Normalisation for the fundamental crossed complex
of a simplicial set
Ronald Brown*
r.brown@bangor.ac.uk
University of Wales, Bangor, Dean St.,
Bangor
Gwynedd LL57 1UT
U.K.
Rafael Sivera
Rafael.Sivera@uv.es
Departamento de Geometr'ia y Topolog'ia
Universitat de Val`encia,
46100 Burjassot, Valencia
Spain
November 23, 2006
Dedicated to the memory of Saunders Mac Lane
Abstract
Crossed complexes are shown to have an algebra sufficiently rich to mod*
*el
the geometric inductive definition of simplices, and so to give a purely a*
*lge-
braic proof of the Homotopy Addition Lemma (HAL) for the boundary of a
simplex. This leads to the fundamental crossed complex of a simplicial set.
The main result is a normalisation theorem for this fundamental crossed co*
*m-
plex, analogous to the usual theorem for simplicial abelian groups, but mo*
*re
complicated to set up and prove, because of the complications of the HAL
and of the notion of homotopies for crossed complexes. We start with some
historical background, and give a survey of the required basic facts on cr*
*ossed
complexes. 1
Introduction
Crossed complexes are analogues of chain complexes but with nonabelian features
in dimensions 1 and 2. So one aim of the use of crossed complexes is to increase
the_power_of_methods_analogous_to those of chain complexes.
*Brown was supported by a Leverhulme Emeritus Fellowship, 2002-2004, which al*
*so supported
collaboration1of the authors.
MATHSCICLASS: 18D10, 18G30, 18G50, 20L05, 55N10, 55N25, 55U10,55U99
KEYWORDS: crossed complex, simplicial set, normalisation, homotopy addition lem*
*ma, groupoid,
crossed module, fundamental crossed complex
1
Crossed complexes can incorporate information on presentations of groups, or
groupoids. Thus another aim is to bring features of the fundamental group nearer
to the centre of the toolkit of algebraic topology. From the early 20th century*
*, the
fundamental group ss1(X, p) has played an anomalous r^ole in algebraic topology.
This invariant of a pointed space has the properties:
oit is nonabelian;
oit can under some circumstances be calculated precisely as nonabelian group
by a van Kampen theorem;
opresentations of it are important;
oit models all pointed, connected, weak homotopy 1-types.
Yet higher dimensional tools (homology groups, homotopy groups) were generally
abelian. An exception was the use of crossed modules, as developed by J.H.C.
Whitehead in [47], and independently by Peiffer, [41], and Reidemeister, [42]. *
*They
were shown in [37] to model all pointed, connected, weak homotopy 2-types (there
called 3-types).
A possible resolution of this anomaly - nonabelian invariant in dimension 1,
abelian invariants in dimensions greater than 1 - has also appeared with the tr*
*an-
sition from groups to groupoids. The fundamental groupoid ss1(X, A) on a set
A of base points was shown to have computational and conceptual utility, [7, 8].
Groupoids were found to have 2-dimensional nonabelian generalisations, for exam*
*ple
crossed modules and forms of double groupoids, with computational and conceptual
utility [24, 23]. Whitehead's crossed modules derived from homotopy theory could
in some cases be calculated precisely as non abelian structures by a 2-dimensio*
*nal
van Kampen type theorem, whose proof used a relative homotopy double groupoid,
[13, 25].
These results were generalised in [15] to all dimensions, using crossed comp*
*lexes,
whose definition and basic theorems we recall in section 2. It is important tha*
*t these
results are proved without the use of traditional tools such as homology and si*
*m-
plicial approximation, but by working directly with homotopically defined funct*
*ors.
Surveys of the use of crossed complexes are in [9, 10]. We also mention the work
of Huebschmann, [33], on the representation of group cohomology by crossed n-
fold extensions, and the use in [43] of Whitehead's methods from CHII to model
algebraically filtrations of a manifold given by Morse functions.
We can also see from the use of the Homotopy Addition Lemma in proofs of
the Hurewicz and relative Hurewicz Theorems, [45], that the category Crslies at
the transition between homology and homotopy. For further work on crossed com-
plexes, see for example [1, 3, 4, 40, 44]. The works with Baues refer to cross*
*ed
complexes as `crossed chain complexes', and, as with Huebschmann, are in the one
vertex case. It can be argued that the category Crsgives a linear approximation
to homotopy theory: that is, crossed complexes can incorporate presentations of
the fundamental group(oid) and its actions, but do not incorporate, say, higher
dimensional Whitehead products, or composition operators. The tensor product of
crossed complexes (see later in 2.2) allows, analogously to work on chain compl*
*exes,
for corresponding notions of an `algebra' and so for the modelling of more stru*
*cture,
as in [11, 3, 4].
2
The main result of this paper, Theorem 11.7, extends the techniques of cross*
*ed
complexes by proving for crossed complexes an analogue of a basic normalisation
theorem for the traditional chain complex associated with a simplicial set, due*
* orig-
inally to Eilenberg and Mac Lane, [30]. We use the Homotopy Addition Lemma to
construct two crossed complexes associated to a simplicial set K, the unnormali*
*sed
K and the normalised K, of which the first is free on all simplices of K and*
* the
second is free on the nondegenerate simplices. It is important to have both cro*
*ssed
complexes for applications of acyclic model theory; this is analogous to the ap*
*pli-
cation of acyclic models to the usual singular chain complex by Eilenberg and M*
*ac
Lane in [28, 29]. Indeed it was this relation with acyclic model theory which m*
*oti-
vated this investigation, and which we plan to deal with elsewhere. As an examp*
*le
of a corollary from our normalisation theorem and properties of crossed complex*
*es
we obtain the well known fact that the projection kKk ! |K|, from the thick to
the standard geometric realisation of a simplicial set K, is a homotopy equival*
*ence
(Corollary 1.3).
To make this paper self-contained, we give a fairly full account of the nece*
*ssary
properties of crossed complexes, so that this paper can form an introduction to*
* their
use. The structure of this paper is as follows: Section 1 gives an introduction*
* to the
Homotopy Addition Lemma: this Lemma is essential for describing the fundamental
crossed complex of a simplicial set. Section 2 gives an account of the basic re*
*sults
on crossed complexes that are needed. Section 3 states the main relation with
chain complexes with a groupoid of operators; this is useful for understanding *
*many
constructions on crossed complexes. Sections 4, 5, 6 give brief accounts of gen*
*erating
crossed complexes, normal subcrossed complexes and free crossed complexes; the
first two topics are not easily available in the literature. Sections 7, 8 giv*
*e specific
rules for the crossed complex constructions of cylinders and homotopies, and th*
*en
cones, and so allow the algebraic deduction of the Homotopy Addition Lemma.
Section 9 defines the (non normalised) fundamental crossed complex of a simplic*
*ial
set. Section 10 normalises this crossed complex at "0. Section 11 gives the f*
*ull
normalisation theorem.
1 The Homotopy Addition Lemma
The normalisation theorem for simplicial abelian groups (see, for example, Eile*
*nberg-
Mac Lane [30, Theorem 4.1], Mac Lane [36, xVIII.6]), is of importance in homolo*
*g-
ical algebra and in geometric applications of simplicial theory. It is based on*
* the
formula, fundamental in much of simplicial based algebraic topology and homolog-
ical algebra, that if x has dimension n then
Xn
@x = (-1)i@ix, (1)
i=0
which can be interpreted intuitively as: `the boundary of a simplex is the alte*
*rnating
sum of its faces'. The setting for this formula is the theory of chain complexe*
*s: these
are sequences of morphisms of abelian groups (or R-modules) @ : An ! An-1 such
that @@ = 0. For simplicial abelian groups, each @iis a morphism of abelian gro*
*ups
and the formula (1)is just the alternating sum of morphisms. Thus we have a cha*
*in
complex (A, @). Further, if (DA)n is for n > 0 the subgroup of An generated by
degenerate elements, then DA is a contractible subchain complex of (A, @). This*
* is
the normalisation theorem.
3
In homotopy, rather than homology, theory, there is another and more compli-
cated basic formula, known as the Homotopy Addition Lemma (HAL) (or theorem)
[5, x12], [32], [45, Theorem IV-6.1, p174 ff.]. It has roughly the same import *
*as (1),
namely it gives `the boundary of a simplex', but it takes account also of:
oa set of base points (the vertices of the simplex);
ononabelian structures in dimensions 1 and 2;
ooperators of dimension 1 on dimensions > 2.
From our standpoint, the set of base points is taken account of through the use
of groupoids in dimension 1, while the boundary from dimension 3 to dimension 2
uses crossed modules of groupoids. This leads to basic formulae, which with our
conventions are as follows:
In dimension 2 we have a groupoid rule:
ffi2x = -@1x + @2x + @0x, (HAL2)
which is represented by the diagram
?2?``>
>>>
c >>b
x >>> (HAL2-diagram)
>>
0________a_______//1
and the easy to understand formula (HAL2) says that ffi2x = -c + a + b. Note th*
*at
we use additive notation throughout for group or groupoid composition.
In dimension 3 we have the nonabelian rule:
ffi3x = (@3x)u3- @0x - @2x + @1x. (HAL3)
Understanding of this is helped by considering the diagram
3OOGGWW/
ffflfl//
ff|l/
fffflflffl///
ffflf|flfl///
x := dfflf2flffl??``>>e// (HAL3-diagram)
ffflfl >>>///
fflfflc b> //
fflffl >>>//
ffl >
0 _______a________//1
In (HAL3) , we have an exponent u3, given by f = @20x. This is necessitated by
our convention that each n-simplex x has as base point its last vertex @n0x. Th*
*us
the base point of the above 3-simplex x is 3, while the base point of @3x is 2.
The exponent f relocates @3x to have base point at 3, and so yields a well defi*
*ned
formula. Given the labelling in (HAL3-diagram) we have the groupoid formula
-f + (-c + a + b) + f - (-e + b + f) - (-d + a + e) + (-d + c + f) = 0.
4
This is a translation of the rule ffi2ffi3 = 0, provided we assume ffi2(yf) = -*
*f +ffi2y+f,
which is the first rule for a crossed module.
In dimension n > 4 we have the abelian rule, but still with operators:
n-1X
ffinx = (@nx)un + (-1)n-i@ix for n >,4 (HAL> 4)
i=0
where unx = @n-10x. We have some difficulty in drawing a diagram for this! Thes*
*e,
or analogous, formulae underly much nonabelian cohomology theory.
The rule ffin-1ffin = 0 is straightforward to verify for n > 4, through work*
*ing in
abelian groups, but for n = 4 we require the second crossed module rule, that f*
*or
x, y of dimension 2
-y + x + y = xffi2y.
A consequence is that Kerffi2, is central. Hence so also is Imffi3, since we ha*
*ve verified
ffi2ffi3 = 0. The type of argument that is used for the n = 4 case, [45], and w*
*hich we
shall use later, is the simple:
Lemma 1.1 If fl = ff + fi is a central element in a group, then fl = fi + ff.
Proof
fl - ff - fi = -ff + fl - fi = 0. (centrality)
2
The above formulae are not exactly as will be found generally in the literat*
*ure,
but they follow from our conventions given in section 7 for crossed complexes a*
*nd
their `cylinder object' I C and cone object Cone(C). Thus the chain complex
and homological boundary formula (1)becomes a much more complicated, but still
fundamental, result in homotopy theory, as the formulae in the Homotopy Addition
Lemma. Yet formulae of these type occur frequently in mathematics, for example *
*in
the cohomology of groups, [22], in differential geometry, [35], and in the coho*
*mology
of stacks, [6].
The formal structure required for this HAL is known as a crossed complex of
groupoids, and such structures form the objects of a category which we write Cr*
*s.
We give more details on this category in section 2. It is complete and cocom*
*plete.
It contains `free' objects, satisfying the universal property that a morphism f*
* : F !
C from a free crossed complex is defined by its values on a free basis, subject*
* to
certain geometric conditions. Note that in the formulae for the HAL, the @iare *
*not
morphisms, but x and all @ix are elements of a free basis.
Our first aim here is to show how the HAL for a simplex fits neatly into an
algebraic pattern in crossed complexes, using a cone construction Cone(B) for a
crossed complex B. We define algebraically and inductively an `algebraic' or `c*
*rossed
complex simplex' a n by
a 0 = {0}, a n = Cone(a n-1). (2)
Our conventions for the tensor product, [16], ensure that this definition yield*
*s alge-
braically precisely the HAL given above.
Our main result is an application to the (non normalised) fundamental crossed
complex, written here K, of a simplicial set K. This is defined to be the fr*
*ee
5
crossed complex on the elements of Kn, n > 0, with boundary given by the homoto*
*py
addition lemma HAL. Thus K contains basis elements which are degenerate
simplices, of the form "iy for some y. We may also construct K as a coend as
follows. Let be the simplicial operator category, so that a simplicial set K*
* is a
functor op! Sets. Let be the subcategory of generated by the injective
maps, i.e. those which correspond to simplicial face operators. Then we can s*
*ee
the unnormalised fundamental crossed complex of K as the coend in the category
of crossed complexes Z
,n
K = Kn x a n.
We are also interested in the normalised crossed complex, defined as the coend
Z ,n
K = Kn x a n.
In Section 11 we complete the proof of:
Theorem 1.2 (Normalisation theorem) For a simplicial set K, the quotient
morphism p : K ! K is a homotopy equivalence with section q : K ! K,
and the quotient crossed complex K is free.
This has application to the usual thick and standard geometric realisations *
*of a
simplicial set K, defined respectively as coends:
Z ,n
kKk = Kn x n,
Z ,n
|K| = Kn x n,
where n is the geometric simplex. Then the normalisation theorem together with
standard properties of crossed complexes, implies:
Corollary 1.3For a simplicial set K, the projection kKk ! |K| from the thick to
the standard geometric realisation is a homotopy equivalence.
Proof We use the Generalised van Kampen Theorem of [15] to give natural iso-
morphisms
K ~= (kKk*), K ~= (|K|*).
It is immediate that the projection induces an isomorphism of fundamental group*
*oids
and of the homologies of the universal covers at all base points. *
* 2
We assume work on groupoids as in [8, 31].
2 Basics on crossed complexes
Crossed complexes, but called group systems, were first defined, in the one ver*
*tex
case, in 1946 by Blakers in [5], following a suggestion of Eilenberg. He combin*
*ed into
a single structure the fundamental group ss1(X, p) and the relative homotopy gr*
*oups
ssn(Xn, Xn-1, p), n > 2 associated to a filtered space X*, but only in the redu*
*ced
case, i.e. when X0 is a singleton {p}. We now call this structure the fundament*
*al
crossed complex (X*) of the filtered space (see below).
6
Blakers' concept was taken up in J.H.C. Whitehead's deep paper `Combinatorial
homotopy theory II' (CHII) [47], in the reduced and free case, under the term `*
*homo-
topy system'; this paper is much less read than the previous paper `Combinatori*
*al
homotopy I' (CHI) [46], which introduced the basic concept of CW -complex. We
give below a full definition of the category Crsof crossed complexes: our viewp*
*oint,
following that of CHII, is that Crsshould be seen as a basic category for appli*
*cations
in algebraic topology, with better realisability properties, [18, 21], than the*
* more
usual chain complexes with a group of operators, [17].
We use relative homotopy theory to construct the functor
: FTop ! Crs (3)
where FTop is the category of filtered spaces, whose objects
X* : X0 X1 . . .Xn . . .X1
consist of a compactly generated topological space X1 and an increasing sequenc*
*e of
subspaces Xn, n > 0. The morphisms f : X* ! Y* of FTop are maps f : X1 ! Y1
such that for all n > 0 f(Xn) Yn.
The functor is given on a filtered space X* by
8
>ss1(X1, X0) ifn = 1, (4)
:ssn(Xn, Xn-1, X0) ifn > 2.
Here ss1(X1, X0) is the fundamental groupoid of X1 on the set X0 of base points,
and ssn(Xn, Xn-1, X0) is the family of relative homotopy groups ssn(Xn, Xn-1, p)
for all p 2 X0.
If we write Cn = ( X*)n, then we find that there is a structure of a family *
*of
groupoids over C0 with source and target maps s, t:
ffin ffi2
. ._.__//Cn____//_Cn-1___//_._._._//_C2___//_C1 (5)
t|| |t| t|| s||t||
fflffl| fflffl| fflffl|ffflffl|flffl|
C0 C0 C0 C0
in which: for n > 2, Cn is totally disconnected, i.e. s = t; C1 operates (on t*
*he
right) on Cn, n > 2, and on the family of vertex groups of C1 by conjugation; a*
*nd
the axioms are, in addition to the usual operation rules, that:
CC1) sffi2 = tffi2, ffin-1ffin = 0;
CC2) ffin is an operator morphism;
CC3) ffi2 : C2 ! C1 is a crossed module;
CC4) for n > 3, Cn is abelian and ffi2C2 operates trivially on Cn.
7
It will be convenient to write all group and groupoid compositions additively, *
*and
the operations as xa. Thus if a : p ! q in dimension 1, then p = sa, q = ta, an*
*d if
further b : q ! r then a + b : p ! r. If further n > 2 and x 2 Cn(p), then tx =*
* p
and xa 2 Cn(q).
These laws for C = X* reflect basic facts in relative homotopy theory, and
also define the objects of the category Crsof crossed complexes. The morphisms
f : C ! D of crossed complexes consist of groupoid morphisms f : Cn ! Dn, n > 1,
preserving all the structure.
A crossed complex C has a fundamental groupoid ss1C defined to be C1=(ffi2C2*
*),
whose set of components is written ss0C, and also called the set of components *
*of
C. It also has a family of homology groups given for n > 2 by
Hn(C, p) = Ker(ffin : Cn(p) ! Cn-1(p))=Im (ffin+1(Cn+1(p) ! Cn(p)),
which can be seen to be a module over ss1C. A morphism f : C ! D of crossed
complexes induces a morphism of fundamental groupoids and homology groupoids,
and is called a weak equivalence if it induces an equivalence of fundamental gr*
*oupoids
and isomorphisms H*(C, p) ! H*(D, fp) for all p 2 C0.
If X* is the skeletal filtration of a CW -complex X, then (see [47]) there *
*are
natural isomorphisms
ss1( X*) ~=ss1(X, X0), Hn( X*, p) ~=Hn(Xep),
where eXpis the universal cover of X based at p. It follows from this and White*
*head's
theorem from [46] that if f : X ! Y is a cellular map of CW -complexes X, Y whi*
*ch
induces a weak equivalence f : X* ! Y*, then f is a homotopy equivalence.
The following additional facts on crossed complexes were found in a sequence*
* of
papers by Brown and Higgins:
2.1 ([15])The functor : FTop ! Crs from the category of filtered spaces to
crossed complexes preserves certain colimits.
2.2 ([16])The category Crs is monoidal closed, with an exponential law of the
form
Crs(A B, C) ~=Crs(A, CRS(B, C)). (exponential law)
2.3 ([12])The category Crshas a unit interval object written {0} ' I, which is
essentially just the indiscrete groupoid on two objects 0, 1, and so has in dim*
*ension
1 only one element ' : 0 ! 1. For a crossed complex B, this gives rise to a cyl*
*inder
object
Cyl(B)= (B ' I B),
and so a homotopy theory for crossed complexes.
The first result is a kind of Generalised van Kampen Theorem, GvKT, and has
consequences which include the relative Hurewicz Theorem, and nonabelian results
in dimension 2, [25], not obtained by other means. The proof of the GvKT is
independent of standard methods in algebraic topology, and uses cubical higher
homotopy groupoids.
8
We need that the category FTop of filtered spaces is monoidal closed with an
exponential law
FTop(X* Y*, Z*) ~=FTop(X*, FTOP (Y*, Z*)). (6)
S
Here (X* Y*)n = p+q=nXp x Yq. A standard example of a filtered space is a
CW -complex with its skeletal filtration, and among the CW -complexes we have t*
*he
n-ball En with its cell structure
8
>e0 [ e1 ifn = 1, (7)
:e0 [ en-1 [ en ifn > 1.
The complications of the cell structure of Em x En are modelled in the tensor
product of crossed complexes, as shown by:
2.4 ([18])There is for filtered spaces X*, Y* a natural transformation
j : X* Y* ! (X* Y*),
which is an isomorphism if X*, Y* are the skeletal filtrations of CW -complexes*
*, [18],
and more widely, [2].
Remark 2.5 From the early days, basic results of relative homotopy theory have
been proved by relating the geometries of cells and cubes. This geometric relat*
*ion
was translated into a relation between algebraic theories in several papers, pa*
*rtic-
ularly [14, 16], which give an equivalence of monoidal closed categories betwee*
*n a
category of `cubical !-groupoids' and the category Crs. Many constructions and
proofs are clear in the former category, but both categories are required for s*
*ome
results. For example the natural transformation j of 2.4 is easy to see in the *
*cubical
category, [18]. For a survey on crossed complexes and their uses, see [10]. A b*
*ook
in preparation, [19], is planned to give a full account of all these main prope*
*rties,
and make the theory more accessible and hopefully more usable.
The exponential law for crossed complexes, (exponential law), involves an `i*
*n-
ternal hom' crossed complex CRS(B, C). This is in dimension 0 simply Crs(B, C),
the set of morphisms B ! C; in dimension 1 is the groupoid of homotopies be-
tween morphisms; and in higher dimensions consists of `higher homotopies'. The
full structure of this is quite complicated. This complexity is also reflected *
*in the
structure of the tensor product A B of crossed complexes A, B: it is generate*
*d in
dimension n by elements a b where a 2 Al, b 2 Bk, l + k = n, but the full lis*
*t of
structure and laws is again quite complex (see [16]).
Let I denote the groupoid with two objects 0, 1 and exactly one arrow ' : 0 *
*! 1.
We can obtain from this a crossed complex also written I by extending trivially*
* in
higher dimensions than 1. This crossed complex, which is isomorphic to E1*, can
be given the structure of a unit interval object
{0} ' I
in the category Crs(see for example [34]). This allows us to define homotopies *
*of
morphisms B ! C of crossed complexes as morphisms I B ! C, or, equivalently,
as morphisms I ! CRS(B, C). The detailed structure of this cylinder object I *
*C,
[34], will be given in section 7.
An important result on crossed complexes is the following:
9
2.6 ([18])There is a classifying space functor B : Crs ! Top and a homotopy
classification theorem
[X, BC] ~=[ X*, C]
for a CW -complex X with its skeletal filtration, and crossed complex C.
This result is a homotopy classification theorem in the non simply connected ca*
*se,
and includes many classical results. It relates to Whitehead's comment in [47] *
*that
crossed complexes have better realisation properties than chain complexes with a
group of operators. It is also relevant to nonabelian cohomology, and cohomology
with local coefficients, as discussed in [18]. See also [26, 38, 39].
3 Crossed complexes and chain complexes
As is clear from the definition, crossed complexes differ from chain complexes *
*of
modules over groupoids only in dimensions 1 and 2. It is useful to make this re*
*la-
tionship more precise. We therefore define a category Chn of such chain complex*
*es
which is monoidal closed, give a functor r : Crs! Chn, and state that this func*
*tor
is monoidal and has a right adjoint. The results of this section are taken from*
* [17],
which develops results from [47]. See also [20] for the low dimensional and red*
*uced
case.
We first define the category Mod of modules over groupoids. We will often wr*
*ite
G0 for Ob G for a groupoid G. The objects of the category Mod are pairs (G, M)
where G is a groupoid and M is a family M(p), p 2 G0 of disjoint abelian groups
on which G operates. The notation for this will be the same as for the operatio*
*ns
of C1 on Cn for n > 3 in the definition of a crossed complex. The morphisms of
Mod are defined analogous to those for the category Crs. Instead of writing (G*
*, M)
we often say M is a G-module.
The category Mod is monoidal closed. We define here only the tensor product:
for modules (G, M), (H, N) we set
(G, M) (H, N) = (G x H, M N)
using the product of groupoids and with
(M N)(p, q) = M(p) N(q),
the usual tensor product of abelian groups, and action the product action (m
n)(g,h)= mg nh.
A particular module we need is (G, Z! G), also written Z! G, for a groupoid *
*G.
If p 2 G0, then Z! G(p) is the free abelian group on the elements of G with fin*
*al
point p, and with operations induced by the right action of G. Note that in con*
*trast
to the single object case, i.e. of groups, we obtain a module and not an analog*
*ue
of a ring. A set J defines a discrete groupoid on J also written J and so a mod*
*ule
(J, Z! J): when the set J is understood, we abbreviate this module to Z! . The
augmentation map in this context is given as usual by the sum of the coefficien*
*ts. It
is a module morphism " : (G, Z! G) ! (G0, Z! ) and its kernel is the augmentati*
*on
module (G, I! G) which we abbreviate to I! G.
Let _ : G ! H be a groupoid morphism which is bijective on objects, and
let (H, N) be a module. We need the generalisation to groupoids of the universal
10
derivations of Crowell [27]. A _-derivation d : G ! N assigns to each g 2 G
with final point p an element d(g) 2 N(_p) satisfying the rule that d(g0+ g) =
d(g0)_g + d(g) whenever g0+ g is defined in G. The _-derivation d is universal
if given any other _-derivation d0 : G ! L where L is an H-module, there is a
unique H-module morphism f : L ! N such that fd0= d. The construction of the
universal _-derivation is straightforward, and is written ff : G ! D_.
3.1 Let C be a crossed complex, and let OE : C1 ! G be a cokernel of ffi2 of C*
*. Then
there are G-morphisms
Cab2@2-!DOE-@1!Z! G (8)
such that the diagram
OE
. ._._//_Cnffin//_Cn-1//_._._.//C3ffi3//_C2ffi2//_C1//_G
| | |
= || = || = || |ff2| |ff1| ff0| (9)
fflffl| fflffl| fflffl|fflffl||fflffl fflffl|
. ._._//_Cn__//_Cn-1__//_._._.//C3__//Cab2__//DOE__//Z! G
@n @3 @2 @1
commutes and the lower line is a chain complex over G. Here ff2 is abelianisati*
*on,
ff1 is the universal OE-derivation, ff0 is the G-derivation x 7! x - 1q for x 2*
* G(p, q),
as a composition G ! I! G ! Z! G, and @n = ffin for n > 4. Further
(i)the sequence (8)is exact at DOEand the image of @1 is the augmentation mod*
*ule
I! G;
(ii)if C1 is a free groupoid on a generating graph X1 and _ is surjective, the*
*n DOE
is the free H-module on the basis at p 2 H0 of elements x of X1 such that
_tx = p;
(iii)if C1 is free, then ff2 is injective on Kerffi2.
3.2 (i)The bottom row of diagram (9)defines a functor r : Crs! Chn, which
has a right adjoint. Hence r preserves colimits.
(ii)The functor r preserves tensor products: there is a natural equivalence for
crossed complexes A, B
r(A) r(B) ~=r(A B).
This last result shows that the major unusual complications of the tensor produ*
*ct
of crossed complexes occur in dimensions 1 and 2. These cases are analysed in [*
*16].
Remark 3.3 These results show the close relation of crossed complexes and these
chain complexes, but the functor r loses information. Whitehead remarks in [CHI*
*I]
that (using our terminology) these chain complexes have less good realisation p*
*rop-
erties even than free crossed complexes. Indeed, the problem of which 2-dimensi*
*onal
free chain complexes are realisable by a crossed complex is known to be hard.
11
4 Generating structures
Let C be a crossed complex, and let R* be a family of subsets Rn Cn for all
n > 0. We have to explain what is meant by the subcrossed complex of C
generated by R*.
A formal definition of B = is easy: it is the smallest sub-crossed comp*
*lex
D of C such that Rn Dn for all n > 0, and so is also the intersection of all *
*such
D. A direct construction is as follows. We set
[
B0 = R0 [ sR1 [ tRn.
n>1
Let B1 be the subgroupoid of C1 generated by R1[ ffi2(R2) and the identities at*
* B0.
Let B2 be the subcrossed B1-module of C2 generated by R2. For n > 3, let Bn be
the sub-B1-module of Cn generated by Rn [ ffi(Rn+1) and the identities at eleme*
*nts
of B0.
Note that this definition is inductive. The usual property of a generating s*
*truc-
ture holds: thus if R* generates C, i.e. = C, and f, g : C ! D are two cro*
*ssed
complex morphisms which agree on R*, then f = g. This is proved by induction.
We say the family R* is a generating structure for a subcrossed complex B of
C if for each n > 0 the boundaries in C of elements of Rn lie in the subcrossed
complex generated by the Ri for i < n, and R* generates B.
5 Normal subcrossed complexes
We assume work on normal subgroupoids, as in [8, 31].
Definition 5.1A subcrossed complex A of a crossed complex C is called normal
in C if:
N1) A is wide in C, i.e. A0 = C0;
N2) A1 is a totally disconnected normal subgroupoid of C1;
N3) for n > 2, An is C1-invariant, i.e. a 2 An, x 2 C1 and ax is defined implies
ax 2 An;
N4) for n > 2, if a 2 A1, x 2 Cn, and xa is defined then x - xa 2 An. 2
Note that N3) implies that A2 is a normal subgroupoid of C2 since -x+a+x = affi*
*2x.
The above are necessary and sufficient conditions for A to be the kernel of a
morphism C ! D of crossed complexes which is injective on objects for some D,
as the following proposition shows.
Proposition 5.2 If A is a normal subcrossed complex of the crossed complex C,
then the family of quotients Cn=An, n > 1 inherits the structure of crossed com*
*plex,
which we call the quotient crossed complex C=A.
We leave the proof to the reader.
12
Let R* be a family of subsets of the crossed complex C as in the previous se*
*ction,
and such that R1 is totally disconnected, i.e. just a family of subsets of vert*
*ex groups
of the groupoid C1. We say that R* normally generates a subcrossed complex A
of C if A is the smallest wide normal subcrossed complex of C containing R*, and
then we say A is the normal closure of R* in C, and write A = <>. We also s*
*ay
R* is a normal structure in C if for each n > 0 the boundaries of elements of Rn
are in the normal closure of the Ri for i < n.
We consider how to construct A = <>. In dimension 1, this is the normal
closure of R1 [ ffi2(R2) as extended to the groupoid case in [8, 31], i.e we ta*
*ke the
`consequences' of R1 [ ffi2(R2) in C1. Suppose this A1 has been constructed.
Proposition 5.3 For n > 2, An = <>n is generated as a group (abelian if
n > 3) by the elements
rc, x - xa for allr 2 Rn [ ffin+1(Rn+1), c 2 C1, x 2 Cn, a 2 A1.
Proof Clearly these elements belong to An. We now prove the set of these is
C1-invariant.
This is clear for the set of elements of the form rc as above. Suppose then *
*x, c, a
are as above. Then
(x - xa)c= xc- xa+c
= xc- (xc)-c+a+c,
which implies what we want since -c + a + c 2 A1 by normality.
It follows that the group generated by these elements is C1-invariant. In di*
*men-
sion 2, this implies normality, by the crossed module rules. 2
6 Free crossed complexes
Write F(n) for the crossed complex freely generated by one generator cn in dime*
*nsion
n. So F(0) is a singleton in all dimensions; F(1) is essentially the groupoid I*
* ; and for
n > 2, F(n) is in dimensions n and n - 1 an infinite cyclic group with generato*
*rs cn
and fficn respectively, and otherwise trivial. Let S(n - 1) be the subcrossed c*
*omplex
of F(n) generated by fficn. Thus S(-1) is empty.
If En*and Sn-1*denote the skeletal filtrations of the standard n-ball and (n*
*-1)-
sphere respectively, then a basic result in algebraic topology is that
En*~=F(n), Sn-1*~=S(n - 1).
This is also a consequence of the result indicated in 2.1.
We now define a particular kind of morphism j : A ! F of crossed complexes
which we call a morphism of relative free type. Let A be any crossed complex. A
sequence of morphisms jn : F n-1! F nmay be constructed inductively as follows.
Set F -1= A. Supposing F n-1is given, choose any family of morphisms as in the
13
top row of the diagram
F (f~)
~2 n S(n - 1)__________//F n-1
| ||
| jn|
F fflffl| fflffl|
~2 nF(n) _____________//F n
and form the pushout in Crsto obtain jn : F n-1! F n. Let F = colimnF n, and
let j : A ! F be the canonical morphism. The image xn~of the element cn in the
summand indexed by ~ is called a basis element of F relative to A, and we may
conveniently write
F = A [ {xn~}~2 n,n>0.
We now give some useful results on this notion.
Proposition 6.1Given two morphism of relative free type, so is their composite.
Proposition 6.2 If in a pushout square
A _____//A0
| |
| |
fflffl| fflffl|
F _____//F 0
the morphism A ! F is of relative free type, so is the morphism A0! F 0.
Proposition 6.3 If in a commutative diagram
A0 ____//_A1___//_._._.//_An____//. . .
| | |
| | |
fflffl| fflffl| fflffl|
F 0____//_F_1__//_._._.//_F_n___//. . .
each vertical morphism is of relative free type, so is the induced morphism col*
*imnAn !
colimnF n.
In particular:
Corollary 6.4 If in a sequence of morphisms of crossed complexes
F 0! F 1! . .!.F n! . . .
each morphism is of relative free type, so are the composites F 0! F nand the
induced morphism F 0! colimnF n.
A crossed complex F is free on R* if in the first place R* generates F , and
secondly morphisms on F to any crossed complex can be defined inductively by
their values on R*.
So in the first instance we have R0 = F0, and F1 is the free groupoid on the
graph (R1, R0, s, t). We assume this concept as known; it is fully treated in [*
*8, 31]
14
Secondly, R2 comes with a function w : R2 ! F1 given by the restriction of f*
*fi2.
We require that the inclusion R2 ! F2 makes F2 the free crossed F1-module on R2.
By this stage, the fundamental groupoid ss1F is defined; we require that for
n > 3, Fn is the free ss1F -module on Rn.
A standard fact, due in the reduced case to Whitehead in [CHII], is that if *
*X*
is the skeletal filtration of a CW -complex, then X* is the free crossed compl*
*ex
on the characteristic maps of the cell structure of X*. This may be proved using
the relative Hurewicz theorem, and is also a consequence of the Generalised van
Kampen Theorem of [15].
Proposition 6.5 If C is a free crossed complex on R*, then a morphism f : C ! D
is specified by the values fx 2 Dn, x 2 Rn, n > 0 provided only that the follow*
*ing
geometric conditions hold:
sfx = fsx, x 2 R1, tfx = ftx, x 2 Rn, n > 1, ffifx = fffix, x 2(Rn,1n0>)2.
We refer also to [12, 18] for more details on free crossed complexes. It is pro*
*ved in
[12] that a weak equivalence of free crossed complexes is a homotopy equivalenc*
*e.
We now illustrate some of the difficulties of working with free crossed modu*
*les
by giving a proposition and a counterexample due essentially to Whitehead, [48].
Theorem 6.6 Let C be the free crossed complex on R*, and suppose S* R*
generates a subcrossed complex B of C. Let F be the free crossed complex on S*.
Then the induced morphism F ! C is injective if the induced morphism ss1B ! ss1C
is injective.
Proof First of all, we know that a subgroupoid of a free groupoid is free. Also*
* in
dimensions > 2 Cn is the free ss1C-module on the basis Rn. So injectivity, under
the given condition, is clear in this case.
Thus the only problem is in dimension 2, and here we generalise an argument
of Whitehead, [48]. We use the functor r : Crs! Chn given in section 3.
The abelianised groupoids F2ab, Cab2are respectively free ss1F, ss1C-modules*
* on
the bases S2, R2. Since the induced morphism on ss1 is injective, so also is t*
*he
induced morphism F2ab! Cab2. But the morphism C2 ! Cab2is injective on Kerffi2 :
C2 ! C1, since C1 is a free groupoid, by (ii) of 3.1. So F ! C is injective in
dimension 2. 2
Example 6.7 Let X = Y = {x}, R = {a, b}, S = {b} where a = x, b = 1. The
group presentations , determine free crossed modules ffiS : C(S)*
* !
F (X), ffiR : C(R) ! F (X). The inclusion i : S ! R determines C(i) : C(S) !
C(R). Now F (X) = F (Y ) = C, the infinite cyclic group, while C(S) is abelian
and is the free C-module on the generator b. Also in C(R), ab = ba since ffiR b*
* = 1.
Hence
C(i)(bx) = (C(i)(b))ffiRa= a-1ba = b = C(i)(b),
and so C(i) is not injective.
Of course the geometry of this example is the cell complex K = E2 _ S2 and
the subcomplex S1 _ S2. 2
15
7 Cylinder and homotopies
It is useful to write out first all the rules for the cylinder Cyl(C) = I C, *
*as a
reference. For full details of the tensor product, see [16, 10].
Let C be a crossed complex. The cylinder I C is generated by elements
0 x, 1 x of dimension n and ' x, (-') x of dimension (n + 1) for all n *
*> 0
and x 2 Cn, with the following defining relations for a 2 I:
Source and target
t(a x)= ta tx for alla 2 I, 2 C
s(a x)= a sx ifa = 0, 1, n = 1 ,
s(a x)= sa x ifa = ', -', n = 0 .
Relations with operations
a xc= (a x)ta c ifn > 2, c 2 C1.
Relations with additions
( ta y
a (x + y)= (a x) + a y,ifa = ', -', n = 1,
a x + a y, ifa = 0, 1, n > 1 or ifa = ', -', n > 2,
(
(-') x= -(' x) ifn = 0,
-(' x)(-') tx ifn > 1.
Boundaries
8
> 2;
ffi(a =x)>-ta x - a sx + sa x + a txifa = ', -', n = 1;
:-(a ffix) - (ta x) + (sa x)aitxfa = ', -', n > 2.
2
Now we can translate the rules for a cylinder into rules for a homotopy. Thu*
*s a
homotopy f0 ' f of morphisms f0, f : C ! D of crossed complexes is a pair (h, f)
where h is a family of functions hn : Cn ! Dn+1 with the following properties:
thn(x)= tf(x) for allx 2 C; (11)
h1(x + y)= h1(x)fy + h1(y) ifx, y 2 C1 andx + y is defined; (12)
hn(x + y)= hn(x) + hn(y) ifx, y 2 Cn, n > 2 and x + y is defined;(13)
hn(xc)= (hnx)fc ifx 2 Cn, n > 2, c 2 C1, and xc is defined.(14)
Then f0, f are related by
8
>(h0sx) + (fx) + (ffi2h1x) - (h0tx)ifx 2 C1, (15)
:{fx + hn-1ffinx + ffin+1hnx}-(h0tx)ifx 2 Cn, n > 2.
Remark 7.1 Part of the force of this statement is that if (h, f) satisfy prope*
*rties
(11-14), then f0 defined by (15) is a morphism of crossed complexes.
16
The following is a substantial result:
Proposition 7.2 ([18])If F, F 0are free crossed complexes, on bases B*, B0*, th*
*en
F F 0is the free crossed complex on the basis B B0.
The proof in [18] uses the inductive construction of free complexes as successi*
*ve
pushouts given in section 6; the exponential law and the symmetry of show that
preserves colimits on either side, and this gives an inductive proof, analogo*
*us to
a corresponding result for CW -complexes.
A consequence, which may also be proved directly, is:
Proposition 7.3If f : C ! D is a morphism of crossed complexes and C is a
free crossed complex on R*, then a homotopy (h, f) : f0 ' f : C ! D is specified
by the values hx 2 Dn+1, x 2 Rn, n > 0 provided only that the following geometr*
*ic
conditions hold:
thx = tfx, x 2 Rn, n > 0. (16)
Proof The main special fact we need here is that an f-derivation on a free grou*
*poid
is uniquely defined by its values on a free basis. But this follows easily from*
* the fact
that an f-derivation h1 : C1 ! D2 corresponds exactly to a section of a semidir*
*ect
product construction F1 n C2 ! F1. 2
8 Cones and the HAL
Definition 8.1Let C be a crossed complex. The cone Cone(C) is defined by the
pushout
{1} C_______//_{v}
| |
| |
fflffl| fflffl|
I C______//Cone(C).
We call v the vertex of the cone. 2
Because the cone is formed from the cylinder by shrinking the end at 1 to a
point, the rules for the cylinder now simplify nicely.
Proposition 8.2 If C is a crossed complex, then the cone Cone (C) on C is gen-
erated by elements 0 x, ' x, x 2 Cn of dimensions n, n + 1 respectively, an*
*d v of
dimension 0 with the following rules, for all a 2 I:
Source and target
(
t(a x)= 0 tx, ifa = 0,
v otherwise.
Relations with operations
a xc= a x if n > 2, c 2 C1.
17
Relations with additions
a (x + y)= a x + a y.
and
(
(-') x= -(' x) ifn = 0,
-(' x)-' tx ifn > 1.
Boundaries
ffin(0 x)= 0 ffinx ifn > 2.
(
ffin+1(' =x)-' sx + 0 x + ' txifn = 1,
-(' ffinx) + (0 x)'itxfn > 2.
Proposition 8.3 Let F be a free crossed complex on a basis B*. Then Cone(F )
is the free crossed complex on v in dimension 0, and elements 0 b, ' b for *
*all
b 2 B*, with boundaries given by proposition 8.2.
Proof This follows from proposition 6.2. 2
We use the above to work out the fundamental crossed complex of the simplex
a n in an algebraic fashion. We define a 0 = {0}, a n inductively by
a n = Cone(a n-1).
The vertices of a 1 = I are ordered as 0 < 1. Inductively, we get vertices v0, *
*. .,.vn
of a n with vn = v being the last introduced in the cone construction, the other
vertices vi being (0, vi). The fact that our algebraic formula corresponds to *
*the
topological one follows from facts stated earlier on the tensor product and on *
*the
GvKT.
We now define inductively top dimensional generators of the crossed complex
a n by, in the cone complex:
oe0 = v, oe1 = ', oen = (' oen-1), n > 2,
with oe0 being the vertex of a 0.
Next we need conventions for the faces of oen. We define inductively
( n-1
@ioen = ' @ioe ifi < n,
0 oen-1 ifi = n.
Theorem 8.4 (Homotopy Addition Lemma) The following formulae hold, where
un = ' vn-1:
ffi2oe2= -@1oe2 + @2oe2 + @0oe2, (17)
ffi3oe3= (@3oe3)u3- @0oe3 - @2oe3 + @1oe3, (18)
while for n > 4
n-1X
ffinoen= (@noen)un + (-1)n-i@ioen. (19)
i=0
18
Proof For the case n = 2 we have
ffi2oe2= ffi2(' ')
= -' 0 + 0 ' + ' 1
= -@1oe2 + @2oe2 + @0oe2.
For n = 3 we have:
ffi3oe3= ffi3(' oe2)
= (0 oe2)' v2- ' ffi2oe2
= (0 oe2)u3- ' (-@1oe2 + @2oe2 + @0oe2)
= (@3oe3)u3- @0oe3 - @2oe3 + @1oe3.
We leave the general case to the reader, using the inductive formula
ffin+1oen+1= (0 oen)' vn- ' ffinoen.
The key points that make it easy are the rules on operations and additions of
Proposition 8.2. 2
Corollary 8.5 The formula for the boundary of a simplex is as given by the HAL
in section 1.
Proof We use the fact that for n > 2, the geometric n-simplex n may be regarded
as the cone Cone( n-1). Our previous results thus give an isomorphism
n*~=Cone( n-1*).
Since 1*= E1*, the HAL now follows from theorem 8.4. 2
9 The unnormalised fundamental crossed complex
of a simplicial set
Definition 9.1We define (K) the (unnormalised) fundamental crossed complex
of the simplicial set K as the free crossed complex having the elements of Kn as
generators in dimension n and boundary maps given by the Homotopy Addition
Lemma. In detail this gives the crossed complex (K) as follows:
1. The objects are the vertices of K: (K)0 = K0;
2. The groupoid (K)1 is the free groupoid associated to the directed graph K*
*1.
So it has a free generator x : @1x ! @0x for each x 2 K1;
3. The crossed module (K)2 ! (K)1 is the free (K)1-crossed module
generated by the map ffi2 : K2 ! (K)1 given by
ffi2x = -@1x + @2x + @0x
for all x 2 K2.
19
4. For all n > 3, (K)n is the free (K)1-module with generators Kn and
boundary given by
( u
3- @0x - @2x + @1x ifn = 3,
ffinx = (@3x) P n-1
(@nx)un + i=0(-1)n-i@ix ifn > 4.
This construction is natural, giving a fundamental crossed complex functor of s*
*im-
plicial sets
: Simp! Crs. 2
Remark 9.2 There are two notions of realisation of a simplicial set K, usually
written kKk, and |K|. In the first the only identifications are along faces, an*
*d in
the second the degenerate simplices are also factored out. Each realisation is*
* a
CW -complex with its skeletal filtration, and the Generalised van Kampen Theorem
of [15], shows that there is a canonical isomorphism K ~= (kKk*).
10 0-normalisation
We first contrast with the usual case of a simplicial abelian group A, where the
simplicial operators @i, "i are morphisms of abelian groups. The associated cha*
*in
complex (A, @) is then An in dimension n > 0 with boundary
Xn
@ = (-1)i@i.
i=0
Let (DA)n for n > 0 be the subgroup of An generated by the degenerate elements.
It is an easy calculation from the rules for simplicial operators that @(DA)n
(DA)n-1 and so (DA, @) is a subchain complex of (A, @).
In the nonabelian case, we have more problems, but the formulae cope well wi*
*th
them. For the rest of this section, K is a simplicial set.
Proposition 10.1 Let E* be the set of degenerate elements in K, together with t*
*he
elements of E0. Then E* is a normal structure in K.
Proof By the rules @i"i= @i+1"i= 1, and the Homotopy Addition Lemma, we get
immediate cancellation in ffin"iy for 0 < i < n-1 but not necessarily for i = 0*
*, n-1,
because of the operators, and the nonabelian structures in dimensions 1,2. Thus
terms of concern are:
ffi2"0y= -y + "0@1y + y,
ffi3"0y= ("0@2y)@0y+ (-y - "0@1y + y),
2y
ffi3"2y= (y)"0@0 - "1@0y - y + "1@0y,
2y
= (y"0@0 - y) + (y - "1@0y - y) + "1@0y,
and for n > 4
n-1y
ffin"n-1y= y"0@0 - y + terms involving "n-2.
This proves the result in view of the definitions in section 5. *
* 2
20
Definition 10.2We define a normal subcrossed complex E0K of K to be K0 in
dimension 0 and in higher dimensions to be normally generated by the degenerate
elements "0y. 2
Definition 10.3We define the 0-normalised crossed complex of K to be
0NK = ( K)=E0K.
Our first result is:
Theorem 10.4 The projection p0 : K ! 0NK has a section q such that qp0 '
1.
The proof will occupy the rest of this section.
We first need a lemma, which will be used later as well.
Lemma 10.5 Let h1 : ( K)1 ! ( K)2 be a derivation. Then for x 2 K2 we
have
h1ffi2x= -(h1@1x)ffi2x+ (h1@2x)@0x+ h1@0x.
Proof
h1ffi2x= h1(-@1x + @2x + @0x)
= (h1(-@1x + @2x))@0x+ h1@0x
= ((h1(-@1x))@2x)@0x+ (h1@2x)@0x+ h1@0x
= -(h1@1x)ffi2x+ (h1@2x)@0x+ h1@0x. 2
Lemma 10.6 If h : _ ' 1 : K ! K is given by h0 = "0 in dimension 0, and
in dimension 1 by h1 is "0 or "1 on the free basis given by K1, then _ is given*
* in
dimensions 0, 1 by (
_x = x ifdimx = 0,
x - "0@0x ifdimx = 1,
and hence _"0y = 0y for all y 2 K0.
Proof The case dimx = 0 is clear. For the case dimx = 1 and for h1 = "0 we have
_x = "0sx + x + ffi2(-"0x) - "0tx
= "0@1x + x - (-x + "0@1x + x) - "0tx
= x - "0@0x.
and for h1 = "1 we have
_x = 0sx+ x + ffi2(-"1x) - 0tx
= x + (x - x - "0@0x)
= x - "0@0x. 2
Now we define simultaneously a morphism _ : K ! K and a homotopy
h : _ ' 1 such that _(E0K) is trivial.
21
Proposition 10.7 (0-normalisation)Let K be a simplicial set. Then a homo-
topy (h, 1) on K may be defined on generators from K by hn = (-1)n"0, yieldi*
*ng
h : _ ' 1 where _ is given on generators by
8
>x - "0@0x if dimx = 1,
:(x - "0@0x)-"0txif dimx > 1.
This _ satisfies
1.- _("0x) = 0txfor all x 2 K.
2.- The induced morphism ~_: 0NK ! K satisfies p0_~= 1 and _ = ~_p0 ' 1.
Thus p0 is a homotopy equivalence.
Proof To verify the formula for _ requires working out a formula for _"0ffinx -
ffin+1"0x, where _"0is the derivation or operator morphism defined by "0 on gen*
*era-
tors, and we also have to use the crossed module rules.
Thus for x 2 K2, we have by Lemma 10.5:
_" ffi2x @0x
0ffi2x= -("0@1x) + ("0@2x) + "0@0x
while
2" x
ffi3"0x= (@3"0x)@0 0 - x - "0@1x + x
= ("0@2x)@0x+ (-"0@1x)ffi2x
= (-"0@1x)ffi2x+ ("0@2x)@0x, by centrality of ffi3"0x
From this we get
-ffi3"0x + _"0ffi2x= "0@0x.
More easily, we have for n > 3 and x 2 Kn
n-1x Xn n+1-i
ffin+1"0x= ("0@nx)@0 + (-1) @i"0x
i=2
and
n-1X
_" @n-1x n-i
0ffinx= ("0@nx) 0 + (-1) "0@ix
i=0
so that
_" n
0ffinx - ffin+1"0x= (-1) "0@0x.
With these computations we get h : _ ' 1 where _ is the morphism given in the
statement. Hence _("n0v) = 0v for all n > 1, and in fact _"0x = 0txfor all x 2 *
*K.
From this we easily deduce that _( 0K) is the trivial subcomplex on K0. The
morphism _ then defines a morphism ~_: 0NK ! K satisfying ~_p0 = 1.
The homotopy _"0gives also p0_~' 1. Thus ~_is a homotopy equivalence (actual*
*ly
a deformation retract). 2
22
Remark 10.8 Let v be a vertex of the simplicial set K. Then in K we have
ffi2("20v)= "0v
and so "0v acts trivially on ( K)n for n > 3. Further for n > 3
(
ffin("n0v)= 0n-1 ifn is odd, 2
"0 v ifn is even.
Proposition 10.9 The crossed complex 0NK is isomorphic by _~to the (free)
subcrossed complex of K on the elements of K not of the form "0y for y 2
Kn-1, n > 1.
Proof This follows from theorem 6.6. 2
Remark 10.10 An advantage of working in the 0-normalised complex is that cer-
tain awkward exponents, which would vanish or not appear in the usual abelian
case, now disappear in the 0-normalised complex. For example if y 2 K1 we have
ffi2"1y= -@1"1y + @2"1y + @0"1yffi2"0y= -@1"0y + @2"0y + @0"y
= -y + y + "0@0y = -y + "0@1y + y
= 0ty mod "0. = 0sy mod "0. 2
Remark 10.11 There is another way of proceeding, by first reducing in K all
degeneracies of the vertices. Let K0 denote also the simplicial set on the vert*
*ices of
K, and also the discrete crossed complex on the object set K0. Then the inclusi*
*on
K0 ! K0 is a strong deformation retract, as is easily seen from Remark 10.8,
with retraction r0 : K ! K0, say. So we may form the pushout
r0
K0 ______//K0
| |
| |
fflffl| |fflffl
K __r__// 0K
Then r is also a strong deformation retract, by the methods of the homotopy the*
*ory
of crossed complexes, [12]. We can then apply the previous methods to 0K to
factor out the 0-degeneracies. We leave details and comparisons to the reader.
11 Normalisation
Now we are able to define, in analogy with Mac Lane [36, xVIII.6], some further
homotopies on 0N(K) to obtain the normalisation theorem. We can model more
closely the classical case on this 0-normalised crossed complex. Note that if x*
* 2 Kn
we write also x for the corresponding elements of both K and 0NK.
Definition 11.1For any k > 0 we define a subcrossed complex DkK 0NK as
follows:
o (DkK)0 = ( 0NK)0 = K0.
23
o(DkK)1 is trivial, i.e. consists only of identities.
o(DkK)n is normally generated by "iy for y 2 Kn-1, i 6 k and i 6 n - 1.
S S
Also, we define the degeneracy subcomplex DK = k DkK, i.e. (DK)n = k(DkK)n
for all n 2 N. 2
Now we define a sequence of homotopies from the identity to morphisms of
crossed complexes sending DkK into Dk-1K and leaving fixed the elements up to
dimension k -1. Then, the composition of these morphisms is well defined and ki*
*lls
all the degeneracy subcomplex. Let us formalise this sketch.
Definition 11.2For any k > 0 we define a homotopy (ok, 1) : 0NK ! 0NK
given on the free basis x 2 Kn by
(
okx= 0tx ifn < k, 2
(-1)n+k"kx ifn > k.
Therefore, for any k > 0 the homotopy ok defines a morphism of crossed complex,
OEk : 0NK ! 0NK such that ok : OEk ' 1. Clearly OE0 = _. For n > 1 this map is
given when x 2 Kn by
(
OEkx= x _ ifn < k,
x + (-1)k+n-1"kffinx + (-1)k+nffin+1"kxifk 6 n.
where _"iis the extension of "ion the basis to a derivation or operator morphis*
*m as
appropriate.
Proposition 11.3 OEk : 0NK ! 0NK satisfies
(i)OEkDjK DjK when j < k, and
(ii)OEkDkK Dk-1K.
Proof
(i) By the definition of OEk we have to prove the inclusion only in the case k *
*6 n.
In this case the generators of (DjK)n are elements "ix for i 6 min{j, n - 1}, s*
*o the
definition of OEk is
OEk"ix = "ix + (-1)k+n-1"kffin"ix + (-1)k+nffin+1"k"ix.
Therefore, since "ix 2 DjK, which is a subcrossed complex, we have that ffin"ix*
* 2
DjK. So ffin"ix can be written as a combination of "py with y 2 Kn-2, p 6
min{j, n - 2}. Therefore, since we have
"k"p = "p"k-1 ifk > p
we have that "kffin"ix 2 DjK.
On the other hand, for the same reason we have ffin+1"k"i 2 DjK. Therefore,
OEk"ix 2 DjK.
(ii) Now let us prove OEkDkK Dk-1K. Since (DkK)1 is trivial we have to prove
this inclusion only for generators of dimension n > 2.
24
We first deal with the case n = 2. Suppose then x 2 K2. Then
ffi3"1x= (@3"1x)@0x- "0@0x - x + x
= (@3"1x)@0x mod "0.
_" ffi2x @0x
1ffi2x= (-"1@1x) + ("1@2x) + "1@0x
so that mod "0 and by centrality
OE1x= x + _"1ffi2x - ffi3"1x
= x - ("1@1x)ffi2x+ "1@0x.
Now it is clear that, mod "0, x = "1y implies OE1x = 0.
Let "iy 2 (DkK)n, where i 6 min{k, n - 1}. If i < k then "iy 2 Dk-1K and so
OEk"iy 2 Dk-1K by (i). It only remains to prove OEk"ky 2 Dk-1K for y 2 Kn-1.
We have already done the case of n 6 2. In general
OEk"ky= "ky + (-1)k+n-1"kffin"ky + (-1)k+nffin+1"k"ky,
for y 2 Kn-1 with n > 2, and, in this case, (Dk-1K)n is abelian. We can write,
n-1" y n-1X n-j
"kffin"ky= "k(@n"ky)@0 k + (-1) "k@j"ky
j=0
and
n" " y Xn n+1-j
ffin+1"k"ky= (@n+1"k"ky)@0 k k + (-1) @j"k"ky.
j=0
Therefore OEk(DkK) Dk-1K follows from
8
><"k-1"k-1@jy ifj < k
"k@j"ky= >"ky ifj = k, k + 1
:"k"k@j-1y ifj > k + 1
and on the other hand,
8
><"k-1"k-1@jy ifj < k
@j"k"ky= >"ky ifj = k, k + 1, k + 2
:"k"k@j-1y ifj > k + 2.
2
Now we define OE = OE0OE1. .O.Ek . .:. 0NK ! 0NK.
Notice that since OEkx = x for k > dimx, this composite is finite in each di*
*men-
sion.
Proposition 11.4 OEDK = 0.
25
Proof We have (DK)0 = 0 and for n > 0, (DK)n is generated by "iy where
y 2 Kn-1 and i 6 n - 1. Therefore,
OE"iy = OE0OE1. .O.En"iy
If i = n - 1 we have that "iy 2 (DnK)n. So,
OEn"iy 2 Dn-1K, OEn-1OEn"iy 2 Dn-2K, . .,. OE0. .O.En"iy 2 D0K.
If i < n - 1 we have that "iy 2 (DiK)n. Therefore, since OEjDiK DiK for i < j
we have OEi+1. .O.En"iy 2 DiK. So, as above , OE0. .O.En"iy 2 D0K. 2
Definition 11.5We define the normalised fundamental crossed complex of the sim-
plicial set K by
0NK
K = ______DK.
Theorem 11.6 The quotient morphism
p : 0NK ! K
is a homotopy equivalence with a section q. Further, K has free generators giv*
*en
by the images of the non degenerate elements of K.
This follows as for the 0-normalised case in the previous section. Putting the *
*two
results together gives:
Theorem 11.7 The quotient morphism
p : K ! K
is a homotopy equivalence with a section q. Further, K has free generators giv*
*en
by the images of the non degenerate elements of K.
The crossed complex K homotopy equivalent to K can be described as
freely generated by the non degenerate simplices of K, with boundary maps given
by the HAL, forgetting the degenerate parts. In this sense, we have two alterna*
*tive
descriptions of K, one as just given and another in terms of coends as describ*
*ed
in section 1. The latter is used in classifying space results in [18].
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