Crossed complexes and higher homotopy groupoids
as non commutative tools for higher dimensional
local-to-global problems
Ronald Browny
October 13, 2008
MATHEMATICS SUBJECT CLASSIFICATION: 01-01,16E05,18D05,18D35,55P15,55Q05
Abstract
We outline the main features of the definitions and applications of cros*
*sed complexes and cu-
bical !-groupoids with connections. These give forms of higher homotopy gr*
*oupoids, and new
views of basic algebraic topology and the cohomology of groups, with the a*
*bility to obtain some
non commutative results and compute some homotopy types in non simply conn*
*ected situations.
Contents
Introduction *
* 1
1 Crossed modules *
* 4
2 The fundamental groupoid on a set of base points *
* 6
3 The search for higher homotopy groupoids *
* 9
4 Main results *
* 14
5 Why crossed complexes? *
* 16
_____________________________________
This is a revised version (2008) of a paper published in Fields Institute Co*
*mmunications 43 (2004) 101-130, which was
an extended account of a lecture given at the meeting on `Categorical Structure*
*s for Descent, Galois Theory, Hopf algebras
and semiabelian categories', Fields Institute, September 23-28, 2002. The autho*
*r is grateful for support from the Fields
Institute and a Leverhulme Emeritus Research Fellowship, 2002-2004, and to M. H*
*azewinkel for helpful comments on a
draft. This paper is to appear in Michiel Hazewinkel (ed.), Handbook of Algebra*
*, volume 6, Elsevier, 2008/2009.
ySchool of Computer Science, Bangor University, Dean St., Bangor, Gw*
*ynedd LL57 1UT, U.K. email:
r.brown@bangor.ac.uk
1
6 Why cubical !-groupoids with connections? *
* 17
7 The equivalence of categories *
* 17
8 First main aim of the work: Higher Homotopy van Kampen Theorems *
* 19
9 The fundamental cubical !-groupoid aeX of a filtered space X *
* 20
10 Collapsing *
* 22
11 Partial boxes *
* 23
12 Thin elements *
* 23
13 Sketch proof of the HHvKT *
* 24
14 Tensor products and homotopies *
* 25
15 Free crossed complexes and free crossed resolutions *
* 27
16 Classifying spaces and the homotopy classification of maps *
* 28
17 Relation with chain complexes with a groupoid of operators *
* 30
18 Crossed complexes and simplicial groups and groupoids *
* 31
19 Other homotopy multiple groupoids *
* 32
20 Conclusion and questions *
* 33
Introduction
An aim is to give a survey and explain the origins of results obtained by R. Br*
*own and P.J. Higgins
and others over the years 1974-2008, and to point to applications and related a*
*reas. These results
yield an account of some basic algebraic topology on the border between homolog*
*y and homotopy; it
differs from the standard account through the use of crossed complexes, rather *
*than chain complexes,
as a fundamental notion. In this way one obtains comparatively quickly1 not onl*
*y classical results
such as the Brouwer degree and the relative Hurewicz theorem, but also non comm*
*utative results on
second relative homotopy groups, as well as higher dimensional results involvin*
*g the fundamental
group, through its actions and presentations. A basic tool is the fundamental c*
*rossed complex X
of the filtered space X , which in the case X0 is a singleton is fairly classic*
*al; applied to the skeletal
_____________________________________
1This comparison is based on the fact that the methods do not require singula*
*r homology or simplicial approximation.
2
filtration of a CW-complex X, gives a more powerful version of the usual cell*
*ular chains of the
universal cover of X, because it contains non-Abelian information in dimensions*
* 1 and 2, and has
good realisation properties. It also gives a replacement for singular chains b*
*y taking X to be the
geometric realisation of a singular complex of a space.
One of the major results is a homotopy classification theorem (4.1.9) which*
* generalises a classical
theorem of Eilenberg-Mac Lane, though this does require results on geometric re*
*alisations of cubical
sets.
A replacement for the excision theorem in homology is obtained by using cub*
*ical methods to prove
a Higher Homotopy van Kampen Theorem (HHvKT)2 for the fundamental crossed compl*
*ex functor
on filtered spaces. This theorem is a higher dimensional version of the van Kam*
*pen Theorem (vKT)
on the fundamental group of a space with base point, [vKa33]3, which is a class*
*ical example of a
non commutative local-to-global theorem,
and was the initial motivation for the work described here. The vKT determines *
*completely the fun-
damental group ss1#X, x# of a space X with base point which is the union of ope*
*n sets U, V whose
intersection is path connected and contains the base point x; the `local inform*
*ation' is on the mor-
phisms of fundamental groups induced by the inclusions U " V ! U, U " V ! V. Th*
*e importance
of this result reflects the importance of the fundamental group in algebraic to*
*pology, algebraic geom-
etry, complex analysis, and many other subjects. Indeed the origin of the funda*
*mental group was in
Poincar'e's work on monodromy for complex variable theory.
Essential to this use of crossed complexes, particularly for conjecturing a*
*nd proving local-to-global
theorems, is a construction of a cubical higher homotopy groupoid, with propert*
*ies described by an
algebra of cubes. There are applications to local-to-global problems in homoto*
*py theory which are
more powerful than available by purely classical tools, while shedding light on*
* those tools. It is hoped
that this account will increase the interest in the possibility of wider applic*
*ations of these methods and
results, since homotopical methods play a key role in many areas.
Background in higher homotopy groups
Topologists in the early part of the 20th century were well aware that:
o the non commutativity of the fundamental group was useful in geometric app*
*lications;
o for path connected X there was an isomorphism
H1#X# ,#ss1#X, x#ab;
o the Abelian homology groups Hn#X# existed for all n > 0.
Consequently there was a desire to generalise the non commutative fundamental g*
*roup to all dimen-
sions.
In 1932 ~Cech submitted a paper on higher homotopy groups ssn#X, x# to the *
*ICM at Zurich, but
it was quickly proved that these groups were Abelian for n > 2, and on these gr*
*ounds ~Cech was
persuaded to withdraw his paper, so that only a small paragraph appeared in the*
* Proceedings [Cec32].
_____________________________________
2We originally called this a generalised van Kampen Theorem, but this new ter*
*m was suggested in 2007 by Jim Stasheff.
3An earlier version for simplicial complexes is due to Seifert.
3
We now see the reason for this commutativity as the result (Eckmann-Hilton) tha*
*t a group internal
to the category of groups is just an Abelian group. Thus, since 1932 the vision*
* of a non commutative
higher dimensional version of the fundamental group has been generally consider*
*ed to be a mirage.
Before we go back to the vKT, we explain in the next section how nevertheless w*
*ork on crossed
modules did introduce non commutative structures relevant to topology in dimens*
*ion 2.
Work of Hurewicz, [Hur35], led to a strong development of higher homotopy g*
*roups. The fun-
damental group still came into the picture with its action on the higher homoto*
*py groups, which I
once heard J.H.C. Whitehead remark (1957) was especially fascinating for the ea*
*rly workers in ho-
motopy theory. Much of Whitehead's work was intended to extend to higher dimens*
*ions the methods
of combinatorial group theory of the 1930s - hence the title of his papers: `Co*
*mbinatorial homotopy,
I, II' [W:CHI , W:CHII]. The first of these two papers has been very influentia*
*l and is part of the basic
structure of algebraic topology. It is the development of work of the second pa*
*per which we explain
here.
The paper by Whitehead on `Simple homotopy types' [W:SHT ], which deals wit*
*h higher dimen-
sional analogues of Tietze transformations, has a final section using crossed c*
*omplexes. We refer to
this again later in section 15.
It is hoped also that this survey will be useful background to work on the *
*van Kampen Theorem
for diagrams of spaces in [BLo87a], which uses a form of higher homotopy groupo*
*id which is in an
important sense much more powerful than that given here, since it encompasses n*
*-adic information;
however current expositions are still restricted to the reduced (one base point*
*) case, the proof uses
advanced tools of algebraic topology, and the result was suggested by the work *
*exposed here.
1 Crossed modules
In the years 1941-50, Whitehead developed work on crossed modules to represent *
*the structure of the
boundary map of the relative homotopy group
ss2#X, A, x# ! ss1#A, x# *
* (1)
in which both groups can be non commutative. Here is the definition he found.
A crossed module is a morphism of groups ~ # M ! P together with an action #*
*m, p# 7! mp of the
group P on the group M satisfying the two axioms
CM1) ~#mp# # p 1#~m#p
CM2) n 1mn # m~n
for all m, n 2 M, p 2 P.
Standard algebraic examples of crossed modules are:
(i)an inclusion of a normal subgroup, with action given by conjugation;
(ii)the inner automorphism map O # M ! AutM, in which Om is the automorphism n*
* 7! m 1nm;
(iii)the zero map M ! P where M is a P-module;
4
(iv)an epimorphism M ! P with kernel contained in the centre of M.
Simple consequences of the axioms for a crossed module ~ # M ! P are:
1.1 Im ~ is normal in P.
1.2 Ker~ is central in M and is acted on trivially by Im~, so that Ker~ inherit*
*s an action of M= Im~.
Another important algebraic construction is the free crossed P-module
@ # C#!# ! P
determined by a function ! # R ! P, where P is a group and R is a set. The grou*
*p C#!# is generated
by elements #r, p# 2 R P with the relations
#r, p# 1#s, q# 1#r, p##s, qp 1#!r#p#;
the action is given by #r, p#q # #r, pq#; and the boundary morphism is given by*
* @#r, p# # p 1#!r#p,
for all #r, p#, #s, q# 2 R P.
A major result of Whitehead was:
Theorem W [W:CHII] If the space X # A [ fe2rgr2R is obtained from A by attachin*
*g 2-cells by maps
fr # #S1, 1# ! #A, x#, then the crossed module of (1)is isomorphic to the free *
*crossed ss1#A, x#-module on
the classes of the attaching maps of the 2-cells.
Whitehead's proof, which stretched over three papers, 1941-1949, used transv*
*ersality and knot
theory - an exposition is given in [Bro80]. Mac Lane and Whitehead [MLW50 ] use*
*d this result as part
of their proof that crossed modules capture all homotopy 2-types (they used the*
* term `3-types').
The title of the paper in which the first intimation of Theorem W appeared w*
*as `On adding relations
to homotopy groups' [Whi41 ]. This indicates a search for higher dimensional vK*
*Ts.
The concept of free crossed module gives a non commutative context for chain*
*s of syzygies. The
latter idea, in the case of modules over polynomial rings, is one of the origin*
*s of homological algebra
through the notion of free resolution. Here is how similar ideas can be applied*
* to groups. Pioneering
work here, independent of Whitehead, was by Peiffer [Pei49] and Reidemeister [R*
*ei49]. See [BHu82 ]
for an exposition of these ideas.
Suppose P # hX j !i is a presentation of a group G, so that X is a set of ge*
*nerators of G and
! # R ! F#X# is a function, whose image is called the set of relators of the pr*
*esentation. Then we have
an exact sequence
1 #i!N#!R# #OE!F#X# #! G #! 1
where N#!R# is the normal closure in F#X# of the set !R of relators. The above *
*work of Reidemeister,
Peiffer, and Whitehead showed that to obtain the next level of syzygies one sho*
*uld consider the free
crossed F#X#-module @ # C#!# ! F#X#, since this takes into account the operatio*
*ns of F#X# on its
normal subgroup N#!R#. Elements of C#!# are a kind of `formal consequences of t*
*he relators', so that
the relation between the elements of C#!# and those of N#!R# is analogous to th*
*e relation between
the elements of F#X# and those of G. It follows from the rules for a crossed mo*
*dule that the kernel
of @ is a G-module, called the module of identities among relations, and someti*
*mes written ss#P#;
there is considerable work on computing it [BHu82 , Pri91, HAM93 , ElK99, BRS99*
*]. By splicing to @
5
a free G-module resolution of ss#P# one obtains what is called a free crossed r*
*esolution of the group
G. We explain later (Proposition 15.3) why these resolutions have better realis*
*ation properties than
the usual resolutions by chain complexes of G-modules. They are relevant to the*
* Schreier extension
theory, [BrP96].
This notion of using crossed modules as the first stage of syzygies in fact*
* represents a wider tradi-
tion in homological algebra, in the work of Fr"olich and Lue [Fro61, Lue81].
Crossed modules also occurred in other contexts, notably in representing el*
*ements of the cohomol-
ogy group H3#G, M# of a group G with coefficients in a G-module M [McL63 ], and*
* as coefficients in
Dedecker's theory of non Abelian cohomology [Ded63 ]. The notion of free crosse*
*d resolution has been
exploited by Huebschmann [Hue80 , Hue81b, Hue81a] to represent cohomology class*
*es in Hn#G, M#
of a group G with coefficients in a G-module M, and also to calculate with thes*
*e.
The HHvKT can make it easier to compute a crossed module arising from some *
*topological situa-
tion, such as an induced crossed module [BWe95 , BWe96 ], or a coproduct crosse*
*d module [Bro84],
than the cohomology class in H3#G, M# the crossed module represents. To obtain*
* information on
such a cohomology element it is useful to work with a small free crossed resolu*
*tion of G, and this is
one motivation for developing methods for calculating such resolutions. However*
*, it is not so clear
what a calculation of such a cohomology element would amount to, although it is*
* interesting to know
whether the element is non zero, or what is its order. Thus the use of algebrai*
*c models of cohomology
classes may yield easier computations than the use of cocycles, and this somewh*
*at inverts traditional
approaches.
Since crossed modules are algebraic objects generalising groups, it is natu*
*ral to consider the prob-
lem of explicit calculations by extending techniques of computational group the*
*ory. Substantial work
on this has been done by C.D. Wensley using the program GAP [GAP02 , BWe03].
2 The fundamental groupoid on a set of base points
A change in prospects for higher order non commutative invariants was suggested*
* by Higgins' paper
[Hig64], and leading to work of the writer published in 1967, [Bro67]. This sho*
*wed that the van
Kampen Theorem could be formulated for the fundamental groupoid ss1#X, X0# on a*
* set X0 of base
points, thus enabling computations in the non-connected case, including those i*
*n Van Kampen's orig-
inal paper [vKa33]. This successful use of groupoids in dimension 1 suggested t*
*he question of the
use of groupoids in higher homotopy theory, and in particular the question of t*
*he existence of higher
homotopy groupoids.
In order to see how this research programme could progress it is useful to c*
*onsider the statement
and special features of this generalised van Kampen Theorem for the fundamental*
* groupoid. If X0 is a
set, and X is a space, then ss1#X, X0# denotes the fundamental groupoid on the *
*set X"X0 of base points.
This allows the set X0 to be chosen in a way appropriate to the geometry. For e*
*xample, if the circle
S1 is written as the union of two semicircles E# [ E , then the intersection f*
* 1, 1g of the semicircles
is not connected, so it is not clear where to take the base point. Instead one*
* takes X0 # f 1, 1g,
and so has two base points. This flexibility is very important in computations,*
* and this example of
S1 was a motivating example for this development. As another example, you might*
* like to consider
the difference between the quotients of the actions of Z2 on the group ss1#S1, *
*1# and on the groupoid
ss1#S1, f 1, 1g# where the action is induced by complex conjugation on S1. Rele*
*vant work on orbit
6
groupoids has been developed by Higgins and Taylor [HiT81, Tay88], (under usefu*
*l conditions, the
fundamental groupoid of the orbit space is the orbit groupoid of the fundamenta*
*l groupoid [Bro06,
11.2.3]).
Consideration of a set of base points leads to the theorem:
Theorem 2.1 [Bro67] Let the space X be the union of open sets U, V with inters*
*ection W, and let X0 be
a subset of X meeting each path component of U, V, W. Then
(C) (connectivity) X0 meets each path component of X and
(I) (isomorphism) the diagram of groupoid morphisms induced by inclusions
ss1#W, X0#_i_//_ss1#U, X0#
j|| j0|| *
* (2)
fflffl| fflffl|
ss1#V, X0#i0//_ss1#X, X0#
is a pushout of groupoids.
From this theorem, one can compute a particular fundamental group ss1#X, x0*
*# using combinatorial
information on the graph of intersections of path components of U, V, W, but fo*
*r this it is useful to
develop the algebra of groupoids. Notice two special features of this result.
(i) The computation of the invariant you may want, a fundamental group, is obta*
*ined from the com-
putation of a larger structure, and so part of the work is to give methods for *
*computing the smaller
structure from the larger one. This usually involves non canonical choices, e.g*
*. that of a maximal
tree in a connected graph. The work on applying groupoids to groups gives many *
*examples of this
[Hig64, Hig71, Bro06, DiV96].
(ii) The fact that the computation can be done is surprising in two ways: (a) T*
*he fundamental group is
computed precisely, even though the information for it uses input in two dimens*
*ions, namely 0 and 1.
This is contrary to the experience in homological algebra and algebraic topolog*
*y, where the interaction
of several dimensions involves exact sequences or spectral sequences, which giv*
*e information only up
to extension, and (b) the result is a non commutative invariant, which is usual*
*ly even more difficult
to compute precisely.
The reason for the success seems to be that the fundamental groupoid ss1#X,*
* X0# contains informa-
tion in dimensions 0 and 1, and so can adequately reflect the geometry of the i*
*ntersections of the path
components of U, V, W and of the morphisms induced by the inclusions of W in U *
*and V.
This suggested the question of whether these methods could be extended succ*
*essfully to higher
dimensions.
Part of the initial evidence for this quest was the intuitions in the proof*
* of this groupoid vKT, which
seemed to use three main ideas in order to verify the universal property of a p*
*ushout for diagram (2).
So suppose given morphisms of groupoids fU , fV from ss1#U, X0#, ss1#V, X0# to *
*a groupoid G, satisfying
fU i # fV j. We have to construct a morphism f # ss1#X, X0# ! G such that fi0# *
*fU , fj0# fV and prove
f is unique. We concentrate on the construction.
ffl One needs a `deformation', or `filling', argument: given a path a # #I,*
*`I# ! #X, X0# one can write
a # a1 # # an where each ai maps into U or V, but ai will not necessarily hav*
*e end points in X0.
So one has to deform each ai to a0iin U, V or W, using the connectivity conditi*
*on, so that each a0i
7
has end points in X0, and a0 # a01# # a0nis well defined. Then one can constr*
*uct using fU or fV
an image of each a0iin G and hence of the composite, called F#a# 2 G, of these *
*images. Note that we
subdivide in X and then put together again in G (this uses the condition fU i #*
* fV j to prove that the
elements of G are composable), and this part can be summarised as:
ffl Groupoids provided a convenient algebraic inverse to subdivision. Note *
*that the usual exposition
in terms only of the fundamental group uses loops, i.e. paths which start and *
*finish at the same
point. An appropriate analogy is that if one goes on a train journey from Bango*
*r and back to Bangor,
one usually wants to stop off at intermediate stations; this breaking and cotin*
*uing a journey is better
described in terms of groupoids rather than groups.
Next one has to prove that F#a# depends only on the class of a in the funda*
*mental groupoid. This
involves a homotopy rel end points h # a ' b, considered as a map I2 ! X; subdi*
*vide h as h # #hij# so
that each hijmaps into U, V or W; deform h to h0# #h0ij# (keeping in U, V, W) s*
*o that each h0ijmaps
the vertices to X0 and so determines a commutative square4 in one of ss1#Q, X0#*
* for Q # U, V, W.
Move these commutative squares over to G using fU , fV and recompose them (this*
* is possible again
because of the condition fU i # fV j), noting that:
ffl in a groupoid, any composition of commutative squares is commutative. H*
*ere a `big' composition
of commutative squares is represented by a diagram such as
ffl__//_ffl//_ffl//_ffl//_ffl//_ffl//_ffl
| | | | | | |
| | | | | | |
fflffl|fflffl|fflffl|fflffl|fflffl|fflffl|fflffl|ffl//*
*_ffl//_ffl//_ffl//_ffl//_ffl//_ffl
| | | | | | |
| | | | | | | *
* (3)
fflffl|fflffl|fflffl|fflffl|fflffl|fflffl|fflffl|ffl//*
*_ffl//_ffl//_ffl//_ffl//_ffl//_ffl
OO OO
|| || | | | | |
|| || | | | | |
fflffl|fflffl|fflffl|fflffl|fflffl|fflffl|fflffl|ffl//*
*_|ffl//_|ffl//_ffl//_ffl//_ffl//_ffl
and one checks that if each individual square is commutative, so also is the bo*
*undary square (later
called a 2-shell) of the compositions of the boundary edges.
Two opposite sides of the composite commutative square in G so obtained are ide*
*ntities, because
h was a homotopy relative to end points, and the other two sides are F#a#, F#b#*
*. This proves that
F#a# # F#b# in G.
Thus the argument can be summarised: a path or homotopy is divided into sma*
*ll pieces, then
_____________________________________ *
* i j
4We need the notion of commutative square in a category C. This is a quadrupl*
*e acd of arrows in C, called `edges' of
*
* b
the square, such that ab # cd, i.e. such that these compositions are defined an*
*d agree. The commutative squares in C form
a double category C in that they compose `vertically'
i c j i b j i c j
abd ffi1 a0ed0 # aa0dd0e
and `horizontally' i
acdj i c0j i cc0j
b ffi2 d bf0# abb0f
This notion of C was defined by C. Ehresmann in papers and in [Ehr83]. Note th*
*e obvious geometric conditions for these
compositions to be defined. Similarly, one has geometric conditions for a recta*
*ngular array #cij#, 1 6 i 6 m, 1 6 j 6 n,
of commutative squares to have a well defined composition, and then their `mult*
*iple composition', written #cij#, is also a
commutative square, whose edges are compositions of the `edges' along the outsi*
*de boundary of the array. It is easy to give
formal definitions of all this.
8
deformed so that these pieces can be packaged and moved over to G, where they a*
*re reassembled.
There seems to be an analogy with the processing of an email.
Notable applications of the groupoid theorem were: (i) to give a proof of *
*a formula in van
Kampen's paper of the fundamental group of a space which is the union of two co*
*nnected spaces
with non connected intersection, see [Bro06, 8.4.9]; and (ii) to show the topol*
*ogical utility of the
construction by Higgins [Hig71] of the groupoid f #G# over Y0 induced from a gr*
*oupoid G over X0
by a function f # X0 ! Y0. (Accounts of these with the notation Uf#G# rather th*
*an f #G# are given in
[Hig71, Bro06].) This latter construction is regarded as a `change of base', an*
*d analogues in higher di-
mensions yielded generalisations of the Relative Hurewicz Theorem and of Theore*
*m W, using induced
modules and crossed modules.
There is another approach to the van Kampen Theorem which goes via the theo*
*ry of covering
spaces, and the equivalence between covering spaces of a reasonable space X and*
* functors ss1#X# !
Set [Bro06]. See for example [DoD79 ] for an exposition of the relation with tr*
*aditional Galois theory,
and [BoJ01] for a modern account in which Galois groupoids make an essential ap*
*pearance. The paper
[BrJ97] gives a general formulation of conditions for the theorem to hold in th*
*e case X0 # X in terms
of the map U t V ! X being an `effective global descent morphism' (the theorem *
*is given in the
generality of lextensive categories). This work has been developed for toposes,*
* [BuL03]. Analogous
interpretations for higher dimensional Van Kampen theorems are not known.
The justification of the breaking of a paradigm in changing from groups to *
*groupoids is several fold:
the elegance and power of the results; the increased linking with other uses of*
* groupoids [Bro87]; and
the opening out of new possibilities in higher dimensions, which allowed for ne*
*w results and calcu-
lations in homotopy theory, and suggested new algebraic constructions. The impo*
*rtant and extensive
work of Charles Ehresmann in using groupoids in geometric situations (bundles, *
*foliations, germes,
. .).should also be stated (see his collected works of which [EH84 ] is volume *
*1 and a survey [Bro07]).
3 The search for higher homotopy groupoids
Contemplation of the proof of the groupoid vKT in the last section suggested th*
*at a higher dimensional
version should exist, though this version amounted to an idea of a proof in sea*
*rch of a theorem.
Further evidence was the proof by J.F. Adams of the cellular approximation theo*
*rem given in [Bro06].
This type of subdivision argument failed to give algebraic information apparent*
*ly because of a lack
of an appropriate higher homotopy groupoid, i.e. a gadget to capture what might*
* be the underlying
`algebra of cubes'. In the end, the results exactly encapsulated this intuition.
One intuition was that in groupoids we are dealing with a partial algebraic *
*structure5, in which
composition is defined for two arrows if and only if the source of one arrow is*
* the target of the other.
This seems to generalise easily to directed squares, in which two such are comp*
*osable horizontally if
and only if the left hand side of one is the right hand side of the other (and *
*similarly vertically).
However the formulation of a theorem in higher dimensions required specifica*
*tion of the topologi-
_____________________________________
5The study of partial algebraic operations was initiated in [Hig63]. We can n*
*ow suggest a reasonable definition of `higher
dimensional algebra' as dealing with families of algebraic operations whose dom*
*ains of definitions are given by geometric
conditions.
9
cal data, the algebraic data, and of a functor
# (topological data)! (algebraic data)
which would allow the expression of these ideas for the proof.
Experiments were made in the years 1967-1973 to define some functor from *
*spaces to some
kind of double groupoid, using compositions of squares in two directions, but t*
*hese proved abortive.
However considerable progress was made in work with Chris Spencer in 1971-3 on *
*investigating
the algebra of double groupoids [BSp76a ], and showing a relation to crossed mo*
*dules. Further ev-
idence was provided when it was found, [BSp76b ], that group objects in the cat*
*egory of groupoids
(or groupoid objects in the category of groups, either of which are often now c*
*alled `2-groups') are
equivalent to crossed modules, and in particular are not necessarily commutativ*
*e objects. It turned
out this result was known to the Grothendieck school in the 1960s, but not publ*
*ished.
We review next a notion of double category which is not the most general bu*
*t is appropriate in
many cases. It was called an edge symmetric double category in [BMo99 ].
In the first place, a double category, K, consists of a triple of category *
*structures
#K2, K1, @1 , @#1, ffi1, "1#, #K2, K1, @2 , @#2, ffi2, *
*"2#
#K1, K0, @ , @# , ffi, "#
as partly shown in the diagram
@2
K2 ________//_//_K1 *
* (4)
#
|| @2 ||
|| ||
@#1|@1||| @# ||@||
|| ||
ffflffl|flffl|fflffl|fflffl|@//_
K1 ________//_K0
@#
The elements of K0, K1, K2 will be called respectively points or objects, edges*
*, squares. The maps
@ ,@i , i # 1, 2, will be called face maps, the maps "i # K1 #! K2, i # 1, 2, *
*resp. " # K0 #! K1 will
be called degeneracies. The boundaries of an edge and of a square are given by *
*the diagrams
___//_@1_
| | ___/2/
@ ___//__@# @2fflffl||fflffl|@#2|1fflffl| *
* (5)
|__//_|_
@#1
The partial compositions, ffi1, resp. ffi2, are referred to as vertical resp. *
* horizontal composition of
squares, are defined under the obvious geometric conditions, and have the obvio*
*us boundaries. The
axioms for a double category also include the usual relations of a 2-cubical se*
*t (for example @ @#2#
@# @1 ), and the interchange law. We use matrix notation for compositions as
~ ~
a a b
c # a ffi1 c, # a ffi2 b,
10
and the crucial interchange law6 for these two compositions allows one to use m*
*atrix notation
~ ~ " # ~~ ~ ~ ~~
a b a b a b
c d # c d # c d
for double composites of squares whenever each row composite and each column co*
*mposite is defined.
We also allow the multiple composition #aij# of an array #aij# whenever for all*
* appropriate i, j we have
@#1aij# @1 ai#1,j, @#2aij# @2 ai,j#1. A clear advantage of double categories an*
*d cubical methods is
this easy expression of multiple compositions which allows for algebraic invers*
*e to subdivision, and so
applications to local-to-global problems.
The identities with respect to ffi1 (vertical identities)_are given by "1 a*
*nd will be denoted by ||.
Similarly, we have horizontal identities denoted by __._Elements of the form "1*
*"#a# # "2"#a# for a 2 K0
are called double degeneracies and will be denoted by |_|.
A morphism of double categories f # K ! L consists of a triple of maps fi #*
* Ki ! Li, #i # 0, 1, 2#,
respecting the cubical structure, compositions and identities.
Whereas it is easy to describe a commutative square of morphisms in a categ*
*ory, it is not possible
with this amount of structure to describe a commutative cube of squares in a do*
*uble category. We first
of all define a cube, or 3-shell, i.e. without any condition of commutativity, *
*in a double category.
Definition 3.1Let K be a double category. A cube (3-shell) in K,
ff # #ff1 , ff#1, ff2 , ff#2, ff3 , ff#3#
consists of squares ffi 2 K2 #i # 1, 2, 3# such that
@oei#ffoj# # @oj 1#ffoei#
for oe, o # and 1 6 i < j 6 3.
It is also convenient to have the corresponding notion of square, or 2-shell*
*, of arrows in a category.
The obvious compositions also makes these into a double category.
It is not hard to define three compositions of cubes in a double category so*
* that these cubes form
a triple category: this is done in [BKP05 ], or more generally in Section 5 of *
*[BHi81a]. A key point is
that to define the notion of a commutative cube we need extra structure on a do*
*uble category. Thus
this step up a dimension is non trivial, as was first observed in the groupoid *
*case in [BHi78a]. The
problem is that a cube has six faces, which easily divide into three even and t*
*hree odd faces. So we
cannot say as we might like that `the cube is commutative if the composition of*
* the even faces equals
the composition of the odd faces', since there are no such valid compositions.
The intuitive reason for the need of a new basic structure in that in a 2-di*
*mensional situation we
also need to use the possibility of `turning an edge clockwise or anticlockwise*
*'. The structure to do
this is as follows.
A connection pair on a double category K is given by a pair of maps
, # # K1 #! K2
_____________________________________
6The interchange implies that a double monoid is simply an Abelian monoid, so*
* partial algebraic operations are essential
for the higher dimensional work.
11
whose edges are given by the following diagrams for a 2 K1:
__a_//_ ___//__
| fflO| | fflO| ___*
*//
| fflOfflO| | fflO| 2
#a# # a fflffl||fflO1||# ff__|lffl|fflOfflO|# __| 1ff*
*lffl|
|//o//fflO|ooo_ ||//o/fflO||oo/o_
1
__1____/o/o/o/o _///o/ooo//__
|fflOfflO| |fflOfflO| __ ___/2/
# #a# # 1 ||fflOafflffl||# |||_fflOafflffl||# | fflffl|
|fflOfflO| |fflOfflO| 1
|___//|_ |__//_|_
a a
This `hieroglyphic' notation, which was introduced in [Bro82], is useful for ex*
*pressing the laws these
connections satisfy. The first is a pair of cancellation laws which read
~__~
| __ |_ __|
__|# __, # ||,
which can be understood as `if you turn right and then left, you face the same *
*way', and similarly the
other way round. They were introduced in [Spe77]. Note that in this matrix nota*
*tion we assume that
the edges of the connections are such that the composition is defined.
Two other laws relate the connections to the compositions and read
~__ __~ ~ ~
| ___ __ __|||_
|| | # | , __ __|# __|.
These can be interpreted as `turning left (or right) with your arm outstretched*
* is the same as turning
left (or right)'. The term `connections' and the name `transport laws' was beca*
*use these laws were
suggested by the laws for path connections in differential geometry, as explain*
*ed in [BSp76a ]. It was
proved in [BMo99 ] that a connection pair on a double category K is equivalent *
*to a `thin structure',
namely a morphism of double categories # K1 ! K which is the identity on the*
* edges. The proof
requires some `2-dimensional rewriting' using the connections.
We can now explain what is a `commutative cube' in a double category K with*
* connection pair.
Definition 3.2Suppose given, in a double category with connections K, a cube (3*
*-shell)
ff # #ff1 , ff#1, ff2 , ff#2, ff3 , ff#3#.
We define the composition of the odd faces of ff to be
~ __ ~
| ff __|
@oddff# ff 1#__ *
* (6)
3 ff2 __
and the composition of the even faces of ff to be
~__ #~
__ ff ff
@evenff# |_ ff2# 3 *
* (7)
1 __|
12
We define ff to be commutative if it satisfies the Homotopy Commutativity Lemma*
* (HCL), i.e.
@ oddff # @evenff. (*
*HCL)
This definition can be regarded as a cubical, categorical (rather than groupoid*
*) form of the Homotopy
Addition Lemma (HAL) in dimension 3.
You should draw a 3-shell, label all the edges with letters, and see that t*
*his equation makes sense
in that the 2-shells of each side of equation (HCL) coincide. Notice however th*
*at these 2-shells do
not have coincident partitions along the edges: that is the edges of this 2-she*
*ll in direction 1 are
formed from different compositions of the type 1 ffi a and a ffi 1. This defini*
*tion is discussed in more
detail in [BKP05 ], is related to other equivalent definitions, and it is prove*
*d that compositions of
commutative cubes in the three possible directions are also commutative. These *
*results are extended
to all dimensions in [Hig05]; this requires the full structure indicated in sec*
*tion 9 and also the notion
of thin element indicated in section 12.
The initial discovery of connections arose in [BSp76a ] from relating cross*
*ed_modules to double
groupoids. The first example of a double groupoid was the double groupoid |_|G*
* of commutative
squaresiinja group G. The first step in generalising this construction was to *
*consider quadruples
c
a d of elements of G such that abn # cd for some element n of a subgroup N *
*of G. Experiments
b
quickly showed that for the two compositions of such quadruples to be valid it *
*was necessary and
sufficient that N be normal in G. But in this case the element n is determined *
*by the boundary, or 2-
shell, a, b, c, d. In homotopy theory we requireisomethingjmore general. So we *
*consider a morphism
c
~ # N ! G of groups and and consider quintuples n # a bd such that ab~#n# # cd*
*. It then turns
out that we get a double groupoid if and only if ~ # N ! G is a crossed module.*
* The next question
is which double groupoids arise in this way? It turns out that we need exactly *
*double groupoids with
connection pairs, though in this groupoid case we can deduce from # using*
* inverses in each
dimension. This gives the main result of [BSp76a ], the equivalence between the*
* category of crossed
modules and that of double groupoids with connections and one vertex.
These connections were also used in [BHi78a] to define a `commutative cube'*
* in a double groupoid
with connections using the equation
2 __ 1 __3
| a0 |
c1 # 4 b0 c0 b15
|_ a1 __|
*
* __
representing one face of_a cube in terms of the other five and where the other *
*connections |_, |
are obtained from __|, | by using the two inverses in dimension 2. As you might*
* imagine, there are
problems in finding a formula in still higher dimensions. In the groupoid case,*
* this is handled by a
homotopy addition lemma and thin elements, [BHi81a], but in the category case a*
* formula for just a
commutative 4-cube is complicated, see [Gau01 ].
The blockage of defining a functor to double groupoids was resolved after*
* 9 years in 1974 in
discussions with Higgins, by considering the Whitehead Theorem W. This showed t*
*hat a 2-dimensional
universal property was available in homotopy theory, which was encouraging; it *
*also suggested that
a theory to be any good should recover Theorem W. But this theorem was about re*
*lative homotopy
groups. This suggested studying a relative situation X # X0 ` X1 ` X. On looki*
*ng for the simplest way
13
to get a homotopy functor from this situation using squares, the `obvious' answ*
*er came up: consider
maps #I2, @I2, @@I2# ! #X, X1, X0#, i.e. maps of the square which take the edg*
*es into X1 and the
vertices into X0, and then take homotopy classes of such maps relative to the v*
*ertices of I2 to form a
set ae2X . Of course this set will not inherit a group structure but the surpri*
*se is that it does inherit
the structure of double groupoid with connections - the proof is not entirely t*
*rivial, and is given in
[BHi78a] and the expository article [Bro99]. In the case X0 is a singleton, the*
* equivalence of such
double groupoids to crossed modules takes aeX to the usual second relative hom*
*otopy crossed module.
Thus a search for a higher homotopy groupoid was realised in dimension 2. C*
*onnes suggests in
[Con94 ] that it has been fashionable for mathematicians to disparage groupoids*
*, and it might be that
a lack of attention to this notion was one reason why such a construction had n*
*ot been found earlier
than 40 years after Hurewicz's papers.
Finding a good homotopy double groupoid led rather quickly, in view of the *
*previous experience,
to a substantial account of a 2-dimensional HHvKT [BHi78a]. This recovers Theor*
*em W, and also
leads to new calculations in 2-dimensional homotopy theory, and in fact to some*
* new calculations of
2-types. For a recent summary of some results and some new ones, see the paper *
*in the J. Symbolic
Computation [BWe03 ] - publication in this journal illustrates that we are inte*
*rested in using general
methods in order to obtain specific calculations, and ones to which there seems*
* no other route.
Once the 2-dimensional case had been completed in 1975, it was easy to conj*
*ecture the form of
general results for dimensions > 2. These were proved by 1979 and announcements*
* were made in
[BHi78b] with full details in [BHi81a, BHi81b]. However, these results needed *
*a number of new
ideas, even just to construct the higher dimensional compositions, and the proo*
*f of the HHvKT was
quite hard and intricate. Further, for applications, such as to explain how the*
* general behaved on
homotopies, we also needed a theory of tensor products, found in [BHi87], so th*
*at the resulting theory
is quite complex. It is also remarkable that ideas of Whitehead in [W:CHII] pla*
*yed a key role in these
results.
4 Main results
Major features of the work over the years with Philip Higgins and others can be*
* summarised in the
following diagram of categories and functors:
Diagram 4.1
j j
_________filtered_spaces______________________*
*_____________________________________________________________________________*
*____________________________Kfilteredoo_
C # r ffi________________________________________________*
*_______________________________________wwwwKKKKcubical;sets;wwOO
_____________________________________wwaeKKww|
_________________________________wwKKKwww|
_____________________________wwwwwKKKKwww |U|
""___________________________--wwwwBwwK%%ww|
operator r w ~ cubical
chain ____//_crossedcomplexesoo_________//!-groupoidsoo_
complexes fl with connections
in which
14
4.1.1 the categories FTopof filtered spaces, !-Gpd of cubical !-groupoids with *
*connections, and Crs
of crossed complexes are monoidal closed, and have a notion of homotopy u*
*sing and a unit
interval object;
4.1.2 ae, are homotopical functors (that is they are defined in terms of homo*
*topy classes of certain
maps), and preserve homotopies;
4.1.3 ~, fl are inverse adjoint equivalences of monoidal closed categories;
4.1.4 there is a natural equivalence flae ' , so that either ae or can be us*
*ed as appropriate;
4.1.5 ae, preserve certain colimits and certain tensor products;
4.1.6 the category of chain complexes with (a groupoid) of operators is monoida*
*l closed, r preserves
the monoid structures, and is left adjoint to ;
4.1.7 by definition, the cubical filtered classifying space is B2 # j jffiU wh*
*ere U is the forgetful functor
to filtered cubical sets7 using the filtration of an !-groupoid by skelet*
*a, and j j is geometric
realisation of a cubical set;
4.1.8 there is a natural equivalence ffi B2 ' 1;
4.1.9 if C is a crossed complex and its cubical classifying space is defined as*
* B2C # #B2 C## , then
for a CW-complex X, and using homotopy as in 4.1.1 for crossed complexes,*
* there is a natural
bijection of sets of homotopy classes
#X, B2C# ,## X , C#.
Recent applications of the simplicial version of the classifying space ar*
*e in [Bro08b, PoT07,
FMP07 ].
Here a filtered space consists of a (compactly generated) space X# and an *
*increasing sequence of
subspaces
X # X0 ` X1 ` X2 ` ` X# .
With the obvious morphisms, this gives the category FTop. The tensor product i*
*n this category is the
usual [
#X Y #n # Xp Yq.
p#q#n
The closed structure is easy to construct from the law
FTop#X Y , Z # ,#FTop#X , FTOP#Y , Z ##.
An advantage of this monoidal closed structure is that it allows an enrichment*
* of the category FTop
over either crossed complexes or !-Gpd using or ae applied to FTOP #Y , Z #.
The structure of crossed complex is suggested by the canonical example, th*
*e fundamental crossed
complex X of the filtered space X . So it is given by a diagram
_____________________________________
7Cubical sets are defined, analogously to simplicial sets, as functors K # *
*op! Setwhere is the `box' category with
objects In and morphisms the compositions of inclusions of faces and of the va*
*rious projections In ! Irfor n > r. The
geometric realisation jKj of such a cubical set is obtained by quotienting the*
* disjoint union of the sets K#In# In by the
relations defined by the morphisms of . For more details, see [Jar06], and fo*
*r variations on the category to include for
example connections, see [GrM03]. See also section 9.
15
Diagram 4.2
_____//Cnffin//_Cn_1__//______//C2ffi2//_C1
|t| t|| |t| s|t|||
fflffl| fflffl| fflffl|ffflffl|flffl|
C0 C0 C0 C0
in which in this example C1 is the fundamental groupoid ss1#X1, X0# of X1 on th*
*e `set of base points'
C0 # X0, while for n > 2, Cn is the family of relative homotopy groups fCn#x#g *
*# fssn#Xn, Xn 1, x# j x 2
X0g. The boundary maps are those standard in homotopy theory. There is for n > *
*2 an action of the
groupoid C1 on Cn (and of C1 on the groups C1#x#, x 2 X0 by conjugation), the b*
*oundary morphisms
are operator morphisms, ffin 1ffin # 0, n > 3, and the additional axioms are sa*
*tisfied that
4.3 b 1cb # cffi2b, b, c 2 C2, so that ffi2 # C2 ! C1 is a crossed module (of g*
*roupoids);
4.4 if c 2 C2 then ffi2c acts trivially on Cn for n > 3;
4.5 each group Cn#x# is Abelian for n > 3, and so the family Cn is a C1-module.
Clearly we obtain a category Crsof crossed complexes; this category is not so f*
*amiliar and so we give
arguments for using it in the next section.
As algebraic examples of crossed complexes we have: C # C#G, n# where G is a*
* group, commuta-
tive if n > 2, and C is G in dimension n and trivial elsewhere; C # C#G, 1 # M,*
* n#, where G is a group,
M is a G-module, n > 2, and C is G in dimension 1, M in dimension n, trivial el*
*sewhere, and zero
boundary if n # 2; C is a crossed module (of groups) in dimensions 1 and 2 and *
*trivial elsewhere.
A crossed complex C has a fundamental groupoid ss1C # C1= Imffi2, and also f*
*or n > 2 a family
fHn#C, p#jp 2 C0g of homology groups.
5 Why crossed complexes?
ffl They generalise groupoids and crossed modules to all dimensions. Note th*
*at the natural context
for second relative homotopy groups is crossed modules of groupoids, rather tha*
*n groups.
ffl They are good for modelling CW-complexes.
ffl Free crossed resolutions enable calculations with small CW-complexes and*
* CW-maps, see section
15.
ffl Crossed complexes give a kind of `linear model' of homotopy types which *
*includes all 2-types.
Thus although they are not the most general model by any means (they do not con*
*tain quadratic
information such as Whitehead products), this simplicity makes them easier to h*
*andle and to relate
to classical tools. The new methods and results obtained for crossed complexes*
* can be used as a
model for more complicated situations. This is how a general n-adic Hurewicz Th*
*eorem was found
[BLo87b].
ffl They are convenient for calculation, and the functor is classical, inv*
*olving relative homotopy
groups. We explain some results in this form later.
16
ffl They are close to chain complexes with a group(oid) of operators, and r*
*elated to some classical
homological algebra (e.g. chains of syzygies). In fact if SX is the simplicia*
*l singular complex of a
space, with its skeletal filtration, then the crossed complex #SX# can be cons*
*idered as a slightly non
commutative version of the singular chains of a space.
ffl The monoidal structure is suggestive of further developments (e.g. cros*
*sed differential algebras)
see [BaT97 , BaBr93]. It is used in [BGi89] to give an algebraic model of homot*
*opy 3-types, and to
discuss automorphisms of crossed modules.
ffl Crossed complexes have a good homotopy theory, with a cylinder object, *
*and homotopy colim-
its, [BGo89 ]. The homotopy classification result 4.1.9 generalises a classical*
* theorem of Eilenberg-
Mac Lane. Applications of (the simplicial version) are given in for example [FM*
*07 , FMP07 , PoT07].
ffl They have an interesting relation with the Moore complex of simplicial *
*groups and of simplicial
groupoids (see section 18).
6 Why cubical !-groupoids with connections?
The definition of these objects is more difficult to give, but will be indicate*
*d in section 9. Here we
explain why these structures are a kind of engine giving the power behind the t*
*heory.
ffl The functor ae gives a form of higher homotopy groupoid, thus confirming*
* the visions of the early
topologists.
ffl They are equivalent to crossed complexes.
ffl They have a clear monoidal closed structure, and a notion of homotopy, f*
*rom which one can
deduce those on crossed complexes, using the equivalence of categories.
ffl It is easy to relate the functor ae to tensor products, but quite diffic*
*ult to do this directly for .
ffl Cubical methods, unlike globular or simplicial methods, allow for a simp*
*le algebraic inverse to
subdivision, which is crucial for our local-to-global theorems.
ffl The additional structure of `connections', and the equivalence with cros*
*sed complexes, allows for
the sophisticated notion of commutative cube, and the proof that multiple compo*
*sitions of commutative
cubes are commutative. The last fact is a key component of the proof of the HHv*
*KT.
ffl They yield a construction of a (cubical) classifying space B2C # #B2 C##*
* of a crossed complex
C, which generalises (cubical) versions of Eilenberg-Mac Lane spaces, including*
* the local coefficient
case. This has convenient relation to homotopies.
ffl There is a current resurgence of the use of cubes in for example combina*
*torics, algebraic topology,
and concurrency. There is a Dold-Kan type theorem for cubical Abelian groups w*
*ith connections
[BrH03 ].
7 The equivalence of categories
Let Crs, !-Gpd denote respectively the categories of crossed complexes and !-gr*
*oupoids: we use the
latter term as an abbreviation of `cubical !-groupoids with connections'. A maj*
*or part of the work
consists in defining these categories and proving their equivalence, which thus*
* gives an example of
17
two algebraically defined categories whose equivalence is non trivial. It is ev*
*en more subtle than that
because the functors fl # Crs ! ! Gpd, ~ # ! Gpd ! Crs are not hard to define*
*, and it is easy
to prove fl~ ' 1. The hard part is to prove ~fl ' 1, which shows that an !-gro*
*upoid G may be
reconstructed from the crossed complex fl#G# it contains. The proof involves us*
*ing the connections
to construct a `folding map' # Gn ! Gn , with image fl#G#n, and establishing *
*its major properties,
including the relations with the compositions. This gives an algebraic form of *
*some old intuitions of
several ways of defining relative homotopy groups, for example using cubes or c*
*ells.
On the way we establish properties of thin elements, as those which fold do*
*wn to 1, and show
that G satisfies a strong Kan extension condition, namely that every box has a *
*unique thin filler. This
result plays a key role in the proof of the HHvKT for ae, since it is used to s*
*how an independence of
choice. That part of the proof goes by showing that the two choices can be seen*
*, since we start with a
homotopy, as given by the two ends @n#1x of an #n # 1#-cube x. It is then shown*
* by induction, using
the method of construction and the above result, that x is degenerate in direct*
*ion n # 1. Hence the
two ends in that direction coincide.
Properties of the folding map are used also in showing that X is actually*
* included in aeX ; in
relating two types of thinness for elements of aeX ; and in proving a homotopy *
*addition lemma in aeX .
Any !-Gpd G has an underlying cubical set UG. If C is a crossed complex, th*
*en the cubical set
U#~C# is called the cubical nerve N2C of C. It is a conclusion of the theory th*
*at we can also obtain
N2C as
#N2C#n # Crs# In, C#
where In is the usual geometric cube with its standard skeletal filtration. The*
* (cubical) geometric
realisation jN2Cj is also called the cubical classifying space B2C of the cross*
*ed complex C. The filtration
C of C by skeleta gives a filtration B2C of B2C and there is (as in 4.1.6) a *
*natural isomorphism
#B2C # ,#C. Thus the properties of a crossed complex are those that are univer*
*sally satisfied by
X . These proofs use the equivalence of the homotopy categories of Kan8 cubica*
*l sets and of CW-
complexes. We originally took this from the Warwick Masters thesis of S. Hintze*
*, but it is now available
with different proofs from Antolini [Ant96] and Jardine [Jar06].
As said above, by taking particular values for C, the classifying space B2C*
* gives cubical versions
of Eilenberg-Mac Lane spaces K#G, n#, including the case n # 1 and G non commut*
*ative. If C is
essentially a crossed module, then B2C is called the cubical classifying space *
*of the crossed module,
and in fact realises the k-invariant of the crossed module.
Another useful result is that if K is a cubical set, then ae#jKj # may be i*
*dentified with ae#K#, the free
!-Gpd on the cubical set K, where here jKj is the usual filtration by skeleta.*
* On the other hand, our
proof that #jKj # is the free crossed complex on the non-degenerate cubes of K*
* uses the generalised
HHvKT of the next section.
It is also possible to give simplicial and globular versions of some of the*
* above results, because
the category of crossed complexes is equivalent also to those of simplicial T-c*
*omplexes [Ash88] and
of globular #-groupoids [BHi81c]. In fact the published paper on the classifyin*
*g space of a crossed
complex [BHi91] is given in simplicial terms, in order to link more easily with*
* well known theories.
_____________________________________
8The notion of Kan cubical set K is also called a cofibrant cubical set. It i*
*s an extension condition that any partial r-box in
K is the partial boundary of an element of Kr. See for example [Jar06], but the*
* idea goes back to the first paper by D. Kan
in 1958.
18
8 First main aim of the work: Higher Homotopy van Kampen Theorems
These theorems give non commutative tools for higher dimensional local-to-globa*
*l problems yielding a
variety of new, often non commutative, calculations, which prove (i.e. test) th*
*e theory. We now explain
these theorems in a way which strengthens the relation with descent, since that*
* was a theme of the
conference at which the talk was given on which this survey is based.
We suppose given an open cover U # fU~g~2 of X. This cover defines a map
G
q # E # U~ ! X
~2
and so we can form an augmented simplicial space
~C#q# # E X E X E____//_//_//_E_/X/E_//_Eq//_X
where the higher dimensional terms involve disjoint unions of multiple intersec*
*tions U of the U~.
We now suppose given a filtered space X , a cover U as above of X # X# , and*
* so an augmented
simplicial filtered space ~C#q # involving multiple intersections U of the ind*
*uced filtered spaces.
We still need a connectivity condition.
Definition 8.1A filtered space X is connected if and only if the induced maps *
*ss0X0 ! ss0Xn are
surjective and ssn#Xr, Xn, # # 0 for all n > 0, r > n and 2 X0.
Theorem 8.2 (MAIN RESULT (HHvKT)) If U is connected for all finite intersecti*
*ons U of the ele-
ments of the open cover, then
(C) (connectivity) X is connected, and
(I) (isomorphism) the following diagram as part of ae#~C#q ##
ae#q #
ae#E X E_#__//_//_aeE//_aeX . *
*(cae)
is a coequaliser diagram. Hence the following diagram of crossed complexes
#q #
#E X E #____//_//__E//_ X . *
*(c )
is also a coequaliser diagram.
So we get calculations of the fundamental crossed complex X .
It should be emphasised that to get to and apply this theorem takes just the*
* two papers [BHi81a,
BHi81b] totalling 58 pages. With this we deduce in the first instance:
o the usual vKT for the fundamental groupoid on a set of base points;
o the Brouwer degree theorem (ssnSn # Z);
o the relative Hurewicz theorem;
19
o Whitehead's theorem that ssn#X [ fe2~g, X# is a free crossed module;
o an excision result, more general than the previous two, on ssn#A[B, A, x# *
*as an induced module
(crossed module if n # 2) when #A, A " B# is #n 1#-connected.
The assumptions required of the reader are quite small, just some familiarity w*
*ith CW-complexes.
This contrasts with some expositions of basic homotopy theory, where the proof *
*of say the relative
Hurewicz theorem requires knowledge of singular homology theory. Of course it i*
*s surprising to get
this last theorem without homology, but this is because it is seen as a stateme*
*nt on the morphism of
relative homotopy groups
ssn#X, A, x# ! ssn#X [ CA, CA, x# ,#ssn#X [ CA, x#
and is obtained, like our proof of Theorem W, as a special case of an excision *
*result. The reason for
this success is that we use algebraic structures which model the underlying pro*
*cesses of the geometry more
closely than those in common use. These algebraic structures and their relation*
*s are quite intricate,
as befits the complications of homotopy theory, so the theory is tight knit.
Note also that these results cope well with the action of the fundamental g*
*roup on higher homotopy
groups.
The calculational use of the HHvKT for X is enhanced by the relation of *
* with tensor products
(see section 15 for more details).
9 The fundamental cubical !-groupoid aeX of a filtered space X
Here are the basic elements of the construction.
In: the n-cube with its skeletal filtration.
Set RnX # FTop#In, X #. This is a cubical set with compositions, connection*
*s, and inversions.
For i # 1, . .,.n there are standard:
face maps @i # RnX ! Rn 1X ;
degeneracy maps "i# Rn 1X ! RnX
connections i # Rn 1X ! RnX
compositions a ffiib defined for a, b 2 RnX such that @#ia # @i b
inversions i# Rn ! Rn.
The connections are induced by flffi# In ! In 1 defined using the monoid str*
*uctures max, min#
I2 ! I. They are essential for many reasons, e.g. to discuss the notion of comm*
*utative cube.
These operations have certain algebraic properties which are easily derived *
*from the geometry and
which we do not itemise here - see for example [AABS02 ]. These were listed fir*
*st in the Bangor thesis
of Al-Agl [AAl89]. (In the paper [BHi81a] the only basic connections needed are*
* the #i, from which
the i are derived using the inverses of the groupoid structures.)
Now it is natural and convenient to define f j g for f, g # In ! X to mean *
*f is homotopic to g
through filtered maps an relative to the vertices of In. This gives a quotient *
*map
p # RnX ! aenX # #RnX = j#.
20
The following results are proved in [BHi81b].
9.1 The compositions on RX are inherited by aeX to give aeX the structure of*
* cubical multiple groupoid
with connections.
9.2 The map p # RX ! aeX is a Kan fibration of cubical sets.
The proofs of both results use methods of collapsing which are indicated in *
*the next section.
The second result is almost unbelievable. Its proof has to give a systematic me*
*thod of deforming a
cube with the right faces `up to homotopy' into a cube with exactly the right f*
*aces, using the given
homotopies. In both cases, the assumption that the relation j uses homotopies r*
*elative to the vertices
is essential to start the induction. (In fact the paper [BHi81b] does not use *
*homotopy relative to
the vertices, but imposes an extra condition J0, that each loop in X0 is contra*
*ctible X1, which again
starts the induction. This condition is awkward in applications, for example to*
* function spaces. A full
exposition of the whole story is in preparation, [BHS09 ].)
An essential ingredient in the proof of the HHvKT is the notion of multiple *
*composition. We have
discussed this already in dimension 2, with a suggestive picture in the diagram*
* (3). In dimension
n, the aim is to give algebraic expression to the idea of a cube In being subdi*
*vided by hyperplanes
parallel to the faces into many small cubes, a subdivision with a long history *
*in mathematics.
Let #m# # #m1, . .,.mn# be an n-tuple of positive integers and
OE#m# # In ! #0, m1# #0, mn#
be the map #x1, . .,.xn# 7! #m1x1, . .,.mnxn#. Then a subdivision of type #m# o*
*f a map ff # In ! X
is a factorisation ff # ff0ffi OE#m#; its parts are the cubes ff#r#where #r# # *
*#r1, . .,.rn# is an n-tuple of
integers with 1 6 ri6 mi, i # 1, . .,.n, and where ff#r## In ! X is given by
#x1, . .,.xn# 7! ff0#x1 # r1 1, . .,.xn # rn 1#.
We then say that ff is the composite of the cubes ff#r#and write ff # #ff#r#*
*#. The domain of ff#r#is
then the set f#x1, . .,.xn# 2 In # ri 1 6 xi 6 ri, 1 6 i 6 ng. This ability to*
* express `algebraic inverse
to subdivision' is one benefit of using cubical methods.
Similarly, in a cubical set with compositions satisfying the interchange law*
* we can define the mul-
tiple composition #ff#r## of a multiple array #ff#r## provided the obviously ne*
*cessary multiple incidence
relations of the individual ff#r#to their neighbours are satisfied.
Here is an application which is essential in many proofs, and which seems ha*
*rd to prove without
the techniques involved in 9.2.
Theorem 9.3 (Lifting multiple compositions)Let #ff#r## be a multiple compositio*
*n in aenX . Then
representatives a#r#of the ff#r#may be chosen so that the multiple composition *
*#a#r## is well defined in
RnX .
Proof: The multiple composition #ff#r## determines a cubical map
A # K ! aeX
21
where the cubical set K corresponds to a representation of the multiple composi*
*tion by a subdivision
of the geometric cube, so that top cells c#r#of K are mapped by A to ff#r#.
Consider the diagram, in which is a corner vertex of K,
_________//RX.
| ">>
| " |
| A0 " |
| "" p|
| " |
| " |
fflffl|" fflffl|
K ___A____//_aeX
Then K collapses to , written K & . (As an example, see how the subdivision i*
*n the diagram (3)
may be collapsed row by row to a point.) By the fibration result, A lifts to A0*
*, which represents #a#r##,
as required. *
* 2
So we have to explain collapsing.
10 Collapsing
We use a basic notion of collapsing and expanding due to J.H.C. Whitehead, [W:S*
*HT ].
Let C ` B be subcomplexes of In. We say C is an elementary collapse of B, B *
*&e C, if for some
s > 1 there is an s-cell a of B and #s 1#-face b of a, the free face, such th*
*at
B # C [ a, C " a # `an b
(where `adenotes the union of the proper faces of a).
We say B1 collapses to Br, written B1 & Br, if there is a sequence
B1 &e B2 &e &e Br
of elementary collapses.
If C is a subcomplex of B then
B I & #B f0g [ C I#
(this is proved by induction on dimension of B n C).
Further, In collapses to any one of its vertices (this may be proved by indu*
*ction on n using the
first example). These collapsing techniques allows the construction of the exte*
*nsions of filtered maps
and filtered homotopies that are crucial for proving 9.1, that aeX does obtain*
* the structure of multiple
groupoid.
However, more subtle collapsing techniques using partial boxes are required *
*to prove the fibration
theorem 9.2, as partly explained in the next section.
22
11 Partial boxes
Let C be an r-cell in the n-cube In. Two #r 1#-faces of C are called opposite*
* if they do not meet.
A partial box in C is a subcomplex B of C generated by one #r 1#-face b of*
* C (called a base of B)
and a number, possibly zero, of other #r 1#-faces of C none of which is oppos*
*ite to b.
The partial box is a box if its #r 1#-cells consist of all but one of the *
*#r 1#-faces of C.
The proof of the fibration theorem uses a filter homotopy extension property*
* and the following:
Proposition 11.1 (Key Proposition)Let B, B0 be partial boxes in an r-cell C of *
*In such that B0 ` B.
Then there is a chain
B # Bs & Bs 1 & & B1 # B0
such that
(i)each Biis a partial box in C;
(ii)Bi#1# Bi[ aiwhere aiis an #r 1#-cell of C not in Bi;
(iii)ai" Biis a partial box in ai.
The proof is quite neat, and follows the pictures. Induction up such a chain of*
* partial boxes is one of
the steps in the proof of the fibration theorem 9.2. The proposition implies th*
*at an inclusion of partial
boxes is what is known as an anodyne extension, [Jar06].
Here is an example of a sequence of collapsings of a partial box B, which il*
*lustrate some choices
in forming a collapse B & 0 through two other partial boxes B1, B2.
e e e e e e e
B B1 B2
The proof of the fibration theorem gives a program for carrying out the defo*
*rmations needed to do
the lifting. In some sense, it implies computing a multiple composition can be *
*done using collapsing
as the guide.
Methods of collapsing generalise methods of trees in dimension 1.
12 Thin elements
Another key concept is that of thin element ff 2 aenX for n > 2. The proofs he*
*re use strongly results
of [BHi81a].
We say ff is geometrically thin if it has a deficient representative, i.e. a*
*n a # In ! X such that
a#In# ` Xn 1.
23
We say ff is algebraically thin if it is a multiple composition of degenera*
*te elements or those com-
ing from repeated (including 0) negatives of connections. Clearly any multiple *
*composition of alge-
braically thin elements is thin.
Theorem 12.1 (i) Algebraically thin is equivalent to geometrically thin.
(ii) In a cubical !-groupoid with connections, any box has a unique thin fil*
*ler.
Proof The proof of the forward implication in (i) uses lifting of multiple comp*
*ositions, in a stronger
form than stated above.
The proofs of (ii) and the backward implication in (i) use the full force of*
* the algebraic relation
between !-groupoids and crossed complexes. *
* 2
These results allow one to replace arguments with commutative cubes by argum*
*ents with thin
elements.
13 Sketch proof of the HHvKT
The proof goes by verifying the required universal property. Let U be an open c*
*over of X as in Theorem
8.2.
We go back to the following diagram whose top row is part of ae#~C#q ##
____@0___//_ ae#q #
ae#E X E_#_______//_ae#E_#______//_aeX *
*(cae)
@1 KKK O
KK O
KKK Of0
f KKKKK O
K%%Kfflffl
G
To prove this top row is a coequaliser diagram, we suppose given a morphism f *
*# ae#E # ! G of cubical
!-groupoids with connection such that f ffi @0 # f ffi @1, and prove that there*
* is a unique morphism
f0# aeX ! G such that f0ffi ae#q # # f.
To define f0#ff# for ff 2 aeX , you subdivide a representative a of ff to gi*
*ve a # #a#r## so that each
a#r#lies in an element U#r#of U; use the connectivity conditions and this subdi*
*vision to deform a into
b # #b#r## so that
b#r#2 R#U#r##
and so obtain
fi#r#2 ae#U#r##.
The elements
ffi#r#2 G
24
may be composed in G (by the conditions on f), to give an element
`#ff# # #ffi#r##2 G.
So the proof of the universal property has to use an algebraic inverse to subdi*
*vision. Again an analogy
here is with sending an email: the element you start with is subdivided, deform*
*ed so that each part is
correctly labelled, the separate parts are sent, and then recombined.
The proof that `#ff# is independent of the choices made uses crucially prop*
*erties of thin elements.
The key point is: a filter homotopy h # ff j ff0in RnX gives a deficient eleme*
*nt of Rn#1X .
The method is to do the subdivision and deformation argument on such a homo*
*topy, push the little
bits in some
aen#1#U~#
(now thin) over to G, combine them and get a thin element
o 2 Gn#1
all of whose faces not involving the direction #n # 1# are thin because h was *
*given to be a filter
homotopy. An inductive argument on unique thin fillers of boxes then shows that*
* o is degenerate in
direction #n # 1#, so that the two ends in direction #n # 1# are the same.
This ends a rough sketch of the proof of the HHvKT for ae.
Note that the theory of these forms of multiple groupoids is designed to ma*
*ke this last argument
work. We replace a formula for saying a cube h has commutative boundary by a st*
*atement that h is
thin. It would be very difficult to replace the above argument, on the composit*
*ion of thin elements,
by a higher dimensional manipulation of formulae such as that given in section *
*3 for a commutative
3-cube.
Further, the proof does not require knowledge of the existence of all coequ*
*alisers, not does it give
a recipe for constructing these in specific examples.
14 Tensor products and homotopies
The construction of the monoidal closed structure on the category !-Gpd is base*
*d on rather formal
properties of cubical sets, and the fact that for the cubical set In we have Im*
* In #, Im#n . The
details are given in [BHi87]. The equivalence of categories implies then that t*
*he category Crsis also
monoidal closed, with a natural isomorphism
Crs#A B, C# ,#Crs#A, CRS #B, C##.
Here the elements of CRS #B, C# are in dimension 0 the morphisms B ! C, in dime*
*nsion 1 the left
homotopies of morphisms, and in higher dimensions are forms of higher homotopie*
*s. The precise
description of these is obtained of course by tracing out in detail the equival*
*ence of categories. It
should be emphasised that certain choices are made in constructing this equival*
*ence, and these choices
are reflected in the final formulae that are obtained.
25
An important result is that if X , Y are filtered spaces, then there is a *
*natural transformation
j # aeX aeY ! ae#X Y #
#a# #b#7! #a b#
where if a # Im ! X , b # In ! Y then a b # Im#n ! X Y . It not hard to *
*see, in this cubical
setting, that j is well defined. It can also be shown using previous results th*
*at j is an isomorphism if
X , Y are the geometric realisations of cubical sets with the usual skeletal f*
*iltration.
The equivalence of categories now gives a natural transformation of crossed*
* complexes
j0 # X Y ! #X Y #. *
* (8)
It would be hard to construct this directly. It is proved in [BHi91] that j0is *
*an isomorphism if X , Y
are the skeletal filtrations of CW-complexes. The proof uses the HHvKT, and the*
* fact that A on
crossed complexes has a right adjoint and so preserves colimits. It is proved i*
*n [BaBr93] that j is an
isomorphism if X , Y are cofibred, connected filtered spaces. This applies in *
*particular to the useful
case of the filtration B2C of the classifying space of a crossed complex.
It turns out that the defining rules for the tensor product of crossed comp*
*lexes which follows from
the above construction are obtained as follows. We first define a bimorphism of*
* crossed complexes.
Definition 14.1A bimorphism ` # #A, B# ! C of crossed complexes is a family of *
*maps ` # Am Bn !
Cm#n satisfying the following conditions, where a 2 Am , b 2 Bn, a1 2 A1, b1 2*
* B1 (temporarily
using additive notation throughout the definition):
(i)
fi#`#a, b## # `#fia, fib# for alla 2 A, b 2 B .
(ii)
`#a, bb1# # `#a, b#`#fia,b1#ifm > 0, n > 2 ,
`#aa1, b# # `#a, b#`#a1,fib#ifm > 2, n > 0 .
(iii)
(
`#a, b# # `#a, b0# ifm # 0, n > 1 orm > 1, n > 2*
* ,
`#a, b # b0## 0
`#a, b#`#fia,b##`#a, b0#ifm > 1, n # 1 ,
(
`#a, b# # `#a0, b# ifm > 1, n # 0 or m > 2, n > *
*1 ,
`#a # a0, b## 0
`#a0, b# # `#a, b#`#a ,fib#ifm # 1, n > 1 .
(iv)
8
>>>`#ffim a, b# # # #m `#a, ffinb# ifm > 2,*
* n > 2 ,
><
`#a, ffinb# `#fia, b# # `#ffa, b#`#a,fib#ifm # 1*
*, n > 2 ,
ffim#n #`#a, b###
>>># #m#1 `#a, fib# # # #m `#a, ffb#`#fia,b## `#ffimia,*
*fb#m > 2, n # 1 ,
>:
`#fia, b# `#a, ffb# # `#ffa, b# # `#a, fib#ifm #*
* n # 1 .
26
(v)
(
`#a, ffinb#ifm # 0, n > 2 ,
ffim#n #`#a, b###
`#ffim a, b#ifm > 2, n # 0 .
(vi)
ff#`#a, b## # `#a, ffb# and fi#`#a, b## # `#a,ifib#fm # 0, n #*
* 1 ,
ff#`#a, b## # `#ffa, b# and fi#`#a, b## # `#fia,ib#fm # 1, n #*
* 0 .
The tensor product of crossed complexes A, B is given by the universal bimor*
*phism #A, B# ! A B,
#a, b# 7! a b. The rules for the tensor product are obtained by replacing `#a*
*, b# by a b in the
above formulae.
The conventions for these formulae for the tensor product arise from the der*
*ivation of the tensor
product via the category of cubical !-groupoids with connections, and the formu*
*lae are forced by our
conventions for the equivalence of the two categories [BHi81a, BHi87].
The complexity of these formulae is directly related to the complexities of *
*the cell structure of the
product Em En where the n-cell En has cell structure e0 if n # 0, e0 [e1 if n *
*# 1, and e0[en 1 [en
if n > 2.
It is proved in [BHi87] that the bifunctor is symmetric and that if a0*
* is a vertex of A then
the morphism B ! A B, b ! a0 b, is injective.
There is a standard groupoid model I of the unit interval, namely the indisc*
*rete groupoid on two
objects 0, 1. This is easily extended trivially to either a crossed complex or *
*an !-Gpd . So using we
can define a `cylinder object' I in these categories and so a homotopy theo*
*ry, [BGo89 ].
15 Free crossed complexes and free crossed resolutions
Let C be a crossed complex. A free basis B for C consists of the following:
B0 is set which we take to be C0;
B1 is a graph with source and target maps s, t # B1 ! B0 and C1 is the free gro*
*upoid on the graph B1:
that is B1 is a subgraph of C1 and any graph morphism B1 ! G to a groupoid G ex*
*tends uniquely to
a groupoid morphism C1 ! G;
Bn is, for n > 2, a totally disconnected subgraph of Cn with target map t # Bn *
*! B0; for n # 2, C2 is
the free crossed C1-module on B2 while for n > 2, Cn is the free #ss1C#-module *
*on Bn.
It may be proved using the HHvKT that if X is a CW-complex with the skeleta*
*l filtration, then
X is the free crossed complex on the characteristic maps of the cells of X . *
*It is proved in [BHi91]
that the tensor product of free crossed complexes is free.
A free crossed resolution F of a groupoid G is a free crossed complex which*
* is aspherical together
with an isomorphism OE # ss1#F # ! G. Analogues of standard methods of homologi*
*cal algebra show
that free crossed resolutions of a group are unique up to homotopy equivalence.
27
In order to apply this result to free crossed resolutions, we need to repla*
*ce free crossed resolutions
by CW-complexes. A fundamental result for this is the following, which goes ba*
*ck to Whitehead
[W:SHT ] and Wall [Wal66], and which is discussed further by Baues in [Bau89, C*
*hapter VI, x7]:
Theorem 15.1 Let X be a CW-filtered space, and let OE # X ! C be a homotopy *
*equivalence to a free
crossed complex with a preferred free basis. Then there is a CW-filtered space *
*Y , and an isomorphism
Y ,#C of crossed complexes with preferred basis, such that OE is realised by *
*a homotopy equivalence
X ! Y .
In fact, as pointed out by Baues, Wall states his result in terms of chain c*
*omplexes, but the crossed
complex formulation seems more natural, and avoids questions of realisability i*
*n dimension 2, which
are unsolved for chain complexes.
Corollary 15.2If A is a free crossed resolution of a group G, then A is realise*
*d as free crossed complex
with preferred basis by some CW-filtered space Y .
Proof We only have to note that the group G has a classifying CW-space BG whos*
*e fundamental
crossed complex #BG# is homotopy equivalent to A. *
* 2
Baues also points out in [Bau89, p.657] an extension of these results which *
*we can apply to the
realisation of morphisms of free crossed resolutions. A new proof of this exten*
*sion is given by Faria
Martins in [FM07a ], using methods of Ashley [Ash88].
Proposition 15.3Let X # K#G, 1#, Y # K#H, 1# be CW-models of Eilenberg - Mac La*
*ne spaces and let
h # X ! #Y # be a morphism of their fundamental crossed complexes with the p*
*referred bases given
by skeletal filtrations. Then h # #g# for some cellular g # X ! Y.
Proof Certainly h is homotopic to #f# for some f # X ! Y since the set of poin*
*ted homotopy classes
X ! Y is bijective with the morphisms of groups A ! B. The result follows from *
*[Bau89, p.657,(**)]
(`if f is -realisable, then each element in the homotopy class of f is -reali*
*sable'). 2
These results are exploited in [Moo01 , BMPW02 ] to calculate free crossed r*
*esolutions of the fun-
damental groupoid of a graph of groups.
An algorithmic approach to the calculation of free crossed resolutions for g*
*roups is given in
[BRS99 ], by constructing partial contracting homotopies for the universal cove*
*r at the same time
as constructing this universal cover inductively. This has been implemented in *
*GAP4 by Heyworth and
Wensley [HWe06 ].
16 Classifying spaces and the homotopy classification of maps
The formal relations of cubical sets and of cubical !-groupoids with connection*
*s and the relation of
Kan cubical sets with topological spaces, allow the proof of a homotopy classif*
*ication theorem:
28
Theorem 16.1 If K is a cubical set, and G is an !-groupoid, then there is a na*
*tural bijection of sets of
homotopy classes
#jKj, jUGj# ,##ae#jKj #, G#,
where on the left hand side we work in the category of spaces, and on the right*
* in !-groupoids.
Here jKj is the filtration by skeleta of the geometric realisation of the cubi*
*cal set.
We explained earlier how to define a cubical classifying space say B2#C# of*
* a crossed complex C
as B2#C# # jUN2Cj # jU~Cj. The properties already stated now give the homotopy*
* classification
theorem 4.1.9.
It is shown in [BHi81b] that for a CW-complex Y there is a map p # Y ! B2 Y*
* whose homotopy
fibre is n-connected if Y is connected and ssiY # 0 for 2 6 i 6 n 1. It foll*
*ows that if also X is a
connected CW-complex with dimX 6 n, then p induces a bijection
#X, Y# ! #X, B Y #.
So under these circumstances we get a bijection
#X, Y# ! # X , Y #. *
* (9)
This result, due to Whitehead [W:CHII], translates a topological homotopy class*
*ification problem to
an algebraic one. We explain below how this result can be translated to a resul*
*t on chain complexes
with operators.
It is also possible to define a simplicial nerve N #C# of a crossed comple*
*x C by
N #C#n # Crs# # n#, C#.
The simplicial classifying space of C is then defined using the simplicial geom*
*etric realisation
B #C# # jN #C#j.
The properties of this simplicial classifying space are developed in [BHi91], a*
*nd in particular an ana-
logue of 4.1.9 is proved.
The simplicial nerve and an adjointness
Crs# #L#, C# ,#Simp#L, N #C##
are used in [BGPT97 , BGPT01 ] for an equivariant homotopy theory of crossed co*
*mplexes and their
classifying spaces. Important ingredients in this are notions of coherence and *
*an Eilenberg-Zilber type
theorem for crossed complexes proved in Tonks' Bangor thesis [Ton93, Ton03]. Se*
*e also [BSi07].
Labesse in [Lab99] defines a crossed set. In fact a crossed set is exactly*
* a crossed module (of
groupoids) ffi # C ! X o G where G is a group acting on the set X, and X o G is*
* the associated actor
groupoid; thus the simplicial construction from a crossed set described by Larr*
*y Breen in [Lab99]
is exactly the simplicial nerve of the crossed module, regarded as a crossed co*
*mplex. Hence the
cohomology with coefficients in a crossed set used in [Lab99] is a special case*
* of cohomology with
coefficients in a crossed complex, dealt with in [BHi91]. (We are grateful to B*
*reen for pointing this
out to us in 1999.)
29
17 Relation with chain complexes with a groupoid of operators
Chain complexes with a group of operators are a well known tool in algebraic to*
*pology, where they
arise naturally as the chain complex C eX of cellular chains of the universal c*
*over eX of a reduced
CW-complex X . The group of operators here is the fundamental group of the spac*
*e X.
J.H.C. Whitehead in [W:CHII] gave an interesting relation between his free c*
*rossed complexes (he
called them `homotopy systems') and such chain complexes. We refer later to his*
* important homotopy
classification results in this area. Here we explain the relation with the Fox *
*free differential calculus
[Fox53].
Let ~ # M ! P be a crossed module of groups, and let G # Coker~. Then there *
*is an associated
diagram
~ OE
M _____//_P_____//_G *
*(10)
| |
h2|| |h1 h0|
fflffl| fflffl| fflffl|
Mab __@2_//DOE@1_//Z#G#
in which the second row consists of (right) G-modules and module morphisms. Her*
*e h2 is simply
the Abelian isation map; h1 # P ! DOEis the universal OE-derivation, that is it*
* satisfies h1#pq# #
h1#p#OEq# h1#q#, for all p, q 2 P, and is universal for this property; and h0 i*
*s the usual derivation
g 7! g 1. Whitehead in his Lemma 7 of [W:CHII] gives this diagram in the case*
* P is a free group,
when he takes DOEto be the free G-module on the same generators as the free gen*
*erators of P. Our
formulation, which uses the derived module due to Crowell [Cro71], includes his*
* case. It is remarkable
that diagram (10)is a commutative diagram in which the vertical maps are operat*
*or morphisms, and
that the bottom row is defined by this property. The proof in [BHi90] follows e*
*ssentially Whitehead's
proof. The bottom row is exact: this follows from results in [Cro71], and is a *
*reflection of a classical
fact on group cohomology, namely the relation between central extensions and th*
*e Ext functor, see
[McL63 ]. In the case the crossed module is the crossed module ffi # C#!# ! F#*
*X# derived from a
presentation of a group, then C#!#ab is isomorphic to the free G-module on R, D*
*OEis the free G-
module on X, and it is immediate from the above that @2 is the usual derivative*
* #@r=@x# of Fox's free
differential calculus [Fox53]. Thus Whitehead's results anticipate those of Fox.
It is also proved in [W:CHII] that if the restriction M ! ~#M# of ~ has a se*
*ction which is a
morphism but not necessarily a P-map, then h2 maps Ker~ isomorphically to Ker@2*
*. This allows cal-
culation of the module of identities among relations by using module methods, a*
*nd this is commonly
exploited, see for example [ElK99] and the references there.
Whitehead introduced the categories CW of reduced CW-complexes, HS of homot*
*opy systems,
and FCC of free chain complexes with a group of operators, together with functo*
*rs
CW #! HS #C!FCC .
In each of these categories he introduced notions of homotopy and he proved *
*that C induces an
equivalence of the homotopy category of HS with a subcategory of the homotopy c*
*ategory of FCC .
Further, C X is isomorphic to the chain complex C eX of cellular chains of the*
* universal cover of X,
so that under these circumstances there is a bijection of sets of homotopy clas*
*ses
# X , Y # ! #C eX, C eY#. *
*(11)
30
This with the bijection (9)can be interpreted as an operator version of the Hop*
*f classification theorem.
It is surprisingly little known. It includes results of Olum [Olu53] published *
*later, and it enables quite
useful calculations to be done easily, such as the homotopy classification of m*
*aps from a surface to
the projective plane [Ell88], and other cases. Thus we see once again that this*
* general theory leads to
specific calculations.
All these results are generalised in [BHi90] to the non free case and to th*
*e non reduced case, which
requires a groupoid of operators, thus giving functors
FTop#! Crs#r! Chain.
(The paper [BHi90] uses the notation for this r.) One utility of the generali*
*sation to groupoids is
that the functor r then has a right adjoint, and so preserves colimits. An exam*
*ple of this preservation
is given in [BHi90, Example 2.10]. The construction of the right adjoint to r*
* builds on a number
of constructions used earlier in homological algebra.
The definitions of the categories under consideration in order to obtain a *
*generalisation of the
bijection (11)has to be quite careful, since it works in the groupoid case, and*
* not all morphisms of
the chain complex are realisable.
This analysis of the relations between these two categories is used in [BHi*
*91] to give an account
of cohomology with local coefficients.
It is also proved in [BHi90] that the functor r preserves tensor products, *
*where the tensor in the
category Chainis a generalisation to modules over groupoids of the usual tensor*
* for chain complexes
of modules of groups. Since the tensor product is described explicitly in dimen*
*sions 6 2 in [BHi87],
and #rC#n # Cn for n > 3, this preservation yields a complete description of th*
*e tensor product of
crossed complexes.
18 Crossed complexes and simplicial groups and groupoids
The Moore complex NG of a simplicial group G is not in general a (reduced) cros*
*sed complex. Let
DnG be the subgroup of Gn generated by degenerate elements. Ashley showed in hi*
*s thesis [Ash88]
that NG is a crossed complex if and only if #NG#n " #DG#n # f1g for all n > 1.
Ehlers and Porter in [EhP97 , EhP99] show that there is a functor C from sim*
*plicial groupoids to
crossed complexes in which C#G#n is obtained from N#G#n by factoring out
#NGn " Dn#dn#1#NGn#1 " Dn#1#,
where the Moore complex is defined so that its differential comes from the last*
* simplicial face operator.
This is one part of an investigation into the Moore complex of a simplicial *
*group, of which the
most general investigation is by Carrasco and Cegarra in [CaC91 ].
An important observation in [Por93] is that if N/G is an inclusion of a norm*
*al simplicial subgroup
of a simplicial group, then the induced morphism on components ss0#N# ! ss0#G# *
*obtains the structure
of crossed module. This is directly analogous to the fact that if F ! E ! B is *
*a fibration sequence then
the induced morphism of fundamental groups ss1#F, x# ! ss1#E, x# also obtains t*
*he structure of crossed
module. This last fact is relevant to algebraic K-theory, where for a ring R th*
*e homotopy fibration
sequence is taken to be F ! B#GL#R## ! B#GL#R### .
31
19 Other homotopy multiple groupoids
A natural question is whether there are other useful forms of higher homotopy g*
*roupoids. It is because
the geometry of convex sets is so much more complicated in dimensions > 1 than *
*in dimension 1 that
new complications emerge for the theories of higher order group theory and of h*
*igher homotopy
groupoids. We have different geometries for example those of disks, globes, si*
*mplices, cubes, as
shown in dimension 2 in the following diagram.
The cellular decomposition for an n-disk is Dn # e0 [ en 1 [ en, and that for g*
*lobes is
Gn # e0 [ e1 [ [ en 1 [ en.
The higher dimensional group(oid) theory reflecting the n-disks is that of cros*
*sed complexes, and that
for the n-globes is called globular !-groupoids.
A common notion of higher dimensional category is that of n-category, which *
*generalise the 2-
categories studied in the late 1960s. A 2-category C is a category enriched in *
*categories, in the sense
that each hom set C#x, y# is given the structure of category, and there are app*
*ropriate axioms. This
gives inductively the notion of an n-category as a category enriched in #n 1#*
*-categories. This is
called a `globular' approach to higher categories. The notion of n-category for*
* all n was axiomatised
in [BHi81c] and called an #-category; the underlying geometry of a family of se*
*ts Sn, n > 0 with
operations
Dffi# Sn ! Si, Ei# Si! Sn, ff # 0, 1; i # 1, . .,.n 1
was there axiomatised. This was later called a `globular set' [Str00], and the *
*term !-category was
used instead of the earlier #-category. Difficulties of the globular approach a*
*re to define multiple
compositions, and also monoidal closed structures, although these are clear in *
*the cubical approach.
A globular higher homotopy groupoid of a filtered space has been constructed in*
* [Bro08a], deduced
from cubical results.
Although the proof of the HHvKT outlined earlier does seem to require cubica*
*l methods, there is
still a question of the place of globular and simplicial methods in this area. *
*A simplicial analogue of
the equivalence of categories is given in [Ash88, NTi89], using Dakin's notion *
*of simplicial T-complex,
[Dak76 ]. However it is difficult to describe in detail the notion of tensor pr*
*oduct of such structures,
or to formulate a proof of the HHvKT theorem in that context. There is a tenden*
*cy to replace the term
T-complex from all this earlier work such as [BHi77, Ash88] by complicial set, *
*[Ver08].
It is easy to define a homotopy globular set aeflX of a filtered space X b*
*ut it is not quite so clear
how to prove directly that the expected compositions are well defined. However *
*there is a natural
graded map
i # aeflX ! aeX *
*(12)
and applying the folding map of [AAl89, AABS02 ] analogously to methods in [BHi*
*81b] allows one
to prove that i of (12)is injective. It follows that the compositions on aeX a*
*re inherited by aeflX to
make the latter a globular !-groupoid. The details are in [Bro08a].
32
Loday in 1982 [Lod82] defined the fundamental catn-group of an n-cube of sp*
*aces (a catn-group
may be defined as an n-fold category internal to the category of groups), and s*
*howed that catn-
groups model all reduced weak homotopy #n#1#-types. Joint work [BLo87a] formula*
*ted and proved a
HHvKT for the catn-group functor from n-cubes of spaces. This allows new local *
*to global calculations
of certain homotopy n-types [Bro92], and also an n-adic Hurewicz theorem, [BLo8*
*7b]. This work
obtains more powerful results than the purely linear theory of crossed complexe*
*s. It yields a group-
theoretic description of the first non-vanishing homotopy group of a certain #n*
* # 1#-ad of spaces, and
so several formulae for the homotopy and homology groups of specific spaces; [E*
*lM08 ] gives new
applications. Porter in [Por93] gives an interpretation of Loday's results usin*
*g methods of simplicial
groups. There is clearly a lot to do in this area. See [CELP02 ] for relation*
*s of catn-groups with
homological algebra.
Recently some absolute homotopy 2-groupoids and double groupoids have been *
*defined, see
[BHKP02 ] and the references there, while [BrJ04] applies generalised Galois th*
*eory to give a new
homotopy double groupoid of a map, generalising previous work of [BHi78a]. It i*
*s significant that
crossed modules have been used in a differential topology situation by Mackaay *
*and Picken [MaP02 ].
Reinterpretations of these ideas in terms of double groupoids are started in [B*
*Gl93].
It seems reasonable to suggest that in the most general case double groupoi*
*ds are still somewhat
mysterious objects. The paper [AN06 ] gives a kind of classification of them.
20 Conclusion and questions
ffl The emphasis on filtered spaces rather than the absolute case is open to qu*
*estion.
ffl Mirroring the geometry by the algebra is crucial for conjecturing and pr*
*oving universal properties.
ffl Thin elements are crucial for modelling a concept not so easy to define *
*or handle algebraically,
that of commutative cubes. See also [Hig05, Ste06].
ffl The cubical methods summarised in section 9 have also been applied in co*
*ncurrency theory, see
for example [GaG03 , FRG06].
ffl HHvKT theorems give, when they apply, exact information even in non comm*
*utative situations.
The implications of this for homological algebra could be important.
ffl One construction inspired eventually by this work, the non Abelian tenso*
*r product of groups, has
a bibliography of 90 papers since it was defined with Loday in [BLo87a].
ffl Globular methods do fit into this scheme. They have not so far yielded n*
*ew calculations in ho-
motopy theory, see [Bro08a], but have been applied to directed homotopy theory,*
* [GaG03 ]. Globular
methods are the main tool in approaches to weak category theory, see for exampl*
*e [Lei04, Str00],
although the potential of cubical methods in that area is hinted at in [Ste06].
ffl For computations we really need strict structures (although we do want t*
*o compute invariants of
homotopy colimits).
ffl No work seems to have been done on Poincar'e duality, i.e. on finding sp*
*ecial qualities of the
fundamental crossed complex of the skeletal filtration of a combinatorial manif*
*old. However the book
by Sharko, [Sha93, Chapter VI], does use crossed complexes for investigating Mo*
*rse functions on a
manifold.
33
ffl In homotopy theory, identifications in low dimensions can affect high d*
*imensional homotopy. So
we need structure in a range of dimensions to model homotopical identifications*
* algebraically. The
idea of identifications in low dimensions is reflected in the algebra by `induc*
*ed constructions'.
ffl In this way we calculate some crossed modules modelling homotopy 2-type*
*s, whereas the corre-
sponding k-invariant is often difficult to calculate.
ffl The use of crossed complexes in ~Cech theory is a current project with *
*Jim Glazebrook and Tim
Porter.
ffl Question: Are there applications of higher homotopy groupoids in other *
*contexts where the
fundamental groupoid is currently used, such as algebraic geometry?
ffl Question: There are uses of double groupoids in differential geometry, *
*for example in Poisson
geometry, and in 2-dimensional holonomy [BrI03]. Is there a non Abelian De Rham*
* theory, using an
analogue of crossed complexes?
ffl Question: Is there a truly non commutative integration theory based on *
*limits of multiple com-
positions of elements of multiple groupoids?
References
[AAl89] F. Al-Agl, 1989, Aspects of multiple categories, Ph.D. thesis, Universi*
*ty of Wales, Bangor. 20,
32
[AABS02] F. Al-Agl, R. Brown and R. Steiner, Multiple categories: the equivalen*
*ce between a globular
and cubical approach, Advances in Mathematics, 170, (2002), 71-118. 20, 32
[AN06] N. Andruskiewitsch and S. Natale, Tensor categories attached to double g*
*roupoids, Adv. Math.
200 (2006) 539-583. 33
[Ant96] R. Antolini, 1996, Cubical structures and homotopy theory, Ph.D. thesis*
*, Univ. Warwick,
Coventry. 18
[Ash88] N. Ashley, Simplicial T-Complexes: a non Abelian version of a theorem o*
*f Dold-Kan, Disserta-
tiones Math., 165 (1988), 11 - 58, (Published version of University of Wal*
*es PhD thesis, 1978).
18, 28, 31, 32
[Bau89] H. J. Baues, Algebraic Homotopy, volume 15 of Cambridge Studies in Adva*
*nced Mathematics,
(1989), Cambridge Univ. Press. 28
[Bro82] Higher dimensional group theory, in Low dimensional topology, London Ma*
*th Soc. Lecture
Note Series 48 (ed. R. Brown and T.L. Thickstun, Cambridge University Pres*
*s, (1982) 215-238.
12
[BaBr93] H. J. Baues and R. Brown, On relative homotopy groups of the product f*
*iltration, the James
construction, and a formula of Hopf, J. Pure Appl. Algebra, 89 (1993), 49-*
*61. 17, 26
[BaT97] H.-J. Baues and A. Tonks, On the twisted cobar construction, Math. Proc*
*. Cambridge Philos.
Soc., 121 (1997), 229-245. 17
34
[BoJ01] F. Borceux and G. Janelidze, Galois Theories, Cambridge Studies in Adva*
*nced Mathematics
72 (2001), Cambridge University Press, Cambridge. 9
[Bro67] R. Brown, Groupoids and van Kampen's Theorem, Proc. London Math. Soc. (*
*3) 17 (1967),
385-340. 6, 7
[Bro80] R. Brown, On the second relative homotopy group of an adjunction space:*
* an exposition of a
theorem of J. H. C. Whitehead, J. London Math. Soc. (2), 22 (1980), 146-15*
*2. 5
[Bro84] R. Brown, Coproducts of crossed P-modules: applications to second homot*
*opy groups and to the
homology of groups, Topology, 23 (1984), 337-345. 6
[Bro87] R. Brown, From groups to groupoids: a brief survey, Bull. London Math. *
*Soc., 19 (1987),
113-134. 9
[Bro06] R. Brown, Topology and groupoids, Booksurge PLC, S. Carolina, 2006, (pr*
*evious editions with
different titles, 1968, 1988). 7, 9
[Bro92] R. Brown, 1992, Computing homotopy types using crossed n-cubes of group*
*s, in Adams Memo-
rial Symposium on Algebraic Topology, 1 (Manchester, 1990), volume 175 of *
*London Math. Soc.
Lecture Note Ser., 187-210, Cambridge Univ. Press, Cambridge. 33
[Bro99] R. Brown, Groupoids and crossed objects in algebraic topology, Homology*
*, homotopy and ap-
plications, 1 (1999), 1-78. 14
[Bro07] R. Brown, Three themes in the work of Charles Ehresmann: Local-to-globa*
*l; Groupoids; Higher
dimensions, Proceedings of the 7th Conference on the Geometry and Topology*
* of Manifolds: The
Mathematical Legacy of Charles Ehresmann, Bedlewo (Poland) 8.05.2005-15.05*
*.2005, Banach
Centre Publications 76 Institute of Mathematics Polish Academy of Sciences*
*, Warsaw, (2007)
51-63. (math.DG/0602499). 9
[Bro08a] R. Brown, A new higher homotopy groupoid: the fundamental globular !-g*
*roupoid of a filtered
space, Homotopy, Homology and Applications, 10 (2008), 327-343. 32, 33
[Bro08b] R. Brown, Exact sequences of fibrations of crossed complexes, homotopy*
* classification of maps,
and nonabelian extensions of groups, J. Homotopy and Related Structures 3 *
*(2008) 331-343. 15
[BGi89] R. Brown and N. D. Gilbert, Algebraic models of 3-types and automorphis*
*m structures for
crossed modules, Proc. London Math. Soc. (3), 59 (1989), 51-73. 17
[BGl93] R. Brown and J. F. Glazebrook, Connections, local subgroupoids, and a h*
*olonomy Lie groupoid
of a line bundle gerbe, Univ. Iagel. Acta Math. XLI (2003) 283-296. 33
[BGo89] R. Brown and M. Golasi'nski, A model structure for the homotopy theory*
* of crossed complexes,
Cah. Top. G'eom. Diff. Cat. 30 (1989) 61-82. 17, 27
[BGPT97] R. Brown, M. Golasi'nski, T. Porter and A. Tonks, Spaces of maps into*
* classifying spaces for
equivariant crossed complexes, Indag. Math. (N.S.), 8 (1997), 157-172. 29
[BGPT01] R. Brown, M. Golasi'nski, T. Porter and A. Tonks, Spaces of maps into*
* classifying spaces for
equivariant crossed complexes. II. The general topological group case, K-T*
*heory, 23 (2001), 129-
155. 29
35
[BHKP02] R. Brown, K. A. Hardie, K. H. Kamps and T. Porter, A homotopy double *
*groupoid of a Haus-
dorff space, Theory Appl. Categ., 10 (2002), 71-93 (electronic).
[BHi77] R. Brown and P. J. Higgins, Sur les complexes crois'es, !-groupo"ides e*
*t T-complexes, C.R. Acad.
Sci. Paris S'er. A. 285 (1977) 997-999. 33
[BHi78a] R. Brown and P. J. Higgins, On the connection between the second relat*
*ive homotopy groups
of some related spaces, Proc.London Math. Soc., (3) 36 (1978) 193-212. 32
[BHi78b] R. Brown and P. J. Higgins, Sur les complexes crois'es d'homotopie ass*
*oci'es `a quelques espaces
filtr'es, C. R. Acad. Sci. Paris S'er. A-B, 286 (1978), A91-A93. 11, 13, 1*
*4, 33
14
[BHi81a] R. Brown and P. J. Higgins, The algebra of cubes, J. Pure Appl. Alg., *
*21 (1981), 233-260.
11, 13, 14, 19, 20, 23, 27
[BHi81b] R. Brown and P. J. Higgins, Colimit theorems for relative homotopy gro*
*ups, J. Pure Appl. Alg,
22 (1981), 11-41. 14, 19, 21, 29, 32
[BHi81c] R. Brown and P. J. Higgins, The equivalence of #-groupoids and crossed*
* complexes,, Cahiers
Top. G'eom. Diff., 22 (1981), 370-386. 18, 32
[BHi87] R. Brown and P. J. Higgins, Tensor products and homotopies for !-groupo*
*ids and crossed
complexes,, J. Pure Appl. Alg, 47 (1987), 1-33. 14, 25, 27, 31
[BHi90] R. Brown and P. J. Higgins, Crossed complexes and chain complexes with *
*operators, Math. Proc.
Camb. Phil. Soc., 107 (1990), 33-57. 30, 31
[BHi91] R. Brown and P. J. Higgins, The classifying space of a crossed complex,*
* Math. Proc. Cambridge
Philos. Soc., 110 (1991), 95-120.
[BrH03] R. Brown and P. J. Higgins, Cubical Abelian groups with connections are*
* equivalent to chain
complexes, Homology, Homotopy and Applications, 5 (2003) 49-52. 18, 26, 27*
*, 29, 31
[BHS09] R. Brown, P. J. Higgins and R. Sivera, Nonabelian algebraic topology (*
*2009). 17
[BHu82] R. Brown and J. Huebschmann, 1982, Identities among relations,*
* in R.Brown and
T.L.Thickstun, eds., Low Dimensional Topology, London Math. Soc Lecture No*
*tes, Cambridge
University Press. 21
[BrI03] R. Brown and I. Icen, Towards two dimensional holonomy, Advances in Mat*
*hematics, 178
(2003) 141-175. 5
[BrJ97] R. Brown and G. Janelidze, Van Kampen theorems for categories of coveri*
*ng morphisms in
lextensive categories, J. Pure Appl. Algebra, 119 (1997), 255-263. 34
[BrJ04] R. Brown and G. Janelidze, A new homotopy double groupoid of a map of s*
*paces, Applied
Categorical Structures 12 (2004) 63-80. 9
[BLo87a] R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, *
*Topology, 26 (1987),
311 - 337. 33
36
[BLo87b] R. Brown and J.-L. Loday, Homotopical excision, and Hurewicz theorems,*
* for n-cubes of spaces,
Proc. London Math. Soc., (3) 54 (1987), 176 - 192. 4, 33
[BKP05] R. Brown, H. Kamps and T. Porter), A homotopy double groupoid of a Hau*
*sdorff space II: a
van Kampen theorem, Theory and Applications of Categories, 14 (2005) 200-2*
*20. 16, 33
[BMPW02] R. Brown, E. Moore, T. Porter and C. Wensley, Crossed complexes, and*
* free crossed resolu-
tions for amalgamated sums and HNN-extensions of groups, Georgian Math. J.*
*, 9 (2002), 623-644.
11, 13
[BMo99] R. Brown and G. H. Mosa, Double categories, 2-categories, thin structu*
*res and connections,
Theory Appl. Categ., 5 (1999), No. 7, 163-175 (electronic). 28
[BrP96] R. Brown and T. Porter, On the Schreier theory of non-abelian extension*
*s: generalisations and
computations, Proceedings Royal Irish Academy 96A (1996) 213-227. 10, 12
[BRS99] R. Brown and A. Razak Salleh, Free crossed resolutions of groups and p*
*resentations of modules
of identities among relations, LMS J. Comput. Math., 2 (1999), 28-61 (elec*
*tronic). 6
[BSi07] R. Brown and R. Sivera), Normalisation for the fundamental crossed comp*
*lex of a simplicial
set, J. Homotopy and Related Structures, Special Issue devoted to the memo*
*ry of Saunders Mac
Lane, 2 (2007) 49-79. 5, 28
[BSp76a] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahi*
*ers Topologie G'eom.
Diff'erentielle, 17 (1976), 343-362. 29
[BSp76b] R. Brown and C. B. Spencer, G-groupoids, crossed modules and the funda*
*mental groupoid of
a topological group, Proc. Kon. Ned. Akad. v. Wet, 79 (1976), 296 - 302. 1*
*0, 12, 13
[BWe95] R. Brown and C. D. Wensley, On finite induced crossed modules, and the*
* homotopy 2-type of
mapping cones, Theory Appl. Categ., 1 (1995) 54-70. 10
[BWe96] R. Brown and C. D. Wensley, Computing crossed modules induced by an in*
*clusion of a normal
subgroup, with applications to homotopy 2-types, Theory Appl. Categ., 2 (1*
*996) 3-16. 6
[BWe03] R. Brown and C. D. Wensley, Computations and homotopical applications *
*of induced crossed
modules, J. Symb. Comp., 35 (2003) 59-72. 6
[BuL03] M. Bunge and S. Lack, Van Kampen theorems for toposes, Advances in Math*
*ematics, 179
(2003) 291 - 317. 6, 14
[CaC91] P. Carrasco and A. M. Cegarra, Group-theoretic Algebraic Models for Ho*
*motopy Types, Jour.
Pure Appl. Algebra, 75 (1991), 195-235. 9
[CELP02] J. M. Casas, G. Ellis, M. Ladra, and T. Pirashvili, Derived functors a*
*nd the homology of n-
types, J. Algebra, 256 (2002) 583-598. 31
[Cec32] E. ^Cech, 1933, H"oherdimensionale Homotopiegruppen, in Verhandlungen d*
*es Internationalen
Mathematiker-Kongresses Zurich 1932, 2 203, International Congress of Math*
*ematicians (4th :
1932 : Zurich, Switzerland, Walter Saxer, Zurich, reprint Kraus, Nendeln, *
*Liechtenstein, 1967.
33
37
[Con94] A. Connes, 1994, Noncommutative geometry, Academic Press Inc., San Dieg*
*o, CA. 3
[Cro71] R. Crowell, The derived module of a homomorphism, Advances in Math., 5 *
*(1971), 210-238.
14
[Dak76] M.K. Dakin, Kan complexes and multiple groupoid structures, PhD. Thesis*
*, University of Wales,
Bangor, (1976). 30
[Ded63] P. Dedecker, Sur la cohomologie non ab'elienne. II, Canad. J. Math., 15*
* (1963), 84-93. 32
[DiV96] W. Dicks and E. Ventura, The group fixed by a family of injective endom*
*orphisms of a free group,
Contemporary Mathematics, 195 American Mathematical Society, Providence, R*
*I, (1996) x+81.
6
[DoD79] A. Douady and R. Douady, 1979, Algebres et theories Galoisiennes, volu*
*me 2, CEDIC, Paris. 7
[EhP97] P. J. Ehlers and T. Porter, Varieties of simplicial groupoids. I. Cross*
*ed complexes, J. Pure Appl.
Algebra, 120 (1997), 221-233. 9
[EhP99] P. J. Ehlers and T. Porter, Erratum to: `Varieties of simplicial groupo*
*ids. I. Crossed complexes' ,
J. Pure Appl. Algebra 120 (1997), no. 3, 221-233; J. Pure Appl. Algebra, 1*
*34 (1999), 207-209.
31
[Ehr83] C. Ehresmann, Cat'egories et Structure, Dunod, Paris (1983). 31
[EH84] C. Ehresmann, OEuvres compl`etes et comment'ees. I-1,2. Topologie alg'e*
*brique et g'eom'etrie
diff'erentielle, With commentary by W. T. van Est, Michel Zisman, Georges *
*Reeb, Paulette Libermann,
Ren'e Thom, Jean Pradines, Robert Hermann, Anders Kock, Andr'e Haefliger, *
*Jean B'enabou, Ren'e
Guitart, and Andr'ee Charles Ehresmann, Edited by Andr'ee Charles Ehresman*
*n, Cahiers Topologie
G'eom. Diff'erentielle,24, (1983) suppll. 1. 8
9
[Ell88]G. J. Ellis, Homotopy classification the J. H. C. Whitehead way, Exposit*
*ion. Math., 6 (1988),
97-110. 31
[ElM08] G. Ellis and R. Mikhailov, A colimit of classifying spaces, arXiv:0804.*
*3581. 33
[ElK99] G. Ellis and I. Kholodna, Three-dimensional presentations for the group*
*s of order at most 30,
LMS J. Comput. Math., 2 (1999), 93-117+2 appendixes (HTML and source code)*
*. 5, 30
[FRG06] L. Fajstrup, M. Rauen, and E. Goubault, Algebraic topology and concurr*
*ency, Theoret. Com-
put. Sci. 357 (2006), 241-278. 33
[FM07] J. Faria Martins, On the homotopy type and fundamental crossed complex *
*of the skeletal filtra-
tion of a CW-complex, Homology, Homotopy and Applications 9 (2007), 295-32*
*9. 17
[FM07a] J. Faria Martins, A new proof of a theorem of H.J. Baues, Preprint IST*
*, Lisbon, (2007) 16pp.
28
[FMP07] J. Faria Martins and T. Porter, On Yetter's invariant and an extension*
* of the Dijkgraaf-Witten
invariant to categorical groups, Theory and Applications of Categories, 18*
* (2007) 118-150. 15,
17
38
[Fox53] R. H. Fox, Free differential calculus I: Derivations in the group ring,*
* Ann. Math., 57 (1953),
547-560. 30
[Fro61] A. Fr"ohlich, Non-Abelian homological algebra. I. Derived functors and *
*satellites., Proc. London
Math. Soc. (3), 11 (1961), 239-275.
[GAP02] The GAP Group, 2002, Groups, Algorithms, and Programming, version 4.3,*
* Technical report,
http://www.gap-system.org. 6
[Gau01] P. Gaucher, Combinatorics of branchings in higher dimensional automata*
*, Theory Appl. Categ.,
8 (2001) 324-376. 6
[GaG03] P. Gaucher and E. Goubault, Topological deformation of higher dimensio*
*nal automata, Ho-
mology Homotopy Appl. 5 (2003), 39-82. 13
33
[GrM03] M. Grandis and L. Mauri, Cubical sets and their site, Theory Applic. C*
*ategories, 11 (2003)
185-201. 15
[HWe06] A. Heyworth and C. D. Wensley, IdRel - logged rewriting and identities*
* among relators, GAP4
2006. 28
[Hig63] P. J. Higgins, Algebras with a scheme of operators, Math. Nachr., 27 (1*
*963) 115-132. 9
[Hig64] P. J. Higgins, Presentations of Groupoids, with Applications to Groups,*
* Proc. Camb. Phil. Soc.,
60 (1964) 7-20. 6, 7
[Hig71] P. J. Higgins, 1971, Categories and Groupoids, van Nostrand, New York. *
*Reprints in Theory
and Applications of Categories, 7 (2005) pp 1-195. 7, 9
[Hig05] P. J. Higgins, Thin elements and commutative shells in cubical !-catego*
*ries, Theory Appl.
Categ., 14 (2005) 60-74. 13, 33
[HiT81] P. J. Higgins and J. Taylor, The Fundamental Groupoid and Homotopy Cros*
*sed Complex of an
Orbit Space, in K. H. Kamps et al., ed., Category Theory: Proceedings Gumm*
*ersbach 1981, Springer
LNM 962 (1982) 115-122. 7
[HAM93] C. Hog-Angeloni and W. Metzler, eds., Two-dimensional homotopy and com*
*binatorial group
theory, London Mathematical Society Lecture Note Series, 197 (1993) Cambri*
*dge University
Press, Cambridge. 5
[Jar06] J.F. Jardine, Categorical homotopy theory, Homology, Homotopy Appl., 8 *
*(2006) 71-144. 15,
18, 23
[Hue80] J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Co*
*mment. Math. Helv.,
55 (1980), 302-313. 6
[Hue81a] J. Huebschmann, Automorphisms of group extensions and differentials i*
*n the Lyndon-
Hochschild-Serre spectral sequence, J. Algebra, 72 (1981), 296-334. 6
[Hue81b] J. Huebschmann, Group extensions, crossed pairs and an eight term exa*
*ct sequence, Jour. fur.
reine. und ang. Math., 321 (1981), 150-172. 6
39
[Hur35] W. Hurewicz, Beitr"age zur Topologie der Deformationen, Nederl. Akad. W*
*etensch. Proc. Ser.
A, 38 (1935), 112-119,521-528, 39 (1936) 117-126,213-224. 4
[vKa33] E. H. v. Kampen, On the Connection Between the Fundamental Groups of so*
*me Related Spaces,
Amer. J. Math., 55 (1933), 261-267. 3, 6
[Lab99] J.-P. Labesse, Cohomologie, stabilisation et changement de base, Ast'er*
*isque, vi+161, 257
(1999), appendix A by Laurent Clozel and Labesse, and Appendix B by Lawren*
*ce Breen. 29
[Lei04] T. Leinster, Higher operads, higher categories, London Mathematical Soc*
*iety Lecture Note Se-
ries, 298 Cambridge University Press, Cambridge, 2004. xiv+433 pp. 33
[Lod82] J.-L. Loday, Spaces with finitely many homotopy groups, J.Pure Appl. Al*
*g., 24 (1982), 179-
202. 33
[Lue81] A. S. T. Lue, Cohomology of groups relative to a variety, J. Alg., 69 (*
*1981), 155-174. 6
[MaP02] M. Mackaay and R. F. Picken, Holonomy and parallel transport for Abeli*
*an gerbes, Advances
in Mathematics 170 (2002) 287-339. 33
[McL63] S. Mac Lane, , Homology, number 114 in Grundlehren der math. Wiss. 114*
* Springer, 1963.
6, 30
[MLW50] S. Mac Lane and J. H. C. Whitehead, On the 3-type of a complex, Proc. *
*Nat. Acad. Sci. U.S.A.,
36 (1950), 41-48. 5
[Moo01] E. J. Moore, Graphs of Groups: Word Computations and Free Crossed Reso*
*lutions, Ph.D. thesis,
(2001) University of Wales, Bangor. 28
[NTi89] G. Nan Tie, A Dold-Kan theorem for crossed complexes, J. Pure Appl. Alg*
*ebra, 56 (1989) 177-
194. 32
[Olu53] P. Olum, On mappings into spaces in which certain homotopy groups vanis*
*h, Ann. of Math. (2)
57 (1953) 561-574. 31
[Pei49] R. Peiffer, Uber Identit"aten zwischen Relationen, Math. Ann., 121 (194*
*9), 67-99. 5
[Por93] T. Porter, n-types of simplicial groups and crossed n-cubes, Topology, *
*32 (1993), 5-24. 31, 33
[PoT07] T. Porter and V. Turaev, Formal Homotopy Quantum Field Theories I: Form*
*al maps and crossed
C-algebras, Journal Homotopy and Related Structures, 3 (2008) 113-159. 15,*
* 17
[Pri91]S. Pride, Identities among relations, in A. V. E. Ghys, A. Haefliger, ed*
*., Proc. Workshop on Group
Theory from a Geometrical Viewpoint, International Centre of Theoretical P*
*hysics, Trieste, 1990,
World Scientific (1991) 687-716. 5
[Rei49] K. Reidemeister, "Uber Identit"aten von Relationen, Abh. Math. Sem. Ham*
*burg, 16 (1949), 114
- 118. 5
[Sha93] V.V. Sharko, Functions on manifolds: Algebraic and topological aspects,*
* Translations of mathe-
matical monographs 131 American Mathematical Society (1993). 33
40
[Spe77] C.B. Spencer, An abstract setting for homotopy pushouts and pullbacks, *
*Cahiers Topologie
G'eom. Diff'erentielle, 18 (1977), 409-429. 12
[Ste06] R. Steiner, Thin fillers in the cubical nerves of omega-categories, The*
*ory Appl. Categ. 16 (2006),
144-173. 33
[Str00] R. Street, The petit topos of globular sets, J. Pure Appl. Algebra 154 *
*(2000), 299-315. 32, 33
[Tay88] J. Taylor, Quotients of Groupoids by the Action of a Group, Math. Proc.*
* Camb. Phil. Soc., 103
(1988), 239-249. 7
[Ton93] A. P. Tonks, 1993, Theory and applications of crossed complexes, Ph.D. *
*thesis, University of
Wales, Bangor. 29
[Ton03] A. P. Tonks, On the Eilenberg-Zilber theorem for crossed complexes, J. *
*Pure Appl. Algebra 179
(2003) 199-220. 29
[Ver08] D. Verity, Complicial sets characterising the simplicial nerves of stri*
*ct !-categories, Mem. Amer.
Math. Soc. 193 (2008), no. 905, xvi+184 pp. 32
[Wal66] C. T. C. Wall, Finiteness conditions for CW-complexes II, Proc. Roy. So*
*c. Ser. A, 295 (1966),
149-166. 28
[Whi41] J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Ma*
*th. (2), 42 (1941),
409-428. 5
[W:CHI] J. H. C. Whitehead, Combinatorial Homotopy I, Bull. Amer. Math. Soc., 5*
*5 (1949), 213-245.
4
[W:CHII] J. H. C. Whitehead, Combinatorial Homotopy II, Bull. Amer. Math. Soc.,*
* 55 (1949), 453-
496. 4, 5, 14, 29, 30
[W:SHT] J. H. C. Whitehead, Simple homotopy types, Amer. J. Math., 72 (1950), *
*1-57. 4, 22, 28
41