THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM FERNANDO MURO AND ANDREW TONKS Abstract.We give a small functorial algebraic model for the 2-stage Post- nikov section of the K-theory spectrum of a Waldhausen category and use * *our presentation to describe the multiplicative structure with respect to bi* *exact functors. Introduction Waldhausen's K-theory of a category C with cofibrations and weak equivalences [Wal85] extends the classical notions of K-theory, such as the K-theory of ring* *s, additive categories and exact categories. In this paper we give an algebraic model D*C for the 1-type P1KC of the K- theory spectrum KC. This model consists of a diagram of groups (D0C)ab (D0C)ab iiFPH ; <.,.>|| 3 O " fflffl|@ K1C ___________//D1C___________//D0C_____////K0C. in which the bottom row is exact. The important features of our model are the following: o It is small, as it has generators given by the objects, weak equivalence* *s and cofiber sequences of the category C. o It has minimal nilpotency degree, since both groups D0C and D1C have nilpotency class 2. o It encodes the 1-type in a functorial way, and the homotopy classes of morphisms D*C ! D*D and P1KC ! P1KD are in bijection. From this structure we can recover the homomorphism .j :K0C Z=2 ! K1C, which gives the action of the Hopf map in the stable homotopy groups of spheres, in the following way a . j = , a 2 K0C. The extra structure given by the quadratic map H is used to describe the be- haviour of D* with respect to biexact functors between Waldhausen categories ____________ 1991 Mathematics Subject Classification. 19B99, 16E20, 18G50, 18G55. Key words and phrases. K-theory, Waldhausen category, Postnikov invariant, s* *table quadratic module, crossed complex, categorical group. The first author was partially supported by the project MTM2004-01865 and th* *e MEC post- doctoral fellowship EX2004-0616, and the second by the MEC-FEDER grant MTM2004-* *03629. 1 2 FERNANDO MURO AND ANDREW TONKS C x D ! E. In particular the classical homomorphisms K0C K0D -! K0E, K0C K1D -! K1E, K1C K0D -! K1E, may be obtained from our model D*. The object D*C is a stable quadratic module in the sense of [Bau91 ]. This ob* *ject is defined below by a presentation in terms of generators and relations in the * *spirit of Nenashev, who gave a model for K1 of an exact category in [Nen98 ]. A stable quadratic module is a particular case of a strict symmetric categorical group, * *or more generally of a commutative monoid in the category of crossed complexes, which were first introduced by Whitehead in [Whi49 ]. The monoidal structure for crossed complexes was defined in [BH87 ]. To obtain our presentation of D* we introduce the total crossed complex X of* * a bisimplicial set X, and_show_that_there_is_an_Eilenberg-Zilber-Cartier_equivale* *nce____________________________ __________________________________________________________* *____________ _ss+Diag(X)+_______________________//_o(X)o_ generalizing [DP61 , Section 2] and [Ton03]. This is then applied to the bisimp* *licial set given by the nerve of Waldhausen's wS. construction [Wal78]. 1.The algebraic 1-type D*C of the K-theory spectrum KC We begin by defining the algebraic structure which the model D*C will have. Definition 1.1. A stable quadratic module C* is a diagram of group homomor- phisms Cab0 Cab0<.,.>-!C1 -@!C0 such that given ci, di2 Ci, i = 0, 1, o @ = [d0, c0], o <@(c1), @(d1)> = [d1, c1], o + = 0. Here [x, y] = -x - y + x + y is the commutator of two elements x, y 2 K in any group K, and Kab is the abelianization of K. We will also write <., .>C* and @C* for the structure homomorphisms of C*. Note that the axioms imply that C0 and C1 are groups of nilpotency degree 2. Stable quadratic modules were introduced in [Bau91 , Definition IV.C.1]. Noti* *ce, however, that we adopt the opposite convention for the homomorphism <., .>. As usual, one can define stable quadratic modules in terms of generators and relations in degrees zero and one. We assume the reader has certain familiarity with Waldhausen categories and related concepts. We refer to [Wei] for the basics, see also [Wal85]. Definition 1.2. Let C be a Waldhausen category with distinguished zero object *. Cofibrations and weak equivalences are denoted by ae and ~!, respectively. A generic cofiber sequence is denoted by A ae B i B=A. We define D*C as the stable quadratic module generated in dimension zero by the symbols THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM 3 o [A] for any object in C, and in dimension one by o [A ~!A0] for any weak equivalence, o [A ae B i B=A] for any cofiber sequence, such that the following relations hold. (1) @[A ~!A0] = -[A0] + [A]. (2) @[A ae B i B=A] = -[B] + [B=A] + [A]. (3) [*] = 0. (4) [A 1A!A] = 0. (5) [A 1A!A i *] = 0, [* ae A 1A!A] = 0. (6) For any pair of composable weak equivalences A ~!B ~!C, [A ~!C] = [B ~!C] + [A ~!B]. (7) For any commutative diagram in C as follows A //___//_B____////_B=A |~| |~| ~|| fflffl| fflffl| fflffl| A0 //___//B0___////_B0=A0 we have [A ~!A0] + [B=A ~!B0=A0] +<[A], -[B0=A0] + [B=A]>= -[A0ae B0i B0=A0] +[B ~!B0] +[A ae B i B=A]. (8) For any commutative diagram consisting of four obvious cofiber sequences in C as follows C=B OOOO | | | B=A //___//C=A OOOO OOOO | | | | | | A //____//B//_____//_C we have [B ae C i C=B] +[A ae B i B=A] = [A ae C i C=A] +[B=A ae C=A i C=B] +<[A], -[C=A] + [C=B] + [B=A]>. (9) For any pair of objects A, B in C i1 p2 i2 p1 <[A], [B]> = -[A aeA _ B i B] + [B ae A _ B i A]. 4 FERNANDO MURO AND ANDREW TONKS Here _i1_//_ oi2o_ A op1o_A _ B _p2_//_B are the inclusions and projections of a coproduct in C. Remark 1.3. Notice that relation (7) implies that if (9) holds for a particular* * choice of the coproduct A_B then it holds for any other choice, since two different co* *prod- ucts are canonically isomorphic by an isomorphism which preserves the inclusions and the projections of the factors. Definition 1.4. A morphism f :C* ! D* in the category squad of stable qua- dratic modules is given by group homomorphisms fi:Ci ! Di, i = 0, 1, such that o @D*f1(c1) = f0@C*(c1), o D* = f1C*. The homotopy groups of C* are ss1C* = Ker@ and ss0C* = Coker@. A weak equivalence in squad is a morphism which induces isomorphisms in homo- topy groups. The homotopy category Ho squad is obtained from squad by inverting weak equivalences. Let WCat be the category of Waldhausen categories as above and exact functor* *s. The construction D* defines a functor D*: WCat -! squad. For an exact functor F :C ! D the stable quadratic module morphism D*F :D*C ! D*D is given on generators by (D*F )([A]) = [F (A)], (D*F )([A ~!A0]) = [F (A) ~!F (A0)], (D*F )([A ae B i B=A]) = [F (A) ae F (B) i F (B=A)]. Let Ho Spec0 be the homotopy category of connective spectra. In Lemma 4.18 below we define a functor ~0: Ho Spec0- ! Ho squad together with natural isomorphisms ssi~0X ~=ssiX, i = 0, 1. This functor induces an equivalence of categories ~0: Ho Spec10~-!Ho squad, were Ho Spec10is the homotopy category of spectra with trivial homotopy groups in dimensions other than 0 and 1. The naive algebraic model for the 1-type of the algebraic K-theory spectrum KC of a Waldhausen category C would be ~0KC. However this stable quadratic module is much bigger than D*C and it is not directly defined in terms of the b* *asic structure of the Waldhausen category C. This makes meaningful the following theorem, which is the main result of this paper. THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM 5 Theorem 1.5. Let C be a Waldhausen category. There is a natural isomorphism in Ho squad ~ D*C -=!~0KC. This theorem shows that the model D*C satisfies the functoriality properties claimed in the introduction. It also shows that the exact sequence of the intro* *duc- tion is available. The theorem will be proved in section four. From a local point of view the 1-type of a connective spectrum is determined up to non-natural isomorphism by the first k-invariant. We now establish the connection between this k-invariant and the algebraic model D*C. Definition 1.6. The k-invariant of a stable quadratic module C* is the homomor- phism k :ss0C* Z=2 -! ss1C*, k(x 1) = . Given a connective spectrum X the k-invariant of ~0X coincides with the action of the Hopf map 0 6= j 2 sss1~=Z=2 in the stable stem of the sphere. ss0X sss1_____//ss1X ~=|| |~=| fflffl| k fflffl| ss0~0X Z=2_____//ss1~0X See Lemma 4.18 below. We recall that the action of j coincides with the first Postnikov invariant of X. This is used to derive the following corollary of The* *orem 1.5. Corollary 1.7. The first Postnikov invariant of the spectrum KC .j :K0C Z=2 -! K1C is defined by ~ [A] . j = [oA,A: A _ A =!A _ A], where oA,A is the automorphism which exchanges the factors of a coproduct A _ A in C. Proof.The corollary follows from the commutativity of the following diagram i1 p2 A //___//A _ A___////_A || ~ o| || || = |A,A || || i2 fflffl|p1|| A //___//A _ A___////_A and relations (4), (7) and (9) in Definition 1.2. The next corollary can be easily obtained from the previous one by using again the relations defining D* and matrix arguments as for example in the proof of [Ran85 , Proposition 2.1 (iv)]. Corollary 1.8. Let A be a Waldhausen category which is additive. Then the first Postnikov invariant of the spectrum KA .j :K0A Z=2 -! K1A is defined by ~ [A] . j = [-1A :A =!A]. 6 FERNANDO MURO AND ANDREW TONKS 2. The multiplicative properties of D* In order to describe the multiplicative properties of D*C with respect to bie* *xact functors we would need a symmetric monoidal structure on squad which models the smash product of spectra. Unfortunately such a monoidal structure does not exist and we need to enrich D*C with an extra structure map H, (D0C)ab (D0C)ab ii___H___________________________________* *____________ _______________________ <.,.>|| ____________________________________* *_________________________________ fflffl| @ _____________________________ D1C ___________//D0C, so that the diagrams _ ! __H__// Dsg0C = D0C oo___ (D0C)ab (D0C)ab , <@,@> _ ! __H@_// Dsg1C = D1C oo___ (D0C)ab (D0C)ab , <.,.> are square groups in the sense of [BP99 ]. Definition 2.1. A square group M is a diagram H MeAE Mee P where Me is a group, Mee is an abelian group, P is a homomorphism, H is a quadratic map, i.e. the symbol (x|y)H = H(x + y) - H(y) - H(x), x, y 2 Me, is bilinear, and the following identities hold, a 2 Mee, (P (a)|x)H = 0, (x|P (a))H = 0, P (x|y)H = [x, y], P HP (a) = P (a) + P (a). Note that (.|.)H induces a homomorphism (2.2) (.|.)H :Coker P CokerP -! Mee. Moreover T = HP - 1: Mee- ! Mee is an involution, i.e. a homomorphism with T 2= 1, and : CokerP -! Xee:x 7! (x|x)H - H(x) + T H(x) defines a homomorphism. A morphism f :M ! N in the category of square groups is given by group homomorphisms fe:Me ! Ne, fee:Mee! Neecommuting with H and P . A quadratic pair module f :M ! N is a square group morphism such that Mee= Neeand feeis the identity. Morphisms in the category qpm of quadratic pair modules are defined again by homomorphisms commuting with all operators. THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM 7 A stable quadratic module C* is termed 0-free if C0 = nilis a free group of nilpotency class 2. Here E is the basis. Lemma 2.3. Let C* be a 0-free stable quadratic module with C0 = niland let H :C0 ! Z[E] Z[E] be the unique quadratic map such that H(e) = 0 for any e 2 E and (x|y)H = y x for x, y 2 C0. Then _ ! _H__//_ Csg0 = C0 oo___Cab0 Cab0 , <@,@> _ ! _H@_//_ Csg1 = C1 oo___Cab0 Cab0 , <.,.> are square groups. Moreover, the homomorphism @ :C1 ! C0 defines a quadratic pair module Csg1-! Csg0. The square group Csg0in this lemma will also be denoted by Znil[E] or just Zn* *il if E is a singleton, as in [BJP05 ]. The stable quadratic module D*C defined in the previous section is 0-free. The basis of D0C is the set of objects in C, excluding the zero object *. In partic* *ular Dsg0C and Dsg1C above are square groups and D*C is endowed with the structure of a quadratic pair module. Moreover, the morphisms induced by exact functors are compatible with H, so that D* lifts to a functor D*: WCat -! qpm . The category of square groups is a symmetric monoidal category with the tensor product " defined in [BJP05 ] that we now recall. Definition 2.4. The tensor product M " N of square groups M, N is defined as follows. The group (M " N)e is generated by the symbols x " y, a ~b for x 2 Me, y 2 Ne, a 2 Meeand b 2 Nee, subject to the following relations (1) the symbol a ~b is bilinear and central, (2) x " (y1 + y2) = x " y1 + x " y2 + H(x) ~(y2|y1)H , (3) the symbol x " y is left linear, (x1 + x2) " y = x1 " y + x2 " y, (4) P (a) " y = a ~(y|y)H . (5) T (a) ~T (b) = -a ~b, (6) x " P (b) = (x) ~b. The abelian group (M " N)ee is defined as the tensor product Mee Nee. The homomorphism P :(M " N)ee- ! (M " N)e is P (a b) = a ~b, and H :(M " N)e -! (M " N)ee is the unique quadratic map satisfying H(x " y) = (x) H(y) + H(x) (y|y)H , H(a ~b) = a b - T (a) T (b), (a ~b|-)H = 0, (-|a ~b)H = 0, (a " b|c " d)H= (a|c)H (b|d)H . 8 FERNANDO MURO AND ANDREW TONKS The unit for the tensor product is the square group Znil. Theorem 2.5. Let C x D ! E: (A, B) 7! A ^ B be a biexact functor between Waldhausen categories. Then there are morphisms of square groups 'ij:DsgiC " DsgjD ! Dsgi+jE, for i, j, i + j 2 {0, 1}, defined by '00e([A] " [C])= [A ^ C], '01e([A] " [C ~!C0])= [A ^ C ~!A ^ C0], '01e([A] " [C ae D i D=C])= [A ^ C ae A ^ D i A ^ (D=C)], '10e([A ~!A0] " [C])= [A ^ C ~!A0^ C], '10e([A ae B i B=A] " [C])= [A ^ C ae B ^ C i (B=A) ^ C], 'ijee([A] [A0] [C] [C0])=[A ^ C] [A0^ C0]. such that the following diagram of square groups commutes Dsg1C " Dsg1DQ @"1mmmmmm QQQQ1"@QQ mmmm QQQ vvmmm QQQ(( Dsg0C "_Dsg1DQ_ Dsg1C "_Dsg0D_ ____QQQQ'01QQ_________________________________'10mmmmmm_______* *______________________________ ________QQQ______________________mmmm________________________* *______ ___________QQQ((___________________________vvmmm_____________* *_____________ _________________________DsgE _________________________ __________________________1 _________________________ ___________________________|___________________________ _______________________________________________________@| ______________________ff_________________________________* *___lffl| 1"@ _____________________________________________________@"1* *______________________________________sg ______________________________________________________* *___________D0OEO _____________________________________________________* *________________ ___________________________________________________* *________________________________________'00|| ___&&_____________________________________xx______* *__________________________________| Dsg0C " Dsg0D Now given a biexact functor CxD ! E: (A, B) 7! A^B we recover the classical homomorphisms ~'00:K0C K0D -! K0E, ~'01:K0C K1D -! K1E, ~'10:K1C K0D -! K1E, from 'ijin Theorem 2.5 as follows. Given i, j, i + j 2 {0, 1}, (2.6) '~ij(a b)= 'ije(a " b). Here we use the natural exact sequence K1C ,! D1C -@!D0C i K0C available for any Waldhausen category C to identify K1C with its image in D1C, and we use the same notation for an element in D0C and for its image in K0C. One can use the relations defining the tensor product " of square groups to che* *ck that the homomorphisms ~'ijare well defined by the formula (2.6) above. In the proof of Theorem 2.5 we use a technical lemma about square groups, which measures the failure of " to preserve certain coproducts. THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM 9 Let M . E be the E-fold coproduct of a square group M, for any indexing set E. We know from [BJP05 ] that Znil[E] = Znil. E, so we have canonical morphisms M " 'x: M ! M " Znil[E] for x 2 E. However, the natural comparison morphism ' = (M " 'x)x2E :M . E -! M " Znil[E] is not an isomorphism. Consider the homomorphisms __ :Z[E] Z[E] i Z[E], __ H :^2 Z[E] ! Ker , __ 2 q :Ker i ^ Z[E], __ __ where (e e) = e for e 2 E and (e e0) = 0 if e 6= e02 E, with H(x ^ y) = y x - x y and q(x y) = x ^ y. For any abelian group A we consider ` A H __' (A ^2Z[E]) = A ^2Z[E] AE A Ker . A q This is isomorphic to the square group defined in [BJP05 , Section 1]. The cons* *truc- tion is obviously functorial in A. Lemma 2.7. For any square group M and any set E there is a pushout diagram in the category of square groups (Coker P CokerP ^2Z[E]) _____//_M . E | | | push |' |fflffl fflffl| (Mee ^2Z[E]) ___________//M " Znil[E] which is natural in M and E. Proof.One can check inductively by using [BJP05 , Proposition 5] and [BJP05 , Section 5.6 (6)] that there is a map of central extensions of square groups in * *the sense of [BJP05 , Section 5.5] as follows. "~ Q (Coker P CokerP ^2Z[E]) O_______//M . E______////_EM | | || | ' | || fflffl| " fflffl| Q || (Mee ^2Z[E]) O___________//M " Znil[E]___////EM Here the left-hand morphism is induced by (2.2). The morphism ~ is completely determined by the homomorphism __ ~ee: CokerP CokerP Ker -! (M . E)ee defined by ~ee(a b x y) = P ('x(a)|'y(b))H for a, b 2 CokerP and x 6= y 2* * E. Similarly is determined by the homomorphism __ ee:Mee Ker -! Mee Z[E] Z[E] __ induced by the inclusion Ker Z[E] Z[E]. It is straightforward to check th* *at the square on the left is the desired pushout. 10 FERNANDO MURO AND ANDREW TONKS Proof of Theorem 2.5.It is obvious that 'ijeeis a well-defined abelian group ho- momorphism in all cases. The square group morphism '00 is well-defined as a consequence of [BJP05 , Proposition 34]. Let E be the set of objects of D, excluding *, so that Znil[E] = Dsg0D, and let M = Dsg1C. To see that '10 is well defined by the formulas in the statement we note that it is just the morphism determined, using Lemma 2.7, by the square group morphisms (Mee ^2Z[E]) -,! Dsg1E -i M . E defined as follows. The square group morphism , is completely determined by __ ab ab ,ee= '10ee:(D0C)ab (D0C)ab Ker -! (D0E) (D0E) . For each A 2 E, the component i O 'A : Dsg1C ! Dsg1C . E ! Dsg1E is the unique square group morphism such that i sg Dsg1C _____'A____//_Dsg1C ._E_________//D1 E @ || @|| fflffl| 'A '00 fflffl| Dsg0C ___________//_Dsg0C ._E_________//Dsg1E coincides with the morphism of quadratic pair modules D*( . ^ A): D*C ! D*E induced by the exact functor . ^ A: D ! E. By using this alternative definition of '10 in terms of Lemma 2.7 it is also immediate that the lower right cell in the diagram of the statement is commutat* *ive. To see that '01 is well-defined and that the lower left cell of the diagram i* *n the statement commutes one proceeds similarly, using the fact that " is symmetric. Now we just need to check that the upper cell is commutative. For this it is enough to show that the following equalities hold '01((@[A ~!A0]) " [C ~!C0])='10([A ~!A0] " (@[C ~!C0])), '01((@[A ~!A0]) " [C ae D i D=C])='10([A ~!A0] " (@[C ae D i D=C])), '01((@[A ae B i B=A]) " [C ~!C0])='10([A ae B i B=A] " (@[C ~!C0])), '01((@[A ae B i B=A]) " [C ae D i=D=C])'10([A ae B i B=A] " (@[C ae D i D=C])). This is a tedious but straightforward task which makes use of the laws of stable quadratic modules and the tensor product of square groups, the elementary prop- erties of a biexact functor, and the relations (1), (2), (6), (7) and (8) in De* *finition 1.2. 3. Natural transformations and induced homotopies on D* In this section we define induced homotopies along the functor D* from section * *1. Definition 3.1. Two morphisms f, g :C* ! D* are homotopic f ' g if there exists a function ff: C0 ! D1 satisfying o ff(c0 + d0) = ff(c0) + ff(d0) + D*, o g0(c0) = f0(c0) + @D*ff(c0), o g1(c1) = f1(c1) + ff@C*(c1). Such a function is called a homotopy. THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM 11 The category squad of stable quadratic modules is a 2-category with 2-morphis* *ms given by homotopies; it is indeed a category enriched over groupoids. See [BM05* * , Proposition 7.2] for details. In addition the qutient of the full subcategory s* *quadf squad given by 0-free objects by the homotopy relation is equivalent to the hom* *o- topy category of stable quadratic modules squad f=' -~! Ho squad, compare [BM05 , Proposition 7.7]. The category WCat of Waldhausen categories and exact functors is also a 2- category, where a 2-morphism ": F ) G between two exact functors F, G: C ! D is a natural transformation " given by weak equivalences "(A): F (A) ~!G(A) in D for any object A in C. Theorem 3.2. The construction D*: WCat ! squad defines a 2-functor. Proof.The homotopy D*": D0C ! D1D induced by a 2-morphism ": F ) G in WCat is determined by the formula (D*")([A]) = ["(A): F (A) ~!G(A)]. We leave the details to the reader. Remark 3.3. The homotopies defined in Theorem 3.2 are constructed by using just one kind of degree 1 generators of D*, namely those given by weak equivalences.* * In case we have a cofiber sequence F ae G i H of exact functors F, G, H :C ! D one can define a homotopy using the other class of degree 1 generators, given by co* *fiber sequences, to give a direct proof of the additivity theorem [Wal85, Proposition* * 1.3.2 (4)] for the algebraic model of the 1-type D*. 4. Proof of Theorem 1.5 In this section we use the notions of crossed module and crossed complex in the category of groups and in the category of groupoids. There are different but equivalent ways of presenting these objects, and the functors between them, de- pending on a series of conventions such as using left or right actions, choice * *of basepoint of an n-simplex, etc. In this paper we adopt the conventions which are compatible with [Ton03]. As examples of a crossed complex we can mention the fundamental crossed complex ssCW Y of a CW -complex Y and the fundamental crossed complex ssX of a simplicial set X; these are related by the natural ide* *n- tification ssX = ssCW |X|where |. |denotes the geometric realization functor f* *rom simplicial sets to CW -complexes. See [Ton03] for further details and reference* *s. Definition 4.1. A crossed module of groups is a group homomorphism @ :M ! N such that N acts (on the right) on M and the following equations are satisfied * *for m, m02 M and n 2 N. (4.2) @(mn) = -n + @(m) + n, 0) 0 0 (4.3) m@(m = -m + m + m . Morphisms of crossed modules are defined by commutative squares of group homomorphisms which are compatible with the actions in the obvious way. Such a morphism is a weak equivalence if it induces isomorphisms between the kernels a* *nd cokernels of the homomorphisms @. 12 FERNANDO MURO AND ANDREW TONKS A crossed complex of groups (C, @) is given by a crossed module @2: C2 ! C1 as above, a chain complex of modules Cn, n 3, over Coker@2, and a connecting map @3: C3 ! C2 subject to certain compatibility axioms. Crossed modules and complexes over groups are the `one object' cases of the more general crossed mo* *dules and crossed complexes over groupoids. The category ccplxof crossed complexes is symmetric monoidal with respect to the tensor product , see [BH87 ] or [Ton03, Definition 1.4]. This tensor produ* *ct satisfies the following crucial property: given two CW -complexes Y , Z there i* *s a natural isomorphism ssCW Y ssCW Z ~= ssCW (Y x Z) [BH91 , Theorem 3.1 (iv)] satisfying the usual coherence properties. As examples of monoids in the catego* *ry of crossed complexes we can cite the fundamental crossed complex ssCW M of a CW* * - monoid M, and the crossed cobar construction __CrsX on a 1-reduced simplicial set X, see [BT97 ]. As a consequence of [Ton03] the fundamental crossed complex ssN of a simplicial monoid N is also a monoid in ccplx. For our purposes it will be convenient to have a small model for the fundamen* *tal crossed complex of the diagonal of a bisimplicial set. This is achieved by the following definition. Definition 4.4. The total crossed complex (X) of a bisimplicial set X is the coend Z m,n (4.5) (X) = ss( [m]) ss( [n]) . Xm,n . Here [k] is the k-simplex, k 0, and C . E is the E-fold coproduct of a cross* *ed complex C over an indexing set E; see [Mac71 , IX.6] for more details on coend calculus. Note that if X0,0= {*} then (X) is a crossed complex of groups. The following lemma gives an explicit presentation in terms of generators and relations which* * is suitable for our purposes. Lemma 4.6. Suppose X is a horizontally-reduced bisimplicial set, in the sense t* *hat X0,*= [0]. Then (X) is the crossed complex of groups with one generator xm,n in (X)m+n for each xm,n 2 Xm,n and subject to the following relations: xm,n = 0 if xm,n is degenerate in Xm,n, @2x1,1 = - dv0x1,1+ dv1x1,1, @2x2,0 = - dh1x2,0+ dh0x2,0+ dh2x2,0, @3x1,2 = - dv2x1,2- dv0x1,2+ dv1x1,2, hdvx2,1 @3x2,1 = dh2x2,1+ dh0x2,1d2 1- dv1x2,1- dh1x2,1+ dv0x2,1, h)2x3,0 @3x3,0 = dh2x3,0+ dh0x3,0(d2 - dh3x3,0- dh1x3,0. For m 1 and m + n 4, the boundary relations are abelian: h)m-1(dv)nxm,nmX Xn @m+n xm,n = dh0xm,n(d2 1 + (-1)idhixm,n + (-1)m+j dvjxm,n. i=1 j=0 The last summation is trivial if n = 0; all the other terms are trivial if m = * *1. Proof.This follows by using the presentations for ss [n] and the tensor product* * of crossed complexes in [Ton03, 1.2 and 1.4] for example. THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM 13 The following results are natural generalizations of the Eilenberg-Zilber the* *orem for crossed complexes given in [Ton03]. Theorem 4.7. There is a natural homotopy equivalence (in fact, a strong de- formation retraction) of crossed complexes between the total crossed complex of* * a bisimplicial set X and the fundamental crossed complex of its diagonal, _______________________________ OE0________________________________________________________* *_________________a0 _+ss+Diag(X)________________________//_o(X).o_ b0 Proof.As observed for example in [BF78 , Proposition B.1], the diagonal of a bi* *sim- plicial set X may be expressed as a coend Z m Diag(X) ~= [m] x Xm,*. Since each Xm,* is the coend of [n] . Xm,n, and ss preserves colimits, Z m,n ss Diag(X) ~= ss [m] x [n] . Xm,n Z m,n ~= ss( [m] x [n]) . Xm,n . The result therefore follows from the Eilenberg-Zilber equivalence OE______________________________________________________________* *_________________________________________a _+ss(+[m]_x__[n])______________________//_sso[m]o_ ss [n] b given in [Ton03, Theorem 3.1] (see also [BGPT97 , Section 3]). Theorem 4.8. Given two bisimplicial sets X, Y , there is a natural deformation retraction _______________________________ OE00___________________________________________________________* *______________a00 (4.9) _++(X_x_Y_)_____________________//_o(X)o_ (Y ). b00 Moreover, the following diagram of `shuffle maps' commutes: 0 b0 (4.10) X Y ___________b_________________//ss DiagX ss DiagY b00|| |b| fflffl| b0 ~= fflffl| (X x Y )___________//ss Diag(X x_Y_)_//ss(Diag X x DiagY ) Proof.The natural homotopy equivalence of the objects Z p,p0,q,q0 (X x Y ) ~= ss( [p] x [p0]) ss( [q] x [q0]) . Xp,qx Yp0,q0, Z p,p0,q,q0 (X) (Y ) ~= ss [p] ss [q] ss [p0] ss [q0] . Xp,qx Yp0,q0, is defined using the symmetry ss [q] ss [p0] ~=ss [p0] ss [q] and the Eilen* *berg- Zilber equivalence, see [Ton03]. The commutativity of the diagram (4.10) follows from standard properties of the shuffle map. 14 FERNANDO MURO AND ANDREW TONKS Example 4.11. Suppose X, Y are bisimplicial sets, with x 2 X1,0and y 2 Y1,0 corresponding to generators in degree one of X and Y respectively. Then by [Ton03, 2.6] we have b00(x y) 2 (X x Y )2 given by b00(x y)= -(sh0x, sh1y) + (sh1x, sh0y). The category of crossed modules inherits a monoidal structure from the cate- gory of crossed complexes, since it may be regarded as the full reflective subc* *ategory of crossed complexes concentrated in degrees one and two. We denote the reflect* *ion functor by _ :ccplx! cross. The following lemma illustrates the rigidity of monoids in the category of cr* *ossed modules of groups. Lemma 4.12. (1) Let C be a crossed complex of groups and ~ : C C ! C a unital morphism. Then the induced morphism _~ : _C _C ! _C is a monoid structure. (2) Let f : C ! C0be a morphism of crossed complexes of groups which preserves given unital morphisms ~ : C C ! C and ~0: C0 C0! C0up to some homotopy. Then _f : _C ! _C0 is a strict monoid homomorphism. Proof.(1) Since the only degree 0 element of C is the unit, and ~ is unital, the associativity relation ~(~(a b) c) = ~(a ~(b c)) holds if the degree of* * a, b or c is 0. If not, the total degree is at least 3 and the relation is trivial on _* *C. (2) Write ai, bi, a0i, b0ifor elements of Ci and C0i, i 0. Since all the ma* *ps are unital, ~0(fai fbj) = f~(ai bj) if i or j = 0. It remains to show that this relation holds in the crossed module _C0 for i = j = 1 also. The homotopy will be given by a degree one function h : C C ! C0 satisfying a certain derivation formula and an analogue of @h + h@ = ~0(f f) - f~, see e.g. [Whi49 , BH87 ]. Clearly @h(ai bj) = 0 for {i, j} = {0, 1}, and furthermore0the tensor product relations in C0say that @0(a02 b01) = @0a02 b01- a02+ a02b1. In _C0we can the* *refore deduce that C01acts trivially on the elements a02= h(ai bj) for {i, j} = {0, 1* *}. By the derivation property it now follows that in fact h@(a1 b1) = 0 in _C0, and* * so ~0(fa1 fb1) = f~(a1 b1) here also. Corollary 4.13. (1) Let M x M ! M be a strictly unital multiplication, where M is one of the following: o a reduced simplicial set, o a reduced CW -complex, o a bisimplicial set with M0,0= {point}. Then _ssM, _ssCW M or _ M respectively is a monoid in the category of crossed modules. (2) Let N xN ! N be another such structure and f :M ! N a morphism which preserves multiplication up to a homotopy. Then f induces a strictly multiplica* *tive homomorphism between the respective monoids in the category of crossed modules. Monoids in the category of crossed modules of groups are also termed reduced 2-modules, reduced 2-crossed modules and strict braided categorical groups, see [Con84 , BC97 , BC91 , JS93]. Commutative monoids are similarly termed stable crossed modules, stable 2-modules and strict symmetric categorical groups, see [Con84 , BC97 , BCC93 ]. THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM 15 We recall now the usual definition of these concepts, following [BC91 ] and [BCC93 ] up to a change of conventions. Definition 4.14. A reduced 2-module is a crossed module @ :M ! N together with a map <., .>: N x N -! M satisfying the following identities for any m, m02 M and n, n0, n002 N. (1) @ = [n0, n], (2) mn = m + , (3) + <@(m), n> =00,0 (4) = n + , 0 (5) = + n . Moreover, (@, <., .>) is a stable 2-module if (1), (2), (4) and (6) + = 0 are satisfied. By (2), the action of N on M is completely determined by the bracket <., .>. * *The first crossed module axiom (4.2) is now redundant, and (4.3) is equivalent to (7) <@(m), @(m0)> = [m0, m], If we take (2) as a definition it is straightforward to check that it does defi* *ne a group action. Therefore we do not need to require that @ is a crossed module, b* *ut just a homomorphism of groups. Comparing (1), (4), (6), (7) with Definition 1.1 one readily obtains the foll* *owing. Lemma 4.15. The category of stable quadratic modules is a full reflective subca* *t- egory of the category of stable 2-modules, given by those objects N x N <.,.>-!M -@! N which satisfy = 0 for all n, n0, n002 N. The reflection functor will be denoted by OE: s2mod ! squad. Another nice feature of monoids in the category of crossed modules of groups * *is that the property of being commutative is preserved by weak equivalences. Lemma 4.16. Let C ~! D be a morphism of reduced 2-modules which is a weak equivalence. Then C is stable if and only if D is. Proof.The operation <., .> induces a natural quadratic function Coker@ -! Ker@ :x 7! , the k-invariant of C. By using the properties of <., .> it is easy to see that * *C is stable if and only if this quadratic function is indeed a group homomorphism. Therefore the property of being stable is preserved under weak equivalences between reduc* *ed 2-modules. Remark 4.17. One can obtain a stable 2-module from an (n - 1)-reduced simplicial group G, n 2, by using the following truncation of the Moore complex N*G Nn+1G=d0(Nn+2G) -d0!NnG = Gn. The bracket is defined by = [s1(x), s0(y)] + [s0(y), s0(x)], x, y 2 Gn. 16 FERNANDO MURO AND ANDREW TONKS This stable quadratic module will be denoted by ~n+1G. If G is only 0-reduced this formula defines a reduced 2-module ~2G. Compare [Con84 , BC91 , BCC93 ]. If C ~= ~n+1G for an (n - 1)-reduced free simplicial group G, n 2, then the natural morphism C ! OEC is a weak equivalence. This is a consequence of Curtis's connectivity result [Cur65], which implies that we can divide out weig* *ht three commutators in G and still obtain the same ssn and ssn+1, compare also [Bau91 , IV.B]. In order for C to be such a truncation it is enough that the l* *ower- dimensional group of C is free. Indeed suppose that E is the basis of the lower- dimensional group of C. By [Con84 ] there exists an (n - 1)-reduced simplicial group G whose Moore complex is given by C concentrated in dimensions n and n + 1. In particular, Gn = . By "attaching cells" one can construct a free resolution of G (i.e. a cofibrant replacement) given by a weak equivalence G0!~* * G in the category of simplicial groups which is the identity in dimensions n. T* *hen C ~=~n+1G ~=~n+1G0. As a consequence we observe that the reflection OE preserves weak equivalences between objects with a free low-dimensional group. Let Ho Spec0 be the homotopy category of connective spectra of simplicial set* *s, and let Ho Spec10be the full subcategory of spectra with trivial homotopy groups in dimensions other than 0 and 1. Lemma 4.18. There is a functor ~0: Ho Spec0- ! Ho squad together with natural isomorphisms ssi~0X ~=ssiX, i = 0, 1, which induces an equivalence of categories ~0: Ho Spec10~-!Ho squad. Moreover, for any connective spectrum the first Postnikov invariant of X coinci* *des with the k-invariant of ~0X. Proof.Stable quadratic modules, stable crossed modules and stable 2-modules are known to be algebraic models of the (n + 1)-type of an (n - 1)-reduced simplici* *al set X for n 3, see [Bau91 , Con84, BC97 , BCC93 ]. All these approaches are essentially equivalent, and they encode the first k-invariant as stated above. * *For example, if X is an (n - 1)-reduced simplicial set, n 3, then ~nG(X) is such a model for the (n + 1)-type of X. Here we use the Kan loop group G(X). Its projection to stable quadratic modules OE~nG(X) is also a model for the (n+1)-t* *ype of X since G(X) is free, see Remark 4.17 above. The 1-type of a connective spectrum X of simplicial sets is completely determ* *ined by the 4-type of the third simplicial set Y3 of a fibrant replacement (in parti* *cular an -spectrum) Y of X. We can always assume that Y3 is 2-reduced. Therefore we can define the functor ~0 above as follows. Each spectrum X is sent by ~0 to OE~3G(Y3). Lemma 4.19. Given a 1-reduced simplicial set X there is a natural isomorphism of monoids in crossed modules ___CrsX ~=~2G(X). Proof.This lemma is not surprising, since both ___CrsX and ~2G(X) are models for the 2-type of the loop space of X, and moreover they have the same low-dimensio* *nal group . THE 1-TYPE OF A WALDHAUSEN K-THEORY SPECTRUM 17 Using the presentation of __CrsX as a monoid in the category of crossed com- plexes given in [BT97 , Theorem 2.8] and the convention followed by May [May67 , Definition 2.6.3] for the definition of G(X), a natural isomorphism O: ___CrsX * *~= ~2G(X) can be described on the monoid generators as follows. Given x2 2 X2, let O(x2) = x2, and given x3 2 X3, O(x3) = -s1d2(x3) + x3 - s2d3(x3) + s1d3(x3). This is the identity in low-dimensional groups. In order to check that it inde* *ed defines an isomorphism in the upper groups one can use the presentation of __Cr* *sX in [BT97 ], and a computation of the Moore complex of G(X) in low dimensions by using the Reidemeister-Schreier method, see [Kan58 , 18] and [MKS66 ]. In the statement of the following lemma we use the Moore loop complex functor on the category of fibrant simplicial sets. Given a 1-reduced Kan complex X, define X by ( X)n = Ker[dn+1: Xn+1 ! Xn] in the category of pointed sets; compare [Cur71, 2.9], [May67 , Definition 23.3* *]. The face and degeneracy operators are restrictions of the operators in X. If X * *is a simplicial group then so is X. Recall that the natural simplicial map oX : X ! GX given by ( X)n Xn+1 ! i is a homotopy equivalence when X is a 1-reduced Kan complex. The composite of ssnoX with ssn+1X ~=ssn X coincides with the connecting map ffi :ssn+1X ~= ssnGX in the path-loop group fibration GX ! EX ! X. Lemma 4.20. For any 2-reduced Kan complex X there is a natural weak equiva- lence of simplicial groups oe :G( X) ~! G(X). Proof.For all n 0 we have Gn( X) ~=<( X)n+1 - s0( X)n>, ( G(X))n Gn+1(X) ~=. The homomorphisms oen : Gn( X) ! ( G(X))n are the unique possible homo- morphisms compatible with the inclusions ( X)k Xk+1, k 0, in the obvious way. Since oe O o X = oX : X ! GX, the map oe is a weak equivalence. Now we are ready for the proof of the main theorem of this paper. Proof of Theorem 1.5.The coproduct in C gives rise to a -space A in the sense of Segal [Seg74] with A(1) = |DiagNerwS.C| , see [Wal78, Section 4, Corollary]. The spectrum of topological spaces A(1), BA(1), B2A(1),. . . associated to A is an -spectrum since DiagNer wS.C is reduced. The -spectrum defining KC is obtained from the spectrum of A by shifting the dimensions by +1, i.e. KC is gi* *ven by A(1), A(1), BA(1), B2A(1), . ... A particular choice of the coproduct A _ B of any pair of objects A, B in C induces a product in NerwS.C. We choose A _ * = A = * _ A so that this product is strictly unital as in Corollary 4.13. The structure weak equivalence (a) |DiagNerwS.C| -~! BA(1) 18 FERNANDO MURO AND ANDREW TONKS is a morphism of H-spaces up to homotopy. We can replace BA(1) and B2A(1) by homotopy equivalent spaces |Y2|, |Y3| which are realizations of a 1-reduced fibrant simplicial set Y2 and a 2-reduced* * fibrant simplicial set Y3, respectively. As a consequence we obtain a replacement for (* *a) consisting of a homotopy equivalence of pointed CW -complexes (b) |DiagNerwS.C| -~!__FTopY2. Here __FTopY2 is the model for |Y2|in [BT97 , Theorem 2.7]. The CW -complex __FTopY2 is a monoid and the map (b) is in the conditions of the statement of Corollary 4.13. In order to define ~0KC as OE~3G(Y3) we choose an -spectrum Y in the category of simplicial sets representing KC with Y2 and Y3 the simplicial sets chosen ab* *ove. Combining the results above we obtain the following weak equivalences of stab* *le 2-modules. _ NerwS.C ~! _ss DiagNerwS.C (Theorems 4.7 and 4.8) = _ssCW |DiagNerwS.C| ~! _ss CW __FTopY2 (b) ~= ___CrsY2 [BT97 , proof of Proposition 2.11] ~= ~2G(Y2) (Lemma 4.19) ~! ~ ~ ~ 2G( Y3) (Induced by Y2 ! Y3) ! ~2 G(Y3) (Lemma 4.20) = ~3G(Y3). Here we use Lemma 4.16 to derive that not only ~3G(Y3) but all these reduced 2-modules are indeed stable. Since OE preserves weak equivalences between stable 2-modules with free lower- dimensional group, see Remark 4.17, we obtain OE_ NerwS.C -~!~0KC. Finally the formulas in Lemma 4.6 and Example 4.11 together with the laws of a stable quadratic module show that D*C = OE_ NerwS.C. In fact this was how we obtained the definition of D*C. References [Bau91] H.-J. 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