Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions. * Ronald Browny University of Wales, Bangor. February 26, 2006 Abstract This paper illustrates the themes of the title in terms of: van Kampen * *type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotop* *y groupoids. Interaction with work of the writer is explored. 1 1 Introduction It is a pleasure to honour Charles Ehresmann by giving a personal account of so* *me of the major themes in his work which interact with mine. I hope it will be useful to sugges* *t how these themes are related, how the pursuit of them gave a distinctive character to his aims and h* *is work, and how they influenced my own work, through his writings and through other people. Ehresmmann's work is so extensive that a full review would be a great task, * *which to a considerable extent is covered by Andr'ee Ehresmann in her commentaries in the collected wor* *ks [24]. His wide vision is shown by his description of his overriding aim as: `To find the struc* *ture of everything'. `To find structure' is related to the Bourbaki experience and aim, in which he was * *a partner. A description of a new structure is in some sense a development of part of a new language: th* *e aim of doing this contrasts with that of many, who feel that the development of mathematics is ma* *inly guided by the solution of famous problems. The notion of structure is also related to the notion of analogy. It one of * *the triumphs of category theory in the 20th century to make progress in unifying mathematics through the* * finding of analogies between the behaviours of structures across different areas of mathematics. ___________________________________* Expansion of an invited talk given to the 7th Conference on the Geometry and* * Topology of Manifolds: The Mathe- matical Legacy of Charles Ehresmann, Bedlewo 8.05.2005-15.05.2005 (Poland) yPartially supported by a Leverhulme Emeritus Fellowship 2002-2004 1KEYWORDS: Ehresmann, local-to-global; fundamental groupoid; van Kampen theor* *em; holonomy and monodromy; higher homotopy groupoids. MSClass2000: 01A60,53C29,81Q70,22A22,55P15 1 This theme is elaborated in the article [15]. That article argues that many * *analogies in mathemat- ics, and in many other areas, are not between objects themselves but between th* *e relations between objects. Here we mention only the notion of pushout, which we use later in disc* *ussing the van Kampen Theorem. A pushout has the same definition in different categories even though * *the construction of pushouts in these categories may be wildly different. Thus a concentration on t* *he constructions rather than on the universal properties may lead to a failure to see the analogies. Ehresmann developed new concepts and new language which have been very influ* *ential in mathe- matics: I mention only fibre bundles, foliations, holonomy groupoid, germs, jet* *s, Lie groupoids. There are other concepts whose time perhaps is just coming or has yet to come: includ* *ed here might be ordered groupoids, multiple categories. In this direction of developing language, we can usefully quote G.-C. Rota [* *37, p.48]: "What can you prove with exterior algebra that you cannot prove without it* *?" Whenever you hear this question raised about some new piece of mathematics, be assu* *red that you are likely to be in the presence of something important. In my time, I have he* *ard it repeated for random variables, Laurent Schwartz' theory of distributions, ideles an* *d Grothendieck's schemes, to mention only a few. A proper retort might be: "You are right* *. There is nothing in yesterday's mathematics that could not also be proved without i* *t. Exterior algebra is not meant to prove old facts, it is meant to disclose a new wor* *ld. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectur* *es. 2 Local-to-global questions Ehresmann developed many new themes in category theory. One example is structur* *ed categories, with principal examples those of differentiable categories and groupoids, and o* *f multiple categories. His work on these is quite disparate from the general development of category theor* *y in the 20th century, and it is interesting to search for reasons for this. One must be the fact that* * he used his own language and notation. Another is surely that his early training and motivation came fr* *om analysis, rather than from algebra, in contrast to the origins of category theory in the work of* * Eilenberg, Mac Lane and of course Steenrod, centred on homology and algebraic topology. Part of the* * developing language of category theory became essential in those areas, but other parts, such as th* *at of algebraic theories, groupoids, multiple categories, were not used till fairly recently. It seems likely that Ehresmann's experience in analysis led him to the major* * theme of local-to- global questions. I first learned of this term from Dick Swan in Oxford in 1957* *-58, when as a research student I was writing up notes of his Lectures on the Theory of Sheaves [40]. D* *ick explained to me that two important methods for local-to-global problems were sheaves and spectr* *al sequences_he was thinking of Poincar'e duality, which is discussed in the lecture notes, and* * the more complicated Dolbeaut's theorem for complex manifolds. But in truth such problems are centra* *l in mathematics, science and technology. They are fundamental to differential equations and dyna* *mical systems, for example. Even deducing consequences of a set of rules is a local-to-global pro* *blem: the rules are 2 applied locally, but we are interested in the global consequences. My own work on local-to-global problems arose from writing an account of the* * Seifert-van Kampen theorem on the fundamental group. This theorem can be given as follows, as fir* *st shown by R.H. Crowell: Theorem 2.1 [19] Let the space X be the union of open sets U, V with intersect* *ion W , and suppose W, U, V are path connected. Let x0 2 W . Then the diagram of fundamental group * *morphisms induced by inclusions ss1(W, x0)i__//ss1(U, x0) j|| || * * (1) fflffl| fflffl| ss1(V, x0)__//ss1(X, x0) is a pushout of groups. Here the `local parts' are of course U, V put together with intersection W and * *the result describes completely, under the open set and connectivity conditions, the nonabelian fund* *amental group of the global space X. This theorem is usually seen as a necessary part of basic algeb* *raic topology, but one without higher dimensional analogues. In writing the first 1968 edition of the book [7], I noted that to compute t* *he fundamental group of the circle one had to develop something of covering space theory. Although t* *hat is an excellent subject in its own right, I became irritated by this detour. After some time, I* * found work of Higgins on groupoids, [25], which defined free products with amalgamation of groupoids,* * and this led to a more general formulation of theorem 2.1 as follows: Theorem 2.2 [3] Let the space X be the union of open sets U, V with intersecti* *on W , and suppose W, U, V are path connected. Let X0 be a subset of W meeting each path component* * of W . Then the diagram of fundamental group 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. Here ss1(X, X0) is the fundamental groupoid of X on a set X0 of base points: so* * it consists of homotopy classes rel end points of paths in X joining points of X0\ X. In the case X is the circle S1, one chooses U, V to be slightly extended se* *micircles including X0 = {+1, -1}. The point is that in this case W = U \ V is not path connected a* *nd so it is not clear where to choose a single base point. The day is saved by hedging one's bets, an* *d using two base points. The proof of theorem 2.1 uses the same tricks as to prove theorem 2.2, but in a* * broader context. In order to compute fundamental groups from this theorem, one can set up some gene* *ral combinatorial 3 groupoid theory, see [7, 26]. A key feature of this theory is the groupoid I, t* *he indiscrete groupoid on two objects 0, 1, which acts as a unit interval object in the category of gr* *oupoids. It also plays a r^ole analogous to that of the infinite cyclic group C in the category of group* *s. One then compares the pushout diagrams, the first in spaces, the second in groupoids: {0, 1}___//_{0} {0, 1}___//_{0} | | | | | | | | * * (3) fflffl| fflffl| fflffl| |fflffl [0, 1]___//_S1 I ___u__//_C to see how this version of the van Kampen Theorem gives an analogy between the * *geometry, and the algebra provided by the notion of groupoid. This kind of result is seen as `cha* *nge of base' in [5]. The fundamental group is a kind of anomaly in algebraic topology because of * *its nonabelian nature. Topologists in the early part of the 20th century were aware that: o the non commutativity of the fundamental group was useful in applications; o for path connected X there was an isomorphism H1(X) ~=ss1(X, x)ab; o the abelian homology groups existed in all dimensions. Consequently there was a desire to generalise the nonabelian 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 groun* *ds ~Cech was persuaded to withdraw his paper, so that only a small paragraph appeared in the Proceedin* *gs [17]. We now see the reason for the commutativity as the result (Eckmann-Hilton) that a group in* *ternal to the category of groups is just an abelian group. Thus the vision of a non commutative higher* * dimensional version of the fundamental group has since 1932 been generally considered to be a mirag* *e. Theorem 2.2 is also anomalous: it is a colimit type theorem, and so yields c* *omplete information on the fundamental groups which are contained in it, even in the non connected cas* *e, whereas the usual method in algebraic topology is to relate different dimensions by exact sequenc* *es or even spectral sequences, which usually yield information only up to extension. Thus exact seq* *uences by themselves cannot show that a group is given as an HNN-extension: however such a descripti* *on may be obtained from a pushout of groupoids, generalising the pushout of groupoids in diagram 3. It was then found that the theory of covering spaces could be given a nice e* *xposition using the notion of covering morphism of groupoids. Even later, it was found by Higgins a* *nd Taylor [27] that there was a nice theory of orbit groupoids which gave models of orbit spaces. The objects of a groupoid add a `spatial component' to group theory, which i* *s essential in many applications. This is evident in many parts of Ehresmann's work. Another view o* *f this anomalous success of groupoids is that they have structure in two dimensions, 0 with the * *objects and 1 with the 4 arrows. We have a colimit type theorem for this larger structure, and so a good* * model of the geometry. Useful information on fundamental groups is carried by the fundamental groupoid. It is therefore natural to seek for higher homotopy theory algebraic models * *which: o have structure in a range of dimensions; o contain useful information on classical invariants, and o satisfy van Kampen type theorems. That is, we seek nonabelian methods for higher dimensional local-to-global prob* *lems in homotopy theory. We return to this theme in section 4, which gives an indication of some* * of the motivation for the writer for higher dimensional algebra. 3 Holonomy and monodromy Once I had been led to groupoids by Philip Higgins, and having been told by G.W* *. Mackey in 1967 of his work on ergodic groupoids, [31], it was natural to consider topological gro* *upoids and differentiable groupoids, and to seek their properties and applications, [10, 11]. So I came across papers of Ehresmann, [21], and of Jean Pradines, [35], and * *in 1981 I visited Jean in Toulouse under British Council support, to try and understand something of h* *is papers. I saw the grand vision of the whole scheme of generalising the relation between Lie group* *s and Lie algebras to a relation between Lie groupoids and Lie algebroids, which has now become a lar* *ge theory, see for example [30]. We concentrated on his first paper [35], which states theorems bu* *t gives no indication of proofs. What I found remarkable was first of all the beautiful constructions* * Jean explained, which seemed to me clear in principle, and then the fact that he gave for holonomy an* *d monodromy universal properties: these are fairly rare in differential topology. Jean was interested in the monodromy principle, which involves the following* * situation: M(G,;WF); w w w | FF 0 i0w p|| FfF ww | F w | F w i fflffl| F## W ___________//____________77_______________________* *__________________GH ________________________________________________* *____________________________________________________________ _____________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________________________ f Here G is a topological groupoid, W is an open subset of G which contains the s* *et of identities G0 of G, and i is the inclusion. The aim is to find a topological groupoid M(G), c* *alled the monodromy groupoid of G, with morphism of topological groupoids p : M(G, W ) ! G and a `l* *ocal morphism' i0: W ! M(G, W ) such that for any local morphism f : W ! H to a topological gr* *oupoid H, f 5 `locally extends' to f0: M(G, W ) ! H. The existence such M(G, W ), p, i0with t* *his universal property is called the monodromy principle. This idea was expressed in Chevalley's famous book on `Lie groups', [18], fo* *r the case the groupoid G is the trivial groupoid X x X on a manifold X, when M(G, W ) becomes, for suf* *ficiently small W , the fundamental groupoid of X and p : M(G, W ) ! G is the `anchor map' as defin* *ed in Mackenzie's book [30]. Pradines is the first, I believe, to see how to extend it this notio* *n to the case of a general groupoid, as announced in [35]. We have to explain the term `local morphism'. Note that if u, v 2 W and uv i* *s defined in G, this does not necessarily mean that uv 2 W . So the algebraic structure that W inher* *its from G is what is called a `local groupoid structure'. A local morphism is therefore one that pre* *serves the local groupoid structure. It is easy to construct algebraically the groupoid M(G, W ) and p : M(G, W )* * ! G so that it has the required universal property, algebraically, as follows. The source, target * *and identity maps for G induce on W the structure of a reflexive graph. So one forms F (W ), the free g* *roupoid on this reflexive graph, the reflexive condition ensuring that identities in W become identities * *in F (W ). There is a function j : W ! F (X) which sends w 2 W to the corresponding generator [w] 2 F* * (W ) and then one factors the groupoid F (W ) by the relations [u][v] = [uv] for all u, v 2 W suc* *h that uv is defined in W . This defines M(G, W ), and the function j induces the local morphism i0, wh* *ich is injective since it is determined by the inclusion i. The problem is how to topologise M(G, W ) so that it becomes a topological g* *roupoid for which the monodromy principle is satisfied not only algebraically but also with regar* *d to continuity, or differentiability. Pradines solved this by a beautiful holonomy construction, which he explaine* *d to me in 1981 during my visit, and which develops Ehresmann's ideas. He took the view that the pair (M(G, W ), W 0), where W 0= i0(W ), should be* * regarded as a `locally Lie groupoid', and that this raised the general problem of when a loca* *lly topological groupoid is extendible, i.e. is obtained from an open subset containing the identities o* *f a topological groupoid. (The term used in [35] is morceau d'un groupo"ide diff'erentiable.) We therefore start again and consider a groupoid G and a subset W of G such * *that W contains the identities of G and W has the structure of topological space and even of a * *manifold. We ask: what conditions should be put on W which ensure that G can be given the structure of* * topological or Lie groupoid for which W is an open set? In short, we ask is the pair (G, W ) exten* *dible? This is a classical question in the case G is a group, and the answer is giv* *en in books on topological groups. A topology on G is obtained by taking as subbase the sets gU for all U * *open in W and g 2 G. The conditions for this to give a topological group structure on G are fairly m* *ild, and are loosely that the algebraic operations on G should be as continuous on W as can be expec* *ted given that these operations are defined only partially on W . One of the reasons this works is that in a topological group, the left multi* *plication operator Lg : G ! G, given for g 2 G by u 7! gu, is a homeomorphism and so maps open set* *s to open sets. 6 This property of Lg no longer holds when G is a topological groupoid, becaus* *e Lg is not defined on all of G. This reflects the considerable change in moving from groups to gro* *upoids, that is, from algebraic operations always defined to those only partially defined. This is no* *t a loss of information: the wider concept has greater powers of expression. There is also a wide range of examples where (G, W ) is not extendible, many* * coming from the theory of foliations. For example, if F is the foliation of the M"obius Band M * *given by circles going once or twice round the band, and R is the equivalence relation determined by t* *he leaves, then R is a subset of M x M and as such is a topological groupoid. But R is not a submani* *fold of M x M, since it has self intersections. However the foliation structure does determine* * a locally Lie groupoid (R, W ). The argument for both these facts is spelled out in [14]. Given a groupoid ff, fi : G ' G0, then an admissible section s : G0 ! G of f* *f satisfies ff O s = 1 and fi O s : G0 ! G0 is a bijection. We follow Mackenzie in [30] in calling s a* * bisection. We can also regard s as a homotopy 1 ' a where a : G ! G is an automorphism of G. This inte* *rpretation has intuitive value, and is suggestive for analogues in higher dimensions, as appli* *ed in [12]. Now suppose G is as above but G0 is also a topological space. Then a local b* *isection of G is a function s : U ! G with U open in G0 such that ff O s = 1U and fi O s maps U ho* *meomorphically to its image which is also open in G0. There is an `Ehresmannian composition' s * * *t of local bisections. We first make clear that if g : x ! y and h : y ! z in G then their composition* * in G is hg : x ! z. So the composition of local bisections is (s * t)(x) = (sfit(x))(tx) for x 2 G0. This means that in general the domain of s * t is smaller than that* * of t, and may even be empty. This composition makes the set of local bisections into an inverse se* *migroup. Recall that this is a semigroup in which for each element s there is a unique element s0, c* *alled a relative inverse for s, such that s0ss0= s0, ss0s = s. Pseudogroups, a concept first defined by * *Ehresmann in [20], give examples of such structures. We write this inverse semigroup as (G): of course* * it depends on the topology of G0. A left partial `adjoint' operation Lsof the local bisection s on G is define* *d by Ls(g) = (sfig)g, g 2 G. It is easy to prove that if G is a topological groupoid and s is a continuous l* *ocal bisection, then Ls maps open sets of G to open sets of G. Thus the important observation is that f* *or adjoint operations on topological or Lie groupoids it is not enough to rely on the elements of G: * * we need the local continuous or smooth bisections, which are kind of `tubes' rather than `element* *s', to transport the local structure of G. Now suppose given a pair (G, W ) such that G is a groupoid, G0 W G, W is* * a manifold, and the groupoid operations are `as smooth as possible' on W . By a smooth loca* *l bisection of (G, W ) we mean a local bisection s of G such that s takes values in W and is smooth. T* *he set of smooth local bisections forms a subset (r)(W ) where r denotes the class of different* *iability of the manifold M = G0. Pradines' key definition is to form the sub-inverse semigroup (r)(G, W ) of* * (G) generated by (r)(W ). My interpretation is that an element of (r)(W ) can be thought of as* * a local procedure and 7 an element of (r)(G, W ) can be thought of as an iteration of local procedures* *. Thus an iteration of local procedures need not be local, and this is one of the basic intuitions of * *non trivial holonomy. We say that (G, W ) is sectionable if for all w 2 W there is a smooth local * *bisection s in W whose domain includes ffw and with sffw = w. The next step is to form from (r)(G, W ) the associated sheaf of germs J(r)* *(G, W ): the elements of this are written [s]x where s 2 (r)(G, W ) and x 2 dom s. The inverse semi* *group structure on (r)(G, W ) induces a groupoid structure on J(r)(G, W ). This contains a su* *bgroupoid J0 whose elements are germs [s]x such that fisx = x and there is a neighbourhood U of x * *such that s|U 2 (r)(W ); in words, s is an iteration of local procedures about x which is stil* *l a local procedure. It is a proposition that J0(G, W ) is a normal subgroupoid of J(r)(G, W ). The holonomy* * groupoid Hol(G, W ) is defined to be the quotient groupoid J(r)(G, W )=J0. The class of [s]x in the* * holonomy groupoid is written x. There is a projection p : Hol(G, W ) ! G given by x 7! s(x). The intuition is that first of all W embeds in Hol(G, W ), by w 7! ffw, w* *here f is a local smooth bisection such that fffw = w, and second that Hol(G, W ) has enough local secti* *ons for it to obtain a topology by translation of the topology of W . Let s 2 c(G, W ). We define a partial function oes : W ! Hol(G, W ). The do* *main of oes is the set of w 2 W such that fiw 2 dom(s). The value oesw is obtained as follows. Choose * *a smooth bisection f through w. Then we set oesw = fiwffw= ffw. Then oesw is independent of the choice of the local section f. It is proved in * *detail in [2] that these oes form a set of charts for Hol(G, W ) making it into a Lie groupoid with a univer* *sal property. The books [30, 32] argue that the most efficient treatment of the holonomy g* *roupoid of a foliation is via the monodromy groupoid, which is itself defined using the fundamental gr* *oupoid of the leaves. However there are counter arguments. One fact is that their arguments do not so far obtain a monodromy principle,* * which is obtained by the opposite route in [35, 13]. Thus there is loss of a universal principle,* * with its potentiality for enabling analogies. The second problem is that the route through the fundamental groupoid is bas* *ed on paths, and so on the standard notion of a topological space, and its exemplification as a * *manifold. The Pradines' approach gives a clear realisation of the intuitive idea of it* *eration of local procedures, without requiring the notion of path to `carry' these procedures, as happens fo* *r example in the usual process of analytic continuation. It is possible that this idea would lead to * *wider applications of non abelian groupoid like methods for local-to-global problems. For example, th* *e following picture illustrates a chain of local procedures from a to b: a b We would like to be able to define such a chain, and equivalences of such chain* *s, without resource to 8 the notion of `path'. The claim is that a candidate for this lies in the constr* *uctions of Ehresmann and Pradines for the holonomy groupoid. Here are some final questions in this area. Can one use these ideas in other situations to obtain monodromy (i.e. analo* *gues of `universal covers') in situations where paths do not exist but `iterations of local proced* *ures' do? It is even possible that holonomy may exist in wider situations, but not monodromy. How widely useful is the notion of `locally Lie groupoid' as a context for d* *escribing local situations? Is there a locally Lie groupoid (G, W ) where G is an action groupoid? A further point is that Pradines was very keen on having a theory for germs * *of locally Lie groupoids. It is in these terms that the theorems in [35] are stated. Such work is not con* *sidered in [2, 13]. One could consider other structures on W and ask for bisections which preser* *ve these structures. Simple examples for degrees of differentiability, due to Pradines [36], are giv* *en in [2]. One might consider other geometric structures, such as Riemannian, or Poisson. `I,cen an* *d I, in looking for notions of double holonomy, have considered in [12] a double groupoid G and lin* *ear bisections. 4 Higher dimensions After writing out the proofs of theorem 2.2 a number of times to make the expos* *ition clear, it became apparent to me in 1966 that the method of proof ought to extend to higher dimen* *sions, if one had the right gadgets. So this was an idea of a proof in search of a theorem. The first* * search was for a higher dimensional version of the fundamental groupoid on a set of base points. It was at this point that I found Ehresmann's book, [22]. It was difficult t* *o understand, but it did contain a definition of double category, and a key example, the double cate* *gory 2C of commuting squares in a category C. What caused problems for me was to find any constructi* *on of a fundamental double groupoid of a space which really contained 2-dimensional homotopy inform* *ation. In fact a solution to this problem was not published till 2002, [9]. The construction * *of higher homotopy groupoids went in 1975 in a different direction, using pairs of spaces, and the* *n filtered spaces, in work with Philip Higgins, which is surveyed in [6]. As an intermediate step, I decided in 1970 to investigate the notion of doub* *le groupoid purely algebraically, to see whether the putative homotopy double groupoid functor wou* *ld take values in a category of some interest. Work with Chris Spencer, [16], found a relationship between double groupoids* * and the crossed modules introduced by J.H.C. Whitehead to discuss the second relative homotopy * *group and its boundary @ : ss2(X, A.a) ! ss1(A, a). We found a functor from crossed modules * *to a certain kind of double groupoid, which was later called edge symmetric, in that the groupoid* *s of vertical and of horizontal edges are the same. The next question was what kind of double groupo* *ids arose in this way? At the same time we were looking at conjectured proofs of a vaguely formulat* *ed 2-dimensional van 9 Kampen type theorem. It was clear that we needed to generalise commutative squa* *res to commutative cubes in the context of double groupoids, in such a way that any composite of c* *ommutative cubes is commutative. A square in a groupoid __a_//_ ___/2/ c|| |b| fflffl| fflffl|fflffl|//_1 d is commutative if and only if ab = cd, or, equivalently, a = cdb-1. We wanted a* * similar expression to the last in the case of a cube. If we fold flat five of the faces of a cube we * *get a net such as the first of the following: _____ _____ | | |- | ||- || | 1 | || 1|| ____|____|___ ____||__||____ __/2/_ | | | | | | | | |-2 | | | |-2 | | | 1fflffl| |___|____|___| |___|___|___|__ | | || || | | || || |____| ||__||_ which is not a composable set of squares. However in the second diagram we have* * noted by double lines adjacent pairs of edges which are the same. Therefore we must assume we have ca* *nonical elements, called connections, to fill the corner holes as in the diagram: _________________|_|_________________ ____| | ______ ____|-1|| |____ ____|___|_______ | | | | |-2 | | | |___|___|___|_______ ____| | ____ ____||_ | __|____ ____|___|____________________________ The rules for connections include the transport law: |________|| |__ | __| _____ || | __| | __ | |___|___|_= | | | | | __| |____| || || | | |___|___|_ borrowed from a law for path connections found in Virsik's paper [41]. Virsik was a student of Ehresmann. He left Czechoslovakia at the time of the* * Russian invasion, and went to Australia. There he met Kirill Mackenzie, and suggested to him Lie * *groupoids and Lie algebroids as a PhD topic. Hearing of Kirill's work, I managed to get British C* *ouncil support for him to visit Bangor in 1986, and this led to his continuing to work in the UK. In the paper [41], Virsik introduces the notion of path connection on a prin* *cipal bundle p : E ! B as follows. Let G be the Ehresmann groupoid EE-1 ' B of the principal bundle. T* *hus G(b, b0) can be identified with the bundle maps Eb! Eb0of the fibres over b, b0respectively. Le* *t B be the category of Moore paths on B, and let G be the category of Moore paths on G. This may b* *e combined easily to give the structure of a double category. 10 Virsik defines a path connection for the bundle E to be a function : B ! * * G satisfying two conditions. One is invariance under reparametrisation, and the other is kno* *wn as the transport law. Chris Spencer and I were amazed when this law was found to be exactly righ* *t as a law on the connections we needed for the boundary of a cube, and these have been central i* *n work on cubical higher groupoids ever since. So we found an equivalence between crossed modules* * and what we then called special double groupoids with special connections, [16]. Ehresmann had earlier shown in [23] that a 2-category gave rise to a double * *category of quintettes. Spencer showed in [38] that this gave an equivalence between 2-categories and e* *dge symmetric double categories with connections. This has been generalised to all dimensions in [1]* *, and there has been recent further work on commutative cubes by Higgins in [28] and Steiner [39]. What has yet to be accomplished is to find relations between the higher orde* *r connections in differential geometry and the geometry of cubes. One of the advantages of cubes for local-to-global questions is that they ha* *ve an easy to define notion of multiple composition. Analogous ideas in the globular case present co* *nceptual and technical difficulties. Multiple compositions allow easily an algebraic inverse to subdiv* *ision, and this is a key to certain local-to-global results, as outlined for example in [6]. However the al* *gebraic relations between the cubical, globular, and (in the groupoid case) crossed complexes, are an ess* *ential part of the picture. It is possible to dream of a unification of all these themes in a way which * *I believe Ehresmann would have favoured, but there still seems quite a way to go to realise this. W* *ork is in progress with Jim Glazebrook and Tim Porter on aspects of this, [8], and many others are work* *ing on stacks, gerbes and higher order groupoids. 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