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