THE 1TYPE OF A WALDHAUSEN KTHEORY SPECTRUM
FERNANDO MURO AND ANDREW TONKS
Abstract.We give a small functorial algebraic model for the 2stage Post
nikov section of the Ktheory spectrum of a Waldhausen category and use *
*our
presentation to describe the multiplicative structure with respect to bi*
*exact
functors.
Introduction
Waldhausen's Ktheory of a category C with cofibrations and weak equivalences
[Wal85] extends the classical notions of Ktheory, such as the Ktheory of ring*
*s,
additive categories and exact categories.
In this paper we give an algebraic model D*C for the 1type 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 1type 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. Ktheory, Waldhausen category, Postnikov invariant, s*
*table quadratic
module, crossed complex, categorical group.
The first author was partially supported by the project MTM200401865 and th*
*e MEC post
doctoral fellowship EX20040616, and the second by the MECFEDER 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_EilenbergZilberCartier_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 1type D*C of the Ktheory 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 1TYPE OF A WALDHAUSEN KTHEORY 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 1type of the algebraic Ktheory 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 1TYPE OF A WALDHAUSEN KTHEORY 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 1type of a connective spectrum is determined
up to nonnatural isomorphism by the first kinvariant. We now establish the
connection between this kinvariant and the algebraic model D*C.
Definition 1.6. The kinvariant of a stable quadratic module C* is the homomor
phism
k :ss0C* Z=2 ! ss1C*, k(x 1) = .
Given a connective spectrum X the kinvariant 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 fflfflp1
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
(xy)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,
(xP (a))H = 0,
P (xy)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! (xx)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 1TYPE OF A WALDHAUSEN KTHEORY SPECTRUM 7
A stable quadratic module C* is termed 0free if C0 = nilis a free group of
nilpotency class 2. Here E is the basis.
Lemma 2.3. Let C* be a 0free 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 (xy)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 0free. 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) ~(y2y1)H ,
(3) the symbol x " y is left linear, (x1 + x2) " y = x1 " y + x2 " y,
(4) P (a) " y = a ~(yy)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) (yy)H ,
H(a ~b) = a b  T (a) T (b),
(a ~b)H = 0,
(a ~b)H = 0,
(a " bc " d)H= (ac)H (bd)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 1TYPE OF A WALDHAUSEN KTHEORY SPECTRUM 9
Let M . E be the Efold 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 lefthand 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 welldefined abelian group ho
momorphism in all cases. The square group morphism '00 is welldefined 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 welldefined 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 1TYPE OF A WALDHAUSEN KTHEORY SPECTRUM 11
The category squad of stable quadratic modules is a 2category with 2morphis*
*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 0free 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 2morphism ": 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 2functor.
Proof.The homotopy D*": D0C ! D1D induced by a 2morphism ": 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 1type 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 nsimplex, 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 Xwhere . 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 1reduced 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 ksimplex, k 0, and C . E is the Efold 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 horizontallyreduced 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)m1(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 1TYPE OF A WALDHAUSEN KTHEORY SPECTRUM 13
The following results are natural generalizations of the EilenbergZilber 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 EilenbergZilber 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
2modules, reduced 2crossed modules and strict braided categorical groups, see
[Con84 , BC97 , BC91 , JS93]. Commutative monoids are similarly termed stable
crossed modules, stable 2modules and strict symmetric categorical groups, see
[Con84 , BC97 , BCC93 ].
THE 1TYPE OF A WALDHAUSEN KTHEORY 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 2module 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 2module 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 2modules, 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 2modules 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 kinvariant 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
2modules.
Remark 4.17. One can obtain a stable 2module 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 0reduced
this formula defines a reduced 2module ~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 lowdimensional 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 kinvariant of ~0X.
Proof.Stable quadratic modules, stable crossed modules and stable 2modules 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 kinvariant 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 1type of a connective spectrum X of simplicial sets is completely determ*
*ined
by the 4type 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 2reduced. 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 1reduced 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 2type of the loop space of X, and moreover they have the same lowdimensio*
*nal
group .
THE 1TYPE OF A WALDHAUSEN KTHEORY 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 lowdimensional 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 ReidemeisterSchreier 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 1reduced 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 1reduced Kan complex. The composite of ssnoX with ssn+1X ~=ssn X
coincides with the connecting map ffi :ssn+1X ~= ssnGX in the pathloop group
fibration GX ! EX ! X.
Lemma 4.20. For any 2reduced 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 Hspaces up to homotopy.
We can replace BA(1) and B2A(1) by homotopy equivalent spaces Y2, Y3
which are realizations of a 1reduced fibrant simplicial set Y2 and a 2reduced*
* 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 Y2in [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
2modules.
_ 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
2modules are indeed stable.
Since OE preserves weak equivalences between stable 2modules 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. Baues, Combinatorial Homotopy and 4Dimensional Complexes, Walter*
* de
Gruyter, Berlin, 1991.
[BC91] M. Bullejos and A. M. Cegarra, A 3dimensional nonabelian cohomology of*
* groups
with applications to homotopy classification of continuous maps, Canad.*
* J. Math. 43
(1991), no. 2, 265296.
[BC97] H.J. Baues and D. Conduch'e, On the 2type of an iterated loop space, *
*Forum Math.
9 (1997), no. 6, 721738.
[BCC93] M. Bullejos, P. Carrasco, and A. M. Cegarra, Cohomology with coefficien*
*ts in sym
metric catgroups. An extension of EilenbergMac Lane's classification *
*theorem, Math.
Proc. Cambridge Philos. Soc. 114 (1993), no. 1, 163189.
[BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of spaces, spe*
*ctra, and
bisimplicial sets, Geometric applications of homotopy theory (Proc. Con*
*f., Evanston,
Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 19*
*78, pp. 80130.
[BGPT97]R. Brown, M. Golasi'nski, T. Porter, and A. Tonks, Spaces of maps into *
*classifying
spaces for equivariant crossed complexes, Indag. Math. (N.S.) 8 (1997),*
* no. 2, 157172.
THE 1TYPE OF A WALDHAUSEN KTHEORY SPECTRUM 19
[BH87] R. Brown and P. J. Higgins, Tensor products and homotopies for !groupo*
*ids and
crossed complexes, J. Pure Appl. Algebra 47 (1987), no. 1, 133.
[BH91] _____, The classifying space of a crossed complex, Math. Proc. Cambridg*
*e Philos.
Soc. 110 (1991), no. 1, 95120.
[BJP05] H.J. Baues, M. Jibladze, and T. Pirashvili, Quadratic algebra *
* of square
groups, MPIM Preprint, http://www.mpimbonn.mpg.de/preprints/send?bid=2*
*884,
http://arxiv.org/abs/math.CT/0601777, 2005.
[BM05] H.J. Baues and F. Muro, Secondary homotopy groups, Preprint MPIM, 2005.
[BP99] H.J. Baues and T. Pirashvili, Quadratic endofunctors of the category o*
*f groups, Ad
vances in Math. 141 (1999), 167206.
[BT97] H.J. Baues and A. Tonks, On the twisted cobar construction, Math. Proc*
*. Cambridge
Philos. Soc. 121 (1997), no. 2, 229245.
[Con84] D. Conduch'e, Modules crois'es g'en'eralis'es de longueur 2, J. Pure Ap*
*pl. Algebra 34
(1984), no. 23, 155178.
[Cur65] E. B. Curtis, Some relations between homotopy and homology, Ann. of Mat*
*h. 82
(1965), 386413.
[Cur71] _____, Simplicial homotopy theory, Advances in Math. 6 (1971), 107209 *
*(1971).
[DP61] A. Dold and D. Puppe, Homologie nichtadditiver Funktoren, Anwendungen,*
* Ann.
Inst. Fourier 11 (1961), no. 6, 201312.
[JS93] A. Joyal and R. Street, Braided tensor categories, Adv. Math. 102 (1993*
*), no. 1, 2078.
[Kan58] D. M. Kan, A combinatorial definition of homotopy groups, The Annals of*
* Mathemat
ics 67 (1958), no. 2, 282312.
[Mac71] S. MacLane, Categories for the Working Mathematician, Graduate Texts in*
* Mathe
matics, no. 5, Springer Verlag, New YorkHeidelbergBerlin, 1971.
[May67] J. P. May, Simplicial objects in algebraic topology, The University of *
*Chicago Press,
Chicago and London, 1967.
[MKS66] W. Magnus, A. Karras, and D. Solitar, Combinatorial Group Theory, Inter*
*science,
New York, 1966.
[Nen98] A. Nenashev, K1 by generators and relations, J. Pure Appl. Algebra 131 *
*(1998), no. 2,
195212.
[Ran85] A. Ranicki, The algebraic theory of torsion. I. Foundations, Algebraic *
*and geometric
topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126*
*, Springer,
Berlin, 1985, pp. 199237.
[Seg74] G. Segal, Categories and cohomology theories, Topology 13 (1974), 2933*
*12.
[Ton03] A. P. Tonks, On the EilenbergZilber theorem for crossed complexes, J. *
*Pure Appl.
Algebra 179 (2003), no. 12, 199220.
[Wal78] F. Waldhausen, Algebraic Ktheory of topological spaces. I, Algebraic a*
*nd geometric
topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1*
*976), Part 1,
Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 19*
*78, pp. 35
60.
[Wal85] F. Waldhausen, Algebraic Ktheory of spaces, Algebraic and geometric to*
*pology (New
Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Be*
*rlin, 1985,
pp. 318419.
[Wei] C. Weibel, An introduction to algebraic ktheory, Book in progress avai*
*lable at
http://math.rutgers.edu/~weibel/Kbook.html.
[Whi49] J. H. C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 5*
*5 (1949),
453496.
MaxPlanckInstitut f"ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Email address: muro@mpimbonn.mpg.de
London Metropolitan University, 166220 Holloway Road, London N7 8DB, UK
Email address: a.tonks@londonmet.ac.uk