Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems* Ronald Browny December 22, 2002 University of Wales, Bangor, Maths Preprint 02.26 MATHEMATICS SUBJECT CLASSIFICATION: 01-01,16E05,18D05,18D35,55P15,55Q05 Abstract We outline the main features of the definitions and applications of cro* *ssed complexes and cubical !- groupoids with connections. These give forms of higher homotopy groupoids,* * and new views of basic algebraic topology and the cohomology of groups, with the ability to obtai* *n some non commutative results and compute some homotopy types. Contents Introduction * * 1 1 Crossed modules * * 4 2 The fundamental groupoid on a set of base points * * 5 3 The search for higher homotopy groupoids * * 7 4 Main results * * 9 5 Why crossed complexes? * * 11 *_____________________________________ This is an extended account of a lecture given at the meeting on `Categorica* *l Structures 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. yMathematics Division, School of Informatics, University of Wales, Dean St., * *Bangor, Gwynedd LL57 1UT, U.K. email: r.brown@bangor.ac.uk 1 6 Why cubical !-groupoids with connections? * * 11 7 The equivalence of categories * * 12 8 Main aim of the work: colimit, or local-to-global, theorems * * 13 9 The fundamental cubical !-groupoid %(X*) of a filtered space X* * * 14 10 Collapsing * * 16 11 Partial boxes * * 16 12 Thin elements * * 17 13 Sketch proof of the GSVKT * * 17 14 Tensor products and homotopies * * 18 15 Free crossed complexes and free crossed resolutions * * 20 16 Classifying spaces and homotopy classification of maps * * 21 17 Relation with chain complexes with a groupoid of operators * * 23 18 Crossed complexes and simplicial groups and groupoids * * 24 19 Other homotopy multiple groupoids * * 24 20 Conclusion and questions * * 25 Introduction An aim is to give a survey of results obtained by R. Brown and P.J. Higgins and* * others over the years 1974-2002, and to point to applications and related areas. This work gives an account of s* *ome basic algebraic topology which differs from the standard account through the use of crossed complexes, r* *ather than chain complexes, as a fundamental notion. In this way one obtains comparatively quickly not onl* *y classical results such as the Brouwer degree and the relative Hurewicz theorem, but also non commutative * *results on second relative homotopy groups, as well as higher dimensional results involving the action of * *and also presentations of the fundamental group. For example, the fundamental crossed complex X* of the skel* *etal filtration of a CW - complex X is a useful generalisation of the usual cellular chains of the univer* *sal cover of X. It also gives a replacement for singular chains by taking X to be the geometric realisation of * *a singular complex of a space. A replacement for the excision theorem in homology is obtained by using cub* *ical methods to prove a 2 colimit theorem for the fundamental crossed complex functor on filtered spaces.* * This colimit theorem is a higher dimensional version of a classical example of a non commutative local-to* *-global theorem, which itself was the initial motivation for the work described here. This Seifert-Van Kampen* * Theorem (SVKT) determines completely the fundamental group ß1(X, x) of a space X with base point which is* * the union of open sets U, V whose intersection is path connected and contains the base point x; the `local * *information' is on the morphisms of fundamental groups induced by the inclusions U \ V ! U, U \ V ! V . The impo* *rtance of this result reflects the importance of the fundamental group in algebraic topology, algebra* *ic geometry, complex analysis, and many other subjects. Indeed the origin of the fundamental group was in Poin* *car'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 theo- rems, is a construction of higher homotopy groupoids, with properties described* * by an algebra of cubes. There are applications to local-to-global problems in homotopy theory which are more * *powerful than purely classical tools, while shedding light on those tools. It is hoped that this account will * *increase the interest in the possibility of wider applications of these methods and results, since homotopical methods p* *lay a key role in many areas. Higher homotopy groups Topologists in the early part of the 20th century were well aware that: the* * non commutativity of the funda- mental group was useful in geometric applications; for path connected X there w* *as an isomorphism H1(X) ~=ß1(X, x)ab; and the abelian homology groups existed in all dimensions. Consequently there w* *as a desire to generalise the non commutative fundamental group to all dimensions. In 1932 ~Cech submitted a paper on higher homotopy groups ßn(X, x) to the I* *CM at Zurich, but it was quickly proved that these groups were abelian for n > 2, and on these grounds ~* *Cech was persuaded to withdraw his paper, so that only a small paragraph appeared in the Proceedings [44]. We * *now see the reason for this commutativity as the result (Eckmann-Hilton) that a group internal to the categ* *ory of groups is just an abelian group. Thus the vision of a non commutative higher dimensional version of the f* *undamental group has since 1932 been generally considered to be a mirage. Before we go back to the SVKT, w* *e explain in the next section how nevertheless work on crossed modules did introduce non commutative structur* *es relevant to topology in dimension 2. Of course higher homotopy groups were strongly developed following on from * *the work of Hurewicz (1935). The fundamental group still came into the picture with its action on t* *he higher homotopy groups, which J.H.C. Whitehead once remarked (1957) was especially fascinating for the * *early workers in homotopy theory. Much of Whitehead's work was intended to extend to higher dimensions th* *e methods of combinatorial group theory of the 1930s - hence the title of his papers: `Combinatorial homot* *opy, I, II' [82, 83]. The first of these two papers has been very influential and is part of the basic structur* *e of algebraic topology. It is the development of work of the second paper which we explain here. The paper by Whitehead on `Simple homotopy types' [84], which deals with hi* *gher dimensional analogues of Tietze transformations, has a final section using crossed complexes. We refe* *r 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 [33], which uses a form of homotopy groupoid which is in one sense* * much more powerful than that given here, since it encompasses n-adic information, but in which current * *expositions are still restricted to the reduced (one base point) case. 3 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. 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 Ø : M ! AutM, in which Øm is the automorphism n* * 7! m-1nm; (iii) the zero map M ! P where M is a P -module; (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~ inheri* *ts an action of M= Im~. Another important construction is the free crossed P -module @ : C(!) ! P determined by a function ! : R ! P , where R is a set. The group C(!) is genera* *ted by R x 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 x P . A major result of Whitehead was: Theorem W [83] If the space X = A[{e2r}r2Ris obtained from A by attaching 2-cel* *ls 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 trans* *versality and knot theory - an exposition is given in [9]. Mac Lane and Whitehead [70] used this result as par* *t 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 * *was `On adding relations to homotopy groups' [81]. This indicates a search for higher dimensional SVKTs. 4 The concept of free crossed module gives a non commutative context for chai* *ns of syzygies. The latter idea, in the case of modules over polynomial rings, is one of the origins of homologi* *cal algebra through the notion of free resolution. Here is how similar ideas can be applied to groups. Pioneer* *ing work here, independent of Whitehead, was by Peiffer [74] and Reidemeister [77]. See [29] for an expositio* *n of these ideas. Suppose P = is a presentation of a group G, where ! : R ! F (X) is a * *function, allowing for repeated relators. Then we have an exact sequence 1 -i!N(!R) -ffi!F (X) -! G -! 1 where N(!R) is the normal closure in F (X) of the set !R of relations. The abov* *e 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 operations of * *F (X) on its normal subgroup N(!R). Elements of C(!) are a kind of `formal consequence of the relators', so * *that the relation between the elements of C(!) and those of N(!R) is analogous to the relation between the el* *ements of F (X) and those of G. The kernel ß(P) of @ is a G-module, called the module of identities amon* *g relations, and there is considerable work on computing it [29, 76, 60, 51, 36]. By splicing to @ a free* * G-module resolution of ß(P) one obtains what is called a free crossed resolution of the group G. These reso* *lutions have better realisation properties than the usual resolutions by chain complexes of G-modules, as expla* *ined later. This notion of using crossed modules as the first stage of syzygies in fact* * represents a wider tradition in homological algebra, in the work of Frölich and Lue [54, 67]. Crossed modules also occurred in other contexts, notably in representing el* *ements of the cohomology group H3(G, M) of a group G with coefficients in a G-module M [69], and as coefficien* *ts in Dedecker's theory of non abelian cohomology [47]. The notion of free crossed resolution has been ex* *ploited by Huebschmann [61, 63, 62] to represent cohomology classes in Hn(G, M) of a group G with coef* *ficients in a G-module M, and also to calculate with these. Our results can make it easier to compute a crossed module arising from som* *e topological situation, such as an induced crossed module [39, 40], or a coproduct crossed module [10], than* * the cohomology element in H3(G, M) it represents. To obtain information on such an element it is usefu* *l to work with a small free crossed resolution 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 wou* *ld amount to, although it is interesting to know whether the element is non zero, or what is its order. Thus* * the use of algebraic models of cohomology classes may yield easier computations than the use of cocycles, a* *nd this somewhat inverts traditional approaches. Since crossed modules are algebraic objects generalising groups, it is natu* *ral to consider the problem of explicit calculations by extending techniques of computational group theory. Su* *bstantial work on this has been done by C.D. Wensley using the program GAP [56, 41]. 2 The fundamental groupoid on a set of base points A change in prospects for higher order non commutative invariants was derived f* *rom work of the writer pub- lished in 1967 [8], and influenced by Higgins' paper [58]. This showed 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 in Van Kampen's original paper [64].* * This use of groupoids in 5 dimension 1 suggested the question of the use of groupoids in higher homotopy t* *heory, and in particular the question of the existence of higher homotopy groupoids. In order to see how this research programme could go it is useful to consid* *er the statement and special features of this generalised Van Kampen Theorem for the fundamental groupoid. F* *irst, if X0 is a set , and X is a space, then ß1(X, X0) denotes the fundamental groupoid on the set X \ X0 of b* *ase points. This allows the set X0 to be chosen in a way appropriate to the geometry. For example, if the c* *ircle S1 is written as the union of two semicircles E+ [ E-, then the intersection {-1, 1} of the semicircles is* * not connected, so it is not clear where to take the base point. Instead one takes X0 = {-1, 1}, and so has two ba* *se 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 quotient* *s of the actions of Z2 on the group ß1(S1, 1) and on the groupoid ß1(S1, {-1, 1}) where the action is induced* * by complex conjugation on S1. Relevant work on orbit groupoids has been developed by Higgins and Taylor [* *59, 78]. Consideration of a set of base points leads to the theorem: Theorem 2.1 [8] Let the space X be the union of open sets U, V with intersectio* *n 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|| || * * (2) fflffl| fflffl| ß1(V, X0)____//_ß1(X, X0) is a pushout of groupoids. From this theorem, one can compute a particular fundamental group ß1(X, x0)* * using combinatorial infor- mation on the graph of intersections of path components of U, V, W , but for th* *is it is useful to develop the algebra of groupoids. Notice two special features of this result. (i) The computation of the invariant you want, the fundamental group, is obtain* *ed from the computation 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 i* *n a connected graph. The work on applying groupoids to groups gives many examples of this [58, 57, 12]. (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 dimensions, nam* *ely 0 and 1. This is contrary to the experience in homological algebra and algebraic topology, where the inte* *raction of several dimensions involves exact sequences or spectral sequences, which give information only up * *to extension, and (b) the result is a non commutative invariant, which is usually even more difficult to compute* * precisely. The reason for the success seems to be that the fundamental groupoid ß1(X, * *X0) contains information in dimensions 0 and 1, and so can adequately reflect the geometry of the intersect* *ions of the path components of U, V, W and 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 SVKT, which seemed to use three main ideas in order to verify the universal property of a p* *ushout for diagram (2)and given morphisms fU, fV from ß1(U, X0), ß1(V, X0) to a groupoid G, satisfying fUi = fV* * j: 6 o A deformation or filling argument. Given a path a : (I, `I) ! (X, X0) one* * can write a = a1 + . .+.an where each aimaps into U or V , but aiwill not necessarily have end points in X* *0. So one has to deform each ai to a0iin U, V or W , using the connectivity condition, so that each a0ihas end * *points in X0, and a0= a01+. .+.a0n is well defined. Then one can construct 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 tog* *ether again in G (this uses the condition fUi = fV j to prove that the elements of G are composable), and this * *part can be summarised as: o Groupoids provided a convenient algebraic inverse to subdivision. Next one has to prove that F (a) depends only on the class of a in the fund* *amental groupoid. This involves a homotopy rel end points h : a ' b, considered as a map I2 ! X; subdivide h as* * h = [hij] so that each hij maps into U, V or W ; deform h to h0= [h0ij] (keeping in U, V, W ) so that each* * h0ijmaps the vertices to X0and so determines a commutative square in one of ß1(Q, X0) for Q = U, V, W . Move t* *hese commutative squares over to G using fU, fV and recompose them (this is possible again because of th* *e condition fUi = fV j), noting that: o in a groupoid, any composition of commutative squares is commutative. Two opposite sides of the composite commutative square in G so obtained are ide* *ntities, because h was a homotopy rel end points, and the other two 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 deformed so that these pieces can be packaged and moved over to G, where they are reassembl* *ed. 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 connected s* *paces with non connected intersection, see [12, 8.4.9]; and (ii) to show the topological utility of the * *construction by Higgins [57] of the groupoid f*(G) over Y0 induced from a groupoid G over X0 by a function f : X0 !* * Y0. (Accounts of these with the notation Uf(G) rather than f*(G) are given in [57, 12].) This la* *tter construction is regarded as a `change of base', and analogues in higher dimensions yielded generalisatio* *ns of the Relative Hurewicz Theorem and of Theorem 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[12]. See for example [48] for an exposition of the relation with Galois theory. The paper [3* *1] gives a general formulation of conditions for the theorem to hold in the 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 [42]. Analogous interpretations for higher dimensional Va* *n 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 gro* *upoids [11]; and the opening out of new possibilities in higher dimensions, which allowed for new results and ca* *lculations in homotopy theory, and suggested new algebraic constructions. 3 The search for higher homotopy groupoids Contemplation of the proof of the SVGKT in the last section suggested that a hi* *gher dimensional version should exist, though this version amounted to an idea of a proof in search of a theore* *m. In the end, the results exactly 7 encapsulated this intuition. One intuition was that in groupoids we are dealing with a partial algebraic* * structure, 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 easily to generalise to directed squares, in which two are composable 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 specific* *ation of the three elements of a functor : (topological data)! (higher order groupoids) 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 these pro* *ved abortive. However con- siderable progress was made in work with Chris Spencer in 1971-3 on investigati* *ng the algebra of double groupoids [37, 38], and showing a relation to crossed modules. Further evidence* * was provided when it was found [38] that group objects in the category of groupoids are equivalent to cr* *ossed modules, and in particular are not necessarily commutative objects. (It turned out this result was known t* *o the Grothendieck school in the 1960s, but not published.) A key discovery was that a category of double groupoids with one vertex and* * what we called `special connections' [37] is equivalent to the category of crossed modules. Using these* * connections we could define what we called a `commutative cube' in such a double groupoid. The key equation* * for this was: 2 __ -1 __3 | a0 | c1 = 4-b0 c0 b15 |_ a1 __| which corresponded to folding flat five faces of a cube and filling in the corn* *ers with new `canonical' elements which we called `connections', because a crucial `transport law' which was borr* *owed from a paper on path connections in differential geometry, and can be written ~__ __~ | ___ __ || | = | . The connections provide a structure additional to the usual compositions, ident* *ities, inverses, which can be expressed intuitively by saying that in the 2-dimensional algebra of squares no* *t only can you move forward and backwards, and reverse, but you can also turn left and right. For more details * *see for example [35, 2]. As you might imagine, there are problems in finding a formula in still higher dimensio* *ns. In the groupoid case, this is handled by a homotopy addition lemma and thin elements [22], but in the cate* *gory case a formula for just a commutative 4-cube is complicated [55]. The blockage of defining a functor to double groupoids was resolved in 19* *74 in discussions with Higgins, by considering Whitehead's Theorem W. This showed that a 2-dimensional universa* *l property was available in homotopy theory, which was encouraging; it also suggested that a theory to b* *e any good should recover Theorem W. But this theorem was about relative homotopy groups. This suggested * *studying a relative situation X* : X0 X1 X. On looking for the simplest way to get a homotopy functor fro* *m this situation using squares, the `obvious' answer came up: consider maps (I2, @I2, @@I2) ! (X, X1, * *X0), i.e. maps of the square which take the edges into X1and the vertices into X0, and then take homotopy cl* *asses rel vertices of such maps to form a set %2(X*). Of course this set will not inherit a group structure but* * the surprise is that it does inherit 8 the structure of double groupoid with connections - the proof is not entirely t* *rivial, and is given in [20] and the expository article [14]. In the case X0 is a singleton, the equivalence of * *such double groupoids to crossed modules takes %(X*) to the usual relative homotopy crossed module. Thus a search for a higher homotopy groupoid was realised in dimension 2. I* *t might be that a lack of attention to the notion of groupoid (as suggested in [45]) was one reason why s* *uch a construction had not been found earlier. Finding a good homotopy double groupoid led rather quickly, in view of the * *previous experience, to a sub- stantial account of a 2-dimensional SVKT [20]. This recovers Theorem W, and als* *o leads to new calculations in 2-dimensional homotopy theory, and in fact to some new calculations of 2-typ* *es. For a recent summary of some results and some new ones, see the paper in the J. Symbolic Computation [4* *1] - publication in this journal illustrates that we are interested 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, and announcements were made in [21] with full detai* *ls in [22, 23]. However, these results needed a number of new ideas, even just to construct the higher dimensi* *onal compositions, and the proof of the Generalised SVKT 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, so* * that the resulting theory is quite complex. In the next section we give a summary. 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 filteredHspaces ss99scHHHcHH % sssssss HHHHH sssss HHHHHH ssssss HHHHH yysss|s|sOsU*s B HHH##HHH cubical __________fl________//crossed !-groupoidsoo__________________ complexes with connections ~ in which 1.1.1the categories FTopof filtered spaces, !-Gpd of !-groupoids, and Crsof cro* *ssed complexes are monoidal closed, and have a notion of homotopy using and a unit interval object; 1.1.2%, are homotopical functors (that is they are defined in terms of homoto* *py classes of certain maps), and preserve homotopies; 1.1.3~, fl are inverse adjoint equivalences of monoidal closed categories; 1.1.4there is a natural equivalence fl% ' , so that either % or can be used * *as appropriate; 1.1.5%, preserve certain colimits and certain tensor products; 9 1.1.5by definition, the cubical filtered classifying space is B2 = | | O U* whe* *re U* is the forgetful functor to filtered cubical sets using the filtration of an !-groupoid by skeleta, an* *d | | is geometric realisation of a cubical set; 1.1.6there is a natural equivalence O B2 ' 1; 1.1.7if C is a crossed complex and its cubical classifying space is defined as * *B2C = (B2 C)1 , then for a CW -complex X, and using homotopy as in 1.1.1 for crossed complexes, there* * is a natural bijection of sets of homotopy classes [X, B2C] ~=[ (X*), C]. Here a filtered space consists of a (compactly generated) space X1 and an i* *ncreasing sequence of subspaces X* : X0 X1 X2 . . .X1 . With the obvious morphisms, this gives the category FTop. The tensor product in* * this category is the usual [ (X* Y*)n = Xpx 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 FTopover either crossed complexes or !-Gpd using or % applied to FTOP (Y*, Z*). The structure of crossed complex is suggested by the canonical example, the* * fundamental crossed complex (X*) of the filtered space X*. So it is given by a diagram 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 C1is the fundamental groupoid ß1(X1, X0) of X1on the `* *set of base points' C0 = X0, while for n > 2, Cn is the family of relative homotopy groups {Cn(x)} = {ßn(Xn,* * Xn-1, x) | x 2 X0}. The boundary maps are those standard in homotopy theory. There is for n > 2 an acti* *on of the groupoid C1 on Cn (and of C1 on the groups C1(x), x 2 X0 by conjugation), the boundary morphisms * *are operator morphisms, ffin-1ffin = 0, n > 3, and the additional axioms are satisfied that 4.3 b-1cb = cffi2b, b, c 2 C2, so that ffi2 : C2 ! C1 is a crossed module (of g* *roupoids), 4.4 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. 10 As algebraic examples of crossed complexes we have: C = C(G, n) where G is * *a group, commutative if n > 2, and C is G in dimension n and trivial elsewhere; C = C(G : M, n), whe* *re G is a group, M is a G-module, n > 2, and C is G in dimension 1, M in dimension n, trivial elsewhe* *re, and zero boundary if n = 2; C is a crossed module (of groups) in dimensions 1 and 2 and trivial else* *where. A crossed complex C has a fundamental groupoid ß1C = C1= Imffi2, and also f* *or n > 2 a family {Hn(C, p)|p 2 C0} of homology groups. 5 Why crossed complexes? o They generalise groupoids and crossed modules to all dimensions. Note that* * the natural context for second relative homotopy groups is crossed modules of groupoids, rather than groups. o They are good for modelling CW -complexes. o Free crossed resolutions enable calculations with small CW -models of K(G* *, 1)s and their maps (White- head, Wall, Baues). o Crossed complexes give a kind of `linear model' of homotopy types which i* *ncludes all 2-types. Thus although they are not the most general model by any means (they do not contain * *quadratic information such as Whitehead products), this simplicity makes them easier to handle and to rela* *te to classical tools. The new methods and results obtained for crossed complexes can be used as a model for m* *ore complicated situations. This is how a general n-adic Hurewicz Theorem was found [32]. o They are convenient for calculation, and the functor is classical, invo* *lving relative homotopy groups. We explain some results in this form later. o They are close to chain complexes with a group(oid) of operators, and rel* *ated to some classical homo- logical algebra (e.g. chains of syzygies). In fact if SX is the simplicial sing* *ular complex of a space, with its skeletal filtration, then the crossed complex (SX) can be considered as a slig* *htly non commutative version of the singular chains of a space. o The monoidal structure is suggestive of further developments (e.g. cross* *ed differential algebras) see [7, 6]. It is used in [15] to give an algebraic model of homotopy 3-types, and * *to discuss automorphisms of crossed modules. o Crossed complexes have a good homotopy theory, with a cylinder object, an* *d homotopy colimits. The homotopy classification result 1.1.7 generalises a classical theorem of Eilenbe* *rg-Mac Lane. o They have an interesting relation with the Moore complex of simplicial gr* *oups 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 later. Here we explain why we need to introduce such new structures. o The functor % gives a form of higher homotopy groupoid, thus confirming t* *he visions of the early topolo- gists. o They are equivalent to crossed complexes. 11 o They have a clear monoidal closed structure, and a notion of homotopy, fr* *om which one can deduce those on crossed complexes, using the equivalence of categories. o It is easy to relate the functor % to tensor products, but quite difficul* *t to do this directly for . o Cubical methods, unlike globular or simplicial methods, allow for a simpl* *e algebraic inverse to subdivi- sion, which is crucial for our local-to-global theorems. o The additional structure of `connections', and the equivalence with cross* *ed complexes, allows for the sophisticated notion of commutative cube, and the proof that multiple compositi* *ons of commutative cubes are commutative. The last fact is a key component of the proof of the GSVKT. o They yield a construction of a (cubical) classifying space B2C = (B2 C)1 * *of a crossed complex C, which generalises (cubical) versions of Eilenberg-Mac Lane spaces, including th* *e local coefficient case. This has convenient relation to homotopies. o There is a current resurgence of the use of cubes in for example combinat* *orics, algebraic topology, and concurrency. There is a Dold-Kan type theorem for cubical abelian groups with c* *onnections [28]. 7 The equivalence of categories Let Crs, !-Gpd denote respectively the categories of crossed complexes and !-gr* *oupoids. A major part of the work consists in defining these categories and proving their equivalence, which* * thus gives an example of two algebraically defined categories whose equivalence is non trivial. It is even m* *ore subtle than that because the functors fl : Crs! !- Gpd, ~ : !- Gpd ! Crsare not hard to define, and it is ea* *sy to prove fl~ ' 1. The hard part is to prove ~fl ' 1, which shows that an !-groupoid G may be reconstructed* * from the crossed complex fl(G) it contains. The proof involves using the connections to construct a `fol* *ding map' : Gn ! Gn , and establishing its major properties, including the relations with the composition* *s. This gives an algebraic form of some old intuitions of several ways of defining relative homotopy groups, for e* *xample using cubes or cells. 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 condition, namely that every box has a unique thin fille* *r. This result plays a key role in the proof of the GSVKT for %, since it is used to show an independence of choic* *e. 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 constr* *uction and the above result, that x is degenerate in direction n + 1. Hence the two ends in that direction coinci* *de. Properties of the folding map are used also in showing that (X*) is actual* *ly included in %(X*); in relating two types of thinness for elements of %(X*); and in proving a homotopy addition* * lemma in %(X*). 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 N2 C of C. It is a conclusion of the theory that we ca* *n also obtain N2 C as (N2 C)n = Crs( (In*), C) where In*is the usual geometric cube with its standard skeletal filtration. The* * (cubical) geometric realisation |N2 C| is also called the cubical classifying space B2C of the crossed complex * *C. The filtration C* of C by skeleta gives a filtration B2C* of B2C and there is (as in 1.1.6) a natural iso* *morphism (B2C*) ~=C. Thus the properties of a crossed complex are those that are universally satisfied by* * (X*). These proofs use the equivalence of the homotopy categories of cubical Kan complexes and of CW -comp* *lexes. We originally took 12 this from the Warwick Masters thesis of S. Hintze, but it is now available with* * a different proof from Antolini [3]. 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 commutati* *ve. 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 %(|K|*) may be id* *entified with %(K), the free !-Gpd on the cubical set K, where here |K|* is the usual filtration by skeleta. On th* *e other hand, our proof that (|K|*) is the free crossed complex on the non-degenerate cubes of K uses the g* *eneralised SVKT 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 -complexes [4]* * and of globular 1-groupoids [24]. In fact the published paper on the classifying space of a crossed complex* * [27] is given in simplicial terms, in order to link more easily with well known theories. 8 Main aim of the work: colimit, or local-to-global, 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) the theory* *. We now explain these theorems in a way which strengthens the relation with descent. We suppose given an open cover U = {U~}~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 xX E xX E ____//_//_//_E_xX/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 = X1 , an* *d so an augmented simplicial filtered space ~C(q*) involving multiple intersections U* of the induced filter* *ed spaces. We still need a connectivity condition. Definition 8.1A filtered space X* is connected if and only if for all n > 0 the* * induced map ß0X0 ! ß0Xn is surjective and for all r > n > 0 and 2 X0, ßn(Xr, Xn, ) = 0. Theorem 8.2 (MAIN RESULT (GSVKT)) If U* is connected for all finite intersecti* *ons U of the elements of the open cover, then (C) (connectivity) X* is connected, and (I) (isomorphism) the following diagram as part of %(C~(q*)) %(q*) %(E*xX* E*) ____//_//_%(E*)//_%(X*). * * (c%) 13 is a coequaliser diagram. Hence the following diagram of crossed complexes (q*) (E*xX* 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 the two* * papers [22, 23], of 58 pages together. With this we deduce in the first instance: o the usual SVKT for the fundamental groupoid on a set of base points; o the Brouwer degree theorem (ßnSn = Z); o the relative Hurewicz theorem; o Whitehead's theorem that ßn(X [ {e2~}, X) is a free crossed module; o a more general excision result on ßn(A[B, A, x) as an induced module (cros* *sed 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 con- trasts with some expositions of basic homotopy theory, where the proof of say t* *he relative Hurewicz theorem requires knowledge of singular homology theory. Of course it is surprising to g* *et this last theorem without homology, but this is because it is seen as a statement on the morphism of rela* *tive 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 geometry and underlying pro* *cesses more closely than those in common use. Note also that these results cope well with the action of the fundamental g* *roup on higher homotopy groups. The calculational use of the GSVKT for (X*) is enhanced by the relation of* * with tensor products (see section 15 for more details). 9 The fundamental cubical !-groupoid %(X*) of a filtered space X* Here are the basic elements of the construction. In*: the n-cube with its skeletal filtration. Set Rn(X*) = FTop(In*, X*). This is a cubical set with compositions, connec* *tions, 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* 14 compositions a Oib defined for a, b 2 RnX* such that @+ia = @-ib inversions -i: Rn ! Rn. The connections are induced by flffi: In ! In-1 defined using the monoid st* *ructures max, min: I2 ! I. They are essential for many reasons, e.g. to discuss the notion of commutative * *cube. These operations have certain algebraic properties which are easily derived* * from the geometry and which we do not itemise here - see for example [2]. These were listed first in the Ba* *ngor thesis of Al-Agl [1]. (In the paper [22] the only connections needed are the +i, from which the -iare d* *erived using the inverses of the groupoid structures.) Definition 9.1 p : Rn(X*) ! %n(X*) = (Rn(X*)= ) is the quotient map, where f g 2 Rn(X*) means filter homotopic (i.e. through* * filtered maps) rel vertices of In. The following results are proved in [23]. 9.2 The compositions on R are inherited by % to give %(X*) the structure of cub* *ical multiple groupoid with connections. 9.3 The map p : R(X*) ! %(X*) 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 method of def* *orming a cube with the right faces `up to homotopy' into a cube with exactly the right faces, using the give* *n homotopies. In both cases, the assumption that the relation uses homotopies rel vertices is essential to sta* *rt the induction. (In fact the paper [23] does not use homotopy rel vertices, but imposes an extra condition J0, tha* *t each loop in X0 is contractible X1. A full exposition of the whole story is in preparation.) Here is an application which is essential in many proofs. Theorem 9.4 (Lifting multiple compositions)Let [ff(r)] be a multiple compositio* *n in %n(X*). Then repre- sentatives a(r)of the ff(r)may be chosen so that the multiple composition [a(r)* *] is well defined in Rn(X*). Proof: The multiple composition [ff(r)] determines a cubical map A : K ! %(X*) 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, *_____________//|R(X*). | _=|= | __ | | _ | | 0 _ | | A _ |p | _ | | _ _ | | _ | | _ | fflffl|_ fflffl| K _____A______//_%(X*) 15 Then K collapses to *, written K & *. By the fibration result, A lifts to A0, w* *hich 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. 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 that B = C [ a, C \ a = `a\ b (where `adenotes the union of the proper faces of a). We say B1 & Br, B1 collapses to Br, if there is a sequence B1 &e B2 &e . .&.eBr of elementary collapses. If C is a subcomplex of B then B x I & (B x {0} [ C x I) (this is proved by induction on dimension of B \ C). Further In collapses to any one of its vertices (this may be proved by indu* *ction on n using the first exam- ple). These collapsing techniques are crucial for proving 9.2, that %(X*) does * *obtain the structure of multiple groupoid, since it allows the construction of the extensions of filtered maps a* *nd filtered homotopies that are required. However, more subtle collapsing techniques using partial boxes are required* * to prove the fibration theorem 9.3, as partly explained in the next section. 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 o* *f C (called a base of B) and a number, possibly zero, of other (r - 1)-faces of C none of which is opposite 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 propert* *y and the following: Proposition 11.1Key Proposition: Let B, B0be 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 16 (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 a such a chain of partial boxes is one of the steps in the pro* *of of the fibration theorem 9.3. The proof of the fibration theorem gives a program for carrying out the def* *ormations 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 %n(X*) for n > 2. The proofs h* *ere use strongly results of [22]. We say ff is geometrically thin if it has a deficient representative, i.e. * *an a : In*! X* such that a(In) Xn-1. We say ff is algebraically thin if it is a multiple composition of degenera* *te elements or those coming from repeated negatives of connections. Clearly any composition of algebraically thi* *n elements is thin. Theorem 12.1 (i) algebraically thin geometrically thin. (ii) In a cubical !-groupoid with connections, any box has a unique thin fi* *ller. 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) uses the full force * *of the algebraic relation between !-groupoids and crossed complexes. * * 2 These results allow one to replace arguments with commutative cubes by argu* *ments with thin elements. 13 Sketch proof of the GSVKT We go back to the following diagram whose top row is part of %(C~(q*)) __@0_____//_ %(q*) %(E*xX* E*) _________//_%(E*)_______//_%(X*) * * (c%) @1 LLL Ø LL Ø LLL Øf0 fLLLLLL Ø LL&&fflffl G To prove this top row is a coequaliser diagram, we suppose given a morphism f * *: %(E*) ! G of cubical !-groupoids with connection such that f O @0 = f O @1, and prove that there is * *unique f0 : %(X*) ! G such that f0O %(q*) = f. 17 To define f0(ff) for ff 2 %(X*), you subdivide a representative a of ff to * *give a = [a(r)] so that each a(r) lies in an element U(r)of U; use the connectivity conditions and this subdivisi* *on to deform a into b = [b(r)] so that b(r)2 R(U(r)*) and so obtain fi(r)2 %(U(r)*). The elements ffi(r)2 G 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, deformed so th* *at 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 ff0in Rn(X*) gives a deficient element of * *Rn+1(X*). The method is to do the subdivision and deformation argument on such a homo* *topy, push the little bits in some %n+1(U~*) (now thin) over to G, combine them and get a thin element ø 2 Gn+1 all of whose faces not involving the direction (n + 1) are thin because h was g* *iven to be a filter homotopy. An inductive argument on unique thin fillers of boxes then shows that ø is degener* *ate 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 GSVKT for %. 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 statemen* *t that h is thin. It would be very difficult to replace the above argument, on the composition of thin ele* *ments, by a higher dimensional manipulation of formulae such as that given in section 3 for a commutative 3-cu* *be. 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 18 [25]. The equivalence of categories implies then that the category Crsis also m* *onoidal 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 dimen* *sion 1 the left homotopies of morphisms, and in higher dimensions are forms of higher homotopies. The prec* *ise description of these is obtained of course by tracing out in detail the equivalence of categories. It s* *hould be emphasised that certain choices are made in constructing this equivalence, and these choices are reflec* *ted in the final formulae that are obtained. An important result is that if X*, Y* are filtered spaces, then there is a * *natural transformation j : %(X*) %(Y*)! %(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 se* *e, in this cubical setting, that j is well defined. It can also be shown using previous results that j is a* *n isomorphism if X*, Y* are the geometric realisations of cubical sets with the usual skeletal filtration. The equivalence of categories now gives a natural transformation of crossed* * complexes j0 : (X*) (Y*) ! (X* Y*). * * (3) It would be hard to construct this directly. It is proved in [27] that j0is an * *isomorphism if X*, Y*are the skeletal filtrations of CW -complexes. The proof uses the GSVKT, and the fact that A -* * on crossed complexes has a right adjoint and so preserves colimits. It is proved in [6] that j is an iso* *morphism 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 c* *omplexes. Definition 14.1A bimorphism ` : (A, B) ! C of crossed complexes is a family of * *maps ` : Am x 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,ib0)fm > 1, n = 1 , ( `(a, b) + `(a0, b) ifm > 1, n = 0 orm > 2, n > 1 , `(a + a0,=b) 0 `(a0, b) + `(a, b)`(ai,fib)fm = 1, n > 1 . 19 (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)+i`(ffimfa,mb* *)> 2, n = 1 , >: - `(fia, b) - `(a, ffb) + `(ffa, b) + `(a,ifib)fm = n* * = 1 . (v) ( `(a, ffinb)ifm = 0, n > 2 , ffim+n (`(a,=b)) `(ffim a,ib)fm > 2, n = 0 . (vi) ff(`(a, b)) = `(a, ffb) and fi(`(a, b)) =i`(a,ffib)m = 0, n =* * 1 , ff(`(a, b)) = `(ffa, b) and fi(`(a, b)) =i`(fia,fb)m = 1, n =* * 0 . The tensor product of crossed complexes A, B is given by the universal bimo* *rphism (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 de* *rivation of the tensor product via the category of cubical !-groupoids with connections, and the formulae are * *forced by our conventions for the equivalence of the two categories [22, 25]. The complexity of these formulae is directly related to the complexities of* * the cell structure of the product Em x En where the n-cell En has cell structure e0 if n = 0, e0 [ e1 if n = 1, a* *nd e0[ en-1[ en if n > 2. It is proved in [25] that the bifunctor - - is symmetric and that if a0 i* *s 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 indis* *creet 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 theory. 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 extends* * 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 GSVKT that if X* is a CW -complex with the skele* *tal filtration, then (X*) is the free crossed complex on the characteristic maps of the cells of X*. It i* *s proved in [27] that the tensor product of free crossed complexes is free. 20 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 homological algeb* *ra show that free crossed resolutions of a group are unique up to homotopy equivalence. 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 back to W* *hitehead [84] and Wall [80], and which is discussed further by Baues in [5, Chapter VI, x7]: Theorem 15.1 Let X* be a CW -filtered space, and let OE : ßX* ! 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 ßY* ~= C of crossed complexes with preferred basis, such that OE is realised by a homo* *topy equivalence X* ! Y*. In fact, as pointed out by Baues, Wall states his result in terms of chain * *complexes, but the crossed complex formulation seems more natural, and avoids questions of realisability in dimens* *ion 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 [5, p.657] an extension of these results which we * *can apply to the realisation of morphisms of free crossed resolutions. Proposition 15.3Let X = K(G, 1), Y = K(H, 1) be CW -models of Eilenberg - Mac L* *ane spaces and let h : (X*) ! (Y*) be a morphism of their fundamental crossed complexes with the* * preferred 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 poi* *nted homotopy classes X ! Y is bijective with the morphisms of groups A ! B. The result follows from * *[5, p.657,(**)] (`if f is -realisable, then each element in the homotopy class of f is -realisable'). * * 2 These results are exploited in [71, 34], to calculate free crossed resoluti* *ons of the fundamental groupoid of a graph of groups. An algorithmic approach to the calculation of free crossed resolutions for * *groups is given in [36], by con- structing partial contracting homotopies for the universal cover at the same ti* *me as constructing this universal cover inductively. 16 Classifying spaces and 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 classification theo* *rem: 21 Theorem 16.1 If K is a cubical set, and G is an !-groupoid, then there is a nat* *ural bijection of sets of homotopy classes [|K|, |UG|] ~=[%(|K|*), G], where on the left hand side we work in the category of spaces, and on the right* * in !-groupoids. Here |K|* 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) = |UN2 C| = |U~C|. The properties already stated now give the homotopy cl* *assification theorem 1.1.7. It is shown in [23] that if Y is a connected CW -complex, then there is a m* *ap p : Y ! B2 Y* whose homotopy fibre is n-connected if ßiY = 0 for 2 6 i 6 n - 1. It follows that i* *f 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*]. * * (4) This result, due to Whitehead [83], translates a topological homotopy classific* *ation problem to an algebraic one. We explain below how this result can be translated to a result on chain co* *mplexes 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 by B (C) = |N (C)|. The properties of this simplicial classifying space are developed in [27], and * *in particular an analogue of 1.1.7 is proved. The simplicial nerve and an adjointness Crs( (L), C) ~=Simp(L, N (C)) are used in [17, 18] for an equivariant homotopy theory of crossed complexes an* *d their classifying spaces. Im- portant ingredients in this are notions of coherence and an Eilenberg-Zilber ty* *pe theorem for crossed complexes proved in Tonk's Bangor thesis [79]. Labesse in [65] defines a crossed set. In fact a crossed set is exactly a c* *rossed module ffi : C ! XoG where G is a group acting on the set X, and X o G is the associated actor groupoid; t* *hus the simplicial construction from a crossed set described by Larry Breen in [65] is exactly the nerve of the* * crossed module, regarded as a crossed complex. Hence the cohomology with coefficients in a crossed set used* * in [65] is a special case of cohomology with coefficients in a crossed complex, dealt with in [27]. (We are * *grateful to Breen for pointing this out to us in 1999.) 22 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*Xe*of cellular chains of the universal cover e* *X*of a reduced CW -complex X*. The group of operators here is the fundamental group of the space X. J.H.C. Whitehead in [83] gave an interesting relation between his free cros* *sed complexes (he called them `homotopy systems') and such chain complexes. We refer later to his important h* *omotopy classification results in this area. Here we explain the relation with the Fox free differential calcu* *lus [53]. Let ~ : M ! P be a crossed module of groups, and let G = Coker~. Then there* * is an associated diagram ~ ffi M ______//P_____//G * * (5) | | h2|| h1| |h0 fflffl| fflffl| fflffl| Mab _@2_//_Dffi@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 ! Dffiis the universal OE-derivation, that is it satisfies * *h1(pq) = h1(p)ffiq+ h1(q), for all p, q 2 P , and is universal for this property; and h0is the usual derivation g * *7! g-1. Whitehead in his Lemma 7 of [83] gives this diagram in the case P is a free group, when he takes Dffito * *be the free G-module on the same generators as the free generators of P . Our formulation, which uses the derive* *d module due to Crowell [46], includes his case. It is remarkable that diagram (5)is a commutative diagram in* * which the vertical maps are operator morphisms, and that the bottom row is defined by this property. The pr* *oof in [26] follows essentially Whitehead's proof. The bottom row is exact: this follows from results in [46], * *and is a reflection of a classical fact on group cohomology, namely the relation between central extensions and th* *e Ext functor, see [69]. In the case the crossed module is the crossed module ffi : C(!) ! F (X) derived from a* * presentation of a group, then C(!)abis isomorphic to the free G-module on R, Dffiis the free G-module on X, a* *nd it is immediate from the above that @2 is the usual derivative (@r=@x) of Fox's free differential calcul* *us [53]. Thus Whitehead's results anticipate those of Fox. It is also proved in [83] that if the restriction M ! ~(M) of ~ has a secti* *on which is a morphism but not necessarily a P -map, then h2 maps Ker~ isomorphically to Ker@2. This allows ca* *lculation of the module of identities among relations by using module methods, and this is commonly exploi* *ted, see for example [51] and the references there. Whitehead introduced the categories CW of reduced CW -complexes, HS of homo* *topy systems, and FCC of free chain complexes with a group of operators, together with functors 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 category of F* *CC. Further, C X* is isomorphic to the chain complex C*Xe*of cellular chains of the universal cover * *of X, so that under these circumstances there is a bijection of sets of homotopy classes [ X*, Y*] ! [C*Xe*, C*eY*]. * * (6) This with the bijection (4)can be interpreted as an operator version of the Hop* *f classification theorem. It is surprisingly little known. It includes results of Olum [73] published later, an* *d it enables quite useful calculations 23 to be done easily, such as the homotopy classification of maps from a surface t* *o the projective plane [52], and other cases. Thus we see once again that this general theory leads to specific * *calculations. All these results are generalised in [26] to the non free case and to the n* *on reduced case, which requires a groupoid of operators, thus giving functors FTop -! Crs-! Chain. One utility of the generalisation to groupoids is that the functor then has a* * right adjoint, and so preserves colimits. An example of this preservation is given in [26, Example 2.10]. The c* *onstruction of the right adjoint to 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 (6) has to be quite careful, since it works in the groupoid case, and not all morph* *isms of the chain complex are realisable. This analysis of the relations between these two categories is used in [27]* * to give an account of cohomology with local coefficients. 18 Crossed complexes and simplicial groups and groupoids The Moore complex NG of a simplicial group G is in general not a (reduced) cros* *sed complex. Let DnG be the subgroup of Gn generated by degenerate elements. Ashley showed in his thesi* *s [4] that NG is a crossed complex if and only if (NG)n \ (DG)n = {1} for all n > 1. Ehlers and Porter in [49, 50] show that there is a functor C from simplicia* *l 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 [43]. An important observation in [75] 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 the structure o* *f crossed module. This is relevant to algebraic K-theory, where for a ring R the homotopy fibration se* *quence is taken to be F ! B(GL(R)) ! B(GL(R))+. 19 Other homotopy multiple groupoids Although the proof of the GVKT outlined earlier does seem to require cubical me* *thods, there is still a question of the place of globular and simplicial methods in this area. A simplicial ana* *logue of the equivalence of categories is given in [4, 72], using Dakin's notion of simplicial T-complex. H* *owever it is difficult to describe in detail the notion of tensor product of such structures, or to formulate a pr* *oof of the colimit theorem in that context. 24 It is easy to define a homotopy globular set %O (X*) of a filtered space X** * but it is not quite so clear how to prove directly that the expected compositions are well defined. However ther* *e is a natural graded map i : %O (X*) ! %(X*) * * (7) and applying the folding map of [1, 2] analogously to methods in [23] allows on* *e to prove that i of (7)is injective. It follows that the compositions on %(X*) are inherited by %O (X*) t* *o make the latter a globular !-groupoid. Loday in 1982 [66] defined the fundamental catn-group of an n-cube of space* *s, and showed that catn-groups model all reduced weak homotopy (n + 1)-types. Joint work [33] formulated and p* *roved a GSVKT for the catn-group functor from n-cubes of spaces and this allows new local to global c* *alculations of certain homotopy n-types [13]. This work obtains more powerful results than the purely linear th* *eory of crossed complexes, which has however other advantages. Porter in [75] gives an interpretation of L* *oday's results using methods of simplicial groups. There is clearly a lot to do in this area. Recently some absolute homotopy 2-groupoids and double groupoids have been * *defined, see [19] and the references there, and it is significant that crossed modules have been used in * *a differential topology situation by Mackaay and Picken [68]. Reinterpretations of these ideas in terms of double gr* *oupoids are started in [16]. It seems reasonable to suggest that in the most general case double groupoi* *ds are still somewhat mysterious objects. 20 Conclusion and questions o The emphasis on filtered spaces rather than the absolute case is open to ques* *tion. o Mirroring the geometry by the algebra is crucial for conjecturing and pro* *ving universal properties. o Thin elements are crucial as modelling commutative cubes, a concept not s* *o easy to define or handle algebraically. o Colimit theorems give, when they apply, exact information even in non com* *mutative situations. The implications of this for homological algebra could be important. o One construction inspired eventually by this work, the non abelian tensor* * product of groups, has a bibli- ography of 78 papers since it was defined with Loday in 1985. o Globular methods do fit into this scheme, but so far have not yielded new* * results. o For computations we really need strict structures (although we do want to* * compute invariants of homotopy colimits). o In homotopy theory, identifications in low dimensions can affect high dim* *ensional homotopy. So we need structure in a range of dimensions to model homotopical identifications algebra* *ically. 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