ALGEBRAIC COLIMIT CALCULATIONS IN HOMOTOPY THEORY
USING FIBRED AND COFIBRED CATEGORIES
RONALD BROWN AND RAFAEL SIVERA
Bangor University Maths Preprint 06.08
ABSTRACT. Higher Homotopy van Kampen Theorems allow the computation as *
*col-
imits of certain homotopical invariants of glued spaces. One corollary *
*is to describe
homotopical excision in critical dimensions in terms of induced modules *
*and crossed
modules over groupoids. This paper shows how fibred and cofibred categor*
*ies give an
overall context for discussing and computing such constructions, allowin*
*g one result to
cover many cases. A useful general result is that the inclusion of a fi*
*bre of a fibred
category preserves connected colimits. The main homotopical application *
*are to pairs
of spaces with several base points, but we also describe briefly the sit*
*uation for triads. 1
Contents
1 Introduction *
* 1
2 Fibrations of categories *
* 4
3 Cofibrations of categories *
* 8
4 Pushouts and cocartesian morphisms *
*11
5 Groupoids bifibred over sets *
* 13
6 Groupoid modules bifibred over groupoids *
* 14
7 Crossed modules bifibred over groupoids *
* 16
8 Crossed complexes and an HHvKT 19
9 Homotopical excision and induced constructions *
* 23
10 Crossed squares and triad homotopy groups *
*28
1. Introduction
One of our aims is to give the framework of fibred and cofibred categories to c*
*ertain colimit
calculations of algebraic homotopical invariants for spaces with parts glued to*
*gether: the
data here is information on the invariants of the parts, and of the gluing proc*
*ess.
_____________
This work was partially supported by a Leverhulme Emeritus Fellowship 2002-4*
* for the first author.
2000 Mathematics Subject Classification: 55Q99, 18D30, 18A40.
Key words and phrases: higher homotopy van Kampen theorems, homotopical exci*
*sion, colimits,
fibred and cofibred categories, groupoids, modules, crossed modules .
Oc Ronald Brown and Rafael Sivera, . Permission to copy for private use gran*
*ted.
129/09/08
1
2
The second aim is to advertise the possibility of such calculations, which a*
*re based on
various Higher Homotopy van Kampen Theorems (HHvKTs)2, proved in 1978-87. These
are of the form that a homotopically defined functor
: (topological data) ! (algebraic data)
preserves certain colimits [BH78 , BH81b , BL87a ], and where the algebraic dat*
*a contains
non-Abelian information.
The antecedent for dimension 1 of such functors as is the fundamental grou*
*p ss1(X, a)
of based spaces: the van Kampen theorem in its form due to Crowell, [Cro59 ], g*
*ives that
this ss1 preserves certain pushouts.
The next advance was the fundamental groupoid functor ss1(X, A) from spaces *
*X with
a set A of base points to groupoids: groupoids in effect carry information in d*
*imensions
0 and 1. For example, the groupoid van Kampen theorem [Bro67 , Bro06 ] gives t*
*he
fundamental group of the circle as the infinite cyclic group C obtained in the *
*category of
groupoids from the (finite) groupoid I ~=ss1([0, 1], {0, 1}) by identifying 0, *
*1. That is, we
have the analogous pushouts
{0, 1}___//_{0} {0, 1}____//_{0}
| | ss1 | |
| | 7! | | (*
*1)
fflffl| |fflffl fflffl| fflffl|
[0, 1]____//_S1 I _______//_C
the first in the category of spaces, the second in the category of groupoids, a*
*nd ss1 takes
the first with the appropriate sets of base points, to the second. The aim is *
*to have a
similar argument at the module level for ssn, n > 1.
One of the problems of obtaining such results in higher homotopy theory is t*
*hat low
dimensional identifications of spaces usually affect high dimensional homotopy *
*invariants.
To cope with this fact, the algebraic data for the values of the functor must*
* have
structure interacting from low to high dimensions, in order to model how the sp*
*aces are
glued together.
An example of such dimensional interaction is the operation of the fundament*
*al group
on the higher homotopy groups, which fascinated the early workers in homotopy t*
*heory
(private communication, J.H.C. Whitehead, 1957). This operation can be seen as *
*necess-
itated by the dependence of these groups on the base point, but is somewhat neg*
*lected
in algebraic topology, perhaps for perceived deficiencies in modes of calculati*
*ons.
A recognised reason for the difficulty of traditional invariants such as hom*
*otopy groups
in dealing with gluing spaces, and so obtaining colimit calculations, is the fa*
*ilure of
excision. If X is the union of open sets U, V with intersection W then the in*
*clusion of
pairs
" : (V, W ) ! (X, U)
_____________
2This name for certain generalisations of van Kampen's Theorem was recently *
*suggested by Jim
Stasheff.
3
does not induce an isomorphism of relative homotopy groups as it does for singu*
*lar ho-
mology. However there is a residue of homotopical excision which can be deduced*
* from a
HHvKT, as in [BH81b ]; it requires connectivity conditions and gives informatio*
*n in only
the critical dimension, but this information involves the additional structure *
*of operations.
We state it at this stage for a single base point, as follows:
Homotopical Excision Theorem (HET) If U, V and W = U \ V are path connected
with base point a 2 W , and (V, W ) is (n - 1)-connected, n > 3, then (X, U) is*
* (n - 1)-
connected and the excision map
"* : ssn(V, W, a) ! ssn(X, U, a)
presents the ss1(U, a)-module ssn(X, U, a) as induced by the morphism of fundam*
*ental
groups ss1(W, a) ! ss1(U, a) from the ss1(W, a)-module ssn(V, W, a). The same *
*holds for
n = 2 with `module' replaced by `crossed module'.
We show that this theorem implies the classical Relative Hurewicz Theorem, (*
*Corollary
9.11) for which in more usual terms see for example [Whi78 ]. The case n = 2 im*
*plies a deep
theorem of Whitehead on free crossed modules (Corollary 9.8) (of which [Bro80 ]*
* explains
the original proof), and indeed allows the more general calculation of ss2(X [ *
*CA, X, a) as
a crossed ss1(X, a)-module (Corollary 9.13). That result is applied in [BW95 ,*
* BW03 ] to
calculate crossed modules representing some homotopy 2-types of mapping cones o*
*f maps
of classifying spaces of groups.
The notion of inducing used here is both an indication of the existence of s*
*ome uni-
versal properties in higher homotopy theory, and also of the tools needed to ex*
*ploit such
properties as there are. In the case n > 2 the inducing construction is well k*
*nown for
modules over groups: if M is a module over the group G, and f : G ! H is a morp*
*hism
of groups, then the induced module f*(M) is isomorphic to M ZG ZH. For n = 2, *
*the
analogous construction is non-Abelian.
We give the approach of fibred and cofibred categories to this inducing cons*
*truction.
Also the following simple example of algebraic modelling of a gluing process*
* suggests
a further generalisation is needed. Consider the following maps involving the *
*n-sphere
Sn, the first an inclusion and the second an identification:
p n 1
Sn -i! Sn _ [0, 1] -! S _ S . (*
*2)
Here n > 2, the base point 0 of Sn is in the second space identified with 0, an*
*d the map
p identifies 0, 1 to give the circle S1.
This clearly requires an algebraic theory dealing with many base points, and*
* so a
generalisation from groups to groupoids. The geometry of the base points and t*
*heir
interconnections can then play a role in the theory and applications.
This programme has proved successful for the algebraic data of crossed compl*
*exes over
groupoids, or equivalent structures, and for catn-groups, or, equivalently, cro*
*ssed n-cubes
of groups. In both categories there is a HHvKT asserting that a homotopically d*
*efined
functor preserves certain colimits, and this leads to new calculations of hom*
*otopical
invariants.
4
The calculation of colimits in these algebraic categories usually requires w*
*orking up
in dimensions. So it is useful to consider for these hierarchical structures t*
*he forgetful
functors from higher complex structures to lower structures.
The prototype again is groupoids, with the object functor from groupoids to *
*sets.
The fibre of this functor over a set I is written Gpd I: the objects of this c*
*ategory are
groupoids with object set I and morphisms are functors which are the identity o*
*n I.
This fibre has nice properties, and is what is called protomodular; in rough te*
*rms it has
properties analogous to those of the category of groups. In particular, there i*
*s a notion
of normal subgroupoid, and of coproduct, so that colimits can be calculated in *
*Gpd Ias
quotients of a coproduct by a normal subobject.
However the whole category Gpd is of interest for homotopical modelling. The*
* functor
Ob : Gpd ! Sethas the properties of being a fibration and a cofibration in the*
* sense of
Grothendieck, [Gro68 ]. So a function u : I ! J of sets gives rise to a pair o*
*f functors
u* : GpdJ ! GpdI, known often as pullback, and u* : GpdI ! GpdJ, which could be*
* called
`pushin' or `push forward', such that u* is left adjoint to u*. So for G 2 Gpd*
* Iwe have
a morphism u0: G ! u*G over u with a universal property which is traditionally *
*called
`cocartesian', and it is also said that u*G is the object induced from X by u.
It is interesting to see that the main parts of the pattern for groupoids go*
*es over to
the general case. Our categorical results are for a fibration of categories :*
* X ! B. The
first and a main result (Theorem 2.7) is that the inclusion XI ! X of a fibre X*
*I of
preserves colimits of connected diagrams. Our second set of results relates pus*
*houts in X
to the construction of the functors u* : XI ! XJ for u a morphism in B (Proposi*
*tion 4.2).
Finally, we show how the combination of these results uses the computation of c*
*olimits
in B and in each XI to give the computation of colimits in X (Theorem 4.4).
These general results are developed in the spirit of `categories for the wor*
*king math-
ematician' in sections 2 to 4. We illustrate the use of these results for homo*
*topical
calculations not only in groupoids (section 5), but also for crossed modules (s*
*ection 7)
and for modules, in both cases over groupoids. Finally we give a brief account *
*of some
relevance to crossed squares (section 10), as an indication of a more extensive*
* theory, and
in which these ideas need development.
We are grateful to Thomas Streicher for his Lecture Notes [Str99], on which *
*the foll-
owing account of fibred categories is based, and for helpful comments leading t*
*o im-
provements in the proofs. For further accounts of fibred and cofibred categori*
*es, see
[Gra66 , Bor94, Vis04] and the references there. The first paper gives analogie*
*s between
fibrations of categories and Hurewicz fibrations of spaces.
2. Fibrations of categories
We recall the definition of fibration of categories.
2.1. Definition. Let : X ! B be a functor. A morphism ' : Y ! X in X over
u := (') is called cartesian if and only if for all v : K ! J in B and ` : Z !*
* X with
(`) = uv there is a unique morphism _ : Z ! Y with (_) = v and ` = '_.
5
This is illustrated by the following diagram:
__`__________________________________________*
*_____________
_______________________________________________*
*_____________________________________________________________
______________%%__________________________________*
*_________________________________________________________
Z ______//___________Y//_X|
_ ' |
__uv______________||__________________________*
*_______________________________________________________
_____________________|___________________________*
*_____________________________________________________________________________*
*_________________________
______//______$$______fflffl|___________________//_
K v J u I
2
It is straightforward to check that cartesian morphisms are closed under com*
*position,
and that ' is an isomorphism if and only if ' is a cartesian morphism over an i*
*somorphism.
A morphism ff : Z ! Y is called vertical (with respect to ) if and only if *
* (ff) is an
identity morphism in B. In particular, for I 2 B we write XI, called the fibre *
*over I, for
the subcategory of X consisting of those morphisms ff with (ff) = idI.
2.2. Definition. The functor : X ! B is a fibration or category fibred over *
*B if and
only if for all u : J ! I in B and X 2 XI there is a cartesian morphism ' : Y !*
* X over
u: such a ' is called a cartesian lifting of X along u. *
* 2
Notice that cartesian liftings of X 2 XI along u : J ! I are unique up to ve*
*rtical
isomorphism: if ' : Y ! X and _ : Z ! X are cartesian over u, then there exist
vertical arrows ff : Z ! Y and fi : Y ! Z with 'ff = _ and _fi = ' respective*
*ly,
from which it follows by cartesianness of ' and _ that fffi = idY and fiff = i*
*dZ as
_fiff = 'ff = _ = _ idYand similarly 'fiff = ' idY.
2.3. Example. The forgetful functor, Ob : Gpd ! Set, from the category of grou*
*poids
to the category of sets is a fibration. We can for a groupoid G over I and fun*
*ction
u : J ! I define the cartesian lifting ' : H ! G as follows: for j, j02 J set
H(j, j0) = {(j, g, j0) | g 2 G(uj, uj0)}
with composition
(j1, g1, j01)(j, g, j0) = (j1, g1g, j0),
with ' given by '(j, g, j0) = g. The universal property is easily verified. The*
* groupoid
H is usually called the pullback of G by u. This is a well known construction *
*(see for
example [Mac05 , x2.3], where pullback by u is written u##). *
* 2
2.4. Definition. If : X ! B is a fibration, then using the axiom of choice f*
*or classes
we may select for every u : J ! I in B and X 2 XI a cartesian lifting of X alon*
*g u
uX : u*X ! X.
Such a choice of cartesian liftings is called a cleavage or splitting of .
6
If we fix the morphism u : J ! I in B, the splitting gives a so-called reind*
*exing functor
u* : XI ! XJ
defined on objects by X 7! u*X and the image of a morphism ff : X ! Y is u*ff *
*the
unique vertical arrow commuting the diagram:
_____uX______//
u*XO X
O ||
u*ffO ff||
fflfflO fflffl|
u*Y _____________//Y
uY
2
We can use this reindexing functor to get an adjoint situation for each u : *
*J ! I in B.
2.5. Proposition. Suppose : X ! B is a fibration of categories, u : J ! I in*
* B, and
a reindexing functor u* : XI ! XJ is chosen. Then there is a bijection
XJ(Y, u*X) ~=Xu(Y, X)
natural in Y 2 XJ, X 2 XI where Xu(Y, X) consists of those morphisms ff 2 X(Y, *
*X) with
(ff) = u.
Proof This is just a restatement of the universal properties concerned.
In general for composable maps u : J ! I and v : K ! J in B it does not hold*
* that
v*u* = (uv)*
as may be seen with the fibration of Example 2.3. Nevertheless there is a natur*
*al equiv-
alence cu,v: v*u* ' (uv)* as shown in the following diagram in which the full a*
*rrows are
cartesian and where (cu,v)X is the unique vertical arrow making the diagram com*
*mute:
*X
_____vu________//_*
v*u*XO u X
O ||
(cu,v)X~=O |uX
fflfflO |fflffl|
(uv)*X ________________//X
(uv)X
Let us consider this phenomenon for our main examples:
7
2.6. Example. 1.- Typically, for B = @1 : B2 ! B, the fundamental fibration *
*for a
category with pullbacks, we do not know how to choose pullbacks in a functorial*
* way.
2.- In considering groupoids as a fibration over sets, if u : J ! I is a map*
*, we have
a reindexing functor, also called pullback, u* : Gpd I! Gpd J. We notice that *
*v*u*Q is
naturally isomorphic to, but not identical to (uv)*Q. *
* 2
A result which aids understanding of our calculation of pushouts and some ot*
*her
colimits of groupoids, modules, crossed complexes and higher categories is the *
*following.
Recall that a category C is connected if for any c, c02 C there is a sequence *
*of objects
c0 = c, c1, . .,.cn-1, cn = c0such that for each i = 0, . .,.n-1 there is a mor*
*phism ci! ci+1
or ci+1! ci in C. The sequence of morphisms arising in this way is called a zig*
*-zag from
c to c0of length n.
2.7. Theorem. Let : X ! B be a fibration, and let J 2 B. Then the inclusi*
*on
iJ : XJ ! X preserves colimits of connected diagrams.
Proof We will need the following diagrams:
T_(c)__ T (c)
__________________________________FF______________________*
*EE
__________________________________|`(c)FF_________________*
*_________________|`(c)EE
__________________________________FFF______________________*
*_____________EEE
___________________________________(c)FF___________________*
*_________________(c)EE
__________________________________FF________________________*
*__________EE
__________________________fflffl|""F________________________*
*__________________fflffl|""E
T (f)____________________________________________________________*
*Y'_//_Xfl(c)__________________________________Y'_//_X;;
__________________________________OO<_____//Y__'__//______________X??
>_ """
> """
`0 >>OO""""~
X0
By the given property of _ there is a unique morphism _0: Y ! X0 in XJ such *
*that
_0_ = `0. By the cartesian property of ~, there is a unique morphism '0in XJ su*
*ch that
~'0= '. Then
~_0_ = ~`0= ` = '_ = ~'0_.
11
By the cartesian property of ~, and since _0_, '0_ are over uv, we have _0_ = '*
*0_. By
the given property of _, and since '0, _0 are in XJ, we have '0= _0. So ' = ~_0*
*, and
this proves uniqueness.
But we have already checked that ~_0_ = `, so we are done.
The following Proposition allows us to prove that a fibration is also a cofi*
*bration by
constructing the adjoints u* of u* for every u.
3.8. Proposition. Let : X ! B be a fibration of categories. Let u : J ! I *
*have
reindexing functor u* : XI ! XJ. Then the following are equivalent:
(i)For all Y 2 XJ, there is a morphism uY : Y ! u*Y which is cocartesian over*
* u;
(ii)there is a functor u* : XJ ! XI which is left adjoint to u*.
Proof That (ii) implies (i) is clear, using Proposition 3.7, since the adjoin*
*tness gives
the bijection required for the cocartesian property.
To prove that (i) implies (ii) we have to check that the allocation Y 7! u*(*
*Y ) gives
a functor that is adjoint to u*. As before the adjointness comes from the coca*
*rtesian
property.
We leave to the reader the check the details of the functoriality of u*.
To end this section, we give a useful result on compositions.
3.9. Proposition. The composition of fibrations, (cofibrations), is also a f*
*ibration
(cofibration).
Proof We leave this as an exercise.
4. Pushouts and cocartesian morphisms
Here is a small result which we use in this section and section 9, as it applie*
*s to many
examples, such as the fibration Ob : Gpd ! Set.
4.1. Proposition. Let : X ! B be a functor that has a left adjoint D. Then f*
*or each
K 2 Ob B, D(K) is initial in XK . In fact if u : K ! J in B, then for any X 2 X*
*J there
is a unique morphism "K : DK ! X over u.
Proof This follows immediately from the adjoint relation Xu(DK, X) ~=B(K, X)*
* for
all X 2 Ob XJ.
Special cases of cocartesian morphisms are used in [Bro06 , BH78 , BH81b ], *
*and we
review these in section 9. A construction which arises naturally from the vario*
*us Higher
Homotopy van Kampen theorems is given a general setting as follows:
12
4.2. Theorem. Let : X ! B be a fibration of categories which has a left adj*
*oint D.
Suppose that X admits pushouts. Let v : K ! J be a morphism in B, and let Z 2 X*
*K .
Then a cocartesian lifting _ : Z ! Y of v is given precisely by the pushout in *
*X:
D(v)
D(K) _______//D(J)
| |
"K || |"J| (*
**)
| |
fflffl| |fflffl
Z _________//_Y
_
Proof Suppose first that diagram (*) is a pushout in X. Let u : J ! I in B an*
*d let
` : Z ! X satisfy (`) = uv, so that (X) = I. Let f : D(J) ! X be the adjoint *
*of
u : J ! (X).
D(v)
D(K) ____//_D(J)
33
"K || "J||333
fflffl|_ fflffl|f333
Z SSSSS___//Y___33_
SSSSS _'_33_ (**
**)
SSSSS__33____
` SSSS3ssss))S""_____
X
K ___v___//_J_u__//_I
Then (fD(v)) = uv = (`"K ) and so by Proposition 4.1, fD(v) = `"K . The pu*
*shout
property implies there is a unique ' : Y ! X such that '_ = ` and '"J = f. This*
* last
condition gives (') = u since u = (f) = ('"J) = (') idJ= (').
For the converse, we suppose given f : D(J) ! X and ` : Z ! X such that `"K =
fD(v). Then (`) = uv and so there is a cocartesian lifting ' : Y ! X of u. *
*The
additional condition '"J = f is immediate by Proposition 4.1.
4.3. Corollary. Let : X ! B be a fibration which has a left adjoint and sup*
*pose
that X admits pushouts. Then is also a cofibration.
In view of the construction of hierarchical homotopical invariants as colimi*
*ts from the
HHvKTs in [BH81b ] and [BL87a ], the following is worth recording, as a consequ*
*ence of
Theorem 2.7.
4.4. Theorem. Let : X ! B be fibred and cofibred. Assume B and all fibres X*
*I are
cocomplete. Let T : C ! X be a functor from a small connected category. Then co*
*limT
exists and may be calculated as follows:
(i)First calculate I = colim( T ), with cocone fl : T ) I;
(ii)for each c 2 C choose cocartesian morphisms fl0(c) : T (c) ! X(c), over f*
*l(c) where
X(c) 2 XI;
13
(iii)make c 7! X(c) into a functor X : C ! XI, so that fl0 becomes a natural t*
*ransfor-
mation fl0: T ) X;
(iv)form Y = colimX 2 XI with cocone ~ : X ) Y .
Then Y with ~fl0: T ) Y is colimT .
Proof We first explain how to make X into a functor.
We will in stages build up the following diagram:
j
____________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__________________________________________@
__________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________
______________________________________________________*
*_____________________________________________________________________________*
*___________________________________~(c)
T (c)K_______0_________//X(c)O_____//Y__o___))//__________*
*_____Z
KKK fl (c) O ||
KKK OX(f) |1
T (f)KK%% fflffl fflffl| (*
*3)
T (c0)__0__0//_X(c0)_0__//Y___0__//____________*
*___Z
| fl (c ) ~(c ) o
||
fflffl| _________// _________//_________//_______//
K J 0 I I H
T (f) fl(c ) 1 w
Let f : c ! c0 be a morphism in C, K = T (c), J = T (c0). By cocartesianness*
* of
fl0(c), there is a unique vertical morphism X(f) : X(c) ! X(c0) such that X(f)f*
*l0(c) =
fl0(c0)T (f). It is easy to check, again using cocartesianness, that if furthe*
*r g : c0 ! c00,
then X(gf) = X(g)X(f), and X(1) = 1. So X is a functor and the above diagram sh*
*ows
that fl0 becomes a natural transformation T ) X.
Let j : T ) Z be a natural transformation to a constant functor Z, and let *
*(Z) = H.
Since I = colim( T ), there is a unique morphism w : I ! H such that wfl = (j).
By the cocartesian property of fl0, there is a natural transformation j0: X *
*) Z such
that j0fl0= j.
Since Y is also a colimit in X of X, we obtain a morphism o : Y ! Z in X suc*
*h that
o~ = j0. Then o~fl0= j0fl0= j.
Let o0 : Y ! Z be another morphism such that o0~fl0 = j. Then (o) = (o0) =*
* w,
since I is a colimit. Again by cocartesianness, o0~ = o~. By the colimit proper*
*ty of Y ,
o = o0.
This with Theorem 4.4 shows how to compute colimits of connected diagrams in*
* the
examples we discuss in sections 5 to 10, and in all of which a van Kampen type *
*theorem
is available giving colimits of algebraic data for some glued topological data.
4.5. Corollary. Let : X ! B be a functor satisfying the assumptions of theo*
*rem
4.4. Then X is connected complete, i.e. admits colimits of all connected diagra*
*ms.
5. Groupoids bifibred over sets
We have already seen in Example 2.3 that the functor Ob : Gpd ! Set is a fibr*
*ation.
It also has a left adjoint D assigning to a set I the discrete groupoid on I, a*
*nd a right
14
adjoint assigning to a set I the codiscrete groupoid on I.
It follows from general theorems on algebraic theories that the category Gpd*
* is co-
complete, and in particular admits pushouts, and so it follows from previous re*
*sults that
Ob : Gpd ! Setis also a cofibration. A construction of the cocartesian lifting*
*s of u : I ! J
for G a groupoid over I is given in terms of words, generalising the constructi*
*on of free
groups and free products of groups, in [Hig71, Bro06]. In these references the *
*cocartesian
lifting of u to G is called a universal morphism, and is written u* : G ! Uu(G)*
*. This
construction is of interest as it yields a normal form for the elements of Uu(G*
*), and hence
u* is injective on the set of non-identity elements of G.
A homotopical application of this cocartesian lifting is the following theor*
*em on the
fundamental groupoid. It shows how identification of points of a discrete subse*
*t of a space
can lead to `identifications of the objects' of the fundamental groupoid:
5.1. Theorem. Let (X, A) be a pair of spaces such that A is discrete and the i*
*nclusion
A ! X is a closed cofibration. Let f : A ! B be a function to a discrete space *
*B. Then
the induced morphism
ss1(X, A) ! ss1(B [f X, B)
is the cocartesian lifting of f.
This theorem immediately gives the fundamental group of the circle S1 as the*
* infinite
cyclic group C, since S1 is obtained from the unit interval [0, 1] by identifyi*
*ng 0 and 1, as
shown in the Introduction in diagram (1). The theorem is a translation of [Bro0*
*6 , 9.2.1],
where the words `universal morphism' are used instead of `cocartesian lifting'.*
* Section 8.2
of [Bro06 ] shows how free groupoids on directed graphs are obtained by a gener*
*alisation
of this example.
The calculation of colimits in a fibre GpdI is similar to that in the catego*
*ry of groups,
since both categories are protomodular, [BB04 ]. Thus a colimit is calculated *
*as a quo-
tient of a coproduct, where quotients are themselves obtained by factoring by a*
* normal
subgroupoid. Quotients are discussed in [Hig71, Bro06].
Theorem 4.4 now shows how to compute general colimits of groupoids.
We refer again to [Hig71, Bro06 ] for further developments and applications *
*of the
algebra of groupoids; we generalise some aspects to modules, crossed modules an*
*d crossed
complexes in the next sections.
6. Groupoid modules bifibred over groupoids
Modules over groupoids are a useful generalisation of modules over groups, and *
*also form
part of the basic structure of crossed complexes. Homotopy groups ssn(X; X0), n*
* > 2, of
a space X with a set X0 of base points form a module over the fundamental group*
*oid
ss1(X, X0), as do the homotopy groups ssn(Y, X : X0), n > 3, of a pair (Y, X).
6.1. Definition. A module over a groupoid is a pair (M, G), where G is a grou*
*poid
with set of objects G0, M is a totally disconnected abelian groupoid with the s*
*ame set of
15
objects as G, and with a given action of G on M. Thus M comes with a target fun*
*ction
t : M ! G0, and each M(x) = t-1(x), x 2 G0, has the structure of Abelian group.*
* The
G-action is given by a family of maps
M(x) x G(x, y) ! M(y)
for all x, y 2 G0. These maps are denoted by (m, p) 7! mp and satisfy the usual*
* properties,
0 (pp0) 0 p p 0p
i.e. m1 = m, (mp)p = m and (m + m ) = m + m , whenever these are defined.*
* In
particular, any p 2 G(x, y) induces an isomorphism m 7! mp from M(x) to M(y).
A morphism of modules is a pair (`, f) : (M, G) ! (N, H), where f : G ! H and
` : M ! N are morphisms of groupoids and preserve the action. That is, ` is giv*
*en by a
family of group morphisms
`(x) : M(x) ! N(f(x))
for all x 2 G0 satisfying `(y)(mp) = (`(x)(m))f(p), for all p 2 G(x, y), m 2 M(*
*x).
This defines the category Mod having modules over groupoids as objects and *
*the
morphisms of modules as morphisms. If (M, G) is a module, then (M, G)0 is defin*
*ed to
be G0.
We have a forgetful functor M : Mod ! Gpd in which (M, G) 7! G.
6.2. Proposition. The forgetful functor M : Mod ! Gpd has a left adjoint an*
*d is
fibred and cofibred.
Proof The left adjoint of M assigns to a groupoid G the module written 0 ! G*
* which
has only the trivial group over each x 2 G0.
Next, we give the pullback construction to prove that M is fibred. This is*
* entirely
analogous to the group case, but taking account of the geometry of the groupoid.
So let v : G ! H be a morphism of groupoids, and let (N, H) be a module. We *
*define
(M, G) = v*(N, H) as follows. For x 2 G0 we set M(x) = {x} x N(vx) with additi*
*on
given by that in N(vx). The operation is given by (x, n)p = (y, nvp) for p 2 G(*
*x, y).
The construction of N = v*(M, G) for (M, G) a G-module is as follows.
For y 2 H0 we let N(y) be the abelian group generated by pairs (m, q) with m*
* 2 M, q 2
H, and t(q) = y, s(q) = v(t(m)), so that N(y) = 0 if no such pairs exist. The o*
*peration
0 0 0 0
of H on N is given by (m, q)q = (m, qq ), addition is (m, q) + (m , q) = (m + m*
* , q) and
the relations imposed are (mp, q) = (m, v(p)q) when these make sense. The cocar*
*tesian
morphism over v is given by _ : m 7! (m, 1vt(m)).
6.3. Remark. We will discuss the relation between a module over a groupoid and*
* the
restriction to the vertex groups in section 8 in the general context of crossed*
* complexes.
However it is useful to give the general situation of many base points to descr*
*ibe the
relative homotopy group ssn(X, A, a0) when X is obtained from A by adding n-cel*
*ls at
various base points. The natural invariant to consider is then ssn(X, A, A0) wh*
*ere A0 is
an appropriate set of base points.
We now describe free modules over groupoids in terms of the inducing constru*
*ction.
The interest of this is two fold. Firstly, induced modules arise in homotopy th*
*eory from a
16
HHvKT, and we get new proofs of results on free modules in homotopy theory. Sec*
*ondly,
this indicates the power of the HHvKT since it gives new results.
6.4. Definition. Let Q be a groupoid. A free basis for a module (N, Q) consis*
*ts of
a pair of functions tB : B ! Q0, i : B ! N such that tN i = tB and with the uni*
*versal
property that if (L, Q) is a module and f : B ! L is a function such that tLf =*
* tN then
there is a unique Q-module morphism ' : N ! L such that 'i = f.
6.5. Proposition. Let B be an indexing set, and Q a groupoid. The free Q-modu*
*le
(F M(t), Q) on t : B ! Q0 may be described as the Q-module induced by t : B ! Q
from the discrete free module ZB = (Z x B, B) on B, where B denotes also the di*
*screte
groupoid on B.
Proof This is a direct verification of the universal property.
6.6. Remark. Proposition 3.7 shows that the universal property for a free modu*
*le can
also be expressed in terms morphisms of modules (F M(t), Q) ! (L, R).
7. Crossed modules bifibred over groupoids
Out homotopical example here is the family of second relative homotopy groups o*
*f a pair
of spaces with many base points.
A crossed module over a groupoid, [BH81a ], consists first of a morphism of *
*groupoids
~ : M ! P of groupoids with the same set P0 of objects such that ~ is the ident*
*ity on
objects, and M is a family of groups M(x), x 2 P0; second, there is an action o*
*f P on the
family of groups M so that if m 2 M(x) and p 2 P (x, y) then mp 2 M(y). This ac*
*tion
must satisfy the usual axioms for an action with the additional properties:
CM1) ~(mp) = p-1~(m)p, and
CM2) m-1nm = n~m
for all p 2 P , m, n 2 M such that the equations make sense. These form the obj*
*ects of
the category XMod in which a morphism is a commutative square of morphisms of *
*groups
g
M ____//_N
~ || ||
fflffl|fflffl|
P _____//Q
f
which preserve the action in the sense that g(mp) = (gm)fp whenever this makes *
*sense.
The category XMod is equivalent to the well known category 2-Gpd of 2-group*
*oids,
[BH81c ]. However the advantages of XMod over 2-groupoids are:
o crossed modules are closer to the classical invariants of relative homotop*
*y groups;
o the notion of freeness is clearer in XMod and models a standard topologic*
*al situation,
that of attaching 1- and 2-cells;
17
o the category XMod has a monoidal closed structure which helps to define a*
* notion
of homotopy; these constructions are simpler to describe in detail than th*
*ose for
2-groupoids, and they extend to all dimensions.
We have a forgetful functor 1 : XMod ! Gpd which sends (M ! P ) 7! P . Our*
* first
main result is:
7.1. Proposition. The forgetful functor 1 : XMod ! Gpd is fibred and has a*
* left
adjoint.
Proof The left adjoint of 1 assigns to a groupoid P the crossed module 0 ! P*
* which
has only the trivial group over each x 2 P0.
Next, we give the pullback construction to prove that 1 is fibred. So let f*
* : P ! Q be
a morphism of groupoids, and let : N ! Q be a crossed module. We define M = *
**(N)
as follows.
For x 2 P0 we set M(x) to be the subgroup of P (x) x N(fx) of elements (p, n*
*) such
that fp = n. If p1 2 P (x, x0), n 2 N(fx) we set (p, n)p1 = (p-11pp1, nf(p1))*
*, and let
~ : (p, n) 7! p. We leave the reader to verify that this gives a crossed module*
*, and that
the morphism (p, n) 7! n is cartesian.
The following result in the case of crossed modules of groups appeared in [B*
*H78 ],
described in terms of the crossed module @ : u*(M) ! Q induced from the crossed
module ~ : M ! P by a morphism u : P ! Q.
7.2. Proposition. The forgetful functor 1 : XMod ! Gpd is cofibred.
Proof We prove this by a direct construction.
Let ~ : M ! P be a crossed module, and let f : P ! Q be a morphism of groupo*
*ids.
The construction of N = f*(M) and of @ : N ! Q requires just care to the geomet*
*ry of
the partial action in addition to the construction for the group case initiated*
* in [BH78 ]
and pursued in [BW03 ] and the papers referred to there.
Let y 2 Q0. If there is no q 2 Q from a point of f(P0) to y, then we set N(y*
*) to be
the trivial group.
Otherwise, we define F (y) to be the free group on the set of pairs (m, q) s*
*uch that
0 *
* 0
m 2 M(x) for some x 2 P0 and q 2 Q(fx, y). If q02 Q(y, y0) we set (m, q)q = (m,*
* qq ).
We define @0 : F (y) ! Q(y) to be (m, q) 7! q-1(fm)q. This gives a precrossed m*
*odule
over @ : F ! Q, with function i : M ! F given by m 7! (m, 1) where if m 2 M(x) *
*then
1 here is the identity in Q(fx).
We now wish to change the function i : M ! F to make it an operator morphism.
For this, factor F out by the relations
(m, q)(m0, q)= (mm0, q),
(mp, q)= (m, (fp)q),
whenever these are defined, to give a projection F ! F 0and i0 : M ! F 0. As*
* in
the group case, we have to check that @0 : F ! Q induces @00: F 0! H making thi*
*s a
18
precrossed module. To make this a crossed module involves factoring out Peiffer*
* elements,
whose theory is as for the group case in [BH82 ]. This gives a crossed module m*
*orphism
(', f) : (M, P ) ! (N, Q) which is cocartesian.
We recall the algebraic origin of free crossed modules, but in the groupoid *
*context.
Let P be a groupoid, with source and target functions written s, t : P ! P0*
*. A
subgroupoid N of P is said to be normal in P , written N . P , if N is wide in *
*P , i.e.
N0 = P0, and for all x, y 2 P0 and a 2 P (x, y), a-1N(x)a = N(y). If N is also *
*totally
intransitive, i.e. N(x, y) = ; when x 6= y, as we now assume, then the quotient*
* groupoid
P=N is easy to define. (It may also be defined in general but we will need only*
* this case.)
Now suppose given a family R(x), x 2 P0 of subsets of P (x). Then the norma*
*liser
NP(R) of R is well defined as the smallest normal subgroupoid of P containing a*
*ll the
sets R(x). Note that the elements of NP(R) are all consequences of R in P , i.e*
*. all well
defined products of the form
c = (r"11)a1. .(.r"nn)an, "i= 1, ai2 P, n > 0 (*
*4)
and where ba denotes a-1ba. The quotient P=NP(R) is also written P=R, and calle*
*d the
quotient of P by the relations r = 1, r 2 R.
As in group theory, we need also to allow for repeated relations. So we supp*
*ose given
a set R and a function ! : R ! P such that s! = t! = fi, say. This `base point *
*function',
saying where the relations are placed, is a useful part of the information.
We now wish to obtain `syzygies' by replacing the normal subgroupoid by a `f*
*ree
object' on the relations ! : R ! P . As in the group case, this is done using f*
*ree crossed
modules.
7.3. Remark. There is a subtle reason for this use of crossed modules. A no*
*rmal
subgroupoid N of P (as defined above) gives a quotient object P=N in the catego*
*ry GpdX
of groupoids with object set X = P0. Alternatively, N defines a congruence on P*
* , which
is a particular kind of equivalence relation. Now an equivalence relation is in*
* general a
particular kind of subobject of a product, but in this case, we must take the p*
*roduct
in the category Gpd X. As a generalisation of this, one should take a groupoid*
* object
in the category Gpd X. Since these totally disconnected normal subgroupoids det*
*ermine
equivalence relations on each P (x, y) which are congruences, it seems clear th*
*at a groupoid
object internal to GpdX is equivalent to a 2-groupoid with object set X.
7.4. Definition. A free basis for a crossed module @ : C ! P over a groupoid *
*P is a
set R, function fi : R ! P0 and function i : R ! C such that i(r) 2 C(fir), r 2*
* R, with
the universal property that if ~ : M ! P is a crossed module and ` : R ! M a fu*
*nction
over the identity on P0 such that ~` = @i, then there is a unique morphism of c*
*rossed
P -modules ' : C ! M such that 'i = `.
7.5. Example. Let R be a set and fi : R ! P0 a function. Let id : F1(R) ! F2(*
*R)
be the identity crossed module on two copies of F (R), the disjoint union of co*
*pies C(r) of
the infinite cyclic group C with generator cr 2 C(r). Thus F2(R) is a totally i*
*ntransitive
19
groupoid with object set R. Let i : R ! F1(R) be the function r 7! cr. Let fi :*
* R ! R
be the identity function. Then id: F1(R) ! F2(R) is the free crossed module on *
*i. The
verification of this is simple from the diagram
__`_______________________________________*
*_____________________________________________________________________________*
*_____
_____________________________________________*
*_____________________________________________________________________________*
*_________________________________________________________________________
________________&&______________________________*
*______________________________________________________________________
R ___i//_F1(R)````//M
| f |
id| |~
fflffl| fflffl|
F2(R) ____//_F2(R)
id
The morphism f simply maps the generator cr to `r.
7.6. Proposition. Let R be a set, and ~ : M ! P a crossed module over the grou*
*poid
P . Let fi : R ! P0 be a function. Then the functions i : R ! M such that s~ = *
*t~ = fi
are bijective with the crossed module morphisms (f, g)
f
F1(R) _____//M
| ||
id| |~
fflffl| fflffl|
F2(R) __g__//P
such that sg = fi.
Further, the free crossed module @ : C(!) ! P on a function ! : R ! P such t*
*hat
s! = t! = fi is determined as the crossed module induced from id : F1(R) ! F2(R*
*) by
the extension of ! to the groupoid morphism F2(R) ! P .
Proof The first part is clear since g = ~f and f and i are related by f(cr) =*
* i(r), r 2 R.
The second part follows from the first part and the universal property of in*
*duced
crossed modules as shown in the following diagram:
__`______________________________________*
*_____________________________________________________________________________*
*________________________________
_____________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_________________________________
F1(R)_____//C(!)'```))//M
| f | ||
id| |@ |~
fflffl| fflffl| fflffl|
F2(R)___g__//P__=___//_P
8. Crossed complexes and an HHvKT
Crossed complexes are analogous to chain complexes but also generalise groupoid*
*s to all
dimensions and with their base points and operations relate dimensions 0,1 and *
*n. The
structure and axioms for a crossed complex are those universally satisfied by t*
*he main
20
topological example, the fundamental crossed complex X* of a filtered space X**
*, where
( X*)1 is the fundamental groupoid ss1(X1, X0) and for n > 2 ( X*)n is the fami*
*ly of
relative homotopy groups ssn(Xn, Xn-1, x0) for all x0 2 X0, with associated ope*
*rations of
the fundamental groupoid and boundaries.
Crossed complexes fit into our scheme of algebraic structures over a range o*
*f dimensions
satisfying a HHvKT in that the fundamental crossed complex functor
: (filtered spaces) ! (crossed complexes)
preserves certain colimits. We state a precise version below.
A crossed complex C is in part a sequence of the form
ffin ffin-1 ffi3 ffi2
. ._.___//Cn___//_Cn-1___//_......__//C2___//_C1
where all the Cn, n > 1, are groupoids over C0. Here ffi2 : C2 ! C1 is a crosse*
*d module and
for n > 3 (Cn, C1) is a module. The further axioms are that ffin is an operator*
* morphism
for n > 2 and that ffi2c operates trivially on Cn for c 2 C2 and n > 3. We ass*
*ume the
basic facts on crossed complexes as surveyed in for example [Bro99 , Bro04]. Th*
*e category
of crossed complexes is written Crs. A full exposition of the theory of crossed*
* complexes
will be given in [BHS08 ].
To state the Higher Homotopy van Kampen Theorem for relative homotopy groups,
namely Theorem C of [BH81b , Section 5], we need the following definition:
8.1. Definition. A filtered space X* is said to be connected if the following *
*conditions
hold for each n > 0 :
- '(X*, 0) : If r > 0, the map ss0X0 ! ss0Xr, induced by inclusion, is surjecti*
*ve; i.e. X0
meets all path connected components of all stages of the filtration X*.
-'(X*, n)(for n > 1): If r > n and x 2 X0, then ssn(Xr, Xn, x) = 0. *
* 2
8.2. Theorem. [Higher Homotopy van Kampen Theorem] Let X* be a filtered space
and suppose:
(i)X is the union of the interiors of subspaces U, V ;
(ii)the filtrations U*, V* and W*, formed by intersection with X*, and where *
*W = U \V ,
are connected filtrations. Then
(Conn) the filtration X* is connected, and
(Pushout) the following diagram of morphisms of crossed complexes induced by in*
*clusions
W* _____// U*
| |
| |
fflffl| fflffl|
V* ____//_ X*
is a pushout of crossed complexes.
21
8.3. Remark. The connectivity conclusion is significant, but not as important*
* as the
algebraic conclusion. This theorem is proved without recourse to traditional me*
*thods of
algebraic topology such as homology and simplicial approximation. Indeed, the *
*impli-
cations in dimension 2 are in general nonabelian and so unreachable by the trad*
*itional
abelian methods. Instead the theorem is proved using the construction of aeX*,*
* the cu-
bical homotopy !-groupoid of the filtered space X*, defined in dimension n to b*
*e the
set of filter homotopy classes of maps In*! X*. The properties of this construc*
*tion en-
able the proof of the 1-dimensional theorem van Kampen theorem to be generalise*
*d to
higher dimensions, and the theorem on crossed complexes is deduced using a non *
*trivial
equivalence between the two constructions.
The paper [BH81b ] also proves a more general theorem, in which arbitrary un*
*ions
lead to a coequaliser rather than a pushout. The paper also assumes a J 0 cond*
*ition
on the filtered spaces; but this can be relaxed by the refined definition of ma*
*king filter
homotopies of maps In*! X* to be rel vertices, as has been advertised in [BH91 *
*]. 2
8.4. Remark. Colimits of crossed complexes may be computed from the colimits *
*of
the groupoids, crossed modules and modules from which they are constituted.
8.5. Remark. A warning has to be given that some of the algebra is not as str*
*aight-
forward as that in traditional homological and homotopical algebra. For example*
* in an
abelian category, a pushout of the form
A _____//0
| |
i| |
fflffl|fflffl|
B __p__//C
is equivalent to an exact sequence
p
A -i! B -! C ! 0.
However in the category Mod *of modules over groups a pushout of the form
(M, G)_____//(0, 1)
| |
i | |
fflffl| fflffl|
(N, H)__p__//(P, K)
is equivalent to a pair of an exact sequence of groups
p
G -i! H -! K ! 1,
and an exact sequence of induced modules over K
p
M ZG ZK -i! N ZH ZK -! P ! 0.
This shows that pushouts give much more information in our case, but also shows*
* that
handling the information may be more difficult, and that misinterpretations cou*
*ld lead
to false conjectures or proofs. *
* 2
22
8.6. Proposition. The truncation functor tr1 : Crs! Gpd, C 7! C1, is a bifibra*
*tion.
Proof The previous results give the constructions on modules and crossed modu*
*les.
The functoriality of these constructions give the construction of the boundary *
*maps, and
the axioms for all these follow.
We will also need for later applications (Proposition 9.14) the relation of *
*a connected
crossed complex to the full, reduced (single vertex) crossed complex it contain*
*s, analogous
to the well known relation of a connected groupoid to any of its vertex groups.
Recall that a codiscrete groupoid T is one on which T (x, y) is a singleton *
*for all objects
x, y 2 T0. This is called a tree groupoid in [Bro06 ]. Similarly, a codiscrete *
*crossed complex
T is one in which the groupoid T1 is codiscrete and which is trivial in higher *
*dimensions.
We follow similar conventions for crossed complexes as for groupoids in [Bro*
*06 ]. Thus
if D and E are crossed complexes, and S = D0 \ E0 then by the free product D * *
*E we
mean the crossed complex given by the pushout of crossed complexes
S ______//_E
| |
| j| (*
*5)
fflffl| fflffl|
D __i__//D * E
where the set S is identified with the discrete crossed complex which is S in d*
*imension 0
and trivial in higher dimensions. The following result is analogous to and inde*
*ed includes
standard facts for groupoids (cf. [Bro06 , 6.7.3, 8.1.5]).
8.7. Proposition. Let C be a connected crossed complex, let x0 2 C0 and let T*
* be
codiscrete wide subcrossed complex of C. Let C(x0) be the subcrossed complex of*
* C at the
base point x0. Then the natural morphism
' : C(x0) * T ! C
determined by the inclusions is an isomorphism, and T determines a strong defor*
*mation
retraction
r : C ! C(x0).
Further, if f : C ! D is a morphism of crossed complexes which is the identi*
*ty on
C0 ! D0 then we can find a retraction s : D ! D(x0) giving rise to a pushout sq*
*uare
_r__//_
C C(x0)
| |
f|| |f0 (*
*6)
fflffl| fflffl|
D _s__//_D(x0)
in which f0 is the restriction of f.
Proof Let i : C(x0) ! C, j : T ! C be the inclusions. We verify the universal p*
*roperty
of the free product. Let ff : C(x0) ! E, fi : T ! E be morphisms of crossed com*
*plexes
23
agreeing on x0. Suppose g : C ! E satisfies gi = ff, gj = fi. Then g is determi*
*ned on C0.
Let c 2 C1(x, y). Then
c= (ox)((ox)-1c(oy))(oy)-1 (*
**)
and so
gc = g(ox)g((ox)-1c(oy))g(oy)-1
= fi(ox)ff((ox)-1c(oy))fi(oy)-1.
If c 2 Cn(x), n > 2, then
-1
c= (cox)(ox) (**
**)
and so
-1
g(c)= ff(cox)fi(ox).
This proves uniqueness of g, and conversely one checks that this formula define*
*s a mor-
phism g as required.
In effect, equations (*) and (**) give for the elements of C normal forms in*
* terms of
elements of C(x0) and of T .
This isomorphism and the constant map T ! {x0} determine the strong deformat*
*ion
retraction r : C ! C(x0).
The retraction s is defined by the elements fo(x), x 2 C0, and then the diag*
*ram (6)
is a pushout since it is a retract of the pushout square
__1_//_
C C
f || f|| 2
fflffl|fflffl|
D __1_//_D
9. Homotopical excision and induced constructions
We now interpret the HHvKT (Theorem 8.2) when the filtrations have essentially *
*just
two stages.
9.1. Definition. By a based pair Xo = (X, X1; X0) of spaces we mean a pair (X,*
* X1)
of spaces together with a set X0 X1 of base points. For such a based pair and*
* n > 2
we have an associated filtered space X[n]owhich is X0 in dimension 0, X1 in dim*
*ensions
1 6 i < n and X for dimensions i > n. We write nXo for the crossed complex X[*
*n]o.
This crossed complex is trivial in all dimensions 6= 1, n, and in dimension n i*
*s the family
of relative homotopy groups ssn(X, X1, x0) for x0 2 X0, considered as a module *
*(crossed
module if n = 2) over the fundamental groupoid ss1(X1, X0). Colimits of such c*
*rossed
complexes are equivalent to colimits of the corresponding module or crossed mod*
*ule. 2
The following is clear.
24
9.2. Proposition. If Xo = (X, X1; X0) is a based pair, and n > 2 then the asso*
*ciated
filtered space X[n]ois connected if and only if the based pair Xo is (n - 1)-co*
*nnected, i.e.
if:
o X0 meets each path component of X1 and of X;
o each path in X joining points of X0 is deformable in X rel end points to a*
* path in
X1; and
o ssr(X, X1, x0) = 0 for all x0 2 X0 and 1 < r < n.
The last part of the condition is of course vacuous if n = 2. *
* 2
With this in mind, Theorem 8.2 may be restated as:
9.3. Theorem. Let (X, X1; X0) be a based pair of spaces and suppose:
(i)X is the union of the interiors of subspaces U, V ;
(ii)the based pairs (U, U1; U0), (V, V1; V0) and (W, W1; W0) formed by inters*
*ection with
(X, X1; X0), and where W = U \ V , are (n - 1)-connected. Then
(Conn) (X, X1; X0) is (n - 1)-connected, and
(Pushout) the following diagram of morphisms
n(W, W1; W0) _____// n(U, U1; U0)
| |
| |
fflffl| fflffl|
n(V, V1; V0)_____// n(X, X1; X0)
is a pushout of modules if n > 3 and of crossed modules if n = 2.
Proof The important point is the equivalence between colimits of crossed comp*
*lexes
which for a given n > 2 have Ci= 0 for i 6= 0, 1, n or which are trivial in dim*
*ensions > 2,
and colimits of the corresponding modules or crossed modules.
9.4. Remark. It is not easy to see for Theorem 9.3 a direct proof in terms of *
*modules
or crossed modules, since one needs the intermediate structure between 0 and n *
*to use the
connectivity conditions. The cubical homotopy !-groupoid with connections aeX* *
*is found
convenient for this inductive process in [BH81b ]. There are two reasons for th*
*is: cubical
methods are convenient for constructing homotopies, and also for algebraic inve*
*rses to
subdivision. *
* 2
We now concentrate on excision, since this gives rise to cocartesian morphis*
*ms and so
induced modules and crossed modules.
25
9.5. Theorem. [Homotopy Excision Theorem] Let the topological space X be the u*
*nion
of the interiors of sets U, V , and let W = U \ V . Let n > 2. Let W0 U0 U *
*be such
that the based pair (V, W ; W0) is (n - 1)-connected and U0 meets each path com*
*ponent of
U. Then (X, U; U0) is (n - 1)-connected, and the morphism of modules (crossed i*
*f n = 2)
ssn(V, W ; W0) ! ssn(X, U; U0)
induced by inclusions is cocartesian over the morphism of fundamental groupoids
ss1(W, W0) ! ss1(U, U0)
induced by inclusion.
Proof We deduce this excision theorem from the pushout theorem 9.3, applied t*
*o the
based pair (X, U; U0) and the following diagram of morphisms :
n(W, W ; W0) ____//_ n(U, U; U0) (*
*7)
| |
| |
fflffl| fflffl|
n(V, W ; W0)____// n(X, U; U0)
This is a pushout of modules if n > 3 and of crossed modules if n = 2, by Theor*
*em 9.3.
However
n(W, W ; W0) = (0, ss1(W, W0)), n(U, U; U0) = (0, ss1(U, U0)).
So the theorem follows from Theorem 4.2 and our discussion of the examples of m*
*odules
and crossed modules.
9.6. Corollary. [Homotopical excision for an adjunction] Let i : W ! V be a cl*
*osed
cofibration and f : W ! U a map. Let W0 be a subset of W meeting each path comp*
*onent
of W and V , and let U0 be a subset of U meeting reach path component of U and *
*such that
f(W0) U0. Suppose that the based pair (V, W ) is (n - 1)-connected. Let X = U*
* [f V .
Then the based pair (X, U) is (n - 1)-connected and the induced morphism of mod*
*ules
(crossed if n = 2)
ssn(V, W ; W0) ! ssn(X, U; U0)
is cocartesian over the induced morphism of fundamental groupoids
ss1(W, W0) ! ss1(U, U0).
Proof This follows from Theorem 9.5 using mapping cylinders in a similar mann*
*er to
the proof of a corresponding result for the fundamental groupoid [Bro06 , 9.1.2*
*]. That is,
we form the mapping cylinder Y = M(f)[W . The closed cofibration assumption ens*
*ures
that the projection from Y to X = U [f V is a homotopy equivalence.
26
9.7. Remark. These methods were used in [BH78 ]. 2
9.8. Corollary. [Attaching n-cells] Let the space Y be obtained from the space*
* X by
attaching n-cells, n > 2, at a set of base points A of X, so that Y = X [f~en~*
*, ~ 2 ,
where f~ : (Sn-1, 0) ! (X, A). Then ssn(Y, X; A) is isomorphic to the free ss1*
*(X, A)-
module (crossed if n = 2) on the characteristic maps of the n-cells.
9.9. Remark. The previous corollary for n = 2 was a theorem of J.H.C. Whitehe*
*ad.
An account of Whitehead's proof is given in [Bro80 ]. There are several other *
*proofs in
the literature but none give the more general homotopical excision result, theo*
*rem 9.5. 2
9.10. Example. We can now explain the example in the Introduction. That Sn is
(n - 1)-connected and ssn(Sn, 0) ~= Z follows by induction in the usual way fro*
*m the
homotopical excision theorem and the calculation ss1(S1, 0) ~= Z by the groupoi*
*d van
Kampen theorem. Applying the HET to writing Sn _ [0, 1] as a union of Sn and [0*
*, 1] we
get that ssn(Sn _ [0, 1], [0, 1]; {0, 1}) is the free ss1([0, 1], {0, 1})-modul*
*e on one generator.
Again applying the HET but now identifying 0, 1 we get that ss1(Sn _ S1, 0) is *
*the free
ss1(S1, 0)-module in one generator. *
* 2
9.11. Corollary. [Relative Hurewicz Theorem] Let A ! X be a closed cofibration
and suppose A is path connected and (X, A) is (n - 1)-connected. Then X [ CA is
(n - 1)-connected and ssn(X [ CA, x) is isomorphic to ssn(X, A, x) factored by *
*the action
of ss1(A, x).
We now point out that a generalisation of a famous result of Hopf, [Hop42 ],*
* is a
corollary of the relative Hurewicz theorem. The following for n = 2 is part of*
* Hopf's
result. The algebraic description of H2(G) which he gives for G a group is sho*
*wn in
[BH78 ] to follow from the HHvKT.
9.12. Proposition. [Hopf's theorem] Let (V, A) be a pair of pointed spaces suc*
*h that:
(i)ssi(A) = 0 for 1 < i < n;
(ii)ssi(V ) = 0 for 1 < i 6 n;
(iii)the inclusion A ! V induces an isomorphism on fundamental groups.
Then the pair (V, A) is n-connected, and the inclusion A ! V induces an epimor*
*phism
HnA ! HnV whose kernel consists of spherical elements, i.e. of the image of ssn*
*A under
the Hurewicz morphism !n : ssn(A) ! Hn(A).
Proof That (V, A) is n-connected follows immediately from the homotopy exact *
*se-
quence of the pair (V, A) up to ssn(V ). We now consider the next part of the *
*exact
homotopy sequence and its relation to the homology exact sequence as shown in t*
*he
27
commutative diagram:
ssn+1(V, A)@__//_ssn(A)___//ssn(V_)__//ssn(V, A)
!n+1|| !n || || ||
fflffl| fflffl| fflffl| fflffl|
Hn+1(V, A) ____//_Hn(A)___//_Hn(V_)__//_Hn(V, A)
@0 i*
The Relative Hurewicz Theorem implies that Hn(V, A) = 0, and that !n+1 is surje*
*ctive.
Also @ in the top row is surjective, since ssn(V ) = 0. It follows easily that *
*the sequence
ssn(A) ! Hn(A) ! Hn(V ) ! 0 is exact.
There is a nice generalisation of Corollary 9.8, which so far has been prove*
*d only as a
deduction from a HHvKT.
9.13. Corollary. [Attaching cones] Let A be a space and let S be a set consis*
*ting
of one point in each path component of A. By CA, the cone on A, we mean the un*
*ion
of cones on each path component of A. Let f : A ! X be a map, and let S0 be the
image of S by f. Then ss2(X [ CA, X; S0) is isomorphic to the ss1(X, S0)-crosse*
*d module
induced from the identity crossed module ss1(A, S) ! ss1(A, S) by the induced m*
*orphism
f* : ss1(A, S) ! ss1(X, S0).
The paper [BW03 ] uses this result to give explicit calculations for the cr*
*ossed modules
representing the homotopy 2-types of certain mapping cones.
We now explain the relevance to free crossed modules of Proposition 8.7, lea*
*ving the
module and other cases to the reader.
9.14. Proposition. Let X be a path connected space with base point a, and let*
* Y =
X [f~{e2~} be obtained by attaching cells by means of pointed maps f~ : (S1, 0)*
* ! (X, a~),
determining elements x~ 2 ss1(X, a~), ~ 2 . Let A = {a} [ {a~ | ~ 2 }. Let T *
*be a tree
groupoid in ss1(X, A) determining a retraction r : ss1(X, A) ! ss1(X, a). Then *
*ss2(Y, X, a)
is isomorphic to the free crossed ss1(X, a)-module on the elements r(x~).
Proof We consider the following diagram:
F r
(0 ! ~Z) _______//_(0 ! P_)____//_(0 ! P (a))
| | |
| | | (*
*8)
F fflffl|F fflffl| fflffl|
( ~Z ! ~Z) _____//(C( ) ! P )s__//_(F ! P (a))
The left hand square is the pushout defining the free crossed module C( ) ! P a*
*s an
induced crossed module. The right hand square is the special case of crossed mo*
*dules of
the retraction of Proposition 8.7, and so is also a pushout. Hence the composit*
*e square is
a pushout. Hence the crossed module F ! P (a) is the free crossed module as des*
*cribed.
28
9.15. Remark. An examination of Whitehead's paper [Whi41 ], and the exposition*
* of
part of it in [Bro80 ], shows that the geometrical side of the last proposition*
* is intrinsic to
his approach. Of course a good proportion of Whitehead's work was devoted to ex*
*tending
ideas of combinatorial group theory to higher dimensions in combinatorial homot*
*opy
theory. The argument here is that this extension requires combinatorial groupoi*
*d theory
for good modelling of the geometry. *
* 2
10. Crossed squares and triad homotopy groups
In this section we give a brief sketch of the theory of triad homotopy groups, *
*including
the exact sequence relating them to homotopical excision, and show that the thi*
*rd triad
group forms part of a crossed square which, as an algebraic structure with link*
*s over
several dimensions, in this case dimensions 1,2,3, fits our criteria for a HHvK*
*T. Finally
we indicate a bifibration from crossed squares, so leading to the notion of ind*
*uced crossed
square, which is relevant to a triadic Hurewicz theorem in dimension 3.
A triad of spaces (X : A, B; x) consists of a pointed space (X, x) and two p*
*ointed
subspaces (A, x), (B, x). Then ssn(X : A, B; x) is defined for n > 2 as the set*
* of homotopy
classes of maps
(In : @-1In, @-2In; Jn-11,2) ! (X : A, B; x)
where Jn-11,2denotes the union of the faces of In other than @-1In, @-2In. For *
*n > 3 this set
obtains a group structure, using the direction 3, say, which is Abelian for n >*
* 4. Further
there is an exact sequence
! ssn+1(X : A, B; x) ! ssn(A, C, x) -"! ssn(X, B, x) ! ssn(X : A, B; x)*
*(!9)
where C = A \ B, and " is the excision map. It was the fact that these groups m*
*easure
the failure of excision that was their main interest. However they do not shed*
* light on
the above Homotopical Excision Theorem 9.5.
The third triad homotopy group fits into a diagram of possibly non-Abelian g*
*roups
ss3(X; A, B, x)___//_ss2(B, C, x)
(X; A, B, x) := || || (1*
*0)
fflffl| fflffl|
ss2(A, C, x)_____//ss1(C, x)
in which ss1(C, x) operates on the other groups and there is also a function
ss2(A, C, x) x ss2(B, C, x) ! ss3(X : A, B; x)
known as the generalised Whitehead product.
This diagram has structure and properties which are known as those of a cros*
*sed
square, [GWL81 , BL87a ], explained below, and so this gives a homotopical fun*
*ctor
: (based triads) ! (crossed squares). (1*
*1)
29
A crossed square is a commutative diagram of morphisms of groups
__~_//_
L M
~0|| |~| (1*
*2)
fflffl|fflffl|
N _____//P
together with left actions of P on L, M, N and a function h : M x N ! L satisfy*
*ing a
number of axioms which we do not give in full here. Suffice it to say that the *
*morphisms
in the square preserve the action of P , which acts on itself by conjugation; M*
*, N act on
each other and on L via P ; ~, ~0, ~, and ~~ are crossed modules; and h satis*
*fies axioms
reminiscent of commutator rules, summarised by saying it is a biderivation. Mor*
*phisms of
crossed squares are defined in the obvious way, giving a category XSq of crosse*
*d squares.
Let XMod 2 be the category of pairs of crossed modules ~ : M ! P, : N ! P *
*(with
P and ~, variable), and with the obvious notion of morphism. There is a forg*
*etful
functor : XSq ! XMod 2. This functor has a right adjoint D which completes th*
*e pair
~ : M ! P, : N ! P with L = M xP N and ~, ~0 given by the projections and
h : M x N ! L given by h(m, n) = (nmm-1, n mn-1), m 2 M, n 2 N. More interestin*
*gly,
it has a left adjoint which to the above pair of crossed P -modules yields the *
*`universal
crossed square'
M N _____//N
| |
| | (1*
*3)
fflffl| fflffl|
M ___~___//P
where M N, as defined in [BL87a ], is the nonabelian tensor product of groups*
* which
act on each other.
Then is a fibration of categories and also a cofibration. Thus we have a n*
*otion of
induced crossed square, which according to Proposition 4.2 is given by a pushou*
*t of the
form 0 1
ff fifi
` ' @ A` '
M N N ___ff____fl//_R S S
M P R Q
0 1 | |0 1
@u 1A || |@v| 1A
1 1 || ||1 1
` fflffl|' ` fflffl|'
L N _______________//T S
M P 0 ffifi1 R Q
@ A
ff fl
in the category of crossed squares, given morphisms (ff, fl) : (M ! P ) ! (R ! *
*Q), (fi, fl) :
(N ! P ) ! (S ! Q) of crossed modules.
30
The functor is exploited in [BL87a ] for an HHvKT implying some calculatio*
*ns of
the non-Abelian group ss3(X :A, B; x)3. The applications are developed in [BL8*
*7b ] for
a triadic Hurewicz Theorem, and for the notion of free crossed square, both bas*
*ed on
`induced crossed squares'. Free crossed squares are exploited in [Ell93] for ho*
*motopy type
calculations.
In fact the HHvKT works in all dimensions and in the more general setting of*
* n-cubes
of spaces, although not in a many base point situation. For a recent applicati*
*on, see
[EM08 ].
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School of Computer Science
Bangor University,
Gwynedd LL57 1UT,
U.K.
Departamento de Geometr'ia y Topolog'ia
Universitat de Val`encia,
46100 Burjassot, Valencia
Spain
Email: r.brown@bangor.ac.uk,
Rafael.Sivera@uv.es