G = X . This dia gram co mmu tes since the top- r ight comp osite is adjoint to the map D X ∧ S P → S induced by the diago na l P → X × P follo wed by ev aluation o f D X o n X a nd the trivial map P → ∗ , whereas the slanted map is induced b y the map P → X follow ed by ev aluation o f D X on X : D X ∧ P + / /   D X ∧ X + ∧ P +   D X ∧ X + / / S. Since the map F R ( S P, S P ) → F R ( S P, S ) is a weak equiv alence, the S -alg ebra map D X → E is a weak equiv alence. W e ca n o btain a model for the map R → Ext E ( S, S ) as follows. First, it is conv enien t to c ho ose a cofibr ant S -a lgebra approximation E ′ → D X . Then the t wo-sided bar construction S P ′ = B ( D X , E ′ , S P ) is a semi-cofibr ant D X -mo dule approximation o f S P . F urthermore, E ′ -maps S P → S P induce D X -maps S P ′ → S P ′ . By constr uction, the (left) action of R on S P commutes with the (left) action of D X , making S P ′ an R -mo dule in the categor y o f D X -mo dules, or e q uiv a lently , pro ducing a map o f S -algebr as R → F DX ( S P ′ , S P ′ ). This constructs the S -algebra map; we need to s how that this map is a weak equiv alence. Cons ider the cobar construction C • ( ∗ , X , P ), C n ( ∗ , X , P ) = X × · · · × X | {z } n factors × P , 10 ANDREW J. BL UMBERG AND MICHAEL A. MANDELL with cosimplicial maps induced from the diagonal, the inclusion of the ba s ep o int, and the map P → X . The inclusion of G as the fib er o f the fibr ation P → X induces a weak equiv alence G → T ot C • ( ∗ , X , P ). Likewise, we get a ma p R − → Σ ∞ + T ot C • ( ∗ , X , P ) − → T ot Σ ∞ + C • ( ∗ , X , P ) . Results of Dwyer [8] and Bousfie ld [5] (for D ∗ = π S ∗ ) show that this map is a weak equiv alence, as X is simply-co nnected. Mor eov er, when X is not s imply- connected, the “only if ” part o f Dwyer’s results show t hat no mo del of this map will b e a weak equiv alence. The ma p E ′ → DX induces weak equiv alences E ′ ∧ S · · · ∧ S E ′ − → D ( X × · · · × X ) . T ogether with the weak equiv alence of E ′ -mo dules S P → S , these induce w eak equiv alences Σ ∞ + ( X × · · · × X × P ) ∼ = X + ∧ · · · ∧ X + ∧ S P − → F S ( E ′ ∧ S · · · ∧ S E ′ ∧ S, S P ) − → F S ( E ′ ∧ S · · · ∧ S E ′ ∧ S P , S P ) ∼ = F DX ( D X ∧ S E ′ ∧ S · · · ∧ S E ′ ∧ S S P, S P ) . These maps are compa tible with the cosimplicial structure o n the cobar cons truc- tion and the maps induced by the simplicial structure o n the bar construction B ( D X , E ′ , S P ), a nd induce a weak equiv alence on T ot. Finally , the weak equiv- alence of E ′ -mo dules S P ′ → S P induces a weak equiv alence F DX ( S P ′ , S P ′ ) → F DX ( S P ′ , S P ). This descr ib es the maps in the following diag ram. R / /   F DX ( S P ′ , S P ′ )   F DX ( B ( D X , E ′ , S P ) , S P ) T ot Σ ∞ + C • ( ∗ , X , P ) / / T ot F S ( E ′ ∧ S · · · ∧ S E ′ , S P ) O O W e have sho wn all maps but the to p o ne to b e weak equiv alences, and so it suffices to o bserve that the diagram commutes up to homo to py . A t each co simplicial level, the right-do wn comp osite is adjoint to the map R ∧ S E ′ ∧ S · · · ∧ S E ′ ∧ S S P − → S P induced by the action of E ′ and R on S P . The down-right-up comp osite is adjoint to the c o mpo site ma p R ∧ S E ′ ∧ S · · · ∧ S E ′ ∧ S S P − → R ∧ S S − → S P induced by the augmentation E ′ → S , the w eak eq uiv a lence S P → S and the inclusion o f G in P . A contraction P × I → P o n to the basep oint o f P induces a homotopy fr o m the for mer map to the la tter ma p. 4. Contra v ariant equiv alences in algebraic K -theor y and geometric Sw an theor y of sp aces W e now turn to the adjoint functors Ext R ( − , S ): D R / / D DX o o : E xt DX ( − , S ) and describe our point set model for the Quillen a djunction on the mo del ca tegories M R and M DX . The easies t and mo st obvious p oint-set model for these functor s DERIVED KOSZUL DUALITY AND INVOLUTIONS 11 would b e to use the adjunction F R ( − , S P ) : M R → M DX as in the prev ious se ction, but instead we use an equiv alent functor with b etter multiplicativ e prop erties. The diagona l map G → G × G induces a diagonal map R → R ∧ S R , which is clearly a ma p of S -alg ebras. This endows the category M R of R -mo dules with a symmetric monoidal pro duct, given b y ∧ S on the under lying S -mo dules . As D X is a commutativ e S -algebra , the category M DX has a symmetric monoidal pro duct ∧ DX . The diagonal map S P → S P ∧ S S P on S P makes S P a co co mm utative coalgebr a in the category o f R -mo dules and the diagona l map S P → S P ∧ DX S P mak es S P a co commutativ e coa lgebra in the categor y of D X -mo dules. F or our adjunctions, we need a version of S P that is a commutativ e alge br a in b oth categorie s. Notation 4.1 . Let S P ∨ = F S ( S P, S ) ∼ = S P + , a left ( R ∧ S D X )-mo dule. Here F S (and more generally F R and F DX which we use b elow) denotes the function mo dule construction of [11, § I I I.6.1 ]. The commuting left R -mo dule and D X -mo dule structures on S P ma ke F S ( S P, S ) na tur ally a right ( R ∧ S D X )-mo dule, and we turn it in to a left ( R ∧ S D X )-mo dule using co mm utativity of D X and the anti-in v olution R → R induced by the in verse map G → G . Using the diagonal map o n S P , we get now a map of left ( R ∧ S D X )-mo dules S P ∨ ∧ DX S P ∨ − → S P ∨ , which is easily seen to b e as s o ciative and commutativ e in the appr o priate sense. Moreov er, w e hav e a zigzag of weak eq uiv a lences o f left ( R ∧ S D X )-mo dules rela ting S P and S P ∨ , S P ∨ ← − S P ∨ ∧ S S P − → S P , where we make S P ∨ ∧ S S P a left ( R ∧ D X )-mo dule using the dia g onal R -mo dule structure and the D X -mo dule structure on S P ∨ . The left w ard map is induced b y the map of R -mo dules S P → S , and the right w ard map is induced by the diagona l on S P and ev aluation, S P ∨ ∧ S S P = F S ( S P, S ) ∧ S S P − → F S ( S P, S ) ∧ S S P ∧ S S P − → S ∧ S S P ∼ = S P. This is clearly a map of R -mo dules since each map in the compo site is, and it is a map of D X -mo dules since the D X -mo dule s tr ucture on S P ∨ is adjoint to the map D X ∧ S S P ∨ ∧ S S P − → D X ∧ S S P ∨ ∧ S ( X + ∧ S P ) ∼ = ( D X ∧ X + ) ∧ S ( S P ∨ ∧ S S P ) − → S induced by the diagonal on S P and ev aluation. Using the commuting left R -mo dule a nd D X -mo dule structures o n S P ∨ , we get adjoint functor s F R ( − , S P ∨ ): M R / / M DX o o : F DX ( − , S P ∨ ) betw een the (point-set) catego ries of R - mo dules a nd D X -mo dules mo deling the Ext R ( − , S ) and Ext DX ( − , S ) a djunction on derived ca teg ories. The unit ma ps o f this adjunction are the ma ps X − → F DX ( F R ( X, S P ∨ ) , S P ∨ ) and Y − → F R ( F DX ( Y , S P ∨ ) , S P ∨ ) adjoint to the ( R -mo dule and D X -mo dule) maps M ∧ S F DX ( M , S P ∨ ) − → S P ∨ and N ∧ S F R ( N , S P ∨ ) − → S P ∨ 12 ANDREW J. BL UMBERG AND MICHAEL A. MANDELL induced by ev aluation. Since fibr ations and weak equiv alences in EKMM mo dule categorie s a re detected on the underlying S - mo dules, the functor s F R ( − , S P ∨ ) and F DX ( − , S P ∨ ) conv ert cofibrations and acyclic cofibrations to fibrations and acyclic fibrations. The adjunction ab ove is there fore a Quillen adjunction. W e note that these functor ar e la x sy mmetric mono idal. W e hav e the natural transformatio ns F R ( M 1 , S P ∨ ) ∧ DX F R ( M 2 , S P ∨ ) − → F R ( M 1 ∧ S M 2 , S P ∨ ∧ DX S P ∨ ) − → F R ( M 1 ∧ S M 2 , S P ∨ ) and F DX ( N 1 , S P ∨ ) ∧ S F DX ( N 2 , S P ∨ ) − → F DX ( N 1 ∧ DX N 2 , S P ∨ ∧ DX S P ∨ ) − → F DX ( N 1 ∧ DX N 2 , S P ∨ ) induced b y the multiplication S P ∨ ∧ DX S P ∨ → S P ∨ , whic h is both a map of R - mo dules and o f D X -mo dules. Note that b ecause G is a CW complex, M 1 ∧ S M 2 is in fact a cofibrant R -mo dule when M 1 and M 2 are cofibrant R - mo dules. The lax unit natural transfor mations D X − → F R ( S, S P ∨ ) ∼ = F S ( S P ∧ R S, S ) and S − → F DX ( D X , S P ∨ ) ∼ = S P ∨ are induced by the iden tification P /G = X (for the first map) a nd the map of R -mo dules S P → S (for the second map). When M is a cofibrant R -mo dule approximation to S P (for ex a mple, X = S P ∧ S c for a cofibr ant S - mo dule approximation of S ), we ha ve a weak equiv alence of D X -mo dules, D X − → E = F R ( S P, S P ) − → F R ( M , S P ) ≃ F R ( M , S P ∨ ) . It follo ws th at the left der ived functors of F R ( − , S P ∨ ) and F DX ( − , S P ∨ ) induce an equiv alence b etw een the thick sub categ ories of the homotopy categ ories gener ated by S in D R and by D X in D DX . The latter is the ca tegory of compact ob jects D c DX . As in the introduction, we denote the for mer s ubc a tegory by T R ( S ). Prop ositi o n 4. 2 . The derive d functors Ext R ( − , S ) and Ext DX ( − , S ) induc e in- verse e quivalenc es b etwe e n T R ( S ) and D c DX . Likewise, when N is a cofibrant D X -mo dule appr oximation to S P ′ , we have a weak equiv alence of R - mo dules R − → F DX ( S P ′ , S P ′ ) − → F DX ( S P ′ , S P ) ≃ F DX ( S P ′ , S P ∨ ) − → F DX ( N , S P ∨ ) . It follo ws th at the left der ived functors of F R ( − , S P ∨ ) and F DX ( − , S P ∨ ) induce an equiv alence b etw een the thick sub categ ories of the homotopy categ ories gener ated by S in D DX and by R in D R . The latter is the ca tegory of compact o b jects D c R . As in the intro duction, we denote the former sub categ ory by T DX ( S ). Prop ositi o n 4. 3 . The derive d functors Ext R ( − , S ) and Ext DX ( − , S ) induc e in- verse e quivalenc es b etwe e n D c R and T DX ( S ) . W e obtain W aldhausen categor y structures modeling each of the subcateg ories D c R , T R ( S ) in D R and D c DX , T DX ( S ) in D R as follows. W e consider the full sub- category o f cofibr a nt ob jects in the mo del category of R -mo dules or DX -mo dules whose image in the homoto py category lie s in the s ub ca tegory . (T o make these categorie s small, we can fix a set X o f sufficiently larg e cardina lit y and r estrict to DERIVED KOSZUL DUALITY AND INVOLUTIONS 13 ob jects whose po in t-sets are subsets of X as in [3, 1 .7].) W e denote these W ald- hausen catego ries as M c R , M R ( S ), M c DX , and M DX ( S ), resp ectively . W e then get asso ciated K -theo r y sp ectra, including W aldhausen’s alg ebraic K -theory of X and the g eometric Swan theory of X . Definition 4.4. A ( X ) = K ( R ) = K ( M c R ), G ( X ) = K ( M R ( S )), K ( D X ) = K ( M c DX ). The biexact sma s h pro duct ∧ S makes G ( X ) in to an E ∞ ring symmetric sp ectrum and the biexact sma sh pro duct ∧ DX makes K ( D X ) in to an E ∞ ring symmetric sp ectrum b y Theo rem 2.6. Likewise, the biexact functors ∧ S and ∧ DX make A ( X ) int o a mo dule over G ( X ) and K ( M DX ( S )) into a module ov er K ( D X ). W e next explain how the functors F R ( − , S P ∨ ) and F DX ( − , S P ∨ ) induce weak equiv alences of these E ∞ ring symmetric sp ectra and mo dules. Although the functors F R ( − , S P ∨ ) and F DX ( − , S P ∨ ) ar e not exact (and do not la nd in the mo del W aldhausen categories), we do immediately obtain weak equiv alences o f sp ectra K ( D X ) → G ( X ) and K ( M DX ( S )) → A ( X ), us ing the S ′ • construction, a homotopical v arian t of the S • construction introduced in [3]. Rather than working with pushouts over cofibrations, the S ′ • construction dep ends on a theory of “homotopy co cartesian” squares. The S ′ • construction replaces the cofibrations and pus ho uts in S • with homotopy co cartesian squares. Under mild hypotheses [4, App. A], the natural inclusion S • C → S ′ • C is a weak equiv alence. T o study the pr o ducts and pairings , we take a different appro ach that a llows us to contin ue working only with ob jects that are cofibrant. First conside r the catego ries M R ( S ) and M c DX . F or each q , n 1 , . . . , n q , co nsider the categ ory whose ob jects consist of an ele men t A of S ( q ) n 1 ,...,n q M R ( S ), an e le men t B of S ( q ) n 1 ,...,n q M c DX and weak equiv alences φ i 1 ,j 1 ; ··· ; i q ,j q : A i 1 ,j 1 ; ··· ; i q ,j q − → F DX ( B n 1 − j 1 ,n 1 − i 1 ; ··· ; n q − j q ,n q − i q , S P ∨ ) making the Ar[ n 1 ] × · · · × Ar[ n q ] diagr am commute. A ma p ( A, B , φ ) to ( A ′ , B ′ , φ ′ ) consists o f weak equiv alences A → A ′ , B → B ′ such tha t the comp osite A ∗ − → A ′ ∗ φ ′ − → F DX ( B ′ ∗ , S P ∨ ) − → F DX ( B ∗ , S P ∨ ) is φ ∗ . This forms a multi-simplicial catego ry , where we use the o ppo site ordering in each simplicial dir e c tion on S ( q ) • ,..., • M c DX . T aking the classifying spa ce, we o btain a sequence of s paces T ( q ) with the structure of a sy mmetric sp ectr um. The sma s h pro ducts on M R and M DX and la x symmetric mo noidal trans for- mations ab ov e induce maps T ( p ) ∧ T ( q ) − → T ( p + q ) and a multiplication T ∧ T → T . W e o btain a ma p T → G ( X ) dropping the M c DX data; we also obtain a ma p T → K ( D X ) by dropping the M R ( S ) data a nd using the ca nonical ho meo morphism b etw een the g eometric r ealization of a s implicial set and it s opposite. B oth ma ps preserv e the E ∞ structures; Lemmas 4.6 and 4.7 b elow complete the pro of of Theor em A and the first part of Theor em C by showing that these ma ps ar e a re weak e q uiv alences. The analo gous co ns truction, with M c R and M DX ( S ) in place o f M R ( S ) and M c DX , pro duces a symmetric sp ectrum U that is a mo dule ov er T . The a nalogo us maps U → A ( X ) and U → K ( M DX ( S )) a re T -mo dules maps; again Lemmas 4.6 14 ANDREW J. BL UMBERG AND MICHAEL A. MANDELL and 4.7 s how that these maps are weak equiv alences a nd complete the pro o f of Theorem B and the rema ining pa rt o f Theorem C. Before stating Lemma 4.6, we abstract the construction used to build the pieces of T and U . Co nsider the following construction. Construction 4.5. L e t C b e a W aldhausen catego ry , M b e a pointed clos ed mo de l category and let M b e a closed W aldhaus e n subca tegory of cofibrant ob jects in M , i.e., a W aldhausen catego ry under the cofibra tions and weak e q uiv a lences from M , which is clos e d under weak equiv alences in M . Let F : C → M be a cont rav a riant functor that takes ∗ to ∗ , cofibratio ns to fibrations, and weak equiv alences to weak equiv alences. Define M F to be the following categor y . An o b ject of M F consists of an ob ject A of M , an ob ject B of C and a weak equiv alence φ : A → F B . A map in M F fro m ( A, B , φ ) to ( A ′ , B ′ , φ ′ ) consists of weak equiv ale nc e s A → A ′ and B → B ′ such tha t the comp osite map A − → A ′ φ ′ − → F B ′ − → F B is φ . W e ha ve canonical functors M F → w C and M F → w M obtained b y dr opping the M and C data, res pe ctively . Lemma 4.6. With n otation as ab ove: (i) If every obje ct A of C , F A is we akly e quivale nt in M to an obje ct of M , then the fun ctor M F → w C induc es a we ak e qu ivalenc e on nerves. (ii) If C is a close d Waldhausen sub c ate gory of c ofibr ant obj e cts in a close d mo del c ate gory C and F is a left Qu il len adjoint that induc es an e quivale nc e b etwe en t he ful l su b c ate gories of Ho C and Ho M gener ate d by C and M , then M F → w M also induc e s a we ak e quivalenc e on nerves. Pr o of. F or the firs t statemen t, w e a pply Quillen’s Theo rem A. F or an ob ject B of C , the r e lev an t catego ry F M ↓ B has ob jects the maps φ : A → F C , γ : C → B where A is a cofibra nt ob ject in M , C is an ob ject in C , and φ and γ a re weak equiv alences. The nerve of this category is equiv alen t to the nerve of the s ubca tegory w her e C = B and γ is the identit y . This is the category of cofibrant a pproximations of the fibrant ob ject F C ; work of Dwy er-Kan (cf. [10, 6.12 ]) shows that th e nerve of this c ategory is co nt ractible. F or the second statement, let G deno te the contrav ariant left a djoint of F . Then under the hypotheses o f the seco nd statement, a map A → F B is a weak equiv alence if and only if the adjoin t map B → GA is a weak equiv alence. The seco nd s tatement now follows fro m the fir st.  In the case consider ed ab ov e, we a re lo ok ing a t functors F o f the form S ( q ) • ,..., • M DX ( S ) − → Ar[ • , . . . , • ]( M R ) or S ( q ) • ,..., • M c DX − → Ar[ • , . . . , • ]( M R ) , where we hav e written Ar[ • , . . . , • ]( C ) for the ca teg ory of functors from Ar[ • , . . . , • ] to C (wher e Ar[ n 1 , . . . , n q ] is as in Construction 2.2). Both S ( q ) • ,..., • M DX ( S ) a nd S ( q ) • ,..., • M c DX are clo sed W aldha us en sub categ ories of the c ofibrant ob jects in Ar[ • , . . . , • ]( M DX ). Since every map in Ar[ • , . . . , • ]( M R ) is weakly equiv alen t to a co fibration, a nd a co mm uting square in M R is a ho motopy pushout square if and only if it is a homotopy pullbac k square if and only if it is weakly equiv alen t to a pullback squar e o f fibrations, an easy inductive ar gument prov es the following lemma. DERIVED KOSZUL DUALITY AND INVOLUTIONS 15 Lemma 4.7. The functor F DX ( − , S P ∨ ) induc es e quivalenc es b etwe en: (i) The ful l sub c at e gory of the homotopy c ate gory of Ar[ n 1 , . . . , n q ]( M DX ) gener ate d by obje cts of S ( q ) n 1 ,...,n q M c DX , and (ii) t he ful l sub c ate gory of the homotop y c a te gory of Ar[ n 1 , . . . , n q ]( M R ) gen- er ate d by obje ct s of S ( q ) n 1 ,...,n q M R ( S ) . It also induc es e qu ivalenc es b etwe en: (i) The ful l sub c at e gory of the homotopy c ate gory of Ar[ n 1 , . . . , n q ]( M DX ) gener ate d by obje cts of S ( q ) n 1 ,...,n q M DX ( S ) , and (ii) t he ful l sub c ate gory of the homotop y c a te gory of Ar[ n 1 , . . . , n q ]( M R ) gen- er ate d by obje ct s of S ( q ) n 1 ,...,n q M c R . 5. Cov ariant equiv alences in algebraic K -Theor y and geometric Sw an theor y of sp aces Using the mo dels descr ib ed in Section 3, the gener alized Mor ita theory of [9] admits a p oint-set refinement into adjoint pairs of cov a r iant functors F R ( S P, − ): M R / / M DX o o :( − ) ∧ DX S P and ( − ) ∧ R S P ′ : M R / / M DX o o : F DX ( S P ′ , − ) , forming Quillen adjunctions. Here we switch b etw een left and right mo dules a t will using the co mm utativity of D X and the anti-inv olution on R (induced by the inv erse map on the top olo g ical gr oup G ). Since S is compact in D R , the fir s t adjunction induces an equiv alence be t ween the lo ca liz ing sub catego ry of D R generated by S and D DX , and, in particular, it restricts to an equiv a lence b etw een T R ( S ) and D c DX . In genera l, S is not c o mpact in D DX , but nonetheless the se cond adjoin t pair yields an equiv alence b etw een T DX ( S ) a nd D c R . In this case, one of the functor s in each pair is exact, a nd so W aldhausen’s approximation theorem (or the more ge ne r al form ulations of [4] or [29]) implies that these equiv alences induce e q uiv a lences K ( D X ) − → G ( X ) and A ( X ) − → K ( M DX ( S )) . Combining these equiv alences with the equiv alences of the prev ious section, we obtain s elf-homotopy equiv alences on A ( X ) and G ( X ). W e complete our analysis by identifying these as the standard Spanier-Whitehead dualit y in v olution on A ( X ) and an analogous in volution on G ( X ). (Note that when X is a smo oth manifold, this in volution is generally not compatible with the involution on pseudo-isotopy theory unless X is paralleliza ble [30].) Roughly , the inv olution on A ( X ) is given by the functor which takes a left R -mo dule M to the r ight R - mo dule Ext R ( M , R ), which w e tra nsform in to a left R -mo dule via the anti-in v olution R → R op . F or G ( X ), the inv olution is similar but with Ext S ( M , S ) instead. Because the duality maps are contrav ariant, it is co nv enien t to work with W ald- hausen categ ories mo deling the opp osite categor ies of D c R and T R ( S ). As o bserved in [3 , § 1], M op R has the structure of a W aldhausen catego ry with weak equiv alences the maps opp os ite to the usual weak equiv alences and cofibrations the ma ps opp o- site to the Hurewicz fibrations. Let M op ,c R be the full sub catego ry of ob jects that are opp os ite to co mpact o b jects in D c R , and let M op R ( S ) b e the full sub catego ry 16 ANDREW J. BL UMBERG AND MICHAEL A. MANDELL of M op R opp osite to o b jects in T R ( S ) (again, w e can make these latter tw o W ald- hausen ca teg ories sma ll b y restricting to subsets of a set with high cardinality). The argument for [3 , 1.1] (s e e discussion following [3, 2 .9]) provides weak equiv alences A ( X ) = K ( M c R ) ≃ K ( M op ,c R ) and G ( X ) = K ( M R ( S )) ≃ K ( M op R ( S )) . Essentially the map on S n sends A = { A i,j } to A ′ = { A ′ i,j } whe r e A ′ i,j ≃ A n − j,n − i and the pushouts ov er cofibrations hav e b een replac ed by equiv alen t pullbacks ov er fibrations. The functors F R ( − , R ) : M c R → M op ,c R and F S ( − , S ) : M R ( S ) → M op R ( S ) are then exact. Under the equiv alences ab ov e, the induced maps on K -theory represent the cano nical inv olution. Thus, it now suffices to compar e our comp osite functors to these functors. In the cas e of A ( X ), the comp o site of our equiv alences is the functor M c R → M op ,c R defined as M 7→ F DX ( M ∧ R S P ′ , S P ∨ ) . By a djunction, this is naturally isomorphic to F R ( − , F DX ( S P ′ , S P ∨ )). The w eak equiv alence R → F DX ( S P ′ , S P ∨ ) then induce s a natural weak equiv alence from the duality functor F R ( − , R ). F or G ( X ), the argument ab ove shows that the comp osite map on K ( D X ) → K ( M op ,c DX ) is the functor F R ( − ∧ DX S P, S P ∨ ) and is naturally weakly equiv alen t to the dualit y map F DX ( − , D X ). On the other hand F R ( S P, − ) : M op R ( S ) → M op ,c DX is exact and the following solid arrow diag r am commutes up to natural isomorphism. M R ( S ) F S ( − ,S ) / / M op R ( S ) F R ( S P , − )   M c DX F DX ( − ,S P ∨ ) 8 8 r r r r r r r r r r r r r r r r r F R ( −∧ DX S P ,S P ∨ ) / / ( − ) ∧ DX S P O O M op ,c DX The comp osite of the dotted a rrows is the functor F S (( − ) ∧ DX S P, S ). By the smash-function adjunction, we see that this functor is natur a lly isomorphic to F DX ( − , F S ( S P, S )), w hich is the diag onal a r row since S P ∨ = F S ( S P, S ). References [1] A. A. Beil inson, V .A. Ginzburg, and V. V. 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Dep ar tm ent of Ma thema tics, The University of Texas, Austin, TX 78712 E-mail addr ess : blumberg@math .utexas.edu Dep ar tm ent of Ma thema tics, Indiana Univ ersity, Bloomington, IN 474 05 E-mail addr ess : mmandell@indi ana.edu