Duality in Waldhausen Categories

Forum Math. 1 0 ( 1 9 9 8 ) , 5 3 3 — 6 0 3 Forum Mathematicu m © d e G r u y t e r 1 9 9 8

Duality i n Wa l d h a u s e n Ca t e g o r i e s Michael We i s s and Bruce Williams

(Communicate d b y An d r e w Ra n i c k i )

A b s t r a c t . W e d e v e l o p a th e o r y o f Spanier-Whitehea d du a l i t y i n ca t e g o r i e s wi t h cofibration s a n d we a k equivalence s (Waldhause n categories , fo r short). Th i s in c l u d e s L-theory, the i n v o ¬ l u t i o n o n K - t h e o r y introduced b y [V o ] i n a sp e c i a l ca s e , an d a map Ξ re l a t i n g L-theory to t h e Ta t e sp e c t r u m o f $ ac t i n g o n K-theory. Th e map Ξ i s a distillatio n o f th e lo n g ex a c t Rothenberg se q u e n c e s [Sha], [Ra1], [Ra2], in c l u d i n g an a l o g s in v o l v i n g hi g h e r K-groups. It g o e s b a c k to [WW2] i n s p e c i a l c a s e s . A m o n g the e x a m p l e s c o v e r e d he r e , but not i n [WW2], a r e ca t e g o r i e s o f re t r a c t i v e sp a c e s wh e r e th e notion o f we a k equivalenc e in v o l v e s control. 1 9 9 1 Mathematic s Su b j e c t Classification : 19 D 1 0 , 57 N 6 5 , 5 7 R 6 5 .

0 . Introductio n For an y ri n g R, Qu i l l e n has de f i n e d a n algebraic K-theory Ω - s p e c t r u m , $. Hi s construction i s i n terms o f the category o f finitely ge n e r a t e d pr o j e c t i v e mo d u l e s o v e r R, but i t ca n b e a p p l i e d to an y ex a c t category. I n order to st u d y concordances o f m a n i f o l d s Wa l d h a u s e n generalize d Qu i l l e n ’ s construction to ap p l y to what W a l d ¬ h a u s e n ca l l s ca t e g o r i e s wi t h co f i b r a t i o n s an d we a k eq u i v a l e n c e s . (F o l l o w i n g Thomason w e wi l l ca l l th e m Wa l d h a u s e n categories. ) A n ex a m p l e i s gi v e n b y the c a t e g o r y o f ba s e d compact CW - s p a c e s . More ge n e r a l l y , the ca t e g o r y o f retractive r e l a t i v e CW - s p a c e s $ o v e r a fixed s p a c e X, wi t h compact quotient Y/X, i s an e x a m p l e o f a Wa l d h a u s e n ca t e g o r y ; it s K-theory sp e c t r u m i s kn o w n a s $.

W a l d h a u s e n su p p l i e d se v e r a l po w e r f u l tools al o n g wi t h hi s construction o f $ f o r a Wa l d h a u s e n ca t e g o r y $, su c h a s the additivity theorem, the approximation theorem, and the ge n e r i c fibration theorem. Ev e n i f on e i s primarily in t e r e s t e d i n t h e algebraic K-theory o f ri n g s th e s e tools ha v e important applications [Sta],

1 Bo t h a u t h o r s s u p p o r t e d i n p a r t b y N S F g r a n t . 5 3 4 M . W e i s s , B . W i l l i a m s

[Tho2]. Also i n [CPed] ex c i s i o n in controlled algebraic K-theory o f rings i s proved using Waldhausen’s machinery.

Suppose that R i s a ring with involution (involutory anti-automorphism). The problem o f cl a s s i f y i n g manifolds up to homeomorphism or diffeomorphism lead W a l l to de f i n e algebraic L-groups Ln(R) in terms o f quadratic forms on finitely generated projective (or fr e e , or based fr e e ) modules over R. The L-groups turned out to be the homotopy groups o f a spectrum [Q]. Ranicki [Ra3] associates such L-theory spectra to any additive category with chain duality. (There i s a quadratic L-theory spectrum made using quadratic forms, and a symmetric L-theory spectrum made using symmetric forms.) Specifically , by using additive categories o f “modules parametrized b y a simplicial complex” Ranicki gave an algebraic description o f Quinn’s L-theory assembly map which i s us e d to cl a s s i f y manifolds up to ho m e o ¬ morphism. Also Ranicki’s chain duality setup has been used to construct controlled versions o f L-theory fo r rings [FP]. The assembly map i n L-theory can then also b e id e n t i f i e d with a forget control map. This has been used to prove many cases o f the Novikov conjecture; se e [FRR].

I f one wants to study the spaces o f homeomorphisms or diffeomorphisms o f a manifold, then the L-theory o f rings or ev e n additive categories i s not adequate. One i s fo r c e d to consider L-theory and K-theory o f certain Waldhausen categories equipped with duality, and a certain map Ξ relating the L-theory to the K-theory. For this reason w e need a theory o f duality in Waldhausen categories which, unlike Ranicki’s chain duality theory, allows “nonlinear” cases; and w e need to understand, i n this generality, L-theory, K-theory, and the Ξ-map. Waldhausen categories o f retractive spaces where the notion o f weak equivalence involves control are important i n applications to geometry and should therefore b e the central examples o f such a theory.

T o explain what Ξ i s about, w e note that many o f the classical invariants fo r symmetric forms over a ring R take values i n groups constructed fr o m Ki (R) where i = 0,1, or 2. This suggests that there should b e a connection between the symmetric L-theory spectrum $ and the algebraic K-theory spectrum $. S i n c e the classical constructions o f L-theory are modelled on the pre-Quillen definitions o f “ l o w dimensional K-theory”, the connection i s hidden. In [WW2] w e established the connection b y constructing Ξ, a natural map fr o m $ to the Tate spectrum f o r $ acting on $. A nonlinear version o f this does already appear i n [WW2], but i t i s ve r y limited and there i s no mention o f “control”. But i t was certainly motivated b y our study o f spaces o f homeomorphisms.

In this paper w e introduce the notion o f a Spanier-Whitehead product $ on a Waldhausen category $. This i s a functor $ f r o m $ to pointed spaces. The space $ should be thought o f as the space o f pairings between C and D. I f $ s a t i s f i e s certain axioms listed i n §2, reminiscent o f Spanier-White- D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 3 5 h e a d duality, then fo r ev e r y C i n $ there ex i s t s an essentiall y unique ob j e c t C i n $ wh i c h co m e s w i t h a nondegenerate pairing $. W e think o f C′ a s the dual o f $. W e obtain a n involution o n $ wh i c h , modulo technicalities , i s in d u c e d b y $. W i t h the sa m e hy p o t h e s e s o n $ w e ha v e constructions o f $ (s y m ¬ m e t r i c L-theory sp e c t r u m ) and $ (quadratic L-theory sp e c t r u m ) wh i c h g e n e r ¬ a l i z e Ra n i c k i ’ s constructions fo r ad d i t i v e ca t e g o r i e s wi t h ch a i n duality. There i s a natural transformation Ξ fr o m $ to the T a t e sp e c t r u m fo r $ ac t i n g o n $. There i s al s o a symmetrizatio n map fr o m $ to $.

I n [DWW] w e us e d the fa c t that the as s e m b l y map i n the algebraic K-theory o f s p a c e s ca n b e id e n t i f i e d wi t h a fo r g e t control map to pr o v e an A-theory in d e x t h e o r e m fo r th e A-theory Eu l e r characteristic. Ex p l o i t i n g Poincaré duality an d the r e s u l t s an d ex a m p l e s o f this paper, w e sh a l l i n a fu t u r e paper produce a $- equivariant ve r s i o n wh i c h ca n b e vi e w e d a s an A-theory in d e x th e o r e m fo r the A-theory signature.

S e c t i o n he a d i n g s 1 . Spanier-Whitehea d products 2 . Ax i o m s 3 . First Co n s e q u e n c e s 4 . The involution o n $ 5 . Generating cl a s s e s 6 . Th e ax i o m s an d the $ construction 7 . Th e in v o l u t i o n o n $ 8 . Th e ax i o m s an d parametrization 9 . Sy m m e t r i c L-theory an d the map Ξ 1 0 . St a b l e S W products 11. Quadratic L-theory, and Ξ re v i s i s t e d 1 2 . Naturality

1 . A . Ex a m p l e s re l a t e d to § 1 2 . A . Ex a m p l e s re l a t e d to §2 5 . A . Ex a m p l e s re l a t e d to §5 1 2 . A . Ex a m p l e s re l a t e d to §1 2

G u i d e . W e de f i n e S W products i n §1. I n se c t i o n §2 w e li s t so m e ax i o m s o n a W a l d h a u s e n ca t e g o r y $ wi t h S W product wh i c h are i n fo r c e mo s t o f the ti m e i n s u c c e e d i n g s e c t i o n s . T h e m a i n purpose o f t h e s e a x i o m s i s to e n s u r e that ev e r y ob j e c t i n $ has a sufficientl y unique Spanier-Whitehea d dual. In §3 and §4 w e s h o w that this i s th e ca s e .

M u c h o f the material i n the re m a i n i n g se c t i o n s i s about ve r i f y i n g that S W products and the ax i o m s are hereditary. That is, i f the Wa l d h a u s e n category $ co m e s wi t h 5 3 6 M . W e i s s , B . W i l l i a m s an S W product sa t i s f y i n g the axioms, then certain other Waldhausen categories constructed fr o m $ inherit an S W product, wh i c h again sa t i s f i e s the axioms.

E x a m p l e 1 . Waldhausen constructs the K-theory spectrum $ b y f i r s t constructing a simplicial Waldhausen category $. T o show that an S W product $ on $ s a t i s f y i n g the axioms o f §2 determines an involution on ( a ne w model of) $, w e first have to show that $ determines an S W product on each $, again s a t i s f y i n g the axioms. The S W product on $ appears already i n §1, as an example, but the axioms are first checked i n §5 and §6. Only then are w e ready to produce an involution on $; this i s done in §7.

E x a m p l e 2. A symmetric object in a Waldhausen category $ with S W product i s an object C together with a homotopy fixed point φ o f the symmetry involution on $. I f the underlying point in $ i s nondegenerate, w e call ( C , φ) a symmetric Po i n c a r é object. For us, symmetric L-theory i s the bordism theory o f symmetric Poincaré objects i n a Waldhausen category with S W product. The most elementary type o f bordism (without any symmetric self-duality to begin with) i n the Waldhausen category $ would be a functor fr o m the poset o f nonempty fa c e s o f the standard 1- s i m p l e x $1 to $. The most elementary type o f higher bordism would b e a functor fr o m the poset o f nonempty fa c e s o f the standard n - s i m p l e x Δn to $, with n > 1 . O f course, these notions become more interesting when self-duality conditions are imposed. In any case, to do symmetric L-theory i n $, w e ne e d to introduce a category o f functors $ f r o m the poset o f nonempty fa c e s o f a standard m-simplex to $, fo r each m ≥ 0 . This appears already in §1, as an example. It i s necessary to ch e c k that each $ inherits an S W product fr o m $, which s a t i s f i e s the axioms i f the original one on $ does. W e do this in §8. Only then are w e ready to de f i n e the symmetric L-theory o f $ and the map Ξ; this i s done i n §9.

The discussion i n §9 takes place at the space le v e l . For example, i n §9 w e de f i n e the symmetric L-theory o f $ as a space, not as a spectrum. It turns out that i n order to raise the discussion to spectrum le v e l one needs a spectrum-valued S W product on $. This i s also needed to de f i n e quadratic objects i n $. Namely, suppose that $ comes with a spectrum-valued S W product $. Then w e can ( r e - ) d e f i n e symmetric objects i n $ as pairs ( C , φ) with $ and w e can de f i n e quadratic objects i n $ as pairs ( C , ψ) with $, where the subscript $ indicates a homotopy orbit spectrum. A quadratic object (C, ψ) determines a symmetric object $ where $ i s the norm map. W e c a l l ( C , ψ) a quadratic Po i n c a r é object i f $ i s a symmetric Poincaré object. For us, quadratic L-theory i s the bordism theory o f quadratic Poincaré objects i n a W a l d ¬ hausen category with spectrum-valued S W product.

W e show i n §10 that a space-valued S W product on $ i s always the 0-th term o f an Ω-spectrum-valued S W product. In §11, w e assume that $ comes with an Ω-spectrum-valued S W product satisfying the axioms. W e then introduce the sy m - D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 3 7 m e t r i c L-theory spectrum o f $, im p r o v e Ξ to a map o f spectra, an d de f i n e , at last, t h e quadratic L-theory sp e c t r u m o f $ and the symmetrizatio n map f r o m quadratic L-theory to sy m m e t r i c L-theory.

I n § 1 2 w e e x p l a i n ho w th e s e constructions de p e n d functoriall y o n the Wa l d h a u s e n c a t e g o r y $ wi t h S W product (s a t i s f y i n g the ax i o m s ) .

T h e e x a m p l e s e c t i o n s § l.A, §2.A, § 5 . A a n d § 1 2 . A c a n b e r e a d i n the order i n w h i c h t h e y co m e , af t e r the theoretical part. Another po s s i b i l i t y i s to re a d § n . A a f t e r §n.

0 . 1 . Conventions . A space i s a co m p a c t l y ge n e r a t e d Ha u s d o r f f sp a c e , un l e s s w e s a y o t h e r w i s e . Products and mapping sp a c e s are fo r m e d i n this ca t e g o r y i n the us u a l w a y [MaL]. Ba s e points are nondegenerate un l e s s w e sa y ot h e r w i s e . Th e s e c o n ¬ ventions are taken fr o m [Go1]. T h e ge o m e t r i c realization o f a si m p l i c i a l se t X i s a sp a c e |X|. Th e nerve o f a c a t e g o r y $ i s a certain si m p l i c i a l se t $, and the classifying space o f $ i s $, w h i c h w e us u a l l y shorten to $. Homotopy li m i t s and homotopy co l i m i t s o f diagrams o f sp a c e s are de f i n e d a s i n [Go1] (w h i c h i s to sa y that th e y are de f i n e d a s i n [BK], e x c e p t that w e work wi t h diagrams o f topological sp a c e s rather than diagrams o f si m p l i c i a l se t s ) . W e d i s t i n ¬ g u i s h be t w e e n reduced homotopy co l i m i t s (o f diagrams o f ba s e d sp a c e s ) an d un¬ reduced homotopy co l i m i t s (o f diagrams o f un b a s e d sp a c e s ) . A re d u c e d homotopy c o l i m i t i s the quotient o f the corresponding un r e d u c e d homotopy co l i m i t b y the c l a s s i f y i n g sp a c e o f the in d e x i n g category.

1. Spanier-Whitehead products L e t $ b e a Waldhausen category, that is, a category wi t h co f i b r a t i o n s and we a k equivalence s [Wald2]. Le t $ denote the zero ob j e c t i n $.

1 . 1 . Definition . B y a n SW product o n $ w e sh a l l me a n a fu n c t o r

f r o m $ to the c a t e g o r y o f b a s e d sp a c e s ( s e e 0 . 1 ) w h i c h i s w-invariant, symmetric a n d bilinear (explanation s fo l l o w ) . ● w-Invariance m e a n s that the f u n c t o r takes pairs o f w e a k equivalence s to h o m o ¬ topy equivalences . ● Symmetry m e a n s that th e f u n c t o r co m e s w i t h a natural isomorphis m $ $, w h o s e square i s the identity o n $. ● Bilinearity me a n s (i n the pr e s e n c e o f sy m m e t r y ) that, fo r fixed but arbitrary D, the fu n c t o r $ takes an y cofiber square i n $ to a homotopy pullback square o f sp a c e s . ( A c o f i b e r square i s a commutative pushout square 5 3 8 M . We i s s , B . W i l l i a m s

in which either the horizontal or the vertical arrows are cofibrations.) Bilinearity also means that $ is contractible.

Usually, in studying a specific Waldhausen category $, one is compelled to introduce some other Waldhausen categories. For us it will be important to know whether and how these other Waldhausen categories can be equipped with SW products i f $ i s so equipped.

For example, there is the simplicial Waldhausen category $ constructed from $. Se e [Wald2]. Each $ i s a Waldhausen category whose objects are certain functors $ from the poset of pairs (i,j) with 0 ≤ i,j ≤ m to $. The functors must satisfy two conditions: C(i,j) = * i f j ≤ i, and

$ i s a pushout square wh e r e the horizontal arrows are cofibrations, fo r i,j < m. Up to isomorphism, su c h a fu n c t o r (diagram) i s de t e r m i n e d b y it s top ro w , wh i c h ca n b e a n arbitrary string o f co f i b r a t i o n s

T h e re m a i n i n g information co n s i s t s o f ch o s e n subquotients,

W e s u p p o s e that $ i s eq u i p p e d wi t h an S W product and try to us e this to de f i n e , i n th e si m p l e s t wa y po s s i b l e , an S W product o n the Wa l d h a u s e n ca t e g o r y $, fo r e a c h m ≥ 0 . A go o d fo r m u l a i s

f o r C an d D i n $. (T h e category o f ba s e d sp a c e s i n the se n s e o f 0. 1 i s cl o s e d under homotopy l i m i t s . ) Note that the quadruples (i,j,p,q) s a t i s f y i n g the conditions i + q ≥ m an d j + p ≥ m st i l l fo r m a poset, and $ i s a covariant fu n c t o r , s o that the homotopy li m i t i s at le a s t de f i n e d . Ho w e v e r , the f o r m u l a do e s not s e e m a l l that si m p l e . The justificatio n i s g i v e n i n 1 . 3 an d 1 . 4 b e l o w . For no w , the fo l l o w i n g ma y help. Th e ma i n point he r e i s that, fo r C and D i n $, a n el e m e n t η i n $ d e t e r m i n e s el e m e n t s i n $, o n e f o r ev e r y pair i,j wi t h 0 ≤ i ≤ m and 0 ≤ j ≤ m. Under su i t a b l e conditions o n $, D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 3 9 w e w i l l show i n § 6 that i f η i s nondegenerate (a notion de f i n e d i n § 2), then its im a g e s i n the various $ are also nondegenerate. In other words, the quadruples (i,j,p, q) with i + q = m and j + p = m appearing i n the de f i n i t i o n o f $ are particularly important to us. The quadruples (i,j,p, q) w i t h i + q > m and j + p > m are ne e d e d mostly to gi v e “coherence”.

1 . 2 . Lemma. The functor $ on $ defined above is an SW product.

Proof. Gi v e n C, D i n $ w e map $ isomorphically to $ v i a

using the sy m m e t r y property o f $ as a functor on $. For fixed i,j,p, q the functor $ on $ has the bilinearity property r e ¬ quired i n 1. 1 . Therefore $ i n 1. 2 has the bilinearity property. □

W e ma y as k ho w the S W product i n 1. 2 varies with m. Write $. The category wi t h objects [m] fo r m ≥ 0 and monotone maps as morphisms has an automorphism (conjugation) o f order two w h i c h takes f: [k] → [m] to $ w h e r e rk and rm are the order reversing bijections o f [k] and [m], respectively . In other words, $. Recall also that $ i s a simplicial c a t e ¬ gory. Then, fo r C and D i n $, and a monotone f: [k] → [m], w e have f*C and $ i n $. For example,

$, a n d i f i + q ≥ k a n d j + p ≥ k, then $ and $. It fo l l o w s that w e h a v e a map i n d u c e d b y f,

Let us: [m] → [1] be the unique monotone map wi t h us(s) = 1 and us(s — 1 ) = 0 , f o r 0 < s ≤ m. L e t v : [1] → [m] be g i v e n b y υ ( 0 ) = 0 and υ(1) = m, s o that usυ = i d on [1]. Note $. W e shall us e these monotone maps to compare $ on $ with $ on $, noting that $ and that $ on $ i s the sa m e as $ on $ under this identification.

1 . 3 . Proposition . For C and D in $ and us, ut: [m] → [1] and υ : [1] → [m] as above, the canonical map

$ is a homotopy equivalence if s + t ≤ m + 1. If s + t > m + 1 , then $. 5 4 0 M . W e i s s , B . W i l l i a m s

Proof. Let T be the poset o f al l quadruples (i,j,p,q) such that i + q≥m and j + p ≥ m, and 0 ≤ i,j,p, q ≤ m. Let F be the functor on T taking (i,j,p, q) to $. Let $ consist o f the (i,j,p, q) f o r wh i c h i < s and p < t. Then f o r (i,j, p, q) not i n T0 w e have $ by inspection. Also, any element i n T which i s ≥ an element i n T\T0 belongs to T\T0. It fo l l o w s that the projection

holim F → holim F|| T0 i s a homotopy equivalence (again b y inspection, going back to the definition o f the holim). Suppose no w that s + t ≤ m + 1 . Then fo r $ w e have q = (i + q) — i ≥ m — i ≥ m + 1 — s ≥ t and similarly j ≥ s, s o that F | T0 i s isomorphic to a constant functor with constant value $. The nerve o f T0 i s contractible because there i s a maximal element. This shows that 1. 3 holds wh e n s + t ≤ m + 1 . Next suppose s + t > m + 1 . Let $ consist o f the elements (i,j,p, q) fo r which j < s or q < t. Note that ω = ( s — 1 , m, t — 1 , m) i s the maximal element i n T0. Then $ i s a retract o f T0 ( i n the category o f posets, with maps preserving ≤ as morphisms). Furthermore F|T0 pulls back fr o m $ because F|(T0\T1) i s isomorphic to a constant functor. This leads to maps

w h i c h are ea s i l y se e n to be reciprocal homotopy equivalences. From the definition o f the homotopy inverse limit, there i s a homotopy pullback square

where the vertical arrow on the le f t i s projection, and the one on the right i s the inclusion o f the constant maps. Note that T1 has a final sub-poset consisting o f the (i,j,p, q) with i = s — 1 and p = t — 1 . This final sub-poset has a contractible c l a s ¬ s i f y i n g space, s o that |T1| it s e l f i s contractible. Therefore (f r o m the homotopy pullback square just above) the projection

i s a homotopy equivalence. Si n c e F|T1 has contractible values, w e can conclude holim $ holim $ holim $ holim $. □

W e se e that our formula fo r $ on $ gi v e s good and predictable results when applied to objects o f the f o r m $ with C, D i n $. An arbitrary object E i n $ fits into a natural diagram Duality in Waldhausen categories 541

$ w h e r e the arrows are co f i b r a t i o n s and ea c h co f i b e r Es/Es—1 i s isomorphic to a n o b j e c t o f the fo r m $ fo r so m e Cs i n $. Be c a u s e o f th e bilinearity, i t fo l l o w s that $ i s “c o r r e c t l y de f i n e d ” o n al l o f $.

1 . 4 . Ex a m p l e . Le t $ b e th e ca t e g o r y o f ch a i n co m p l e x e s o f fi n i t e l y ge n e r a t e d fr e e abelian groups, g r a d e d o v e r $ and bounded ( f r o m be l o w and above). The morphisms are th e ch a i n maps; a morphism i s a cofibratio n i f i t i s sp l i t mono i n ea c h de g r e e , a n d a we a k equivalenc e i f i t i s a homotopy eq u i v a l e n c e . Ea c h C i n $ has a dual $. For C, D i n $ w e h a v e ch a i n co m p l e x e s hom(C, D) and $, and an isomorphis m $. W e ca n introduce analogous notions i n $. For C i n $ w e de f i n e C* b y $. For C and D i n $ w e d e f i n e hom(C, D) a s a chain s u b c o m p l e x o f hom(C(0, m), D ( 0 , m)) w h o s e k - c h a i n s ar e the homomorphisms o f graded groups C(0, m) → D(0, m) ra i s i n g de g r e e s b y k, and taking the im a g e o f C(0, i) i n C(0, m) to the im a g e o f D ( 0 , i) i n D ( 0 , m), fo r ea c h i wi t h 0 ≤ i ≤ m. The restriction maps fr o m hom(C, D) to hom(C(m —j, m — i), D(p, q)), de f i n e d w h e n e v e r m —j ≤ p an d m — i ≤ q, de t e r m i n e a ch a i n map

( ● ) $.

(Note that $ i s a covariant fu n c t o r . ) Th e c h a i n map (● ) i s al w a y s a n isomorphism. For the pr o o f , re p l a c e $ b y the ca t e g o r y $ o f gr a d e d f. g . fr e e ab e l i a n groups, bounded fr o m be l o w an d above. Think o f ( ● ) a s a natural transformation be t w e e n bi-additive fu n c t o r s o n $. It i s e n o u g h to ve r i f y that i t i s a n is o m o r p h i s m wh e n C(0, m) and D ( 0 , m) ha v e rank o n e . Th i s ca s e i s e a s y . T h e r e f o r e , i f w e wi s h to ha v e a natural is o m o r p h i s m $ f o r o b j e c t s C, D i n $, w e are fo r c e d to de f i n e

O f course, there ar e other wa y s to sa y the sa m e thing. For ex a m p l e , w e k n o w that $ i n j e c t s i n the ordinary tensor product o f c h a i n c o m p l e x e s $, a n d th e im a g e o f th i s in j e c t i o n i s ea s y to id e n t i f y ; this le a d s to

$. 5 4 2 M . W e i s s , B . W i l l i a m s

I n order to make S W products fr o m tensor products, w e u s e the Kan-Dold functor. The Kan-Dold functor associates to a chain complex E the simplicial abelian group whose n - s i m p l i c e s are the chain maps fr o m the cellular chain complex o f the CW- s p a c e Δn to E. W e w i l l write $ f o r the composition o f the Kan-Dold functor w i t h geometric realization. The functor $ respects finite limits. There are now two slightly di f f e r e n t wa y s o f introducing an S W product i n $. Given C and D i n $, w e might de f i n e $

where the limit is taken over all (i,j, p, q) with i + q ≥ m and j + p ≥ m. Alternatively, we might define

(same conditions on i,j,p, q) which means that w e us e the formula just be f o r e 1. 2 , and an S W product on $ gi v e n by $. These two sl i g h t l y di f f e r e n t definitions are not isomorphic, but they are related b y a natural transformation (the usual map fr o m a li m i t to the corresponding homotopy limit) respecting the symmetry Τ. B y 1 . 3 , the natural transformation i s a homotopy equivalence i n certain cases. The filtration argument gi v e n af t e r the proof o f 1. 3 shows that it i s always a homotopy equivalence.

Another Waldhausen category wh i c h frequently arises i n the study o f a Waldhausen category $ i s $, the parametrization o f $ by a finite simplicial complex X. This i s de f i n e d as fo l l o w s (compare [RaWe], [Ra3]). Let sub(X) b e the poset o f simplicial subcomplexes o f X, v i e w e d as a category. An ob j e c t o f $ i s a functor F fr o m sub ( X ) to $ wh i c h takes al l morphism to cofibrations, takes $ to *, and takes unions to pushouts. (S u c h a functor i s determined up to isomorphism b y its restriction to the fu l l sub-poset o f sub ( X ) consisting o f all the nonempty fa c e s of X.) Any natural transformation F1 → F2 b e t w e e n such functors qu a l i f i e s as a mor¬ phism i n $. It i s a cofibration i f F1(Z) → F2(Z) i s a cofibration fo r al l $, and fo r each pair o f subcomplexes $ o f X, the evident morphism

i s a cofibration. (This ensures that the functor $ i s again i n $; i t i s o f course the co f i b e r o f the cofibration.) The morphism F1 → F2 i s a weak equivalence i f F1(Z) → F2(Z) i s a weak equivalence fo r al l $.

1 . 5 . Definition . An S W product $ on $ gi v e s rise to one on $ b y the formula $ where the homotopy inverse limit i s taken over the poset o f al l nonempty fa c e s $. D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 4 3

2. Ax i o m s For motivation, w e return to the se t t i n g o f 1. 4 . Su p p o s e that C and D are ob j e c t s i n $. A component o f $ w i l l b e re g a r d e d a s nondegenerate i f the corresponding homotopy cl a s s o f chain maps C* → D i s the cl a s s o f a homotopy eq u i v a l e n c e . It i s not a tr i v i a l matter to ge n e r a l i z e this notion to the abstract s e t t i n g o f 1 . 1 . I n this s e c t i o n w e l i s t a number o f axioms, about a W a l d h a u s e n c a t e g o r y $ wi t h S W product, wh i c h wi l l ma k e i t po s s i b l e .

T o b e g i n , w e s u p p o s e a g a i n that $ i s a n abstract Wa l d h a u s e n category. W e s u p p o s e a l s o that $ i s sm a l l (the cl a s s o f ob j e c t s i s a se t ) , although i n our ex a m p l e s $ i s u s u a l l y on l y eq u i v a l e n t to a sm a l l category. Th e first f e w ax i o m s do not in v o l v e a n S W product. The notions cylinder functor, cylinder axiom, saturation axiom fr o m [Wald2] appear i n 2.1; w e ex p l a i n th e m i n the co m m e n t af t e r 2.1.

2 . 1 . Ax i o m s . $ is equipped with a cylinder functor satisfying the cylinder axiom [ W a l d 2 , § 1 . 4 ] . The weak equivalences in $ satisfy the saturation axiom [Wald2, § 1 . 2 ] .

Comment. Th e saturation axiom i s the fo l l o w i n g condition: i f a, b are composable m o r p h i s m s i n $ and tw o o f a , b, ab are we a k equivalences , then s o i s the third. A n ex a m p l e o f a Wa l d h a u s e n ca t e g o r y wi t h cy l i n d e r fu n c t o r i s the ca t e g o r y o f c o m p a c t ba s e d CW - s p a c e s , wh e r e the morphisms are the ba s e d ce l l u l a r maps, th e w e a k eq u i v a l e n c e s are th e homotopy eq u i v a l e n c e s , and the co f i b r a t i o n s are the CW-embeddings . I n this ca s e , the cy l i n d e r fu n c t o r i s th e ru l e as s o c i a t i n g to a m o r p h i s m f: X → Y it s re d u c e d mapping cy l i n d e r T(f), and the canonical maps X → T(f) an d Y → T(f) (f r o n t in c l u s i o n and ba c k in c l u s i o n ) an d T(f) → Y ( c y l i n d e r projection). For the abstract ve r s i o n , w e ne e d tw o other Wa l d h a u s e n ca t e g o r i e s $ and $ a s s o c i a t e d wi t h $. Th e ob j e c t s o f $ are the arrows o f $, the morphisms f r o m f: A → B to g : C → D are the commutative squares

$ i n $, and w e ca l l su c h a mo r p h i s m a co f i b r a t i o n {w e a k equivalence } i f th e tw o v e r t i c a l arrows ar e cofibration s {w e a k equivalences} . Th e ob j e c t s o f $ are the morphisms o f $. Ag a i n , a mo r p h i s m i n $ fr o m f: A → B to g: C → D i s a commutative square o f th e fo r m 5 4 4 M . We i s s , B . W i l l i a m s i n $. If , in this square, both A → C and $ are in $, then the morphism given by the square is a cofibration. If both A → C and B → D are weak equivalences, then the morphism given by the square is a weak equivalence. A cylinder functor on $ is a functor T on $ taking every object f: C →D to a diagram of the form

w i t h pj1 = f and pj2 = idD, and taking morphisms i n $ to natural tr a n s f o r m ¬ ations be t w e e n su c h diagrams. There are tw o additional conditions. Th e fi r s t o f t h e s e (Cyl1) r e q u i r e s thaty j1 a n d j2 as s e m b l e to a n e x a c t fu n c t o r fr o m $ to $,

T h e se c o n d (C y l 2 ) re q u i r e s that j2 : D → T(* → D) b e a n id e n t i t y mo r p h i s m fo r e a c h D i n $. Th e c y l i n d e r fu n c t o r i s s a i d to sa t i s f y th e cylinder axiom i f the c y l i n d e r p r o j e c t i o n p: T(f) → D i s a we a k equivalenc e fo r an y f: C → D. S u p p o s e that $ co m e s wi t h a cy l i n d e r fu n c t o r T. Th e cone o f a mo r p h i s m f: C → D, denoted cone(f), i s the c o f i b e r of j1: C → T(f). W e o f t e n wr i t e cone(C) instead o f cone(idC). The suspension ΣC o f an ob j e c t C i n $ i s the co n e o f $. ( E n d of comment.)

A quasi-morphism fr o m a n ob j e c t C i n $ to an ob j e c t D i n $ i s a diagram o f the f o r m C → D′ ← D wh e r e the arrow ← i s a we a k eq u i v a l e n c e and a cofibration. Diagrams o f this type, wi t h fi x e d C and D, are th e ob j e c t s o f a ca t e g o r y $ w h o s e morphisms are commutative diagrams o f th e fo r m

W r i t e [C, D] fo r th e se t o f components o f $, a n d wr i t e $ f o r t h e cl a s s o f a morphism f: C → D (r e g a r d e d a s a quasi-morphism C → D = D). T h e s e t s [C, D] are the m o r p h i s m se t s i n a n e w c a t e g o r y $, wi t h the s a m e ob j e c t s a s $. Explicitly , the composition l a w [D, E] × [ C , D] → [C, E ] i s d e f i n e d o n r e p r e s ¬ entatives D → E′ ← E and C → D′ ← D a s fo l l o w s . Le t E″ b e the pushout o f D′ ← D → E′ an d fo r m C → E″ ← E (arrow → vi a D′, arrow ← vi a E′). Th i s r e p r e s e n t s a n e l e m e n t i n [C, E], a s required. — C l e a r l y $ i s a fu n c t o r f r o m $ to $. It i s a n ex e r c i s e to sh o w that th e fu n c t o r takes we a k equivalence s to isomorphisms, provided $ sa t i s f i e s 2.1. Th i s i s actually ca r r i e d out i n [Wei]. D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 4 5

2.2. Ax i o m (Simplicity). Let f be a morphism in $ such that [ f ] is an isomorphism. Then f is a weak equivalence.

This axiom ensures that there i s a reasonably close relationship between $ and $. The remaining axioms are about $, and they have a ve r y familiar fo r m reminiscent o f Spanier-Whitehead duality.

2.3. Axiom (Stability). The suspension functor $ is an equivalence of categories.

For the next two axioms, suppose that $ i s equipped with an S W product $. The rule $ can b e regarded as a functor on $.

2.4. Ax i o m (Co-representability). For every B in $, the functor on $ taking C to $ is co-representable, say by an object T(B) in $.

L e t ’ s s p e l l this out. W e must have an isomorphism $ naturally in C. The object T(B) is characterized by this property up to unique isomorphism in $.

2 . 5 . Ax i o m (Involutivity). For every B in $, the canonical morphism from T2(B) to B in $ is an isomorphism.

The canonical morphism i s the image o f id $ under the chain o f bijections $. One can se e that 2. 5 i s not a consequence o f 2. 4 by looking at the most trivial possible S W product, $. Then 2. 4 i s sa t i s f i e d , but 2. 5 i s not unless $ i s equivalent to a category with one object and one morphism.

R e m a r k . When 2. 4 holds, w e can regard T as a functor fr o m $ to $. It i s self-adjoint i n the se n s e that $ i n $, naturally. This fo l l o w s f r o m the symmetry o f $. The canonical natural transformation T2(B) → B i s just the counit o f the adjunction [MaL, IV.1]. It i s therefore an isomorphism fo r al l B i n $ i f and on l y i f T i s an equivalence $.

3. First consequence s

Throughout this section, w e assume that $ sa t i s f i e s the axioms o f §2. The goal here i s to translate statements about $ into statements about a topological category whose objects are those o f $, and where the space o f morphisms fr o m C to D i s the classifyin g space $ o f the category $. This topological category i s essentially a hammock localization i n the se n s e o f Dwyer and Kan, [DwyKa1], [DwyKa2]; se e [W e i , 1. 2 ] fo r more details. 5 4 6 M . We i s s , B . W i l l i a m s

3.1. Quotation [Wei]. There is a composition law in the shape of a functor

$ which on objects i s defined as follows: the pair (D → E′ ← E, C → D′ ← D) is mapped to C → E″ ← E, where E″ is the pushout of D′ ← D → E′. The morphisms C → E″ and E″ ← E are via D′ and E′, respectively. Here we are assuming that pushouts of diagrams F ← G → H in $, where one of the arrows is a cofibration, are canonically defined in $. (The definition of a Waldhausen category requires that they exist, and the universal property makes them unique up to unique isomorphism.) The composition law is associative up to a natural isomorphism of functors. Also, the object $

in $ acts as a two-sided identity, again up to a natural isomorphism.

It i s always possible to replace $ by an equivalent category so that the pushouts needed above are canonically defined and associative, with units. What this means i s that, when $ has been so improved, the composition law just described i s as s o ¬ ciative and has two-sided identities. See [Isb]. W e will use this below, in the proof of 3.5.

3.2. Corollary. Let $ be a quasi-morphism. The corresponding morphism C → D in $ is an isomorphism if and only if f is a weak equivalence. In that case the maps

$ given by composition with the quasi-morphism $ are homotopy equivalences, for arbitrary B and E in $.

Proof. Suppose that the quasi-morphism in question represents an isomorphism in $. Then it follows immediately from 3. 1 that composition with it gives homotopy equivalences. Next, it i s an exercise to show that the composition of our quasi- morphism with

(representing the class [e]) is in the class [f]. But we saw in §2 that [e] is an isomorphism in $, so that C → D′ ← D represents an isomorphism in $ i f and only i f [f] i s an isomorphism in $. Now apply axiom 2.2. □ D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 4 7

3.3. Th e o r e m . For fixed D in $, the contravariant functor $ takes cofiber squares to homotopy pullback squares.

This i s the main result o f [Wei]. It us e s axiom 2. 1 only. A cofiber square i n $ i s a commutative pushout square i n wh i c h either the vertical arrows or the horizontal arrows are cofibrations. With a vi e w to 3. 4 , w e note that $ has a canonical base point, corresponding to the object $

i n $. The base point i s sufficientl y natural, s o that $ becomes a contravariant functor fr o m $ to based spaces.

3.4. Co r o l l a r y . For arbitrary C , D in $, the suspension functor from $ to $ induces a homotopy equivalence $.

R e m a r k . The proof uses axioms 2.1, 2.2, 2. 3 only.

Proof. W e be g i n b y proving that suspension $ induces an isomorphism on πk fo r all k. Suppose inductively that this has been established f o r k ≤ n and al l C and D; the induction start i s o f course axiom 2.3. From 3. 3 w e have the f o l l o w i n g commutative diagram i n the homotopy category o f based spaces:

In more detail: the vertical arrows are extracted fr o m the homotopy pullback squares obtained b y applying $ and $ to the co f i b e r squares

$ respectively . This should make the horizontal arrows i n $ obvious. (Note that w e have us e d Σ cone(C), not cone(ΣC).) Applying the inductive assumption to the upper ro w i n $, one finds that the map in the lo w e r row induces isomorphisms i n πk fo r al l k ≤ n, wh i c h completes the induction step. Therefore the suspension- induced map $ i s bi j e c t i v e on πk fo r all k ≥ 0 and co n ¬ sequently

$ 5 4 8 M . We i s s , B . W i l l i a m s is a homotopy equivalence. Using $ once more, we deduce that 3.4 is correct provided C i s of the form ΣB for some B. But axiom 2.3 implies that we can indeed assume C = ΣB without loss of generality. □

Our main goal in the remainder of this section i s to prove corollary 3. 8 below, which is dual to 3. 3 since it claims linearity of the expression $ in the second variable.

W e begin by comparing $ and $. There are two seemingly different but equally reasonable ways to define a partial stabilization map from $ to $. For the first, start with the cofiber square

$ and apply $ to obtain a commutative square with contractible upper right- hand and lower left-hand terms,

This gives $ as claimed. For the other definition, start with the cofiber square

$ and apply $ to obtain a commutative square with contractible upper right- hand and lower left-hand terms,

This results in $, a homotopy equivalence by 3.3; we pre-compose with $ of 3.4, another homotopy equiva¬ lence. D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 4 9

3 . 5 . Le m m a . The two partial stabilization maps $ are homotopic. (Therefore both are homotopy equivalences.)

Proof. Le t s = sC,D b e th e partial stabilization map de f i n e d first, and le t σ = σC,D b e th e other. Fi x D. A n ea s y Yo n e d a ty p e argument wh i c h w e le a v e to the reader s h o w s that both maps are co m p l e t e l y determined, fo r al l C, b y what th e y do to $. It i s t h e r e f o r e quite enough to v e r i f y that s(idD) and σ ( i d D ) be l o n g to the s a m e component $. Here w e h a v e to b e e x p l i c i t . B y inspection, s(idD) has th e fo l l o w i n g description: Choose a path i n $ f r o m the v e r t e x $ to th e ba s e point (w h i c h i s the ze r o morphism fr o m D to co n e ( D )) , vi e w e d a s a quasi-morphism). The im a g e o f that path under the map fr o m $ to $ i n d u c e d b y cone(D) → ΣD i s a loop, e q u a l to s(idD). (T h e description i s up to contractible ch o i c e , but the de f i n i t i o n o f s i s a l s o up to contractible ch o i c e o n l y . ) Ag a i n b y inspection, σ ( i d D ) has the fo l l o w i n g description: Ch o o s e a path i n $ f r o m the ve r t e x

$ to th e ba s e point. Its im a g e under the map $ i n d u c e d b y the boundary in c l u s i o n D → cone(D) i s a loop, equal to σ ( i d D ) . T o s e e that th e loops are homotopic, w e introduce another loop. Ch o o s e a path i n $ f r o m the identity ve r t e x

$ to th e ba s e point. Its im a g e under the map $ i n d u c e d b y pr o j e c t i o n cone(D) → ΣD an d boundary in c l u s i o n D → cone(D) i s a loop, cl e a r l y homotopic (e v e n equal wi t h appropriate ch o i c e s ) to both s(idD) an d σ ( i d D ) . □

For B, C , D i n $ there i s a slant product $ d e f i n e d b y $. Not su r p r i s i n g l y , this i s in d u c e d b y a map o f s p a c e s $.

T h i s map c a n b e d e s c r i b e d a s f o l l o w s . Le t X0, X1, X2 b e t h e (unreduced ) homotopy c o l i m i t s o f the fu n c t o r s fr o m $ to ba s e d sp a c e s wh i c h take an ob j e c t

$ 5 5 0 M . W e i s s , B . W i l l i a m s

to $ and $ respectively. The obvious natural transformations induce maps

g i v i n g a based map X0 → X2 w e l l d e f i n e d up to contractible choice. (The contractible space by wh i c h w e parametrize the choices i s the space o f retractions r fr o m the reduced mapping cylinder o f X2 → X1 to X2. Each o f these retractions r can b e composed with X0 → X1 and the front inclusion o f the cylinder, gi v i n g a map X0 → X2.) W e obtain the map w e have be e n looking fo r b y projecting fr o m X2 to $ and observing that X0 i s homeomorphic to $.

3.6. Definition . A class i n $ i s nondegenerate i f the corresponding element i n [T(B), C] i s an isomorphism i n $. Equivalently, $ i s nondegenerate i f slant product with [η] i s a bijection $ f o r al l D.

R e m a r k . Axiom 2. 5 i s equivalent to the statement that the symmetry map fr o m $ to $ takes nondegenerate components to nondegenerate c o m ¬ ponents. Proof: Without lo s s o f generality, the nondegenerate component li v e s i n $ and corresponds to $. Apply symmetry to obtain a c l a s s i n $ corresponding to some element o f [T2(B), B]. This element i s exactly the canonical morphism T2(B) → B in $.

3.7. Proposition . If η belongs to a nondegenerate component of $, then the slant product with η is a homotopy equivalence $ for any D.

R e m a r k . The proof us e s only axioms 2.1, 2.2, 2.3, and o f course the S W product $ and the nondegeneracy assumption on η, wh i c h should b e read as i n the second sentence o f 3. 6 .

Proof The proof i s by a bootstrap procedure similar to that used in the proof o f 3 . 4 . First w e show that $ induces isomorphisms on πk, fo r a l l k ≥ 0 . Suppose inductively that this has already been established fo r k ≤ n and a l l D; the induction start (k = 0 ) comes fr o m the nondegeneracy assumption on η. B y applying $ and $ to the co f i b e r square

w e obtain two homotopy pullback squares (b y the second sentence i n 3. 5 and bilinearity o f $) with contractible upper right-hand and lo w e r left-hand vertices, and therefore compatible homotopy equivalences D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 5 1

Compatibility me a n s that the square

$ commutes. Ap p l y i n g the in d u c t i v e assumption to the upper ro w i n the square, w e s e e that $ i n d u c e s isomorphisms o n πk fo r k ≤ n + 1 . S i n c e ev e r y ob j e c t i n $ i s related to a n ob j e c t o f the fo r m ΣD b y a ch a i n o f we a k equivalence s (2 . 3 and 3. 2 ) , this co m p l e t e s the induction step. A s i n the proof o f 3 . 4 , w e m a y co n c l u d e that

$ i s a homotopy equivalenc e fo r al l D. I n particular, the lo w e r ro w i n the square just a b o v e i s a homotopy eq u i v a l e n c e . Bu t then the upper ro w must b e a homotopy equivalence . □

3 . 8 . C o r o l l a r y . For every C in $, the covariant functor $ takes cofiber squares to homotopy pullback squares.

Proof. Fi x C a n d ch o o s e B an d $ i n a nondegenerate component. Th i s i s p o s s i b l e b y 2. 4 an d 2. 5 . Sl a n t product wi t h η i s a map fr o m $ to $, natural i n D. B y 3. 7 i t i s a homotopy equivalence . It i s th e r e f o r e enough to kn o w that $ takes co f i b e r squares to homotopy pullback squares; but this i s part o f the definition , 1. 1 . □

4. Th e in v o l u t i o n o n $ A s s u m e throughout this s e c t i o n that $ wi t h the S W product $ sa t i s f i e s the ax i o m s o f § 2 . A t th e e n d o f this s e c t i o n w e w i l l k n o w that $ has a canonical involution. ( S t r i c t l y sp e a k i n g , w e m u s t re p l a c e $ b y a l a r g e r sp a c e $ mapping to $ b y a homotopy eq u i v a l e n c e , an d i t i s $ w h i c h has the involution. Compare [Vo].) I n later se c t i o n s w e wi l l se e that $ and $ a l s o sa t i s f y the ax i o m s ( s e e 1 . 5 ) . Wh e n c o m b i n e d , th e s e r e s u l t s l e a d to an involution o n the sp e c t r u m $ and ma n y other go o d things.

Write $ f o r the union o f the invertible components o f $. T h e n $ is a full subcategory of $, and by 3.2 its objects are those diagrams $ 5 5 2 M . W e i s s , B . W i l l i a m s

for which f i s a weak equivalence (and e is a weak equivalence and a cofibration, as always). The following key lemma relates the results of §3 to the homotopy type of $.

4.1. Lemma. For C in $, let $ be the functor on $ given by $. Then hocolim $ is contractible.

Proof. B y Thomason’s homotopy colimit theorem [Tho1], hocolim $ is homotopy equivalent to the nerve of a single category $ with the following description. Objects are diagrams

$ where f i s a weak equivalence, e i s a weak equivalence and a cofibration, C is fi x e d but D and D′ are not. Morphisms are commutative diagrams

The objects of the form $ are the objects of a fu l l subcategory $. The inclusion $ has a l e f t adjoint $, so that $. But $ has an initial object. □

R e m a r k . Let $ be the component of $ which contains the object C. It is a consequence of 4.1 that $. To see this note first that $ is empty when D is not in the component of C. Therefore hocolim $ is in ef f e c t a homotopy colimit taken over $. On the other hand, $ takes any morphism in $ to a homotopy equivalence, so that the projection hocolim $ is a quasifibration. Its total space is contractible according to 4.1, and its fiber over the base point is $.

4.2. Proposition. For B in $ let $ be the functor on $ given by $, where $ consists of the nondegenerate components. Then hocolim $ is contractible.

Proof. Essentially this is a consequence of 4.1 together with 3.7. Choose some C and some point $. W e wish to compare the two functors $ and $ on $. To make the comparison easier we introduce a third functor $ defined as follows. For D in $ let $ be the homotopy colimit of the functor on $ whose value on an object $. D u a l i t y i n W a l d h a u s e n ca t e g o r i e s 5 5 3 i s the (contractible) homotopy fiber o f $ over the point f*(η). Projection fr o m the hocolim to $ i s a homotopy equivalence, and it i s also a natural transformation $ o f functors in the variable D. But there i s also a fo r g e t f u l transformation $, and again w e know (f r o m 3. 7 ) that $ i s a homotopy equivalence fo r ev e r y D. (It i s simply the slant product with η restricted to certain components, provided w e make the identification $ T h e r e f o r e

$. □

R e m a r k . The true meaning o f 4.2 i s that ev e r y object B in $ has a (Spanier- Whitehead) dual which i s unique up to contractible choice. Indeed, hocolim $ i s the space o f choices o f duals.

4.3. Corollary. Let $ be the functor on $ defined by $. The composition of the projection map hocolim $ with projection to the first coordinate $ is a homotopy equivalence hocolim $.

Proof. There i s a Fubini principle fo r homotopy colimits o f functors on product categories which, applied to the present case, means that

$ where $ i s the functor de f i n e d i n 4.2. The homeomorphism i s over $. □

4.4. Re m a r k . Homotopy colimits can always b e interpreted as classifyin g spaces o f topological (or simplicial) categories. This i s how they appeared in the literature f o r the first time, i n [Seg]. Specifically , le t $ be the topological category whose objects are triples (B, D, z) where B, D are objects in $ and z i s a point in $. A morphism f r o m (B1, D1, z1) to (B2, D2, z2) i s a pair o f weak equivalences B1 → B2, D1 → D2 so that the induced map $ takes z1 to z2. Then $ i s homeomorphic to hocolim $. From this point o f vi e w , the true meaning o f 4. 3 i s that the fo r g e t f u l (simplicial, continuous) functor

$ induces a homotopy equivalence o f the classifyin g spaces. (Here $ i s a discrete category as always.) Note that $ d e f i n e s an involution on $, where τ i s the symmetry operator from 1. 1 . 554 M. Weiss, B. Williams

5. Generating classes

L e t $ b e a Wa l d h a u s e n ca t e g o r y sa t i s f y i n g ax i o m s 2. 1 and 2. 2 . He r e w e de v e l o p t e c h n i q u e s fo r ch e c k i n g ax i o m 2. 3 and, i f there i s an S W product $, al s o ax i o m s 2 . 4 an d 2. 5 .

L e t $ b e a c l a s s o f o b j e c t s o f $. Le t $ b e the s m a l l e s t f u l l subcategor y o f $ wh i c h ● contains $ and the ze r o ob j e c t ● i s cl o s e d under formation o f mapping co n e s ● i s c l o s e d under isomorphism, that is, $ i n $ an d B i n $ im p l i e s C i n $. W e s a y that $ i s a generating class i f ev e r y ob j e c t o f $ i s isomorphic i n $ (s e e 3 . 2 ) to an ob j e c t i n $.

A covariant or contravariant fu n c t o r fr o m $ to ba s e d sp a c e s i s w-invariant i f it takes we a k eq u i v a l e n c e s to homotopy eq u i v a l e n c e s , an d linear i f , i n addition, i t takes co f i b e r squares to homotopy pu l l b a c k squares and takes $ to a contractible s p a c e .

For e x a m p l e , any o b j e c t D i n $ de t e r m i n e s a contravariant fu n c t o r $ w h i c h i s li n e a r b y 3. 3 . I f $ i s equipped wi t h an S W product $, th e n $ i s a covariant li n e a r fu n c t o r . Le m m a s 5. 1 and 5. 2 be l o w ca n fa c i l i t a t e the an a l y s i s o f su c h functors .

5.1. Lemma. Let $ be a natural transformation between covariant linear functors from $ to based spaces. Let $ be a generating class for $, closed under suspension Σ. If $ is a homotopy equivalence for every C in $, then $ is a homotopy equivalence for every C in $.

Proof. Choose C in $ and m ≥ 0 and a diagram of cofibrations $

s u c h that Ci/Ci—1 i s isomorphic to so m e ob j e c t i n $, fo r 0 < i ≤ m, an d Cm i s isomorphic to C i n $. B y induction, $ i s a homotopy equivalence , and b y hypothesis $ i s a homotopy e q u i v ¬ a l e n c e . Us i n g the commutative diagram

$ D u a l i t y i n W a l d h a u s e n ca t e g o r i e s 5 5 5 where the rows are fibration sequences up to homotopy, we conclude that the middle arrow becomes a homotopy equivalence after Ω has been applied. The argument works equally well for ΣCm, so that

$ is a homotopy equivalence. Linearity of $ and $ now shows that $ is also a homotopy equivalence. □

5.2. Lemma. Let $ be a natural transformation between contravariant linear functors from $ to based spaces. Let $ be a generating class for $ such that $ is a homotopy equivalence for every C in $. Then $ is a homotopy equivalence for every C in $.

Proof. Choose C in $. Arguing inductively, we can assume that C is the mapping cone of a morphism f: C1 → C2 in $ such that the left-hand and middle vertical arrows in the commutative diagram

$ are homotopy equivalences. Since the rows are fibration sequences up to homotopy, it follows that the right-hand vertical arrow is also a homotopy equivalence. □

W e will often use 5. 1 and 5. 2 to establish axiom 2.3 in special cases. For the rest of the section, assume that $ comes with an SW product $. The following lemmas and corollaries are meant to facilitate the verification of 2.4 and 2.5.

Before stating 5.3, we note that any morphism in $ has a mapping cone, well defined up to non-unique isomorphism. Namely, i f the morphism is from C to D, represent it by a quasi-morphism C → D′ ← D (second arrow a weak equivalence and a cofibration); its mapping cone is defined as the mapping cone of C → D′.

5.3. Lemma. Suppose that $ satisfies 2.1, 2.2, 2.3. Let $ be a cofib¬ ration sequence in $. If $ is a co-representable functor on $ for i = 1,2, then it is co-representable for i = 3. A co-representing object is the mapping cone of T(g):T(B1) → T(B2).

Proof. Choose Ci and ηi in $ in nondegenerate components, for i = 1 , 2. Then

$ 5 5 6 M . W e i s s , B . W i l l i a m s b y 3. 7 and the remark following it. Choose a quasi-morphism C1 → C2 i n the component corresponding to that of g: B2 → B1 ( f r o m the cofibration sequence). Replacing C2 by something weakly equivalent i f necessary, w e may assume that the quasi-morphism i s a genuine morphism f: C1 → C2. It turns out that the square

$ commutes up to a homotopy which i s natural i n D. In fa c t the Yoneda li n e o f reasoning reduces this to the assertion that

$ commutes up to homotopy, where i d $ i s the identity vertex. An e q u i ¬ valent assertion i s that f*(η1) and g*(η2) are in the same component o f $. But that i s clear fr o m the construction o f f. Let C3 b e the mapping cone o f f. Bilinearity o f $ shows that there i s a fibration sequence up to homotopy

$ natural i n D, and 3. 3 gi v e s a fibration sequence up to homotopy

$ natural in D. Using the homotopy commutativity of (● ) now, w e conclude that there i s a homotopy equivalence $, w e l l de f i n e d up to homotopy and as such natural i n D. Again the Yoneda reasoning shows that e i s none other than slant product with $, the image under e o f the identity vertex i n $. □

5.4. Corollary. Suppose that $ satisfies 2.1, 2.2, 2.3 and that $ is a co- representable functor on $ for every B in a generating class $. Then $ satisfies 2.4.

Proof. For ev e r y object B i n $, the functor $ i s a co-representable functor on $, by 5.3. □

5.5. Lemma. Suppose that $ satisfies 2.1, 2.2, 2.3, 2.4. Suppose also that the functor $ is linear, for fixed B in $. Let $ be a generating class for $, closed D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 5 7 under Σ. Suppose that whenever B is in $ and C is in $ and $ is in a nondegenerate component, then the slant product with $ is a bijection $ for all D in $. Then $ satisfies 2.5.

Proof. Fi x B i n $ and C i n $ an d $ i n a nondegenerate component. W e note first that slant product w i t h $, a s a map, i s a homotopy equivalenc e $ f o r ev e r y D i n $. This i s pr o v e d ex a c t l y li k e 3. 7 . Li f t i n g the restriction o n D, w e c a n st i l l sa y that sl a n t product wi t h τ(η) i s a natural tr a n s f o r ¬ mation $ o f fu n c t o r s i n th e variable D. (A c t u a l l y , to make i t st r i c t l y natural w e w o u l d ha v e to e n l a r g e $ without c h a n g i n g the homotopy type; s e e the paragraph pr e c e d i n g 3 . 6 . ) The tw o fu n c t o r s are linear b y assumption, and fo r D i n $ the natural tr a n s ¬ formation sp e c i a l i z e s to a homotopy equivalence . Th e r e f o r e b y 5 . 1 i t specialize s to a homotopy equivalenc e fo r arbitrary D i n $. In particular, the slant product map $ i s a bi j e c t i o n fo r arbitrary D, or equivalently , the canonical map T2(B) → B i s an isomorphism i n $. Th i s holds fo r B i n $. L e t $ b e the cl a s s o f al l ob j e c t s B i n $ fo r wh i c h T2 ( B ) → B i s an isomorphis m i n $. W e h a v e s e e n that $. T h e r e f o r e i t i s s u f f i c i e n t to s h o w that i f f: C → D i s a morphism i n $, wi t h C an d D i n $, then the mapping co n e o f f i s i n $. W e ca n regard T a s a contra variant fu n c t o r fr o m $ to $. Le t u s ca l l a diagram B → C → D i n $ ex a c t if , up to an is o m o r p h i s m o f diagrams, i t ca n b e obtained fr o m a cofibratio n se q u e n c e i n $ b y pa s s a g e to $. I f B → C → D i s ex a c t , then [B, A] ← [C, A] ← [D, A] i s an ex a c t se q u e n c e o f abelian groups f o r ev e r y A i n $. (E x a c t n e s s i n the ba s e d se t se n s e fo l l o w s fr o m 3. 3 ; the ab e l i a n group structures are due to the fa c t that $ f o r an y n > 0 , b y 2 . 4 and 3. 3 . ) A n ex t r a c a r e f u l reading o f the proof o f 5 . 3 ( e x e r c i s e ) s h o w s that the contravariant fu n c t o r T takes ex a c t diagrams C → D → i s to ex a c t diagrams T(C) ← T(D) ← T(E). No w fix an ex a c t diagram C → D → E i n $ and fo r m the ladder

$ w h e r e the ve r t i c a l arrows are the canonical morphisms. Su p p o s e that C and D are i n $. Ap p l y i n g [?, A] fo r so m e A i n $ transforms the diagram into a diagram o f a b e l i a n groups wi t h ex a c t rows. Four o f the ve r t i c a l arrows are then isomorphism s o f ab e l i a n groups, and the fifth i s an is o m o r p h i s m b y th e five le m m a . Si n c e this holds fo r arbitrary A i n $, the canonical morphism T2(E) → E mu s t b e an i s o m o r p h i s m i n $, s o that E i s i n $. □ 558 M. Weiss, B. Williams

5 . 6 . L e m m a . Suppose that $ satisfies the axioms of §2. Let $ be a generating class for $. For objects B, C in $ and a class $, the following are equivalent:

( 1 ) [η] is nondegenerate; ( 2 ) slant product with [η] is a bijection $ for every D in $.

Proof As s u m i n g (2 ) , l e t $ b e the c l a s s o f a l l o b j e c t s D i n $ fo r wh i c h slant product w i t h [η] i s a bi j e c t i o n $. W e m u s t sh o w that mapping co n e s o f morphisms be t w e e n ob j e c t s i n $ be l o n g to $. Th i s i s done b y a five le m m a argument, a s i n the proof o f 5. 5 . □

6 . Th e ax i o m s an d th e $ co n s t r u c t i o n

A s s u m e throughout this se c t i o n that $ wi t h the S W product $ sa t i s f i e s the a x i o m s o f § 2 . W e sh a l l prove that $ sa t i s f i e s the axioms.

6.1. Th e o r e m . Let C, D be objects in $. The canonical map

is a homotopy equivalence.

Explanation and proof. The quadruples (i,j,p,q) sa t i s f y i n g the stated conditions f o r m a su b - p o s e t o f [m]4, wh e r e [m] = {0,...,m} as usual. The rule taking a quadruple (i,j,p, q) to $ i s a covariant fu n c t o r . Note that C(m —j, m — i) a n d D(p, q) are o b j e c t s o f $, an d that the conditions o n i,j, p, q are equivalent to m — j ≤ p and m — i ≤ q. F i x D f o r the moment. W e are t h e n d e a l i n g w i t h a natural transformation be t w e e n contravariant linear fu n c t o r s i n th e variable C. Us i n g ax i o m 2. 3 fo r $, on e finds that the ob j e c t s i n $ wh i c h are iterated de g e n e r a c i e s o f ob j e c t s i n $ fo r m a generating cl a s s $. Th e r e f o r e b y 5. 2 i t su f f i c e s to es t a b l i s h 6. 1 i n the sp e c i a l ca s e w h e n C i s i n $. Th e r e f o r e fix $, w h e r e A li v e s i n $ a n d wh e r e us : [m] → [1] i s monotone, onto, wi t h us(s) = 1 and us(s — 1 ) = 0 . The sp e c i a l i z ¬ ation fu n c t o r $ has a le f t ad j o i n t , s o that the c o r r e ¬ sponding specializatio n map

i s a homotopy equivalence . A l l w e h a v e to prove n o w i s that the other specializatio n map Duality i n W a l d h a u s e n ca t e g o r i e s 5 5 9

(●) $

i s al s o a homotopy eq u i v a l e n c e . Here w e st i l l as s u m e $, but no w w e al s o find i t co n v e n i e n t to “u n f i x ” D. I n fa c t domain and codomain o f (● ) are li n e a r f u n c t o r s i n the variable D, b y 3. 8 . Be c a u s e o f 5.1, w e ma y as s u m e without lo s s o f g e n e r a l i t y that $ f o r so m e B i n $ and so m e t. Su m m a r i z i n g , w e ma y a s s u m e $ and $, a n d mu s t sh o w that (● ) i s a homotopy equivalence . T h i s i s done a s i n the proof o f 1. 3 . □

6 . 2 . Co r o l l a r y . For any m ≥ 0 and objects C, D in $, the suspension map from [ C , D] to [ΣC, ΣD] is a bijection.

Proof. W e h a v e $ and $. T h e e x ¬ pressions w h i c h 6 . 1 gi v e s f o r $ a n d $ are homotopy eq u i v a l e n t ( v i a the suspension ) b y 3. 4 . □

6 . 3 . Lemma. $ satisfies 2.3.

Proof Le t $ b e the c l a s s o f al l ob j e c t s i n $ wh i c h c a n b e desuspende d in f i n i t e l y o f t e n i n $. B e c a u s e o f 6. 2 , al l w e h a v e to sh o w i s that $ contains al l ob j e c t s o f $. B u t $ contains the generating cl a s s $ fr o m the proof o f 6.1, and i s c l o s e d under formation o f mapping cones. □

6 . 4 . Le m m a . $ satisfies 2.2.

Proof. Su p p o s e that f: C → D i s a morphism i n $ re p r e s e n t i n g an isomorphis m i n $. Then, f o r an y monotone i n j e c t i v e υ : [1] → [m], the morphism υ*f fr o m υ*C to υ*D re p r e s e n t s an isomorphis m i n $. S i n c e 2. 2 holds fo r $, i t fo l l o w s that ea c h υ*f i s a we a k equivalence , an d this i n turn im p l i e s that f i s a w e a k equivalenc e i n $, b y th e ve r y de f i n i t i o n o f $.

6 . 5 . Lemma. $ satisfies 2.4 and 2.5.

Proof. As s u m e m ≥ 2 . Le t us: [m] → [1] b e the usual monotone su r j e c t i o n wi t h us(s) = 1 , us(s — 1 ) = 0 . B y 5. 4 , to ch e c k 2. 4 i t su f f i c e s to ch e c k that fo r B i n $ and ev e r y s w i t h 0 < s ≤ m, the fu n c t o r $ o n $ i s co-representable. Choose C i n $ and η i n a nondegenerate component o f $. L e t t = m + 1 — s. From 1. 3 w e h a v e a fo r g e t f u l homotopy equivalenc e $ $. Choose $ i n the component corresponding to that o f $. W e es t a b l i s h 2. 4 fo r $ b y sh o w i n g that $ i s i n a nondegenerate component. T o do this w e us e the homotopy commutative diagram 560 M. Weiss, B. Williams

$ w h e r e th e horizontal arrows are slant products (w i t h $ and η, respectively ) an d the v e r t i c a l arrows are specializatio n maps. (I n the lo w e r row, make the identification s $ and $.) W e sa w i n the proof o f 6. 1 that the l e f t - h a n d ve r t i c a l arrow i n (● ) i s a homotopy equivalence . Th e other ve r t i c a l arrow i s a homotopy equivalenc e wh e n D i s an iterated de g e n e r a c y o f a n ob j e c t i n $, b y 1. 3 . It i s th e r e f o r e a homotopy eq u i v a l e n c e i n ge n e r a l , b y 5.1. Th e r e f o r e $ i s nondegenerate. F i n a l l y w e note that the arguments gi v e n fo r nondegenerac y o f $ ap p l y eq u a l l y w e l to $. It fo l l o w s that the canonical morphism $ i s a n i s o m o r ¬ p h i s m i n $. N o w 5. 5 sh o w s that 2. 5 holds fo r $. Note that th e li n e a r i t y h y p o t h e s i s ne e d e d i n 5. 5 i s sa t i s f i e d , thanks to 6. 1 an d 3. 8 . □

6.6. Proposition. For B,C in $, and a class [η] in $, the following conditions are equivalent:

(1) [η] is nondegenerate; (2) the image of [η] in $ is nondegenerate, $; (3) the image of [η] in $ is nondegenerate, $; (4) the image of [η] in $ is nondegenerate $.

Proof. (1 ) $ (2 ) an d (1 ) $ (3 ) : It i s en o u g h to prove (1 ) $ (2 ) . Choose t > 0 and c h o o s e so m e D i n th e im a g e o f $. T h e n D be l o n g s to $, the generating c l a s s u s e d i n the proof o f 6.1. W e h a v e a homotopy commutative diagram

$ w h e r e the ve r t i c a l arrows are specializatio n maps, an d the horizontal on e s are sl a n t product wi t h η and the im a g e o f η i n $, respectively . It i s e n o u g h to sh o w that the tw o ve r t i c a l arrows are homotopy equivalences . (For the i m p l i c a t i o n (1 ) $ (2 ) , note that D(m — t, m) i s arbitrary; fo r (2 ) $ (1 ) , note that D i s arbitrary i n $ an d us e § 5 . ) W e w i l l i n fa c t sh o w that ea c h o f the tw o ve r t i c a l arrows taken b y i t s e l f i s a homotopy equivalence , without as s u m i n g a n y relationship b e t w e e n the ob j e c t s B an d C i n $. Us i n g § 5 , w e ma y as s u m e without lo s s o f g e n e r a l i t y that B, C are al s o i n $. Then 1. 3 takes ca r e o f the right-hand ve r t i c a l arrow, and an argument co m p l e t e l y analogous to 1. 3 (but starting fr o m 6. 1 rather than the de f i n i t i o n o f $ o n $) takes care o f the le f t - h a n d ve r t i c a l arrow. D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 6 1

( 1 ) $ (4 ) : G i v e n B, C and nondegenerate [η] as i n (1 ) and in t e g e r s s, t a s i n (3 ) w i t h 0 < s < t ≤ m, le t w : [2] → [m] b e gi v e n b y w(0) = 0 , w(l) = s, w(2) = t. T h e n $ i s gi v e n b y $, $, $. L e t μ b e the im a g e o f η i n $. T h e n (2 ) holds fo r [μ] be c a u s e i t holds fo r [η], be c a u s e (1 ) holds fo r [η]. Th e r e f o r e (1 ) and (3 ) hold fo r [μ]. I n particular, the im a g e o f [μ] i n π0 o f

$ i s nondegenerate; but o f co u r s e i t eq u a l s the im a g e o f [η] there. □

6 . 7 . Co r o l l a r y . Let w: [m] → [n] be monotone. Suppose that $ is nondegenerate, where B and C are objects in $. Then the image of [η] in $ is nondegenerate.

Proof. Us e the equivalenc e (1 ) $ (4 ) i n 6. 6 . □

7 . Th e in v o l u t i o n o n $ 7 . 1 . Definitions . Waldhausen’ s de f i n i t i o n o f the K-theory sp a c e $ fo r a W a l d ¬ h a u s e n ca t e g o r y $ i s

H e r e $ i s regarded a s a si m p l i c i a l category. Su p p o s e no w that $ co m e s wi t h a n S W product $ and sa t i s f i e s the ax i o m s o f § 2 . Us i n g the notation fr o m the remark fo l l o w i n g 4.3, w e mo d i f y Wa l d h a u s e n ’ s de f i n i t i o n a s f o l l o w s : $.

Explanations . B y § 6 , fo r ea c h m ≥ 0 , the Wa l d h a u s e n category $ co m e s wi t h an S W product an d sa t i s f i e s the ax i o m s o f § 2 . Th e r e f o r e the topological ca t e g o r y $ i s d e f i n e d ; s e e 4 . 4 . B y 6 . 7 , e a c h monotone [m] → [n] d e t e r m i n e s an operator f r o m $ to $, s o that $.$ i s a simplicial to p o l o g i c a l category. B y 4 . 3 and 4. 4 , the fo r g e t f u l fu n c t o r s

$ f o r m ≥ 0 induce homotopy equivalence s o f the classifyin g sp a c e s . Th e y commute with simplicial operators, and so define a (simplicial, continuous) functor $ 5 6 2 M . W e i s s , B . W i l l i a m s which induces a homotopy equivalence o f the classifyin g spaces. Applying Ω, w e obtain a map $ which i s a homotopy equivalence.

Each $ comes with an involution ι, as in 4.4. The involution ι anticommutes with the simplicial operators relating the $ for m ≥ 0. That is, f o r a monotone f: [m] → [n], w e have

( s e e § 1 af t e r 1. 2 ) . Nevertheless, ι determines an involution on $. Here i s an explanation i n abstract terms. First o f all, fo r each m ≥ 0 the order-reversing bijection [m] → [n] determines a linear homeomorphism Δm → Δm which w e i n ¬ dicate by $ for $. Now suppose that Y and Z are simplicial spaces, and $ i s a collection o f maps wh i c h anticommutes with the s i m ¬ plicial operators, i.e.,

$ f o r monotone f : [m] → [n]. De f i n e |g|: |Y| → |Z| by mapping the point with coordinates $ and $ to the point with coordinates $ and $. The special case w e are interested in here i s $ and g = ι. In this case |g| i s an involution, b y inspection.

R e m a r k . One can st i l l se e the similarity with [Vo]. However, our construction i s clearly more general and better adapted to [Wald2].

Waldhausen shows in [Wald2, § 1 . 3 ] that $ i s the zero term o f an Ω-spectrum whose n-th term i s the space

The same argument shows that $ i s the zero term of an Ω-spectrum with involution whose n-th term i s the space with involution

Here $ i s the geometric realization o f the n - s i m p l i c i a l space

$ and the (multi)-simplicial operators anticommute with the involution, as before. D u a l i t y i n W a l d h a u s e n ca t e g o r i e s 5 6 3

7.2. Su m m a r y . $ is an infinite loop space with involution and the forgetful map $ of infinite loop spaces is a homotopy equivalence.

Suppose now that $ and $ are Waldhausen subcategories o f $. What this means f o r $, say, i s that $ i s a Waldhausen category in its own right, with notions o f cofibration and weak equivalence restricted fr o m $, and the inclusion functor $ i s exact. Se e [Wald2, 1. 1 , 1. 2 ] . Suppose moreover that $ and $ are fu l l subcategories o f $, closed under weak equivalence in the following sense: i f C → D i s a weak equivalence in $, and one o f C, D i s i n $ (i n $), then the other i s i n $ (i n $). Then $ and $ also inherit the cylinder functor fr o m $. Suppose fi n a l l y that whenever C and D are objects i n $ such that a nondegenerate class exists i n $, then C belongs to $ i f and only i f D belongs to $. Informally, $ and $ are dual subcategories o f $. L e t $ be the fu l l topological subcategory consisting o f the objects ( C , D, z) with C in $ and D in $. Se e 4.4. Let

$ s o that $ i s an in f i n i t e loop subspace o f $. B y 4. 3 and 4.4, the f o r g e t f u l maps

$ induce homotopy equivalences o f the classifyin g spaces. This leads to the fo l l o w i n g generalization o f 7. 2 , which w e shall ne e d in §9.

7.3. Pr o p o s i t i o n . The forgetful maps $ of infinite loop spaces are homotopy equivalences. There is an involutory homeomorphism

8. Th e axioms an d parametrizatio n The goal i s to ch e c k that the category $ d e f i n e d just before 1. 5 sa t i s f i e s the axioms of § 2 provided $ does. The overall strategy i s similar to the one used in § 6 . In particular, w e re l y on 5. 1 and 5. 2 . Assume throughout this section that $ s a t i s f i e s the axioms o f § 2.

8.1. Pr o p o s i t i o n (compare 6.1). Let C, D be objects in $. The canonical map

(where s and t are faces of X) is a homotopy equivalence. 564 M. Weiss, B. Williams

Explanation and proof. Th e pairs ( s , t) wi t h $ fo r m a po s e t wh e r e ( s , t) ≤ (s′, t') i f $ an d $. Th e ru l e $ i s a covariant fu n c t o r fr o m the p o s e t to sp a c e s . L e t s0 b e a fa c e o f X. W e s a y that an ob j e c t C o f $ i s concentrated on s0 i f $ f o r an y fa c e s not containing s0, and C(s0) → C(s) i s a n isomorphis m i f s do e s contain s0. Th e ob j e c t s i n $ w h i c h are isomorphic to ob j e c t s co n ¬ centrated o n so m e f a c e o f X fo r m a generating cl a s s . T h e r e f o r e , b y 5 . 2 , i t i s e n o u g h to es t a b l i s h 8. 1 i n th e sp e c i a l ca s e wh e r e C i s concentrated o n s0. Note that th e sp e c i a l i z a t i o n fu n c t o r $ f r o m $ to $ has a le f t a d j o i n t λ wh i c h em b e d s $ i n $ a s the f u l l su b c a t e g o r y o f the ob j e c t s c o n c e n ¬ trated o n s0. It fo l l o w s that the specializatio n fu n c t o r $ ( w h e r e C i s concentrated o n s0 but D i s arbitrary) al s o has a le f t adjoint, gi v e n o n o b j e c t s b y

$ w h e r e D" i s the pushout o f λ ( D' ) ← λ(D ( s 0 ) ) → D an d w e ha v e id e n t i f i e d C w i t h λ(C(s0)). Co n s e q u e n t l y the specializatio n map

$ i s a homotopy equivalence . A l l w e h a v e to prove n o w i s that the other specializatio n map

$ i s al s o a homotopy equivalence . Th i s i s true be c a u s e

$. D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 6 5

8.2. Co r o l l a r y (compare 6.2). For objects C, D in $, the suspension map from [C, D] to [ΣC, ΣD] is a bijection. □

8.3. Le m m a (compare 6.3). $ satisfies 2.3.

Proof. Let $ be the class o f all objects i n $ which can be desuspended i n f i n i t e l y often. Be c a u s e o f 8. 2 , all w e have to show i s that $ contains all objects o f $. W e note that ( 1 ) the mapping cone of a morphism in $ with domain and codomain i n $ belongs to $; ( 2 ) i f ΣC belongs to $, then C belongs to $. Finally w e note that i f C i s an object o f $ concentrated on a fa c e s o f X, then C belongs to $. □

8.4. Le m m a (compare 6.4). $ satisfies 2.2. □

For a fa c e $, l e t $ be the class o f al l objects B i n $ such that B(s) i s weakly contractible fo r $. Then $ i s a generating class fo r $, and w e want to use it to check axioms 2. 4 and 2. 5 following the method o f 5. 4 and 5.5. Note that $ f o r B i n $ and arbitrary C in $, f r o m the definition o f $ i n $. It i s this fa c t which makes $ such a convenient generating class. Notation: For a fa c e r o f X, w e denote b y ∂r the union o f the proper fa c e s o f r, a simplicial subcomplex o f X o f dimension |r| — 1 . For C i n $, write C(r/∂r) f o r the co f i b e r C(r)/C(∂r). More generally, i f r and s are fa c e s o f X with $, write C(r/s) fo r the co f i b e r C(r)/C(s).

8.5. Le m m a . For C in $ and D in $, the evaluation functor from $ to $ induces a homotopy equivalence of the classifying spaces.

Proof. B y 5. 2 it su f f i c e s to check this when C i s concentrated on a fa c e s of X. W e have se e n before (proof o f 8. 1 ) that in this case the specialization functor

$ induces a homotopy equivalence o f the classifyin g spaces. It fo l l o w s that |$| i s contractible fo r $, i n agreement with what w e are trying to prove, si n c e then $. In the case r = s w e note that $ and obtain the factorization

Here the first functor induces a homotopy equivalence of the cl a s s i f y i n g spaces, and so does the second because the projection fr o m D(r) to D(r/∂r) i s a weak equivalence i n $. □ 566 M. We i s s , B. Williams

8.6. Proposition . $ satisfies 2.4.

Proof. Gi v e n a f a c e $ and B i n $, choose C i n $ s o that C i s concentrated o n r and su c h that there ex i s t s a nondegenerate cl a s s [η] i n $. Then

Therefore [η] determines $, and it turns out that [$] i s again nondegenerate. Namely, fo r arbitrary D i n $ w e have a commutative square o f se t s

$ i n wh i c h the horizontal arrows are slant products (w i t h [$] and [η], respectively) . The v e r t i c a l arrows are bi j e c t i o n s and the l o w e r horizontal arrow i s also a bi j e c t i o n b y 3. 7 , he r e applied to $. W e have made the id e n t i f i c a t i o n $ $. Summarizing, fo r ea c h B i n $ the fu n c t o r $ on $ i s co-representable. W e complete the proof b y applying 5. 4 . □

For C, D in $ and a fa c e r o f X, w e have the specialization map fr o m $ to $. In the proof o f the ne x t proposition, w e ne e d an enhanced sp e ¬ cialization map o f the fo r m

This i s we l l de f i n e d up to homotopy, a s the composition o f a n obvious map

( w h e r e r i s f i x e d ) wi t h a homotopy in v e r s e o f the inclusion

w h e r e $ i f $ and $. S t r i c t l y speaking, holimsF(s) i s canonically homeomorphic to the sp a c e o f pointed maps $, s o one ne e d s to choose an identificatio n o f r/∂r with $.

8 . 7 . Proposition. $ satisfies 2.5. D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 6 7

Proof. Le t B, C and [$] be as i n the proof o f 8. 6 . B y 5. 5 i t i s su f f i c i e n t to sh o w that the slant product with $ is a bijection $ f o r ev e r y fa c e $ and ev e r y $. The slant product with $ f i t s into a commutative diagram

$ i n wh i c h the lo w e r horizontal arrow i s the slant product with [μ], the im a g e o f [$] under the specialization map

The le f t - h a n d vertical arrow i n the diagram i s a bi j e c t i o n b y 8. 5 . (Here w e us e the restrictive assumption on D.) The right-hand one i s a bi j e c t i o n b y inspection. S o n o w w e ne e d to know that slant product with [μ] (l o w e r horizontal arrow) i s a b i j e c t i o n . I f s does not contain r, then both C(s) and B(s/∂s) are isomorphic to $ i n $, s o that slant product wi t h [μ] i s a bijection fo r trivial reasons. Assume therefore that s contains r. Then $ and B(s/∂s) i s isomorphic to Σ|s|—|r|B(r) i n $. With these identifications w e rewrite the lo w e r row i n the square as

$ and re c o g n i z e (e x e r c i s e ) that i t i s the slant product with $. But this i s a bi j e c t i o n b y 3. 7 , si n c e $ sa t i s f i e s 2. 5 and $ w a s nondegenerate to be g i n with. (S e e the remark af t e r 3. 6 ) . □

8.8. Proposition . For B, C in $ and $ the following are equivalent: ( 1 ) [η] is nondegenerate; (2) for each face $, the image of [η] under the specialization map

$ is nondegenerate.

Proof. W e apply 5. 6 , with $ as in 8. 5 . For D i n $ and $ as above w e have a commutative diagram o f slant products (horizontal arrows) and specialization maps (vertical arrows) 568 M. Weiss, B. Williams

B y 8.5, one of the vertical arrows is a bijection, and we have noted many times before that the other i s a bijection, too. Therefore the upper horizontal arrow is bijective i f and only i f the lower one is; in other words, (1) is equivalent to (2). □

W e need the following application of 8. 8 in §9. It wi l l help us to verify that certain (incomplete) simplicial sets have the Kan extension property.

8 . 9 . Ap p l i c a t i o n . Suppose that X= Δm, that $ is the i-th face (of dimension m — 1) for some i, and $ is the union of all proper faces of X except Y. Suppose that B and C are in $, and a nondegenerate class in $ exists. Then the following are equivalent: (1) $ is a weak equivalence; (2) $ is a weak equivalence.

Proof. For a face r of X, of any dimension, define $ as in 8. 5 and let $ be the class of all objects D in $ for which $ i f t does not contain r, and $ is a weak equivalence if t contains r. Up to weak equivalence, objects in $ are objects concentrated on r. B y 8.8, i f E1 and E2 are objects of $ and a nondegenerate class in $ exists, then E1 is in $ i f and only i f E2 is in $. In other words, $ and $ are “dual” to each other. Now assume that (1) holds. Then we can find a morphism B1 → B in $ such that B1 is in $ and B1(Y) → B(Y) as we l l as B1(X) → B(X) are weak equivalences. The mapping cone of B1 → B i s therefore weakly equivalent to an object in $. In short, B is weakly equivalent to an object in

Then C i s weakly equivalent to an object in

$ and this implies (2). The converse, (2) $ (1), can be proved similarly. □

8.10. Corollary/Definitions. Let $ be a simplicial subcomplex. Restriction of parameters in an exact functor $, and from the definition of $ in $, the restriction functor $ is accompanied by a binatural transformation

$. D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 6 9

B y 8. 8 , this takes nondegenerate components to nondegenerate components. Le t $ b e the in v e r s e im a g e o f $ under the ex a c t fu n c t o r $. Th e n $ inherits fr o m $ the structure o f a Wa l d h a u s e n ca t e g o r y wi t h S W product. It f o l l o w s fr o m 8. 8 that the dual T(B) i n $ o f so m e B i n $ b e l o n g s to $, up to isomorphism, s o that $ s a t i s f i e s ax i o m s 2. 4 and 2. 5 . A x i o m 2. 1 fo r $ i s obvious, and ax i o m s 2. 2 an d 2. 3 fo r $ c a n b e e s t a b l i s h e d a s i n 8.1, 8 . 2 , 8. 3 .

For an y m ≥ 0 an d an y fa c e $ not contained i n Y, w e d e f i n e an ex a c t fu n c t o r $. Note that ps has a le f t in v e r s e qs gi v e n b y (qsD)(t) = D f o r f a c e s $ containing s, and S f o r f a c e s t not containing s. Th e fo l l o w i n g le m m a wi l l b e us e f u l i n § 9 , i n constructing the map Ξ (s e e i n t r o ¬ duction).

8 . 1 1 . Le m m a . The map $ induced by the ps for all faces $ not contained in Y is a homotopy equivalence. Equivalently, the map from $ induced by the qs for all faces s of X not contained in Y is a homotopy equivalence.

Proof (compare [Lü, § 1 0 ] ) . For C i n $ l e t Ci i n $ b e de f i n e d b y $, w h e r e Zi i s the i - s k e l e t o n o f th e su b c o m p l e x $. T h e n ea c h C i n $ f i t s into a natural diagram o f co f i b r a t i o n s

$ w h e r e m i s th e di m e n s i o n o f X. Th e additivity theorem o f [Wald2] no w im p l i e s that the i d e n t i t y map o f $ i s homotopic to the s u m o f the maps in d u c e d by the exact functors $

f o r —1 ≤ i ≤ m. Ea c h Fi fa c t o r s through $ w h e r e the product i s ov e r al l i - s i m p l i c e s $. The co n c l u s i o n i s that the map fr o m $ to $ i n d u c e d b y the ps fo r all f a c e s $ not contained i n Y has a homotopy le f t in v e r s e . Bu t cl e a r l y i t has a homotopy right in v e r s e also, namely, the map from $ induced by the functors qs. □

8 . 1 2 . C o r o l l a r y . The diagram $ is a fibration sequence up to homotopy. □ 570 M. Weiss, B. Williams

9 . Sy m m e t r i c L-theory an d th e ma p Ξ

S u p p o s e throughout this s e c t i o n that $ i s a W a l d h a u s e n ca t e g o r y w i t h S W product, s a t i s f y i n g the ax i o m s o f §2. W e abbreviate $.

9.1. Definition/Notation . A 0- d i m e n s i o n a l symmetric Po i n c a r é object i n $ i s an o b j e c t C i n $, together wi t h a point φ i n $ w h o s e im a g e φ0 i n $ i s i n a nondegenerate component. The se t o f 0-dimensiona l sy m m e t r i c Poincaré o b j e c t s i n $ i s denoted b y $.

F u r t h e r explanations . Th e superscript $ me a n s : fo r m the sp a c e o f homotopy f i x e d points o f the action o f $. Here the action i s o n $, and the generator o f $ acts b y Τ o f 1. 1 . The homotopy fi x e d point φ i s a $- m a p fr o m i s $ to $, and φ0 i s th e va l u e o f φ at the ba s e point o f $.

I n th e ne x t definition , a Δ- s e t i s a si m p l i c i a l se t without de g e n e r a c y operators, i. e . , a contravariant fu n c t o r $ f r o m the ca t e g o r y Δ (w i t h ob j e c t s [m] fo r m ≥ 0 , and monotone in j e c t i o n s a s mo r p h i s m s ) to the category o f sets.

9 . 2 . Definition . $ i s th e realization o f the Δ - s e t $.

(Note that this depends not on l y on $, but al s o ve r y mu c h on $.) On e sh o u l d think o f a 0-dimensiona l sy m m e t r i c Poincaré ob j e c t i n $, f o r ex a m p l e , a s a b o r d i s m be t w e e n tw o 0- d i m e n s i o n a l sy m m e t r i c ob j e c t s i n $. S i m i l a r l y , 0-dimensiona l s y m m e t r i c Poincare ob j e c t s i n $ are to b e thought o f a s m-para- m e t e r bordisms, s o that $ i s the bordism theory o f s y m m e t r i c Poincaré o b j e c t s i n $.

R e m a r k . $ i s (t h e realization of ) a fibrant Δ - s e t , i. e . , on e ha v i n g the Ka n e x t e n s i o n property. Proof: Fi x m > 0 . Le t X = Δm an d de f i n e $ a s i n 8. 9 . W e w i l l so m e t i m e s re g a r d X, Y and Z a s si m p l i c i a l co m p l e x e s , and so m e t i m e s a s Δ-sets. E x t e n d i n g a Δ-map $ to a Δ-map $ i s the sa m e a s li f t i n g a 0-dimensiona l Poincaré ob j e c t ( C , φ) i n $ to a 0-dimensiona l Poincaré ob j e c t (D,ψ) in $, s o that $. Here $ i s the restriction functor. (S e e 8. 1 0 . ) T o construct (D,ψ), w e fi r s t construct D. Note that D(W) = C(W) i s pr e s c r i b e d fo r al l subcomplexe s $. L e t D(X) b e the pushout o f

$ w h e r e cyl denotes ma p p i n g cy l i n d e r s an d p i s a cy l i n d e r projection. It i s go o d to think o f D(X) a s the relative mapping cylinder o f i d : C(Z) → C(Z), re l a t i v e to $. W e d e f i n e $ a s the ba c k in c l u s i o n and $ a s D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 7 1

the front inclusion o f the relative cylinder. This completes the construction o f D; a l l w e really need to know about i t i s summarized in the two properties

( 1 ) the cofibration $ i s a weak equivalence; ( 2 ) the cofibration $ i s also a weak equivalence.

These properties ensure that $ has a li f t $. It remains to show that the class [ψ0] in $ is nondegenerate. Let f: TD → D be a morphism i n $ representing the adjoint o f [ψ0], a morphism i n $. B y 8 . 9 , the cofibrations $ and $ are weak e q u i ¬ valences. B y 8 . 1 0 , the specialization fw : (TD)(W) → D(W) i s a weak equivalence f o r any subcomplex W o f Z, i n particular fo r W= Z. Combining this information with (1 ) and (2 ) above, w e se e that fw: (TD)(W) → D(W) i s a we a k equivalence f o r al l $ s o that f i s a we a k equivalence i n $. □

Each 0-dimensional symmetric Poincaré object ( C , φ) in $ can be v i e w e d as a point i n $ projecting to the vertex ( C , C, φ0) i n $. (The notation comes fr o m §4, i n particular, the pr e f i x x indicates enlarged models designed s o that $ can act.) Further, w e have the inclusions $ and $. Consequently $ and f o r the same reason $ i s contained i n $, and so w e s e e an inclusion map

(The domain o f $ i s the realization o f a Δ-set and the codomain i s the realization o f a Δ-space.) The domain o f $ i s $ and the codomain wi l l eventually (i n 9 . 3 — 9 . 1 4 ) be i d e n t i f i e d with Ω∞ o f the Tate spectrum $ ( p r e f i x x suppressed). This i s the mapping cone o f a certain norm map fr o m a homotopy orbit spectrum to a homotopy fi x e d point spectrum,

More details about Tate spectra are gi v e n later i n this section. The identification o f the codomain o f Ξ with $ goes back to [WW2], although not i n this generality. W e w i l l not use results fr o m [WW2].

9 . 3 . L e m m a . The geometric realization of the Δ - s p e c t r u m $ is contractible.

Proof. B y 8.11, the geometric realization i n question i s homotopy equivalent to the geometric realization o f

$ 5 7 2 M . W e i s s , B . W i l l i a m s

where $ denotes the se t of nonempty subsets o f {0,...,m}. Here the fa c e operator f* induced by a monotone injection f: [m] → [n] i s g i v e n by (f*(x))(s) = x(f(s)) fo r a nonempty $. The geometric realization of any Δ -spectrum [m] → Y[m] has a canonical fil¬ tration (b y skeletons) which leads to a spectral sequence converging to the homotopy groups. Its E1 term including differential i s isomorphic to the chain complex o f graded abelian groups

∙∙∙ → π*Y[m + 1] → π*Y[m] →∙∙∙ where the differentials are alternating sums of homomorphisms induced b y fa c e operators di. In the case o f the Δ -spectrum $, the E1 term i s therefore isomorphic to the tensor product o f the graded abelian group $ with a chain complex D of ungraded abelian groups, $.

The differential in D i s g i v e n by (∂x)(s) = ∑i(— 1)ix(ei(s)) where ei i s the monotone injection from [m] to [m +1] with $. Now al l w e have to show i s that D i s acyclic, because then the E2 term o f our spectral sequence wi l l vanish. Let $ consist o f the $ which contain 0 , and let $ b e the complement o f $. The splittings $ determine a splitting of D as a graded group: $. The differential maps D_ isomorphically to D/D_. Therefore D i s acyclic. □

Call a CW-spectrum V with an action o f a discrete group G induced i f there exists a CW-spectrum U and a G-map $ which i s an ordinary homotopy equivalence.

9.4. Le m m a . Let Φ be the homotopy fiber of the face map $ induced by the functor “restriction to the 0-th vertex”. Then Φ is induced as a spectrum with $-action.

P r o o f . Remember that $ i s short fo r $, a spectrum with $- action. The inclusion of $ i n Φ i s a homotopy equivalence by 8. 1 2 . (Here w e embed Δ0 i n Δm as the 0-th vertex.) S o it i s enough to show that $ i s induced as a spectrum with $-action. Next w e reduce to the case m = 1 , as follows. Let $ be the 0-th face, opposite the 0-th vertex. There i s an isomorphism o f Waldhausen categories fr o m $ to $ where $. The isomorphism takes C in $ to the cofibration i n $, alias object i n $, g i v e n b y

$ D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 7 3 w h e r e s runs through the fa c e s o f Z. The in v e r s e isomorphis m i fr o m $ to $ c o m e s wi t h a binatural in c l u s i o n

$ w h i c h i s a homotopy eq u i v a l e n c e , re s p e c t s the sy m m e t r y operators Τ, an d takes nondegenerate components to nondegenerate components. Th i s i s cl e a r fr o m 1. 5 a n d 8. 8 . T h e r e f o r e the i n c l u s i o n o f $ i n $ i s a homotopy equivalence , an d re s p e c t s the actions o f $. Having r e d u c e d to the c a s e m = 1 , w e f i n a l l y sh o w that $ i s i n d u c e d a s a $- s p e c t r u m . Le t $ b e the 0- t h fa c e (opposite the 0- t h ve r t e x ) an d le t $ b e the unique f a c e o f d i m e n s i o n 1 . L e t $ b e the f u l l subcategor y c o n s i s t i n g o f the ob j e c t s C fo r wh i c h $ i s a w e a k equivalenc e i n $, and l e t $ b e the fu l l su b c a t e g o r y co n s i s t i n g o f th e ob j e c t s D fo r wh i c h D(t) i s we a k l y eq u i v a l e n t to the ze r o ob j e c t o f $. B y 8. 8 , i f C and D are ob j e c t s i n $ a n d a nondegenerate cl a s s i n $ e x i s t s , then C be l o n g s to $ i f a n d on l y i f D be l o n g s to $. W e n o w us e the fu n c t o r s $ and $ d e f i n e d i n 8 . 1 1 . O f th e fu n c t o r s $ and $, the fi r s t in d u c e s a homotopy eq u i v a l e n c e $ b y the approximation th e o r e m [Wald2, 1. 6 ] an d the se c o n d i n d u c e s a nullhomotopic map. O f the fu n c t o r s $ and $, the fi r s t in d u c e s a homotopy eq u i v a l e n c e $ b y the approximation theorem, an d th e s e c o n d in d u c e s a map $ w i t h th e property $, b y the ad - d i t i v i t y theorem. (N a m e l y , f+g i s homotopic to the map in d u c e d b y the ex a c t f u n c t o r $ f r o m $ to $; but $ f o r al l C i n $.) It fo l l o w s that g i s a l s o a homotopy equivalence . W e c o n c l u d e that the composition

( f i r s t map i n d u c e d b y the inclusions , s e c o n d map i n d u c e d b y ps and pt,) i s a homotopy equivalence . Th e r e f o r e , b y 8 . 1 1 ,

$ i s a homotopy equivalence . B y 7. 3 , the duality involution o n $, a s a homotopy c l a s s o f maps, ta k e s the we d g e sm m a n d $ to the we d g e summand $, and takes th e we d g e su m m a n d $ to the we d g e su m m a n d $. □

9 . 5 . Definition . Le t $ b e the fo l l o w i n g minor variation o n $ o b j e c t s o f $ are covariant fu n c t o r s fr o m the poset o f nonempty fa c e s o f Δm to $, an d morphisms are natural transformations be t w e e n su c h fu n c t o r s . A mo r p h i s m f: C → D i s a cofibratio n i f fs: C(s) → D(s) i s a cofibratio n fo r al l fa c e s s, and a w e a k equivalenc e i f fs: C(s) → D(s) i s a we a k equivalenc e fo r al l fa c e s s. 5 7 4 M. We i s s , B . W i l l i a m s

Our reason for introducing $ is this. In some situations it is annoying that the rule $ is not a simplicial spectrum (it is only a Δ -spectrum, since the degeneracy operators are missing). Fortunately the inclusions $ $ are homotopy equivalences (see 9.7 below), and the rule $ is a simplicial spectrum because $ is a simplicial Waldhausen category.

9.6. Lemma. The inclusion $ has the approximation property [Wald2, 1.6].

Proof. The first part, App 1, holds by definition. For the second part, App 2, suppose given C in $ and E in $ and a morphism x : C → E in $. We must find a factorization $

of x such that D is in $ and f i s a cofibration in $ and g is a weak equivalence in $. Suppose that D(s),fs,gs have already been constructed for all faces $ of dimension i. Let $ be a face of dimension i + 1 . Let D1 (t) be the colimit of

This may also be described as a colimit of the objects C(t) and C(s), D(s) for faces $. It follows that there is a unique map u:D1(t) → E(t) extending xt: C(t) → E(t) and the compositions

for faces $. Let D(t) be the mapping cylinder of u. The cylinder projection is a weak equivalence gt: D(t) → E(t) and the inclusion C(t) → D1(t) → D(t) i s a cofibration ft: C(t) → D(t). Proceed in the same way for all other faces of dimension i + 1 . □

9.7. Corollary. The inclusion $ is a homotopy equivalence.

9.8. Corollary. The inclusion $ is an equivalence of categories.

W e introduce an SW product in $ using the formula of 1. 5 , that is, $ $ where the homotopy limit is taken over the faces of Δm.

9.9. Corollary. $ with the SW product defined just above satisfies the axioms of § 2. D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 7 5

Apart fr o m a description o f $, l e m m a s 9.3, 9. 4 and 9. 6 (w i t h corollaries 9 . 7 — 9 . 9 ) are al l w e ne e d i n order to id e n t i f y the codomain o f Ξ i n $ wi t h $. W e turn to a description o f the Ta t e fu n c t o r $. For us, $ w i l l b e a CW-spectru m wi t h a n acton o f the di s c r e t e group G b y ce l l u l a r au t o m o r ¬ phisms, an d $ wi l l b e a CW-spectrum. (S e e the remark af t e r 9. 1 1 . ) CW-spectra and maps be t w e e n CW-spectra are de f i n e d i n [Ad, Part III, §2].

9 . 1 0 . Definitions . ● For CW - s p e c t r a $ and $, le t $ b e the ge o m e t r i c realization o f the s i m p l i c i a l se t wh o s e k- s i m p l i c e s are the maps fr o m $ to $. I f $ and $ are both eq u i p p e d wi t h a n action o f the di s c r e t e group G b y ce l l u l a r maps, then w e h a v e $, the su b s p a c e o f G - m a p s . ● Su p p o s e that $ i s a CW-spectru m wi t h an action o f the di s c r e t e group G b y c e l l u l a r maps. Le t BG b e the standard cl a s s i f y i n g CW - s p a c e fo r G, w i t h un i v e r s a l c o v e r EG. Th e sp a c e s

$ f o r n ≥ 0 fo r m a CW - Ω - s p e c t r u m wh i c h w e c a l l $. Note that $ i s al w a y s an Ω - s p e c t r u m b y de f i n i t i o n , ev e n though $ mi g h t not b e an Ω - s p e c t r u m . W e al s o d e f i n e $. ● Su p p o s e that $ i s a CW-spectru m wi t h an action o f the di s c r e t e group G b y c e l l u l a r maps. For e a c h i ≥ 0 there i s a cofibratio n se q u e n c e $ o f pointed G-spaces, wh e r e $ denotes the join and EGi i s the i - s k e l e t o n of EG. T h i s l e a d s to a fibration se q u e n c e up to homotopy o f spectra,

The right-hand te r m i n this se q u e n c e i s the Ta t e sp e c t r u m $.

9 . 1 1 . Properties . ● A map $ o f CW-spectra wi t h ce l l u l a r G-actions wh i c h i s a n ordinary homotopy equivalenc e in d u c e s a homotopy equivalenc e $. ● The fu n c t o r $ takes homotopy pushout squares o f CW-spectra wi t h c e l l u l a r G-action to homotopy pushout squares o f spectra. It takes $ to $. ● I f G i s finite a n d $ i s a n in d u c e d G-spectrum, then $ i s contractible. ● For finite G, t h e r e i s a ch a i n o f homotopy eq u i v a l e n c e s

$. natural i n the variable $. I n other words, the homotopy fiber o f $ i s related b y a ch a i n o f natural homotopy equivalence s to $. 5 7 6 M . We i s s , B . W i l l i a m s

Proof. The first two properties listed are obvious. For the last two, we assume that G i s finite and use the following fact: there exist a functor V from CW-spectra $ with cellular G-action to CW-spectra and a natural transformation $, the norm map, such that a) $ is related through a chain of natural homotopy equivalences to $; b) $ is a homotopy equivalence i f $ is induced as a G-spectrum.

Such a pair $ is provided by [GM, 5.10], where $ is denoted $ and $ is informally identified with $. W e use it to set up a commutative square

$ where the horizontal arrows are induced by the projections from $ to $. Since $ has a finite G-invariant filtration by CW-subspectra for which the filtration quotients are induced, the norm map for $ is a homotopy eq u i v ¬ alence by property b) of V and $. Therefore the left-hand vertical arrow in the square is a homotopy equivalence. Also, the upper horizontal arrow is a homotopy equivalence by property a) of V and $. As a result we can identify the lower horizontal arrow with the right-hand vertical arrow (the norm map), at least in the homotopy category. The last two properties listed in 9. 1 1 are now restatements of properties a) and b). □

Remark. An early version of the Tate construction appears in [WW2]. More co n ¬ ceptual versions appear in [AdCoDw] and [GM]. The norm map plays a central role in [AdCoDw], but unfortunately property b) which we used in the proof of 9 . 1 1 is not explicitly stated. In the terminology of [GM], the Tate construction is a functor from suitable G-spectra to suitable G-spectra, and our Tate spectrum is the fi x e d point spectrum (in the equivariant sense) of theirs. In [AdCoDw] and [GM], the group G is a compact Lie group, so that our account above i s less general in one respect and more general in another respect (since we allow infinite discrete groups). Allowing infinite groups in this context is an idea going back to Pierre Vogel. However, we only need the case $.

W e return to our business: understanding the codomain of $ in ($).

9.12. Theorem. The following inclusions of spectra are homotopy equivalences:

Proof Abbreviate $. B y 9.4, the homotopy fiber of the 0-th vertex operator from κ(m) to Κ(0) is induced as a spectrum with $-action. Since the D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 7 7

Tate functor $ respects fibration sequences up to homotopy and annihi¬ lates induced spectra, the 0-th vertex operator $ i s a homotopy equivalence for every m. Therefore all face operators in the Δ -spectrum $ are homotopy equivalences, so that the inclusion of $ in $ a homotopy equivalence. Next we look at the cofibration se ¬ quence

B y the last property in 9.11, its last term is homotopy equivalent to the geometric realization of $. This simplifies to

$ which is contractible by 9.3. □

Obviously what we really want i s an infinite loop space version of 9.12, and so what we need is a lemma stating that under certain conditions, Ω∞ commutes with geometric realization. The following i s enough.

9.13. Le m m a [Wald1, Lemma 5.2], [May, Thm. 12.7]. Let $ be a diagram of based simplicial CW-spaces. Suppose that

● g m f m : Xm → Zm is the zero map for each m ≥ 0; ● Xm → Ym → Zm is a fibration sequence up to homotopy for each m ≥ 0; ● Zm is connected for each m ≥ 0.

Then $ is a fibration sequence up to homotopy.

R e m a r k . Let $ be a simplicial CW-spectrum such that $ is 0-connected for all m ≥ 0. Let $ be a functorial Ω-spectrification of $. Lemma 9.13 shows that the geometric realization of $ is again a CW-Ω-spectrum. In particular:

9.14. Theorem/Summary. All arrows in the following diagram of spaces are homotopy equivalences: 578 M. Weiss, B. Williams

Proof. Abbreviate $ a n d $. The la s t arrow i n the diagram i s a homotopy eq u i v a l e n c e si n c e al l fa c e operators i n the Δ -s p a c e $ are homotopy eq u i v a l e n c e s (s e e proof o f 9. 1 2 ) . Using 9 . 6 — 9 . 9 , w e c a n re p l a c e the mi d d l e arrow b y the in c l u s i o n o f ge o m e t r i c realizations o f simplicial sp a c e s

B y 9.13, the diagram

$ i s a fi b r a t i o n se q u e n c e up to homotopy, and it s la s t te r m i s homotopy eq u i v a l e n t to Ω∞ o f the realization o f $, w h i c h i s contractible b y 9 . 1 2 . □

Summarizing , 9. 1 4 i s our li c e n s e to wr i t e

1 0 . St a b l e SW pr o d u c t s

1 0 . 1 . De f i n i t i o n (compare 1. 1 ) . B y a stable SW product o n a Wa l d h a u s e n ca t e g o r y $ w e sh a l l me a n a fu n c t o r

$ D u a l i t y i n W a l d h a u s e n ca t e g o r i e s 5 7 9 from $ to the category of Ω-spectra (details below) which takes pairs of weak equivalences to homotopy equivalences, and which i s symmetric and bilinear in the following sense. ● Symmetry means that the functor comes with an isomorphism $ $, natural in both variables, whose square is the identity on $. ● Bilinearity means (in the presence of symmetry) that, for fi x e d but arbitrary D, the functor $ takes any cofiber square in $ to a homotopy pullback square of Ω-spectra. Bilinearity also means that $ is contractible.

In 10.1, an Ω-spectrum is a collection of based spaces Em for $, together with based maps εm: Em → ΩEm+ 1, for $. (To be even more precise, Σ means smash product on the le f t with [—1, +l]/{ —1, +1}, and Ω is right adjoint to Σ.) A morphism from an Ω-spectrum { E m } to another, say {$}, i s a sequence of maps $ such that $ for all $. The morphism i s a homotopy equivalence i f each fm is a based homotopy equivalence. The m-th term of the Ω-spectrum $ in 10 . 1 will be denoted by $.

10.2. Observation. Let $ be a stable SW product on $. Then for each $, the functor $ is an (unstable) SW product on $. □

10.3. Proposition. Suppose that the SW product $ satisfies the axioms of § 2 for m = 0. Then $ satisfies the axioms of § 2 for all $.

Proof. W e have $, showing that $ i s a co-representable on $ i f and only i f is co-representable. Therefore 2.4 holds for $ i f and only i f it holds for $. Given C in $ and $, write TmC for the object (unique up to unique isomorphism) which co-represents $. Then

$ which shows that $. Therefore $ i s an equivalence of categories i f and only i f Tm _ 1 is an equivalence. In other words (see remark after 2.5), axiom 2.5 holds for $ i f and only i f it holds for $. □

10.4. Proposition. Suppose that the Waldhausen category $ has a cylinder functor and an SW product $ as in 1.1. There exists another SW product $ on $ and a natural transformation $ which is a based homotopy equivalence and respects symmetry. Consequently $ is the zero-th term of a stable SW product $. on $.

Proof. Given C and D in $ let X(C, D) be the reduced homotopy colimit of the commutative diagram of based spaces 5 8 0 M . We i s s , B . W i l l i a m s

(●)

$ w h e r e a l l maps are i n d u c e d b y $, cone(C) → ΣC, $, cone(D) → ΣD. L e t $ b e the re d u c e d homotopy co l i m i t o f the sm a l l e r diagram obtained fr o m the above b y de l e t i n g the term i n the center, $, and al l arrows touching it. Si n c e Y(C, D) i s a reduced homotopy co l i m i t o f contractible s p a c e s , i t i s homotopy eq u i v a l e n t to the re d u c e d homotopy co l i m i t o f th e co r r e ¬ sponding diagram wi t h a point at ea c h ve r t e x . Th e r e f o r e Y(C, D) i s contractible and the in c l u s i o n $ i s a homotopy equivalence , s o that

The sy m m e t r y τ o f $ le a d s to an involutory natural is o m o r p h i s m

$ w h i c h in v o l v e s a diagram flip. W e d e f i n e $. T h e s y m ¬ m e t r y $ i s gi v e n b y $, w h e r e f denote s a ba s e d map f r o m [— l , + l]/{—l,+l} to X(C,D)/Y(C,D) and υ: [—1,+ 1 ] → [—1,+1] i s the re f l e c t i o n at 0 .

L e t V(C, D) b e the re d u c e d homotopy co l i m i t o f th e diagram o f ba s e d sp a c e s

( ● ● )

$ a n d le t $ b e the re d u c e d homotopy co l i m i t o f th e subdiagram obtained b y d e l e t i n g the te r m i n the m i d d l e an d al l arrows touching it. Th e n c l e a r l y

$ w h e r e Ψ i s th e cl a s s i f y i n g sp a c e o f the in d e x i n g ca t e g o r y (w i t h 9 ob j e c t s ) w e are u s i n g , an d ∂Ψ i s the classifyin g sp a c e o f the subcategor y obtained b y de l e t i n g the terminal ob j e c t . D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 8 1

W e want to produce a map j: V(C, D)/W(C, D) → X(C, D)/Y(C, D) with good symmetry properties. T o do s o w e u s e the cofibrations $ c, $ and the zero maps $, C → ΣC, $, D → ΣD to get a map fr o m diagram ($) to diagram (●). Passage to reduced homotopy colimits then gi v e s the map j w e want. B y inspection, the following diagram commutes:

$ where φ : Ψ/∂Ψ → Ψ/∂Ψ i s the involution induced b y the diagram fl i p mentioned earlier. Now Ψ as a space with $-action i s homeomorphic to [ — l, + l]×[—1, + 1], with $ acting b y re f l e c t i o n υ on the first factor and trivially on the second factor. Using this second homeomorphism, w e can write j in the form

$ with adjoint $. S o w e have our natural map fr o m $ to $. It i s a homotopy equivalence and has al l the symmetry properties w e need. □

11. Qu a d r a t i c L-theory, an d Ξ re v i s i t e d

Suppose again that $ i s a Waldhausen category equipped with an SW product $, and that $ and $ sa t i s f y the axioms o f § 2. W e wi l l us e the stabilization $ o f $ g i v e n b y 10 . 4 to improve on §9 in two respects. First, the space $ ( d e f i n e d using $) turns out to be an in f i n i t e loop space, and the map Ξ fr o m it to $ i s a map o f in f i n i t e loop spaces. Second, w e can de f i n e a quadratic L-theory space $, whose realization i s again an in f i n i t e loop space, and a symmetrization map $, which i s a map o f i n f i n i t e loop spaces. Usually $ has greater geometric significanc e than $, and one tends to b e interested in the composition

$ which has enormous geometric significance .

It wi l l be necessary to handle several SW products at the same time, so w e w i l l fo r example write $ to mean $, constructed using the S W product $. 5 8 2 M. W e i s s , B . W i l l i a m s

Notation, pr e l i m i n a r i e s . For a based Δ -space X let $ be the Δ -space given by $ for n > 0, and $. For i≥0 the (i + l)-th face operator $ agrees with the i-th face operator X(n) → X(n — 1), and the 0-th face operator is zero. Compare [Cu]. Note that every simplex in $ has its 0-th vertex and 0-th face (opposite the 0-th vertex) at $. There is a continuous bijection $ provided geometric realizations are taken in the based sense; for example, $. If each X(n) i s a based CW-space and the face operators X(n) → X(m) are cellular, then $. In a similar spirit, we introduce the fu l l subcategory of $ consisting of all objects B for which $ i f s is the 0-th vertex or the 0-th face. It is isomorphic to $, and so we have the inclusion functor $. Explicitly: σ takes B in $ to $ defined by

$ i f t is the 0-th vertex $ i f t is contained in the 0-th face $ otherwise where $ has vertices x1 — 1,...,xr — 1 i f $ has vertices 0, xl,..., xr. The functor $ intertwines the SW products on $ on $ and $ on $ (see 1. 5 ) in the following way. There is a forgetful fibration

$ where t runs through all faces but s only runs through those contained in the 0-th face or equal to the 0-th vertex. The codomain of this fibration i s contractible since it is a homotopy limit of contractible spaces. The fiber over the base point is homeomorphic to $ by inspection. The inclusion of the fiber is therefore a homotopy equivalence $. W e compose

$ to get our intertwining map. (Occasionally we denote it by σ also.) It follows from 8 . 8 that the intertwining map takes nondegenerate components to nondegenerate components; moreover it i s clearly compatible with the symmetries τ of 1.1.

11.1. Proposition. The spaces $ for $ form an Ω-spectrum.

Proof. For the purpose of this proof, think of $ as an (unrealized) Δ-set. Map $ to $ by sending the (m + l)-simplex (C,φ) to the (m + l)-simplex (σC, σφ). Here (C, φ) is a 0-dimensional symmetric Poincaré object in $ with respect to $. B y inspection, the induced homomorphism of D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 8 3 homotopy groups fr o m $ to $ i s an isomorphism fo r m>0. □

Next, w e repeat the ex e r c i s e with $ instead of $. Here again i t w i l l b e important to declare the S W product, so w e write $ instead o f just $, to indicate that $ acts b y an action constructed using the S W product $. Our model fo r $ i s the geometric realization o f the Δ-space $, as explained i n 9. 1 4 .

11.2. Proposition. The spaces |$| for $ form an Ω-spectrum.

Proof. Much as i n the proof o f 11.1, the functors σ induce a map o f Δ -spaces fr o m $ o f $ to $. This gi v e s the s p e c ¬ trum structure, but it does not sh o w that the spectrum s o obtained i s an Ω-spectrum. Using 9. 1 4 , w e reduce to the statement that the spaces

f o r $ form an Ω-spectrum (with structure maps induced b y the functors σ as before). Si n c e al l fa c e operators i n the Δ-spectrum $ are homotopy equivalences, it becomes irrelevant whether w e apply Ω∞ before or af t e r realization. S o al l w e have to sh o w i s that fo r fi x e d $, the map o f Δ-spectra

induced b y the functors σ turns into a homotopy equivalence. For this purpose w e abbreviate $ and $. W e have to sh o w that a certain map $ i s a homotopy equivalence. Si n c e al l fa c e operators i n $ are homotopy equivalences, the canonical map

i s a homotopy equivalence. There i s an analogous map fo r $; i t has the fo r m

and it i s also a homotopy equivalence because al l f a c e operators i n $ are homotopy equivalences. Now i t only remains to sh o w that the map be t w e e n homotopy colimits o f the rows i n the commutative diagram 5 8 4 M . W e i s s , B . W i l l i a m s

$ i s a homotopy equivalence, or equivalently, that

$ i s a fibration sequence up to homotopy. Si n c e the Tate functor respects fibration sequences up to homotopy o f spectra, i t i s more than enough to show that the K-theory functor turns

into a fibration sequence (up to homotopy) o f spectra. But this i s clear f r o m 8.11. □

R e m a r k . The proof o f 11 . 2 also shows that the spectrum described i n 11 . 2 i s homotopy equivalent to $.

For each $, the constructions o f §9 gi v e us a map $ fr o m $ o f 11 . 1 to the space $ o f 11 . 2 . As i varies, these maps constitute a map o f spectra which w e write i n the form

W e turn to the construction o f $, the quadratic L-theory space. It wi l l be necessary to expand the Ω-spectrum $ into an Ω-bispectru m $, with terms $ s o that $. One wa y to achieve this, naturally in C and D, i s to de f i n e $ f o r m > 0 inductively as the n-th term i n a functorial Ω-spectrification o f $.

Given C i n $, a convenient model or replacement fo r $ i s the union $ hofiber[X → Xi] where X i s the space o f $-maps $ and Xi i s the space o f $-maps o f spectra

Both mapping spaces are to be constructed as geometric realizations of si m p l i c i a l sets. Correctness o f this model or replacement fo l l o w s fr o m 9. 1 0 and 9.11; that is, i t has the right homotopy type. It i s convenient because i t comes with a fo r g e t f u l map to $. This i s o f course the norm map, as a map o f in f i n i t e loop spaces. — T h e s e conventions are understood in the next definition. D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 8 5

11.3. Definition/Notation. A 0-dimensional quadratic Poincaré object in $ is an object C in $, together with a vertex ψ in $ whose image under the composition

$ i s in a nondegenerate component. The set of 0-dimensional quadratic Poincaré objects in $ is denoted by $.

11.4. Definition. $ i s the geometric realization of the Δ-set given by $.

R e m a r k . $ is a fibrant Δ-set.

Implicit in definitions 11.3 and 11.4 is a Δ-map $, given by converting all quadratic Poincaré objects in sight into symmetric ones via the norm map. This is the symmetrization map, and for the usual reasons it is an infinite loop map. For example, a delooping of $ is $.

12. Naturality

Let $ be an exact functor between Waldhausen categories equipped with SW products $ and $, respectively. Suppose that the axioms of §2 are satisfied for both $ and $, and that in addition to F we are given a natural transformation

$ commuting with the symmetry operators Τ (see 1. 1 ) and taking nondegenerate components to nondegenerate components. Then the map $ induced by F becomes a $-map between spectra with $-action provided the big models of $ and $ of §7 are used. Also, there are maps of spectra $, $ induced by F and φ, and the diagram

$ commutes. All of this is straightforward, but often the construction of a pair ( F , φ) is laborious. W e shall discuss two examples in 12.A. 5 8 6 M . We i s s , B . W i l l i a m s

1 . A . E x a m p l e s 1 . A . 1 . Ex a m p l e . W e already ha v e the ex a m p l e 1. 4 , wh e r e $ i s the category o f bounded ch a i n co m p l e x e s o f fi n i t e l y ge n e r a t e d abelian groups, and $ $. More g e n e r a l l y , s u p p o s e that $ i s an additive category, an d l e t $ b e the c a t e g o r y o f bounded ch a i n co m p l e x e s i n $. A contravariant additive fu n c t o r $ has a canonical ex t e n s i o n $. Followin g [Ra3], a chain duality o n $ i s a contravariant additive fu n c t o r $ together wi t h a natural transformation e fr o m $ to the in c l u s i o n $ s u c h that fo r ea c h ob j e c t M i n $ i) $, i i ) eM : T2 ( M ) → M i s a homotopy equivalence .

A s s u m e no w that $ i s eq u i p p e d wi t h a ch a i n duality ( T , e). G i v e n ob j e c t s C, D i n $ de f i n e $ ( a ch a i n co m p l e x o f ab e l i a n groups). Ranicki s h o w s that there i s a canonical involutory is o m o r p h i s m $. It f o l l o w s that an S W product o n $ ca n b e de f i n e d b y

1 . A . 2 . Ex a m p l e . Le t $ b e the category o f ba s e d compact CW - s p a c e s (b a s e points a r e understood to b e 0 - c e l l s ) . Morphisms are al l c e l l u l a r maps, the cofibration s are those morphisms wh i c h up to CW - i s o m o r p h i s m are in c l u s i o n s o f su b c o m p l e x e s , a n d the we a k equivalence s are th e morphisms wh i c h are homotopy eq u i v a l e n c e s . For ob j e c t s X, Y i n $ w e l e t $. I n more detail: Le t $ b e the ge o m e t r i c realization o f the si m p l i c i a l s e t wh o s e k- s i m p l i c e s are the ba s e d maps

$ w h e r e $ i s the one-point compactificatio n o f $. Le t $ b e the co l i m i t o f t h e ba s e d sp a c e s $. The conditions i n 1. 1 are ea s i l y ve r i f i e d .

1 . A . 3 . E x a m p l e . F i x a s p a c e B. L e t $ b e the c a t e g o r y o f retractive r e l a t i v e C W - s p a c e s o v e r B, wi t h finitely m a n y ce l l s re l a t i v e to B. I n other words, a n ob j e c t o f $ i s a r e t r a c t i v e sp a c e

( r i = i d ) w h e r e X has th e structure o f a CW - s p a c e re l a t i v e to B, w i t h finitely m a n y c e l l s . A mo r p h i s m i n $, fr o m $ to $, i s a map $ r e l B wh i c h i s c e l l u l a r re l a t i v e to B, an d re s p e c t s the retractions. A m o r p h i s m i s a cofibration if , up to isomorphism, i t i s an in c l u s i o n o f a re l a t i v e CW-subspace . A morphism i s a weak equivalence i f i t be c o m e s an isomorphis m i n D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 8 7 t h e re l a t i v e homotopy ca t e g o r y o f sp a c e s containing B. Th e s e definition s are taken f r o m [Wald2].

For $ i n $ le t $. For $ and $ i n $ l e t $, w h e r e $. T h e union $ i s to b e topologized i n su c h a wa y that ● Z wi t h the ob v i o u s topology i s a n op e n su b s p a c e ● a s e t $ containing ∞ i s a neighborhood o f ∞ i f and on l y i f the i m a g e o f it s co m p l e m e n t under Z → X × Y has cl o s u r e di s j o i n t fr o m B × B.

Points i n $, other than ∞, are tr i p l e s (x, w, y) wh e r e $ an d $ an d ω i s a path [—1,1] → B s u c h that ω(— 1 ) = r(x) an d ω(l) = r(y). W e do not c l a i m that $ i s co m p a c t l y generated Ha u s d o r f f . W e do not cl a i m that the ba s e point i n $ i s nondegenerate. Le t $, u s i n g the conventions o f 1 . A . 2 to make Ω∞Σ∞ pr e c i s e . Note that $ i s a CW - s p a c e .

R e m a r k . T h e s p a c e $ has a canonical filtration w h o s e q-th s t a g e i s the union o f $ f o r al l i,j wi t h i+j = q. He r e Xi i s the re l a t i v e i - s k e l e t o n o f X, st i l l a retractive s p a c e o v e r B. S o m e t i m e s i t i s c o n v e n i e n t to take this filtration into account a n d to r e - d e f i n e $ a s the g e o m e t r i c realization o f the s i m p l i c i a l s e t w h o s e k - s i m p l i c e s ar e filtration preserving ba s e d maps

w h e r e Δk has the sk e l e t o n filtration. Ag a i n , $ w o u l d b e d e f i n e d a s the c o l i m i t o f th e ba s e d sp a c e s $.

Illustration . S u p p o s e that $ a n d $. That is, both X an d Y are s p a c e s obtained fr o m B b y adding a di s j o i n t point. Th e n $ i s homotopy e q u i v a l e n t to $, w h e r e $ i s the sp a c e o f paths i n B f r o m r(x) to r(y), wi t h the compact-open topology. Note that w i t h our definitions , $ i s a CW - s p a c e ev e n though B an d he n c e $ c a n b e pathological.

W e n o w c h e c k that $ i n 1 . A . 3 s a t i s f i e s the conditions i n 1. 1 . Sy m m e t r y i s obvious. For the w-invariance, su p p o s e that f: X → X' i s a we a k equivalenc e i n $ an d Y i s another ob j e c t o f $. Th e n there ex i s t s a map g: X' → X wh i c h i s re l a t i v e to B but perhaps not ov e r B, an d homotopies $, a l s o re l a t i v e to B but perhaps not ov e r B. Us i n g g and the homotopies, on e sh o w s ea s i l y that

i s a ba s e d homotopy eq u i v a l e n c e . Th e r e f o r e $ i d fr o m $ to $ i s a homotopy equivalence . 5 8 8 M . W e i s s , B . Wi l l i a m s

Showing bilinearity i s harder. Si n c e w e have the w-invariance, it i s enough to consider a co f i b e r square i n $ o f the fo r m

(* ) $. where cone(U) and cone(g) are de f i n e d using the cylinder functor i n $. W e must s h o w that fo r any Z i n $ the resulting square

(**)

$ i s a homotopy pullback square. Now (* * ) i s obtained fr o m

(*** ) $ b y applying Ω∞Σ∞, and inspection shows that (* * * ) i s a pushout square o f based spaces, with $ homeomorphic to the reduced cone on $. S o i t i s enough to show that the functor Ω∞Σ∞ takes squares o f based spaces o f the fo r m

$ to homotopy pullback squares. An equivalent but more informal wa y to sa y the s a m e thing: the stable homotopy groups o f the based spaces X, Y and $ cone(X) fit into a long exact sequence. This may se e m li k e a familiar statement. Bu t at this l e v e l o f generality i t i s probably not so familiar, si n c e w e are working with reduced cones and mapping cones, taken i n the category o f “a l l ” based spaces (including based spaces with degenerate base points). Adams proves it i n [Ad, Part III, 3.10]. (Here i s some guidance. W e n e e d III.3.10 o f [Ad] with X and Y equal to suspension spectra o f possibly pathological based spaces, and W equal to the sphere spectrum. S o w e n e e d 3. 1 0 i n the case where the domain i s a CW-spectrum but the codomains are just spectra as i n [Ad, Part III, §2]. Si n c e hi s 3. 1 0 re l i e s on his 3. 8 , 3. 7 and 3. 6 , these must be interpreted i n the same wa y - domains are CW , codomains need not be.) D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 8 9

For the ne x t ex a m p l e s w e introduce control. A s a preliminary, le t B b e a se t an d l e t U, V b e su b s e t s o f B × B. Le t

$ s u c h that $ $.

1 . A . 4 . Definitions . A control structure on a se t B i s a co l l e c t i o n $ o f su b s e t s o f B × B wi t h the fo l l o w i n g properties. ●$. ●$. ●$. ● the diagonal o f B × B i s i n $. A su b s e t W o f B i s $ - b o u n d e d if W×W be l o n g s to $.

E x a m p l e ( b o u n d e d c o n t r o l ) . S u p p o s e that B i s e q u i p p e d wi t h a metric. L e t $ co n s i s t o f a l l s u b s e t s $ f o r w h i c h $. T h i s d e f i n e s a control structure. The $- b o u n d e d su b s e t s o f B are ex a c t l y the su b s e t s o f finite diameter.

E x a m p l e (continuou s co n t r o l ) . Su p p o s e that B i s an op e n de n s e su b s e t o f a sp a c e C. L e t $ co n s i s t o f a l l $ w i t h the f o l l o w i n g property: f o r a n y (Moore-Smith) s e q u e n c e o f points (xα, yα) i n U s u c h that on e o f the (Moore-Smith) se q u e n c e s (x α) and ( y α ) i n B co n v e r g e s to a point i n C\B, the other co n v e r g e s to th e sa m e point i n C\B. Again, this de f i n e s a control structure o n B. The $- b o u n d e d su b s e t s o f B are the on e s wh o s e cl o s u r e i n B ag r e e s wi t h their cl o s u r e i n C. ( A Moore-Smith se q u e n c e i s a map fr o m a di r e c t e d se t ; i f ev e r y point i n C has a countable neighborhood base, ordinary se q u e n c e s in d e x e d b y $ wi l l do.)

I n th e s e ex a m p l e s , B i s a topological sp a c e and the control structure i s compatible w i t h the topology i n the fo l l o w i n g se n s e : Ea c h $ i s contained i n so m e $ w h i c h i s open i n B × B.

S u p p o s e no w that B i s a sp a c e eq u i p p e d wi t h a control structure $, compatible w i t h the topology. L e t p: X → B and q : Y → B b e s p a c e s o v e r B. A map f: X → Y i s $ - c o n t r o l l e d (o r just controlled fo r short) i f the se t $ b e l o n g s to $. A homotopy o f maps fr o m X to Y i s controlled i f i t i s controlled a s a map fr o m X × [0,1] to Y, w h e r e X × [0,1] i s v i e w e d as a s p a c e o v e r B i n the most obvious w a y .

E x a m p l e . Le t $ b e the control structure o n B d e t e r m i n e d a s ab o v e b y an in c l u s i o n B → C, wh e r e B i s op e n de n s e i n C. L e t p: X → B an d q: Y → B b e sp a c e s ov e r B. Th e n a map f : X → Y i s $- c o n t r o l l e d i f an d on l y i f the fo l l o w i n g holds: fo r e v e r y $ and ev e r y neighborhood V o f z i n C, there ex i s t s another neighbor- 5 9 0 M . W e i s s , B . W i l l i a m s hood W o f z i n C su c h that $ implies $ and $ implies $. This i s the definition o f controlled map us e d i n [ACFP] and [CaPe].

1 . A . 5 . Definition . Let $ b e a control structure on Bi, fo r i = 1,2. The product con¬ trol structure $ on $ consists o f those subsets o f $ whose images i n B1 × B1 and B2 × B2 under the appropriate projections belong to $ and $, respectively.

1 . A . 6 . Preliminaries . Suppose that the space B i s equipped with a control structure $, compatible with the topology. Let $ b e the category o f retractive spaces $ f o r wh i c h

● X comes with a structure of finite dimensional CW-space relative to B; ● fo r ev e r y $-bounded $, the number o f ce l l s $ f o r wh i c h $ i s finite; ● the si z e s o f the ce l l s $ are controlled i n the se n s e that $ belongs to $, where r : X → B is the retraction.

The morphisms i n $ are the retractive cellular maps. A morphism i s a cofibration i f , up to isomorphism, it i s the inclusion of a relative CW-subspace. A morphism

$ i s a weak equivalence i f there e x i s t s a map g : Y → X (relative to B but not necessarily over B) and controlled homotopies $, $ relative to B.

It i s not completely obvious that $ sa t i s f i e s the axioms o f a Waldhausen category. Axioms Cof 1, Cof 2, Cof 3 and Weg 1 are obviously sa t i s f i e d , but Weg 2 i s not quite s o easy. However, Weg 2 fo r $ i s a direct consequence o f the fo l l o w i n g controlled homotopy extension property (CHEP) wh i c h can b e established b y induction over skeletons: L e t $ b e a cofibration i n $ (w e assume $). Let W b e any other retractive space ov e r B. Let f : Z → W b e a retractive map and le t {ht: Y → W|0 ≤ t ≤ 1 } be a controlled homotopy o f maps relative to B, with h0 = f on Y. Then there e x i s t s a controlled homotopy { H t : Z → W} o f maps relative to B, extending {ht}, w i t h H0= f.

Note that i f B i s equipped with the trivial control structure, s o that $ consists o f all subsets o f B × B, then $ i n l.A.6 is identical with $ i n l.A.3. In this connection, note also: ● The control structure determined b y a metric on a space B is trivial i f and only i f B has finite diameter. ● Any control structure on a compact space B wh i c h i s compatible with the topology i s trivial (exercise) . D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 9 1

1 . A . 7 . E x a m p l e . W e continue to wo r k i n $ o f 1. A . 6 , but w e as s u m e al s o that B has a countable base f o r its topology. For X, Y i n $ and op e n $, $, l e t

$ w h e r e Z i s the sp a c e o f tr i p l e s (x, ω, y) wi t h $ and $ a n d ω :[ — 1, 1 ] →B s u c h that Ω(—1) = r(x), ω(l) = r(y) and $. The union $ i s to b e topologize d i n su c h a wa y that ● Z wi t h th e ob v i o u s topology i s an open su b s p a c e ● a se t $ containing ∞ i s a neighborhood o f ∞ i f an d on l y i f the i m a g e o f it s co m p l e m e n t under Z → X × Y has cl o s u r e di s j o i n t fr o m B × B, and the im a g e o f it s co m p l e m e n t under Z → X → B i s $-bounded.

W e do not cl a i m that $ i s co m p a c t l y ge n e r a t e d Ha u s d o r f f . Ho w e v e r , us i n g the conventions o f 1. A . 2 and 1. A . 3 to make se n s e o f Ω∞Σ∞, w e find that

$ i s a CW - s p a c e and that $ i s an S W product o n $.

R e m a r k . Su p p o s e that $ i s $-bounded and $. T h e n the se t o f al l $ f o r wh i c h there ex i s t s a $ wi t h $ i s a l s o $-bounded. Th i s sh o w s that, f o r a s u b s e t $ containing ∞, the c o m p l e m e n t o f W has $- b o u n d e d im a g e under Z → X → B if and on l y i f it has $- b o u n d e d im a g e under Z → Y → B. Consequentl y $.

R e m a r k . Al l w e care about wh e n w e de f i n e the neighborhoods o f the ba s e point ∞ i n $ i s this: what i s a ba s e d continuous map f fr o m $ to $ g o i n g to be ? Su c h an f wi l l o f course b e fu l l y de s c r i b e d b y its restriction to $, an o p e n s u b s e t o f $. The restriction i s a continuous map f r o m $, s u b j e c t to a condition: f—1 ( C ) i s compact w h e n ¬ e v e r $ i s the co m p l e m e n t o f a n op e n neighborhood o f th e ba s e point i n $.

Illustration . Su p p o s e that $ and $ w h e r e S and T are di s c r e t e ( t h e r e f o r e countable). Th e n $ i s homotopy eq u i v a l e n t to

$ w h e r e the ho c o l i m i s taken ov e r those $ w h i c h ar e op e n i n B × B, and $ i s the sp a c e o f paths ω: [0,1] → B fr o m r(s) to r(t) su c h that $. 5 9 2 M . We i s s , B . W i l l i a m s

To show that $ in 1 .A.7 satisfies conditions 1.1, we proceed as in 1 .A.3. In particular, the observation that the functor $ from $ to based spaces respects mapping cylinders and mapping cones is crucial. 1.A.8. Va r i a t i o n . To make 1. A . 6 and 1. A . 7 more user-friendly we introduce a larger category $ (with $ as in 1. A . 6 ) whose objects are certain retractive spaces $ with good homotopy theoretic properties, but without a relative CW-structure. More precisely, a retractive space $ is an object of $ i f the following holds.

There exist an object $ in $ and a retractive map f: X → Y which is a weak equivalence (read this as in 1 .A.6).

The notion of weak equivalence (in $) is defined exactly as in l.A.6. A morphism in $, from ($) to ($), is a cofibration if , up to isomorphism, it is the inclusion of a closed subspace $, and has the CHEP described in l.A.6.

With these definitions, $ is a Waldhausen category. W e shall now verify that the inclusion $ has the approximation property [Wald2, 1.6]. This consists of two parts, of which the first stipulates that an arrow in $ be a weak equivalence in $ i f it is a weak equivalence in $. This is true by construction. For the second part, we have to check the following. Given any object X in $ and a morphism f:X → Y in $, there exist a cofibration f1: X → X' in $ and a weak equivalence f2 : X' → Y in $ such that f = f2f1. To construct f1 and f2, we begin with a weak equivalence g: Y1 → Y where Y1 is in $. Since g is a weak equivalence, it is then easy to construct a controlled map u: X → Y1 (relative to B, but not necessarily over B) and a controlled homotopy h (relative to B, but not necessarily over B) from f to g u. B y induction on the skeletons of X, the map u is controlled homotopic (relative to B) to a cellular map; so we may assume without loss of generality that u is cellular to begin with. Let X' be the relative mapping cylinder of u. Define f2 : X' → Y so that it extends the identity on B, agrees with the homotopy h on (X \B) × [0,1), and with g on Y'. There is a unique retraction X' → B such that f2 becomes a morphism in $. Let f1:X→ X' be the front inclusion of the cylinder. Clearly f2 is a weak equivalence and f1 is a cofibration.

Therefore $ induces a homotopy equivalence of the K-theory spectra.

Suppose that X and Y are objects in $. W e define $ literally as in l.A.7. It i s not difficult to show that i f X and Y are in $, and $, $ are weak equivalences with $, $ in $, then the induced map

is a homotopy equivalence. This in turn implies at once that $ with the SW product $ satisfies the conditions of 1.1. D u a l i t y i n W a l d h a u s e n ca t e g o r i e s 5 9 3

1 . A . 9 . Ex a m p l e . W e conclude the li s t o f examples with a twisted version o f l.A.8. Suppose that E → B i s a spherical fibration, with fibers homotopy equivalent to $, and with a distinguished section which i s a fiberwise cofibration [Jm, §22]. For X and Y i n $ (o f 1. A . 8 ) , and open $, $, w e (re-)defin e $ as $, where Z i s the space o f al l quadruples (x, w,y, e) fo r which ● $, $ ● ω:[ — 1,1] → B i s a path such that ω(— 1 ) = r(x), ω(l) = r(y) ● e i s an element o f the fiber o f E → B over ω ( 0 ) .

The topology on $ i s de f i n e d i n such a wa y that ● Z with the obvious topology i s an open subspace ● a se t $ containing ∞ i s a neighborhood o f ∞ i f and only i f the i m a g e o f its complement under Z → X × Y has closure disjoint fr o m B × B, and the image o f its complement under Z → X → B i s $-bounded.

Then w e le t $ and note that $ s o de f i n e d i s an S W product on $. It specializes to $ o f l.A.8 in the case where γ i s the trivial bundle $.

Illustration . Suppose that $ and $ where S and T are discrete. Then $ o f 1. A . 9 i s homotopy equivalent to

w h e r e Er(t) i s the fiber o f E → B over r(t). Apart fr o m that, notation i s as i n the illustration fo l l o w i n g l.A.7.

2.A. Examples

2.A.1. Ex a m p l e . The axioms o f §2 are sa t i s f i e d fo r $ and $ fr o m example l.A.l. This i s obvious.

A l l other examples i n § l.A fa i l to sa t i s f y axiom 2.3. However, this can b e repaired. For simplicity, w e concentrate on $ fr o m 1. A . 3 , the category o f retractive spaces over B which are CW-spaces relative to B with finitely many cells. T o avoid confusion later on, w e denote the suspension functor $ b y ΣB instead o f Σ.

2.A.2. Ex a m p l e . Let $ b e the stabilization o f $ under suspension. In detail, an object of $ i s a pair ( k , X) with $ and X i n $. W e think o f (k, X) as a formal (de)suspension, $, and accordingly de f i n e the se t o f morphisms fr o m ( k , X) to ( l , Y) as 594 M. Weiss, B. Williams

T h e n $ i s a Wa l d h a u s e n ca t e g o r y i n it s ow n right, wi t h a cy l i n d e r fu n c t o r . ( A morphism f: ( k , X) → (l, Y) i s a co f i b r a t i o n i f i t ca n b e re p r e s e n t e d b y a c o f i b r a t i o n $ f o r so m e i. It i s a we a k eq u i v a l e n c e i f it ca n b e r e p r e s e n t e d b y a we a k eq u i v a l e n c e $ f o r so m e i. T o de f i n e the c y l i n d e r o f f: ( k , X) → ( l , Y) find the m i n i m a l i ≥ 0 s u c h that f ca n b e represente d b y so m e $, and fo r m the cy l i n d e r Z o f fi i n $. Th e n ( i , Z) i s the cy l i n d e r o f f i n $.)

2 . A . 3 . L e m m a . $ satisfies axioms 2.1, 2.2 and 2.3.

Proof. 2. 1 an d 2. 3 are ob v i o u s l y sa t i s f i e d . T o prove 2. 2 w e us e [W e i , §3]. Denote o b j e c t s i n $ b y si n g l e letters A, B,C,... fo r br e v i t y . Th e hy p o t h e s i s [W e i , 3. 2 ] c l e a r l y holds fo r $. That is, i f i n a commutative diagram

$ i n $ the ro w s are cofibratio n se q u e n c e s , and tw o o f the ve r t i c a l arrows are we a k equivalences , th e n s o i s the third. (Note that this do e s not ho l d i n $.) Th e r e f o r e b y [W e i , 3.6], a morphism f : C → D i n $ i s a we a k equivalenc e i f and on l y i f i t b e c o m e s in v e r t i b l e i n $ an d has ze r o torsion i n $. Here $ i s K0 o f the fu l l su b c a t e g o r y o f $ co n s i s t i n g o f the ob j e c t s wh i c h be c o m e s isomorphic to the zero ob j e c t i n $. W e no w sh o w that $ i s zero. I f C i s a n ob j e c t i n $ wh i c h be c o m e s isomorphic to $ i n $, then $ ( t h e categorica l coproduct) i s a l s o i s o m o r ¬ p h i c to $ i n $ and has zero torsion i n $. T h e r e f o r e b y [W e i , 3. 6 ] again, the unique morphism $ i s a we a k eq u i v a l e n c e i n $. Bu t then, fr o m t h e d e f i n i t i o n o f weak equivalence i n $, th e unique morphism $ i n $ i s a l s o a we a k equivalence . S o the torsion o f C i s zero. □

H a v i n g st a b i l i z e d $, w e mu s t re d e f i n e th e S W product.

2 . A . 4 . Definition/Observation . For ( k , X) and ( l , Y) i n $ le t

( w h e r e Ωi+j(...) i s d e f i n e d a s the ge o m e t r i c realization o f an appropriate si m p l i c i a l D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 9 5 s e t ) . Th i s i s natural i n ( k , X) and ( l , Y) and has the properties o f an S W product o n $ li s t e d i n 1. 1 .

I n § 5 . A w e w i l l sh o w that $ wi t h the S W product $ sa t i s f i e s ax i o m s 2. 4 and 2. 5 . For no w le t u s lo o k at so m e sp e c i f i c morphism se t s i n $ and $. Note that

$ f o r ob j e c t s X an d Y i n $ (s o that ( k , X) and ( l , Y) are o b j e c t s o f $. T h e r e f o r e i t i s quite en o u g h to look at so m e sp e c i f i c morphism se t s i n $.

2 . A . 5 . Calculation . S u p p o s e that $ and $ a s i n the illustration f o l l o w i n g l.A.3. That is , both X and Y are sp a c e s obtained fr o m B b y adding a d i s j o i n t point. Th e n

(* ) $ w h e r e , a s usual, $ i s the sp a c e o f paths i n B fr o m r(x) to r(y). For the proof w e introduce the sp a c e P o f pairs ( z , ω) w h e r e $ and ω i s a path fr o m r(x) to r(z). Si n c e $, the sp a c e P i s the union o f cl o s e d s u b s p a c e s P0 and P1 wh e r e P0 c o n s i s t s o f the pairs ( z , Ω) wi t h $, and P1 co n s i s t s o f the pairs (z,ω) wi t h $. W e note that $ i s a cofibration, and that

i s a homeomorphism. Also, P0 → P i s a co f i b r a t i o n and P0 i s contractible, s o that P→ P / P 0 i s a homotopy eq u i v a l e n c e . Altogether, $ $. H e n c e w e ma y re p l a c e $ b y P i n (* ) . I n i ) b e l o w w e produce a map ξ fr o m $ to πm(P), i n ii ) w e produce a map ζ f r o m πm(P) to $, and i i i ) w e s h o w that ζ and ξ, are i n v e r s e s o f o n e another.

i ) Le t $ r e p r e s e n t $. S i n c e e i s a co f i b r a t i o n and a we a k equivalence , there ex i s t s a homotopy {ht: Z → Z|0 ≤ t ≤ 1 } r e l im(e) s u c h that h0 = idz an d im(h1) = im(e). Th e n

i s a map fr o m $ to P. Its homotopy cl a s s ξ depends on l y o n α, not o n the ch o s e n representativ e o f α, fo r the fo l l o w i n g reason. W e l o s e n o ge n e r a l i t y b y a l l o w i n g o n l y special quasi-morphism s $, that is , quasi-morphisms f o r wh i c h th e re s u l t i n g map $ i s a cofibratio n i n $. (I f the map i n question i s not a co f i b r a t i o n , re p l a c e Z b y the re l a t i v e mapping cy l i n d e r o f $ and re p l a c e f b y th e fr o n t in c l u s i o n o f the cy l i n d e r . B y inspection, 596 M. Weiss, B. Williams the homotopy cl a s s ξ does not change.) An y tw o sp e c i a l quasi-morphisms re p r e ¬ s e n t i n g α ca n b e co n n e c t e d b y a chain o f morphisms (b e t w e e n sp e c i a l qu a s i - m o r ¬ p h i s m s ) o f the fo r m

$ w h e r e the arrow i n the mi d d l e o f the diagram i s a cofibration. Th i s us e s another m a p p i n g c y l i n d e r construction, ex p l a i n e d i n [W e i , §2]. Ag a i n , upper ro w and lo w e r r o w o f the diagram gi v e ri s e to the sa m e cl a s s i n πm(P). Th i s co m p l e t e s the v e r i f i ¬ cation, s o that w e ma y wr i t e ξ = ξ ( α ) . i i ) W e start wi t h a ba s e d map $ s u c h that the co m p o s i t e map $ i s cellular. Le t Z b e the re d u c e d mapping cy l i n d e r o f that co m ¬ position. Make Z into a retractive sp a c e ov e r B a s fo l l o w s . O n $, u s e the g i v e n retraction. Points o f Z not i n $ c a n b e written i n the fo r m ( a , t) wi t h $ a n d $, a n d w e map those to $. H e r e g2(a) i s a path i n B, t h e se c o n d component o f $. Note that the fr o n t in c l u s i o n o f th e cy l i n d e r e x t e n d s automatically to a map $ o f retractive sp a c e s ov e r B. Th e r e f o r e f r o n t in c l u s i o n and ba c k in c l u s i o n o f th e cy l i n d e r de f i n e a quasi-morphism $ and an el e m e n t ζ ( g ) i n [$]. It i s not hard to se e that i t on l y depends o n th e homotopy cl a s s o f g. i i i ) B y construction, ξζ i s th e id e n t i t y o n πm(P). Su p p o s e that w e start wi t h so m e quasi-morphism $ a s i n i), representing a cl a s s α, then construct f r o m i t a map $ representin g ξ ( α ) , an d then construct fr o m that map a s i n i i ) a quasi-morphism $ representin g ξζ(α). Th e n cl e a r l y there e x i s t s a morphism i n $, g i v e n b y the ve r t i c a l arrows i n

T h e r e f o r e ξζ(α) = α and ξζ i s th e identity. □

R e m a r k . T h e reader ma y f i n d our us e o f arbitrary (c o m p a c t l y ge n e r a t e d Ha u s d o r f f ) s p a c e s i n the calculation 2. A . 5 sc a r y . Here i s a wa y to av o i d pathological sp a c e s . W e note that the tw o si d e s o f ( * ) are invariant under we a k homotopy equivalence s i n the fo l l o w i n g se n s e . Gi v e n a we a k homotopy eq u i v a l e n c e υ: B → C, and X, Y a s i n (* ) , th e maps

$ D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 5 9 7

$ are bi j e c t i o n s . It fo l l o w s im m e d i a t e l y that (* ) i n the sp e c i a l ca s e wh e r e B i s a C W - s p a c e im p l i e s the ge n e r a l ca s e o f (* ) .

2 . A . 6 . E x a m p l e . Th e construction o f $ fr o m $ an d the S W product o n $ fr o m that o n $ g o through without es s e n t i a l ch a n g e s i f w e a s s u m e that B c o m e s wi t h a control structure an d de f i n e $ a s i n 1. A . 6 . , and $ on $ as i n l.A.7. Ev e n more g e n e r a l l y , $ o f 1. A . 6 and $ o f l.A.7 co u l d b e re p l a c e d b y the la r g e r $ o f 1. A . 8 with $ as in 1.A.8; so $ would be defined as the stabilization of $

S t i l l more ge n e r a l l y , w e c a n start wi t h the S W product o f 1 . A . 9 , and us e i t to make a n S W product o n $, a s i n 2. A . 4 . Note i n p a s s i n g that the i n c l u s i o n s $ a n d $ a r e e q u i v a ¬ l e n c e s o f categories.

2 . A . 7 . Calculation . Su p p o s e that $ and $ a s i n the illustration f o l l o w i n g l.A.7. Then, i n the notation us e d there,

The proof re s e m b l e s that o f ( * ) i n 2. A . 4 . Later w e w i l l want to kn o w that th e di r e c t s u m $ ca n b e re p l a c e d b y a product ∏t, s o that w e c a n al s o wr i t e

N a m e l y , fo r ea c h $ and $, there ex i s t on l y fi n i t e l y ma n y $ fo r wh i c h $ i s nonempty. Proof: Fi x $ an d $ and le t $ c o n s i s t o f a l l points $ su c h that $ i s nonempty. Th e n ce r t a i n l y $ and t h e r e f o r e { b } × Ws and Ws × { b } be l o n g to $. Bu t

Ws×Ws = (Ws×{b})({b}×Ws), s o Ws × Ws be l o n g s to $, wh i c h me a n s that Ws i s $-b o u n de d . Si n c e th e se c o n d condition i n 1. A . 6 must ho l d fo r $, the se t o f al l $ fo r wh i c h $ i s f i n i t e . □ 598 M. Weiss, B. Williams

5 . A . Ex a m p l e s 5 . A . 1 . Ex a m p l e . Le t $ b e the category o f 2. A . 2 , the stabilization o f $ i n l.A.3. Equip this wi t h the S W product $ o f 2. A . 4 . Le t $ b e the c l a s s o f al l ob j e c t s wh i c h are isomorphic i n $ to a n ob j e c t o f the f o r m ( n , X) wi t h $ and $ a s i n the illustration fo l l o w i n g l.A.3. Th e n $ i s a generating cl a s s . W e sh a l l us e this fa c t to establish ax i o m s 2. 4 and 2. 5 fo r $.

S u p p o s e that $ and $ are retractive sp a c e s ov e r B. Th e n (m, X) an d ( n , Y) be l o n g to $, and fr o m l.A.3 and illustration an d 2. A . 4 w e h a v e

I n particular, the constant path fr o m r(x) to r(x) i s an el e m e n t ηm i n

T h e component o f ηm i s nondegenerate. T o prove this i t s u f f i c e s b y 5. 6 to sh o w that s l a n t product w i t h [ηm] i s a b i j e c t i o n f r o m [( — m, X), ( n , Y)] to π0 o f $ f o r arbitrary n a n d $ a s above. Now 2 . A . 5 and the observations p r e c e d i n g i t gi v e $ s h o w i n g that [ ( — m, X), ( n , Y)] a n d π0 o f $ a r e i n “abstract” bi j e c t i o n . A n ea s y in s p e c t i o n sh o w s that the slant product wi t h [ηm] i s e x a c t l y this bi j e c t i o n . Now 5. 6 applies, s o [ηm] i s in d e e d nondegenerate.

W e s e e that the conditions o f 5. 4 are me t an d co n c l u d e that $ sa t i s f i e s ax i o m 2. 4 . Further, the sy m m e t r y involution τ takes the component o f $ to the component o f $, w h i c h i s a l s o nondegenerate. T h e r e ¬ f o r e the conditions o f 5. 5 are met, and $ sa t i s f i e s 2.5. □

5 . A . 2 . Ex a m p l e . Si m i l a r reasoning sh o w s that $ and $ o f 2. A . 6 wi t h the S W product $ de f i n e d there sa t i s f y ax i o m s 2. 4 and 2.5. The appropriate generating c l a s s $ co n s i s t s o f al l ob j e c t s isomorphic i n $ o r $ to on e o f ty p e (m, X) w h e r e $ f o r di s c r e t e S. One fi n d s , us i n g 5. 6 , that $ has a nondegenerate component wh i c h remains nondegenerate wh e n τ i s applied. Th e n 5 . 4 ca n b e us e d , and fi n a l l y 5. 5 . □

5 . A . 3 . E x a m p l e . Continuing i n the notation o f 5. A . 2 , su p p o s e that $ and/or $ are eq u i p p e d wi t h th e S W product $ o f l.A.9. He r e γ i s a sp h e r i c a l fi b r a t i o n o n D u a l i t y i n W a l d h a u s e n c a t e g o r i e s 5 9 9

B, wi t h distinguishe d se c t i o n , an d w i t h fibers homotopy eq u i v a l e n t to $, sa y . For $ a s i n 5. A . 2 , on e finds ( a s i n 1. A . 9 , illustration) that

The right-hand si d e contains a n e l e m e n t ηm w h o s e (s, t)-coordinate i s z e r o f o r $, a n d eq u a l to th e constant path fr o m r(s) to r(s) i f s = t. ( S o m e op e n U containing the diagonal mu s t b e s e l e c t e d fo r this to make se n s e , but the component o f ηm d o e s not de p e n d o n the ch o i c e o f U.) Think o f [ηm] a s a component o f $. It i s nondegenerate (u s e 5 . 6 ) and maps under τ to the c o m ¬ ponent [ηn-m] o f $, w h i c h i s a l s o nondegenerate. No w 5 . 4 and 5 . 5 ca n b e us e d a s be f o r e , and the co n c l u s i o n i s that $ and $ wi t h the S W product $ sa t i s f y ax i o m s 2. 4 and 2. 5 . □

1 2 . A . Ex a m p l e s 1 2 . A . 1 . E x a m p l e ( c h a n g e o f c o n t r o l s p a c e ) . Su p p o s e that B, B' are s p a c e s eq u i p p e d w i t h control structures $, $ re s p e c t i v e l y . As s u m e that $ an d $ are compatible w i t h the topologies o f B and B', respectively . Le t f: B → B' b e a (continuous) map w i t h th e properties ( 1 ) $; ( 2 ) $.

T h e n t h e pushout wi t h f (s e e remark be l o w ) i s an e x a c t f u n c t o r $, w h e r e $ an d $ are the Wa l d h a u s e n categories made fr o m $ and $ a s i n 2 . A . 6 . On e finds that li t e r a l l y

$ f o r ob j e c t s C , D i n $, pr o v i d e d the S W products $ o n $ an d $ o n $ are constructed a s i n 2. A . 6 . Ca l l the in c l u s i o n ø. It commutes wi t h the sy m m e t r y operators. It re m a i n s to sh o w that ø takes nondegenerate components to nondegenerate components. W e c a n th i n k o f this a s a statement about $ a n d $, a natural transformation be t w e e n fu n c t o r s o n $. It i s e q u i v a l e n t to the statement that a certain natural transformation

ν : T'F → FT d e f i n e d just be l o w i s a natural isomorphism; he r e T fr o m $ to $ an d T' f r o m $ to $ are the duality functors . Th e natural transformation ev a l u ¬ ated o n C i n $ i s the morphism T'F(C) → FT(C) wh i c h i s the im a g e o f $ under 6 0 0 M . W e i s s , B . W i l l i a m s

Now a five lemma argument shows that v w i l l always be an isomorphism i f it i s an isomorphism fo r C o f the fo r m (0, X), where $ and S i s discrete. (Objects o f this type, and isomorphic objects i n $, and their iterated suspensions and desuspensions, fo r m a generating class i n the se n s e o f § 5 . ) In the case C = (0, X) and $ w e know fr o m 5. A . 2 that TC i s isomorphic to C and T'F(C) i s isomorphic to F(C). In short, w e can proceed b y inspection. □

R e m a r k . In working with several categories o f retractive spaces it i s a good idea to le t the underlying sets o f the base spaces (here B, B') be subsets o f a large se t V1, and to allow only retractive spaces X (here: over B or B') fo r which the underlying s e t o f X? i s contained i n some other large se t V2 disjoint fr o m Vl. Then the pushout o f a retractive space $ along f, fo r example, can be de f i n e d set-theoretically a s $.

12.A.2. Ex a m p l e (linearization) . Let $ b e the Waldhausen category o f l.A.3. For objects X and Y i n $ w e d e f i n e $ as i n the remark fo l l o w i n g l.A.3.

Let B~ → B be a normal covering with translation group π. Denote by $ the category o f bounded chain complexes o f finitely generated fr e e le f t $ π-modules. The standard S W product in $ i s $ (compare 1. 4 and l.A.l), where Ct i s C with a right action o f $. The right action and the le f t action are related vi a the involution $.

W e w i l l use B~ → B to create a pair consisting o f an exact functor F and a natural transformation ø,

$, $.

The functor F takes $ to the reduced cellular chain complex o f X~/B~, where X~ → X i s the covering pulled back from B~ → B using r : X → B. Any m-simplex z i n $ determines a filtration-preserving stable composite map

w h i c h i n turn determines a chain map $ fr o m the cellular chain complex o f Δm to the tensor product $. The geometric realization o f $ i s a natural D u a l i t y i n Wa l d h a u s e n ca t e g o r i e s 6 0 1 map $. (Compare 1. 4 and l.A.l.) It respects the symmetries Τ.

An obvious deficiency of the pair ( F , ø) just constructed is that the domain $ of F is a Waldhausen category which in general fails to satisfy the axioms of § 2. However, we have the stabilization $ from 2.A.2 and 2.A.3. W e can extend F to an exact functor $ by the recipe

Similarly ø extends to a natural transformation of functors on $. Proceed as in 12.A.I to verify that ø takes nondegenerate components to nondegenerate compo¬ nents. □

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R e c e i v e d Oc t o b e r 9 , 19 9 6 ; r e v i s e d an d i n fi n a l fo r m Ju l y 25 , 1 9 9 7 M i c h a e l W e i s s , Dept. o f Mathematics, U n i v e r s i t y o f Notre Dame, Notre D a m e i n 4 6 5 5 6 , U S A E - m a i l ad d r e s s : [email protected] u B r u c e W i l l i a m s , D e p t . o f Mathematics, U n i v e r s i t y o f Notre Dame, Notre D a m e i n 4 6 5 5 6 , U S A E - m a i l ad d r e s s : [email protected] u