Galois theory and a new homotopy double groupoid of a map of spaces Ronald Brown* George Janelidzey August 27, 2002 UWB Maths Preprint 02.18 Abstract The authors have used generalised Galois Theory to construct a homotopy* * double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to constru* *ct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2* *-groupoid, cat1-group or crossed module. An advantage of our construction is that the double groupo* *id can give an algebraic model of a foliated bundle.1 Introduction Our aim is to develop for any map q : M ! B of topological spaces the construct* *ion and properties of a new homotopy double groupoid which has the form of the left hand square in th* *e following diagram, while the right hand square gives a morphism of groupoids: ___s___// i1q æ2(q)______//ß1(M)oo___//_ß1(B) * * (1) ||OO| t |O|O| |O|O| ||| ||| ||| ||| ||| ||| ||| ||| ||| ffflffl|flffl|fflffl|fflffl|fflffl|fflffl||s//_|| Eq (q)_______//Moo___q____//B t where: ß1(M) is the fundamental groupoid of M; Eq(q) is the equivalence relation determined by q; and s, t are the source and target maps of the groupoids. Note that qs = qt and (ß1q)s = (ß1q)t, so that æ2(q) is seen as a double groupo* *id analogue of Eq(q). *_____________________________________ Mathematics Division, School of Informatics, University of Wales, Dean St., * *Bangor, Gwynedd LL57 1UT, U.K. email: r.brown@bangor.ac.uk yMathematics Institute, Georgian Academy of Sciences, Tbilisi, Georgia. 12000 Maths Subject Classification: 18D05, 20L05, 55 Q05, 55Q35 1 This double groupoid contains the 2-groupoid associated to a map defined by* * Kamps and Porter in [16], and hence also includes the cat1-group of a fibration defined by Loday* * in [17], the 2-groupoid of a pair defined by Moerdijk and Svensson in [19], and the classical fundament* *al crossed module of a pair of pointed spaces defined by J.H.C. Whitehead. Advantages of our constru* *ction are: (i) it contains information on the map q, and (ii) we get different results if the topology of M is varied to a finer topolog* *y. In particular, we can apply this construction in the case M is foliated by repl* *acing the topology on M by a finer one so that ß1M is replaced by the fundamental groupoid of the fol* *iation. This applies in particular to the Möbius Band with its standard foliation b* *y circles. We can extract from this double groupoid a small version D(M) with only three vertices, and wh* *ich seems to represent well many properties of the Möbius Band. It has basic vertices, edges and squar* *es as follows: B _`__//_C ___/2/ fflffl| j ||ff |,| 1 fflffl|fflffl| AO__'_//AOOO , ||fi |j| | | C _OE_//_B Note that the vertical groupoid for O1 is an indiscrete groupoid, while the hor* *izontal groupoid for O2 contains a copy of the infinite cyclic group, since there are compositions a1O2a2O2. .O.2an where the aiare alternately (-1ff) and fi. The idea for this double groupoid arose from the Generalised Galois Theory * *of Janelidze [14, 15], which under certain conditions gives a Galois groupoid from a pair of adjoint f* *unctors. The standard fundamental group arises from the adjoint pair between topological spaces and s* *ets given by discrete and ß0, see for example [8]. The adjoint pair between simplicial sets and cros* *sed complexes given by nerve and ß1 was studied in [9] and shown to lead to a Galois double groupoi* *d of a fibration of simplicial sets. We are now giving a topological version of this constructi* *on. We show that if p : E ! B is a Serre fibration then the fundamental groupoid ß1(E) has an addit* *ional compatible groupoid structure arising from the equivalence relation Eq(p) defined by the m* *ap p; these two groupoid structures define a double groupoid which we write fl(p), since it is defined b* *y methods of Galois theory. The double groupoid æ(q) arises by pullback by i where q = pi is the usual fact* *orisation of any map through a homotopy equivalence i and a fibration p. However the proof of the re* *lation of æ(q) with classical notions and compositions is tricky, and so is given in some detail. A* * further reason for this detail is the possibility that a modification of this construction could be use* *d in association with the `thin fundamental groupoids' and their smooth structures in differential geomet* *rical situations, as exemplified by Mackaay and Picken in [18]. Here is some background to the search for higher groupoid models of homotop* *ical structures (for more detailed references, see [3]). Geometers in the early part of the 20th cen* *tury were aware that in the connected case the first homology group was the fundamental group made abelian,* * and that homology 2 groups existed in all positive dimensions. Further, the fundamental group gave * *more information in geometric and analytic contexts than did the first homology group. They were th* *erefore interested in seeking higher dimensional versions of the non abelian fundamental group. E. ~C* *ech submitted to the 1932 ICM at Zurich a paper on higher homotopy groups, using maps of spheres. Ho* *wever these groups were quickly proved to be abelian in dimensions > 1, and on this ground ~Cech w* *as asked to withdraw his paper, so that only a small paragraph appeared [12]. Thus the dream of thes* *e topologists seemed to fail, and was widely felt to be a mirage, although the abelian higher homoto* *py groups became and still are very important. J.H.C. Whitehead in the 1940s introduced the notion of crossed module, usin* *g the boundary of the second relative homotopy group of a pair and the action of the fundamental grou* *p. He and Mac Lane showed that crossed modules classified (connected) homotopy 2-types. Crossed mo* *dules are indeed more complicated than groups, and they make a good candidate for `2-dimensional* * groups'. In the 1960s, Brown introduced the fundamental groupoid of a space on a set* * of base points, and the writing of his 1968 book on topology suggested to him that all of 1-dim* *ensional homotopy theory was better expressed in terms of groupoids rather than groups. This rai* *sed the question of the putative value of groupoids in higher homotopy theory. A relation of certai* *n double groupoids to crossed modules was worked out with C.B. Spencer in the early 1970s, and this s* *howed that double groupoids are indeed more complicated than groups. A definition of a homotopy d* *ouble groupoid of a pair of pointed spaces was made with P.J. Higgins in 1974, and exploited to o* *btain a 2-dimensional Van Kampen type theorem for this double groupoid, and hence for Whitehead's cro* *ssed module of a pair (see [4]). The double groupoid constructed in [4] is edge symmetric and ha* *s a connection, and so is not the same as that constructed here. A classification of certain double groupoids is given in [11], but this doe* *s not yield much information for the double groupoid considered here. Thus there is still a way to go in the* * understanding and in the use of double groupoids. Higher homotopy groupoids were defined by Brown and Higgins for a filtered * *space in [6], and by Loday for an n-cube of spaces in [17]; his catn-groups were shown there to mode* *l connected homotopy (n + 1)-types. These higher groupoid methods yield new calculations in homotop* *y theory through higher order Van Kampen theorems [6, 10], as well as suggesting new algebraic c* *onstructions. 1 Galois groupoids op Later we will be considering the category C = Sets of simplicial sets and the * *fundamental groupoid functor I = ß1 : C ! X from the category C to the category X = Grpdof (small) g* *roupoids. Further, C, an internal category in C, will be the particular simplicial category (actua* *lly groupoid) Eq(p) which is the equivalence relation (in C) determined by p where p : E ! B is a surject* *ive fibration of Kan complexes. Here we give first the general result, using this notation. Let I : C ! X be an arbitrary functor between categories C and X with pullb* *acks, and let i ____//_____//_ j C = C2 ____//_//_C1//_C0oo_ * * (2) be an internal category in C. We recall 3 Proposition 1.1 Suppose the canonical morphisms I(C1xC0C1) ! I(C1) xI(C0)I(C1) * * (3) I(C1xC0C1xC0C1) ! I(C1) xI(C0)I(C1) xI(C0)I(C1) * * (4) are isomorphisms. Then: (a) i j _____// ____//_ I(C) = I(C2)_____////_I(C1)//_I(C0)oo_ * * (5) is an internal category in X; (b) if C is a groupoid, then so is I(C). For a morphism p : E ! B in C, let Eq(p) = ____//_ ____//_ (E xB E) xE (E xB E) E xB E xB E____//_//_E xB_E//_Eoo_ * * (6) be the equivalence relation corresponding to p (=kernel pair of p) regarded as * *an internal groupoid in C. Applying Proposition 1.1 with C = Eq(p), we obtain: Corollary 1.2 Suppose the canonical morphisms I((E xB E) xE (E xB E)) ! I(E xB E) xI(E)I(E xB E) * * (7) I(E xB E) xE (E xB E) xE (E xB E)) ! I(E xB E) xI(E)I(E xB E) xI(E)I(E xB E* *) (8) are isomorphisms. Then I(Eq(p)) = ____//_ ____//_ I((E xB E) xE (E xB E)) I(E xB E xB E)___//_//_I(E xB_E)//_I(E)oo_* * (9) is an internal groupoid in X. This fact, which goes back to A. Grothendieck's observation "the fundamental gr* *oupoids are to be defined as quotients of equivalence relations", is used in categorical Galois t* *heory and its various special cases (see [1], [4], and other references in [1]), to define the Galois* * groupoid of (E, p) as GalI(E, p) = I(Eq(p)). * * (10) In particular this applies to the following situation studied by the authors be* *fore (see Proposition 3.5 in [2]): * * op Proposition 1.3 Let I : C ! X be the fundamental groupoid functor from the cate* *gory C = Sets of simplicial sets to the category X = Grpdof (small) groupoids, and p : E ! B * *a surjective fibration of Kan complexes. Then the morphisms (6) and (7) are isomorphisms and so the Ga* *lois groupoid (9) is well defined. Since the internal groupoids in Grpd are the same as double gr* *oupoids, it is a double groupoid. 4 2 From simplicial sets to topological spaces Consider the diagram oRo__ __I_//_ Top ____//_Setsoopo_Grpd * * (11) dSdIIII OO H tt:: II |y ttt r IIII||tttit t where o Top is the category of topological spaces, R is the geometric realisation * *functor, and S is its right adjoint, usually called the singular complex functor; o I ` H is the adjoint pair used in [2], i.e. I is the fundamental groupoid * *functor, and H the nerve functor; o y is the Yoneda embedding, r and i are the restrictions of R and I respect* *ively along y; explicitly, r is the singular simplex functor and i carries finite ordinals to codiscr* *ete groupoids on the same sets of objects. By the universal property of the Yoneda embedding, the two adjunctions of t* *he row are uniquely (up to isomorphisms) determined by r and i; let us also recall from [9] and [14* *, 15]: Proposition 2.1 (a) The composite IS : Top! Grpdcan be identified with the clas* *sical (geometric) fundamental groupoid functor ß1; (b) For every topological space X, S(X) is a Kan complex; (c) The S-image of a morphism p in Topis a Kan fibration if and only if p i* *tself is a Serre fibration. 3 What is the Galois double groupoid of a Serre fibration? Let p : E ! B be a Serre fibration of topological spaces. By Propositions 1.3 * *and 2.1, the Galois (double) groupoid GalI(S(E), S(p)) is well defined. Moreover, since the functor* * S being a right adjoint preserves pullbacks, we can write GalI(S(E), S(p)) GalSI(E, p) Gali1(E, p) * * (12) and conclude that Gali1(E, p) also is a well-defined double groupoid. We write * *this double groupoid as fl(p) to indicate the relation with Galois theory, and now describe it expli* *citly. The underlying double graph has the following description: o presented as an internal groupoid in Grpd, fl(p) is displayed as ____//_ ____//_ ß1((E xB E) xE (E xB E)) ß1(E xB E xB E)___//_//_ß1(E xB_E)//_ß1(E)* *oo_(13) from which we conclude: 5 o the set of objects in fl(p) = Gali1(E, p) is E; o a horizontal arrow e ! e0is a morphism e ! e0in ß1(E), i.e. a homotopy cla* *ss of a path from e to e0(recall that such a homotopy h : [0, 1] x [0, 1] ! E is required to* * have h(0, t) = e and h(1, t) = e0for every t in [0, 1]); o a vertical arrow e ! e0is just the pair (e, e0) provided p(e) = p(e0); o a square e1__OE//_e2 * * (14) | | | u | fflffl|fflffl| e01___//_e02 OE0 is a homotopy class of a path from (e1, e01) to (e2, e02) in E xB E; its v* *ertical domain OE and vertical codomain OE0are homotopy classes of paths from e1 to e2 and from * *e01to e02respectively, and the horizontal are pairs (e1, e01) and (e2, e02) respectively. Clearly there is no problem with the horizontal composition as well - since* * we know that ß1(ExB E) and ß1(E) are groupoids. The only non-trivial part of our construction is the v* *ertical composition of squares: Given e1_OE_//_e2___2// * * (15) | | fflffl| | u | 1 fflffl|fflffl| e01OE0_//_e02 | | | u0 | fflffl|fflffl| e001__//_e002 OE0 we have to define uO1u0, which must be (an equivalence class of) a path from (e* *1, e001) to (e2, e002) in ExB E. Of course we should choose a representative f of u, a representative f0of u0* *, a homotopy h between the vertical codomain path of f and the vertical domain path of f0, paste them * *together, and take the homotopy class of the resulting path in ExB E. However, how do we know that suc* *h an uO1u0does not depend on the choices we made? The nice consequence of the results above is tha* *t we do not need to prove this. Indeed, since the morphism ß1((E xB E)xE (E xB E)) ! ß1(E xB E)xi1(* *E)ß1(E xB E) is an isomorphism, the pair (u0, u) determines a path v in (E xB E)xE (E xB E), an* *d the desired vertical composite u0u is nothing but the image of v under the functor ß1((ExB E)xE (ExB* * E)) ! ß1(ExB E) induced by the composition map (E xB E) xE (E xB E) ! E xB E of Eq(p). Moreover* *, we do not have to worry about associativity of the horizontal composition, which in fact * *follows from ß1((E xB E) xE (E xB E) xE (E xB E)) ! ß1(E xB E) xi1(E)ß1(E xB E) xi1(E)ß1(E* * xB E) being an isomorphism. 6 Example 3.1 Suppose for example that p : E = (B x F ) ! B is the projection of* * a product, and so is a fibration. Then Eq(p) = E xB E is homeomorphic to the product B x (F x* * F ) and hence ß1(Eq(p)) is easily determined, together with its double groupoid structure. Th* *e vertical edges are triples (b, f- , f+ ), b 2 B, f 2 F , the horizontal edges are pairs (fi, OE) * *2 ß1B x ß1F and the squares are triples (fi, OE-, OE+) 2 ß1(B) x ß1(F ) x ß1(F ) with vertical boundaries given by @1 (fi, OE-, OE+) = (fi, OE ). The horizontal* * composition O2 is that of the fundamental groupoids; the vertical composition O1 reflects that of the product* * groupoid B x (F x F ) in which B is the discrete groupoid and F x F is the codiscrete groupoid. In the next sections we translate the construction for fl(p) where p is a f* *ibration into results for an arbitrary map q : M ! B by the usual factorisation process, and so relate th* *ese ideas to classical constructions. 4 The homotopy double groupoid of an arbitrary continuous map If C is an internal category (or a groupoid) in a category X with pullbacks, an* *d i : M ! C0 a mor- phism into the object-of-object of C, we can always form the "induced internal * *category (respectively groupoid)" i*(C) = ____//_ ____//_ C2xC0xC0xC0(M x M x M) ____//_//_C1xC0xC0(M x_M)_//_Moo_ * * (16) which, in the case X = Sets, can be simply described as the category with objec* *ts all elements of M, and morphisms ä s in C". In particular, for a topological space E and a subspac* *e M one defines the relative fundamental groupoid ß1(E, M) as i*(ß1(E)), where i : M ! E is the inc* *lusion map; in this paper we do not consider ß1(E) as a topological groupoid, but the same construc* *tion of ß1(E, M) can be repeated internally to Top. Let p : E ! B be a Serre fibration and i : M ! E an arbitrary map in Top. U* *sing the morphism ß1(i) : ß1(M) ! ß1(E) of fundamental groupoids, and having in mind that ß1(E) i* *s the object-of- objects of Gali1(E, p) (when it is regarded as an internal groupoid in Grpd), w* *e can construct what we are going to call the relative Galois double groupoid Gali1(E, p, M, i) as Gali1(E, p, M, i) = ß1(i)*Gali1(E, p). * * (17) A good reason for introducing this new double groupoid is that any continuous m* *ap q : M ! B can be presented as a composite q = pi above with i being a homotopy equivalence, and * *so any q determines such a double groupoid equivalent (in an appropriate sense) to the Galois doubl* *e groupoid of a Serre fibration. It is this double groupoid we have displayed in diagram (1). In the next sections we will identify this double groupoid in terms of homo* *topy classes of certain maps and so relate these ideas and the compositions obtained more clearly in te* *rms of classical homotopical ideas. One further reason for doing this is to allow the possibilit* *y of generalisations of these geometric constructions to situations where the appropriate Galois theory* * is not yet obtained, for example in terms of smooth structures and `thin fundamental groupoids', as * *in [18]. This could lead to smooth structures on variations of the constructions given here. 7 5 Geometric interpretation We now start to interpret the previous results geometrically in terms of compos* *itions related to classical notions of relative homotopy groups. To this end, we use standard notation for * *the cubical singular complex KM of a space M. Here KnM is the set of singular n-cubes in M (i.e. con* *tinuous maps In ! M with K0M identified with M). There are standard face maps @i : KnM ! Kn-* *1M and degeneracy maps "i : Kn-1M ! KnM for i = 1, . .,.n (with "1 : K0M ! K1M written* * simply ", and giving for x 2 M the constant path "x at x). There are also for i = 1, . .,* *.n compositions a Oib defined for a, b 2 KnM such that @+ia = @-ib, and inversions -i: Kn ! Kn. Final* *ly we shall later use connections ffi: Kn-1M ! KnM for i = 1, . .,.n, ff = induced by the maps* * flffi: In ! In-1 defined by flffi(t1, t2, . .,.tn) = (t1, t2, . .,.ti-1, A(ti, ti+1), ti* *+2, . .,.tn) where A(s, t) = max(s, t), min(s, t) as ff = -, + respectively. The enrichment * *with connections +ifor the traditional cubical sets was introduced in [5]. The full properties of thes* *e structures are set out in for example [1]. Here we will assume only the obvious geometric properties in t* *he range n = 0, . .,.3. Let q : M ! B be a map of spaces. We recall the standard factorisation q = * *pi where i : M ! Mq is a homotopy equivalence and p : Mq ! B is a fibration. Here Mq = {(x, ~) | ~ : I ! B, ~(0) = q(x)} M x BI and p(x, ~) = ~(1), while i : x 7! (x, "(q(x)) where "(q(x)) is the constant pa* *th at q(x) in B. A point of MqxB Mq is a pair ((x, ~), (x0, ~0)) with ~(1) = ~0(1) as in the following p* *icture: x o q(x) o |~| fflffl|oOO |~0| q(x0)o| x0 o So Mq xB Mq is homeomorphic to the space M0 of triples (x, ~, x0) 2 M x K1B x M* * such that ~(0) = q(x), ~(1) = q(x0). Hence we have a groupoid structure on M0 with object* * set Mq, where the source and target maps s0, t0send (x, ~, x0) to (x, ~1), (x0, -~2) where ~1, ~2* * 2 K1B are respectively the rescaled forms of the first half and the second half of ~. The composition * *O0in this groupoid is, when defined, given by (x, ~, x0) O0(x0, ~0, x00) = (x0, ~1O1~2, x00) where O1 is here the usual composition of paths. Of course this composition is * *defined if and only if ~2 = -1~1. Let j : M xB M ! M0 be given by (x, x0) 7! (x, "(qx), x0). The set of paths* * I ! M0 with end points in Im j can be identified with the subset R2(q) of K1M x K2B x K1M of tr* *iples (f, ff, g) 8 such that qf = @-1ff, qg = @+1ff and @-2ff, @+2ff are constant paths. Thus an e* *lement of R2(q) may be pictured as: _______f |q ____|?_ ||| |||| |||| |||| ||||ff||||||| ___-|2 ||||| ||||| |? _______||||| 1 |6q _______| g where the dotted lines show constant paths. Thus R2(q) fits in the following di* *agram: ______//_ R2(q)______//_K1Moo_OOOO * * (18) |||| ||| |||| ||| |||| ||| ___/2/ |||| ||| fflffl| fffflffl|fflffl|lffl|fflffl|fflffl||//_|1 Eq(q)________//Moo_ The boundary maps are given by: @-1(f, ff,=g)f, @+1(f, ff,=g)g, @-2(f, ff,=g)(f(0), g(0)), @+2(f, ff,=g)(f(1), g(1)). The degeneracy maps "1 : K1M ! R2(q), "2 : Eq(q) ! R2(q) are given by: "1(f)= (f, "1(qf), f), "2(x, x0)= ("(x), "21(q(x)), "(x0)), for (f, ff, g) 2 R2(q), (x, x0) 2 M xB M. Clearly (18)can be considered as a diagram of reflexive graphs. We now exam* *ine compositions on R2(q). The set R2(q) has two partial compositions. The composition O2 is determin* *ed by the usual composition of paths and squares in this direction: (f, ff, g) O2(f0, ff0, g0) = (f O1f0, ff O2ff0, g O1g0). The composition O1 in the direction 1 is given by (f, ff, g) O1(g,=fi,(h)f, ff O1fi, h). * * (19) 9 Note that this definition generalises a construction by Kamps and Porter in* * [16, Section 4.1], in which they assume f(0) = g(0), f(1) = g(1) whereas in our situation we have * *only qf(0) = qg(0), qf(1) = qg(1). Hence they end up with a 2-groupoid, and we end up with a* * double groupoid. Their method of proving the properties of their homotopy 2-groupoid is to assum* *e first that p is a fibration, and then apply this case to an arbitrary map q by converting it to* * a fibration p = ~q. This is analogous to our methods, except that we have used Galois theory, where* *as they use directly properties of fibrations. We now form the quotient of diagram (18)by taking homotopy classes rel vert* *ices of R2(q) and of K1M to yield the diagram: _______// æ2(q)______//ß1(M)oo_OOOO * * (20) ||| ||| ||| ||| ||| ||| ||| ||| ffflffl|flffl|fflffl|fflffl||//_| Eq (q)_______//Moo_ where ß1(M) is the fundamental groupoid of M. It is clear that the horizontal c* *omposition O2 on R2(q) is inherited by æ2(q). Our main result of this section is a direct verification* * that the composition O1 is also inherited, without going through the simplicial Galois theory of the pr* *evious sections. We also have to show that this composition is related to that derived from the equivale* *nce relation structure on K1M0. In fact we prove a stronger result. The set R2(q) @+1x@-1R2(p) is the domain of composition of O1 on R2(q). A homotopy rel vertices on this se* *t is a continuous family ((fu, ffu, gu), (gu, fiu, ku)), 0 6 u 6 1 of elements of this set such that fu(* *0) = f0(0), ku(1) = k0(1), 0 6 u 6 1. We use the notation ßv0for the set of homotopy classes rel vertices. Theorem 5.1 The natural map : ßv0(R2(q) @+1x@-1R2(q)) ! æ2(q) @+1x@-1æ2(q) * * (21) defined by the projections, is a bijection. For the proof we use properties of the connections, and we use the followin* *g notation from [21]. We write: __ __j+1// + __j+1// - | fflffl|for j; __| fflffl|for j j j __/j+1/ __ __j+1// || fflffl|for"j; __ fflffl|for"j+1. j j Thus the thick lines denote degenerate faces. We shall use inversions applied * *to connections, for example __ ___2// -2| , fflffl|, 1 10 __ and write this also as |since it coincides with -1__|. This notation allows us to write some compositions as for example that invo* *lving 3-cubes x, y with @+3x = @-3y as ~ __~ ___/3/ A = ||x|y 2fflffl| which is an abbreviation for ~ ~ "2@-2x +2@-2y x y and makes it transparent what are the faces of A. The direction arrows are omit* *ted when convenient. Proof of theorem 5.1 We define an inverse for . We use square brackets [ ] to denote homotopy classes. Let ([f, ff, g], [h,* * fi, k]) 2 æ2(q) @+1x@-1æ2(q). Then there is a homotopy rel vertices of paths , : g ' h : I2 ! M. We set ([f, ff, g], [h, fi, k]) = [(f, ff, g), (g, (q,) O1fi, * *k)](22) and have to prove is well defined and an inverse to . Suppose we are given homotopies ~ : (f, ff, g)(f0, ff0, g0) * * (23) ~ : (h, fi, k)(h0, fi0, k0) * * (24) ,0: g0' h0. * * (25) Then ~, ~ are given by three component homotopies rel vertices ~1 : f ' f0, ~2 : ff ff0, ~3 : g ' g0, * * (26) ~1 : h ' h0, ~2 : fi fi0, ~3 : k ' k0, * * (27) with the properties that q~1 = @-1~2, q~3 = @+1~2, * * (28) q~1 = @-1~2, q~3 = @+1~1. * * (29) We now use the fact that all homotopies are rel vertices and that the maps * *ff, ff0, fi, fi0: I2 ! B are constant on the edges @-2, @+2. So in the following picture, the dotted lines r* *epresent constant paths, and i is a hollow cube not yet filled in, but has four faces well defined. Not* *e also that the maps I2 ! B given by @2 (~2) and @2 (~2) are constant maps, by our definition of hom* *otopies. 11 _f___ ________~1________________________________________2* *//_,,XXXXXXXX ___________________________________________________* *__________________f0__3fflffl| ___ _______________________________________________* *______1 ____ff____ ________ ________~2_______ff0_______________________________* *__________ __g________________________________________________* *_~3______________________________ ___ _____________________________________0 __,___________________i_______g__________ ___h______________,0_______________________________* *_____________________________ _________________________________~1________________* *________________________________________________________________h0___ _________________________fi________________________* *__________ ________~2________fi0______________________________* *_____________ ___________________________________________________* *______________k______________________ ___________________~3___________________k0__ The maps ,, ,0, ~3, ~1 define a map (I2 x `I) [ (I`x I2) ! E (where I`= {0, 1}) given by (s, t, 0) 7! ,(s, t), (s, t, 1) 7! ,0(s, t) on I2 x* * `I, and by (0, t, u) 7! ~3(t, u), (1, t, u) 7! ~1(t, u) on I`x I2 respectively. By the rel vertices co* *ndition, these maps can be extended by the constant map over I x {0} x I, which is @-2(i). So we now ha* *ve maps defined on 5 faces of I3 and agreeing on their common edges, and so these extend to a map i * *: I3 ! E. However, while this map does agree with the other homotopies, the result will not be a h* *omotopy of the type required since i1 = @+2(i) = (i|(I x {1} x I) is not constant as would be requi* *red. So we have to make a modification to get a homotopy between representatives of the original classe* *s. Intuitively, we move the face i1 of i to the right and down of our composite picture. This modificat* *ion will also change ~2, ~3, but this does not matter for our purposes, since we need to show only t* *hat a homotopy of the required type exists. We let ", "1, "2 denote degenerate elements - the element they act on in th* *e following formulae will be clear from the context, in order to make the compositions properly defined. Our new homotopy (f O ", ff O2"2, g O "), (g O ", ((q,) O1fi) O2"2, k O ")) (f0O ", ff0O2"2, g* *0O "), (g0O ", ((q,0) O1fi0) O2"2, k0O ")) will be given by ~1O2" : f O'"f0O ", * * (30) ~2O2"2 : ff O2"2 ff0O2"2, * * (31) ~3O2" : g O'"g0O ", * * (32) ~02: ((p,) O1fi) O2"2' ((p,0) O1fi0) O2"2, * * (33) ~3O @+2i : k'OkÖ0 ", * * (34) where ~ __~ ___2// ~02= qi~ | fflffl| 2|| 1 12 __ + * * __ where | is given by (s, t, u) 7! (@2 i)(min(s, 1 - t), u). Note that the combi* *nation of | and " = || in the second column of the matrix has the effect of pushing the non constant f* *ace @+2i of i down to be able to combine with ~3. It is clear that compositions with degenerate elements in direction 2 do no* *t change homotopy classes, and so this completes our geometric proof that is well defined. Next we must prove = 1, = 1. Considering the formula (22)for , we see that we can set ([f, ff, g], [g, fi, k]) = [(f, ff, g), (g, (q,) O1fi, * *k)] where now , can be chosen to be a constant homotopy ". It is then easily seen t* *hat [(f, ff, g), (g, (q") O1fi, k)] = [(f, ff, g), (g, fi, k* *)], and so that = 1. To prove = 1 it is sufficient to show that if , : g ' h is a homotopy re* *l vertices, then (g, (q,) O1fi, k) (h, "1(q,) O1fi, k). Such a homotopy is given by (,, ~, "3(* *k)) where ~ ~ ___/3/ ~ = "__| fflffl|. 3(fi) 1 * * 2 Corollary 5.2 The composition O1 on R2(q) is inherited by æ2(q) so that æ(q) be* *comes a double groupoid. Proof The composition O1 on æ2(q) is the composition of the maps æ2(q) @+1x@-1æ2(q) -! ßv0(R2(q) @+1x@-1R2(q)) ! æ2(q) where the second map is induced by the composition on R2(q). It is easy to see * *that the structure O1 gives a groupoid structure on æ2(q). Thus the only part remaining is the interchange law. However we easily find* * that a double com- position can be given as ~ ~ ~ ~ ~ __/~2/_ [f, ff, g][f0, ff0, g0] 0 ff ff0 ff0lffl| [h, fi, k][h0, fi0,=k0]f O f , (q,) O1fi(q,0) O1fi0, k O1k where , : g ' h, ,0: g0' h0. So the interchange law follows from that for singu* *lar squares. 2 Finally, we have to show the relation between the composition O0on K1M0 and* * the composition O1 above. Let (f, ff, g), (g, fi, h) 2 R2(q). We first note that, analogously t* *o the existence of identities in the fundamental groupoid, [f, ff, g] = [f, ff O1("1(qg)), g], [g, fi, h] = [g, ("1(qg))* * O1fi, h]. Hence [f, ff O1fi,[h]f=, ff, g] O1[g, fi, h] =[f, ff O1"1(qg), g] O1[g, "1(qg) O1fi, h] =[(f, ff O1"1(qg), g) O0(g, "1(qg) O1fi, h)] as required. 13 6 Examples In order to study the double groupoid æ(q) we need to have examples of double g* *roupoids with which to compare it, in addition to the product fibration of Example 3.1. As we shall* * see, there are some sub-double groupoids of æ(q) which are familiar, but it is interesting that we * *have little information about the most general form of double groupoids. For example, the methods of [1* *1] give an equivalence between double groupoids satisfying some filler conditions and what are there c* *alled core diagrams, but these do not seem to be helpful in this case. Here we suggest various examples and comparisons for further investigation. Example 6.1 Let i : M ! B be the inclusion of a subspace M of B. Then the equi* *valence relation Eq(i) is discrete, and so æ(i) is a 2-groupoid. Further, if m 2 M then the natu* *ral map j : ß2(B, M, m) ! æ2(i) is injective. Proof We represent ß2(B, M, m) by maps ff : I2 ! B such that the face @-1ff map* *s into M and the other three faces map to the base point m. The homotopy classes of ff which yie* *ld an element [ff] of ß2(B, M, m) are through maps of the same type. Then ff also yields an element <* *ff> of æ2(i), but there the homotopies allow @+1ff to vary in M. We have to prove that the map j : [ff]* * 7! is injective. Suppose then [ff-], [ff+] 2 ß2(B, M, m) and = 2 æ2(i). Let h : I3 * *! B be a homotopy determined by this equality, so that @-3(h) = ff-, @+3(h) = ff+, @2 (h) maps to m. Let ` = @+1(h). The problem is that ` is not constant. So we * *change h to `move' ` to the top face and still give a homotopy h0: ff0-' ff0+where [ff ] = [ff0]. We* * can take ~ ~ ___2// h0= |||h__| 1fflffl| so that the two ends of this homotopy are ~__ ~ @3 (h0)= |_|ff|__||__|, __ where |_|denotes a double identity, as required. * * 2 Note that æ(i) is the homotopy 2-groupoid of a pair discussed by Moerdijk a* *nd Svensson in [19], and is also recovered from the work of [16]. Example 6.2 The double groupoid æ(q) contains a 2-groupoid ____//_ ____//_ æ~(q)___//_ß1(M)___//_M where ~æ2(q) is the subset of æ2(q) of elements u such that @-2u, @+2u are dege* *nerate, that is consist of pairs (x, x). This is essentially the homotopy 2-groupoid of a map discussed* * by Kamps and Porter 14 in [16]. This 2-groupoid contains various cat1-groups of the form considered by* * Loday in [17]. The crossed module of groupoids associated to this 2-groupoid is of the form C ! ß1* *(M) where for each point x 2 M we have C(x) is isomorphic to ß1(Fx, ~x), the fundamental group of * *the homotopy fibre Fx of q over q(x) at the base point ~xdetermined by x. If M is a subspace of B * *and q is the inclusion then C(x) is isomorphic to the familiar relative homotopy group ß2(B, M, x) and* * the crossed module C(x) ! ß1(M, x) is essentially that first studied by J.H.C. Whitehead. However* * we do not have a reconstruction method for æ(q) from ~æ(q), whereas the 2-groupoid can be reco* *nstructed from the crossed module of groupoids it contains, as shown in [7]. * * 2 Example 6.3 Foliations Let F be a foliation on a space M. Thus the leaves of t* *he foliation define an equivalence relation R = R(F). Let q : M ! B be a map of spaces. The folia* *tion defines a finer topology than that given on M to give a space MF in which all leaves of* * the foliation are open components. So we also have a map qF : MF ! B and hence may define the hom* *otopy double groupoid æ(qF ). Where this differs from æ(q) is that in æ(qF ) the `horizontal* *' paths, and the homotopies of paths, all lie in leaves of the foliation. An illustrative example is the Möbius Band M with its projection q : M ! S1* * and foliation F by circles of which the centre one goes once round the Band and the other circles * *go twice round. Then æ(qF ) contains the double groupoid D(M) explained in the Introduction, and whi* *ch seems to be a good discrete algebraic model of the foliated Möbius Band. * * 2 Acknowledgements This work was partially supported by the following grants:INTAS 93-436 `Algebra* *ic K-theory, groups and categories', 97-31961 `Algebraic Homotopy, Galois Theory and Descent', `Alg* *ebraic K-theory, Groups and Algebraic Homotopy Theory'; with Bielefeld, an ARC Grant 965 `Global* * actions and algebraic homotopy', and by the London Mathematical Society fSU Scheme. The first author is also grateful to the Erwin Schrödinger Institute of Mat* *hematical Physics and a Leverhulme Emeritus Fellowship for support to attend a Workshop on Foliations* * in August, 2002. References [1]Al-Agl, F.A., Brown, R. and Steiner, R., `Multiple categories: th* *e equivalence be- tween a globular and cubical approach', Advances in Mathematics (48 pages) * *(to appear) (2002). http://arMiv.org/abs/math.CT/0007009. 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