A van Kampen theorem for the homotopy double groupoid of a Hausdorff space R. Brown*, K.H. Kampsyand T.Porterz January 16, 2004 UWB Math Preprint 04.01 Abstract We show that the homotopy double groupoid of a Hausdorff space defined * *by the authors in a previous paper satisfies a version of the van Kampen theorem* *, and so is a suitable tool for non abelian, 2-dimensional, local-to-global problems. T* *he methods are analogous to those developed by Brown and Higgins for similar theorems fo* *r other higher homotopy groupoids. There is a detailed discussion of commutative cubes in a double categor* *y with con- nections, and a proof of the key result that any composition of commutati* *ve cubes is commutative. 1 Introduction The classical van Kampen theorem for the fundamental group ß1(X, a) of a space * *with base point is a key example (maybe the only one) of a non commutative local-to-globa* *l theorem in dimension 1; it determines this fundamental group when X is the union of two co* *nnected open sets with connected intersection, in terms of the fundamental groups of the par* *ts. In fact van Kampen's paper [16] gave a formula for the case of non connecte* *d intersection, as required for the algebraic geometry applications. His formula follows from * *the version of the theorem given in [2], for the fundamental groupoid ß1(X, A) of a space with* * a set of base points (for the deduction of the formula, see 8.4.9 of [3]). A feature of this * *version is that it gives a complete determination of a non commutative invariant, the fundamental * *group, even in a case requiring interactions of two dimensions (in this case 0 and 1). The* * basic reason for this success, which is contrary to much experience in algebraic topology an* *d homological _____________________________________ *Mathematics Department, Dean St., Bangor, Gwynedd LL57 1UT, UK. email: r.br* *own@bangor.ac.uk yFachbereich Mathematik, FernUniversität in Hagen, D-58084 Hagen, * *Germany. email: heiner.kamps@fernuni-hagen.de zMathematics Department, Dean St., Bangor, Gwynedd LL57 1UT, UK. email: t.po* *rter@bangor.ac.uk 1KEY WORDS: double groupoid, double category, thin structure, connections, c* *ommutative cube, van Kampen theorem MATH SUBJECT CLASSIFICATION: 18D05,20L05,55Q05, 55Q35 1 algebra, seems to be that groupoids have structure in dimensions 0 and 1, and s* *o allow a better algebraic modelling of the geometry of the intersections of open sets. Consideration of this feature and of the structure of the proof of the theo* *rem led in 1966 to the start of a search for a higher dimensional version of the van Kampen the* *orem, involving some higher dimensional version of the fundamental groupoid, which would have a* *n algebraic structure in dimensions 0 to n. The latter construction proved difficult to fin* *d, and to verify its properties, but was successfully completed in a sequence of papers, startin* *g in 1978, of Brown with Higgins and with Loday. Notable features of the theorems and proofs* * were the use of forms of multiple groupoids. All of these higher homotopy groupoids invo* *lved a space structured either as a filtered space or as an n-cube of spaces. The paper [13], published in 2000, showed the existence of an absolute homo* *topy 2-groupoid of a space; this suggested the possibility of an absolute homotopy double group* *oid, as found in [4]. Here we adapt the methods of [5] to prove a van Kampen theorem for thi* *s functor. Since this functor differs in dimension 1 from the standard fundamental groupoi* *d, the result in dimension 1 is also new. A key aspect of a proof of the 1-dimensional van Kampen theorem is the noti* *on of comm- utative square in a groupoid and the easy fact that any composition of such com* *mutative squares is commutative. Commutative squares in a groupoid have an algebraic st* *ructure of double groupoid. So a key aspect in the passage to higher dimensions is the de* *finition of a commutative cube and the proof that any composition of commutative cubes is com* *mutative; that is, the commutative cubes should form a triple category. Here even the definition needed new concepts, namely the notion of connecti* *ons on a cubical set, as first introduced for double groupoids in 1976 in [9]. For convenience, * *we briefly explain these ideas in the early sections of this paper, and show their application to * *commutative cubes, now called 3-shells. The main technical proofs on commutative cubes are deferr* *ed to a final section. The proof in section 4 of the main van Kampen theorem follows the model of * *the arguments for the classical 1-dimensional version, and that in [5], with some extra twist* *s. Given the main machinery, the proof is quite short. It does not need the deformations based on* * connectivity assumptions used in [5]. This convinces us that the proof, like that of [5], sh* *ould extend to still higher dimensions, but no construction is yet available of suitable absolute hi* *gher homotopy groupoids. The validity of the theorem in dimension 2 is an argument for the utility o* *f this particular algebraic structure for 2-dimensional, non commutative, local-to-global problem* *s: the slogan here is that of modelling an `algebraic inverse to subdivision'. In view of th* *e success of the 1-dimensional fundamental group(oid) in a variety of problems, for example in a* *lgebraic and differential geometry, we hope that this will lead to wide investigations of ap* *plications of this type of 2-dimensional homotopical algebra. In a sequel, we plan to show that this homotopy double groupoid captures th* *e weak homo- topy 2-type of a space. 2 1 Double categories, connections and thin structures: definition and notational conventions By a double category K, we will mean what has been called an edge symmetric dou* *ble category in [8]. We briefly recall some of the basic facts about double categories in the ab* *ove sense. In the first place, a double category, K, consists of a triple of category structures (K2, K1, @-1, @+1, +1, "1), (K2, K1, @-2, @+2, +2, "2) (K1, K0, @- , @+ , +, ") as partly shown in the diagram * * @ .........................................* *............................................................................@ K2 ..........K1.............................* *............................................................................@ ............................................* *. + ............................................* *............................@2 ............................................* *............................ - .....................................+......* *............................. -+ @1 ....................................@1......* *..............................@@ ............................................* *............................ .............................................* *............................................................................@ .........................................* *............................................................................@ K1 ..........K0.............................* *............................................................................@ @+ The elements of K0, K1, K2 will be called respectively points or objects, edges* *, squares. The maps @ ,@i , i = 1, 2, will be called face maps, the maps "i : K1 -! K2, i = 1* *, 2, resp. " : K0 -! K1 will be called degeneracies. The compositions, +1, resp. +2, are referred to as vertical resp. horizonta* *l composition of squares. The axioms for a double category include the usual relations of a 2-cu* *bical set and the interchange law. We use matrix notation for compositions as ~ ~ a = a +1 c, a b = a +2 b, c and the interchange law allows one to use matrix notation ~ ~ a b c d for double composites of squares, as in [6]. We also allow as in [6] the multip* *le composition [aij] of an array (aij) whenever for all appropriate i, j we have @+1aij= @-1ai+1,j, * *@+2aij= @-2ai,j+1. The identities with respect to +1 (vertical identities) are_given by "1 and* * will be denoted by .|||||Similarly, we have horizontal identities denoted by __. Elements of the* * form "1ä( ) = "2ä( ) for a 2 K0 are called double degeneracies and will be denoted by . A morphism of double categories f : K ! L consists of a triple of maps fi :* * Ki ! Li, (i = 0, 1, 2), respecting the cubical structure, compositions and identities. A connection pair on a double category K is given by a pair of maps -, + : K1 -! K2 whose edges are given by the following diagrams for a 2 K1: 3 a ...... .... ......................................................* *............................................................................@ ......................................................* *............................................................................@ ......................................................* *............................................................................@ -(a)= a.......................................................* *.........................................................1..................@ ......................................................* *............................................................................@ ......................................................* *............................................................................@ 1 ...1..................................................* *............. .......................................................* *............................................................................@ .......................................................* *....................................................................... .......................................................* *............................................................................@ +(a)= 1.......................................................* *............................................................................@ .......................................................* *............................................................................@ .......................................................* *............................................................................@ a The axioms for the connection pair are listed in [8], Section 4. In particu* *lar, the transport laws ~ __ __~ ~ ~ | ___ __ __|| | = | , __ = __| * * (1) | | | __ __| describe the connections of the composition of two elements, while the laws ~ __~ | __ __ = __, | __| = | |, * * (2) __| allow cancellation of connections. A morphism of double categories with connections is a morphism of double ca* *tegories re- specting connections. Brown and Mosa in [8] have shown that a pair of connections on a double cat* *egory K is equivalent to a thin structure on K, whose definition we recall. We recall the standard language for `commutative squares'; we will later mo* *ve to `comm- utative cubes'. First, if ff is a square in K, then the boundary (2-shell) of ff is the qu* *adruple (@-2ff, @+1ff, @-1ff, @+2ff). We also say ff is a filler of its boundary. This boundary commutes if @-2ff + @* *+1ff = @-1ff + @+2ff. More generally, a 2-shell is defined to be a quadruple (a, b, c, d) of edges su* *ch that @- a = @- c, @+ a = @- b, @+ c = @- d, @+_b = @+ d, and this 2-shell commutes if a + b* * = c + d. If C is a category, then by |_|C we denote the double category of commuting* * squares (2- shells) in C with the obvious double category structure. Then a thin structure * * on a double category K is a morphism of double categories __ : |_|K1 ! K, __ which is the identity on K1. The elements of K2 lying in (|_|K1) are called th* *in squares. The definition implies immediately: (T0) A thin square has commuting boundary. (T1) Any commutative 2-shell has a unique thin filler. (T2) Any composition of thin squares is thin. 4 In the thin structure induced on a double category by a pair of connections* *, any thin square is a certain composite of identities, both horizontal and vertical, and connect* *ions. For an explicit formula, we refer to [8], Theorem 4.3, and we will refer to these thin* * squares as being algebraically thin. Of course, a morphism of double categories with connections* * will preserve (algebraic) thinness. 2 Commutative cubes We need to extend the domain of discourse from double categories to triple cate* *gories in order to explain the notion of commutative cube (3-shell) in a double category with c* *onnections. This notion has been defined for n-shells in the case of cubical !-groupoids with co* *nnections in [6, Section 5], and for the more difficult category case in [1, Section 9]. For the* * convenience of the reader we set up the theory in our low dimensional case. Let D, T be respectively the categories of double categories and of triple * *categories with connections, in the sense of [1]. There is a natural and obvious truncation fun* *ctor tr : T ! D, which forgets the 3-dimensional structure. This functor has a right adjoint cos* *k : D ! T which may be constructed in terms of cubes or 3-shells, which we now define. Definition 2.1 Let K be a double category. A cube (3-shell) in K, ff = (ff-1, ff+1, ff-2, ff+2, ff-3, ff+3) consists of squares ffi 2 K2 (i = 1, 2, 3) such that @ffi(fffij) = @fij-1(ffffi) for oe, ø = 1 and 1 6 i < j 6 3. A cube may be illustrated by the following picture: ff-1 ....... .......... ........... .....................................o.................* *............................................................................@ - .......................................................* *............................................................................@ ff3............................................................* *......................3.....................................................@ .........................................................* *............................................................................@ ......................................1................* *............................................................................@ .......................................................* *............................................................................@ .......................................................* *...........o................................................................@ .................... ..................................* *.. ff-............................................................* *............................................................................@ 2 .................... ..................................* *.. @ ...................o...................................* *.....................................................o.................... ......................................................* *.............................. ...................................................* *............................................................................@ ................................................* *....................................................................... .............................................* *............................................................................@ ..........................................* *............................................................................@ .......................................* *......................................................ff+........... ......................................* *o...........................................................................@ ... ........... .......... ........... ff+1 5 Definition 2.2 We define three partial compositions, +1, +2, +3, of cubes. Let * *ff, fi be cubes in K. (i)If ff+1= fi-1, then we define ff +1 fi = (ff-1, fi+1, ff-2+1 fi-2, ff+2+1 fi+2, ff-3+1 fi-3* *, ff+3+1 fi+3). (ii)If ff+2= fi-2, then we define ff +2 fi = (ff-1+1 fi-1, ff+1+1 fi+1, ff-2, fi+2, ff-3+2 fi-3* *, ff+3+2 fi+3). (iii)If ff+3= fi-3, then we define ff +3 fi = (ff-1+2 fi-1, ff+1+2 fi+1, ff-2+2 fi-2, ff+2+2 fi+* *2, ff-3, fi+3). This is a special case of general definitions in dimension n given in [6]. As explained earlier, we need the notion of a 3-shell ff being `commutative* *'. Intuitively, this says that the composition of the odd faces ff-1, ff-3, ff+2of ff is equal * *to the composition of the even faces ff+1, ff+3, ff-2(think of -,+ as 0,1). The problem of how to ma* *ke sense of such compositions of three squares in a double groupoid, and obtain the right bounda* *ries, is solved using the connections. Definition 2.3 Suppose given a cube (3-shell) in a double category with connect* *ion K ff = (ff-1, ff+1, ff-2, ff+2, ff-3, ff+3). We define the composition of the odd faces of ff to be ~ __ - ~ | ff1 __| @ oddff= - + __ * * (3) ff3 ff2 __ and the composition of the even faces of ff to be ~__ - +~ __ ff2 ff3 @evenff= __ + * * (4) | ff1 __| This definition can be regarded as a cubical, categorical (rather than groupoid* *) form of the Homotopy Addition Lemma (HAL) in dimension 3. Definition 2.4 We define ff to be commutative if @oddff = @evenff. (H* *CL) The reader should draw a 3-shell, label all the edges with letters, and see* * that this equation makes sense in that the 1-shells of each side of equation (HCL) coincide. Notic* *e however that these 1-shells do not have coincident partitions along the edges: that is the e* *dges of this 1-shell in direction 2 are formed from different compositions of the type 1 + a and a +* * 1. 6 The formula in Definition 2.4 is unsymmetrical, and this seems in part a co* *nsequence of trying to express a 3-dimensional idea in a 2-dimensional formula. It also refl* *ects the difficulty of the concept. A formula in dimension 4 is given in [11]. There are other pictures and forms of this which we now explore, and some o* *f which will be used later. First we give a picture in another format which also shows how the boundary* * of each side is made up: .................................................................* *............................................................................@ .................................................................* *............................................................................@ .................................................................* *............................................................................@ .................................................................* *............................................................................@ .................................................................* *............................................................................@ .................................................................* *............................................................................@ .................................................................* *............................................................................@ .................................................................* *............................................................................@ .................................................................* *............................_...............................................@ .................................................................* *............................................................................@ .................................................................* *............................................................................@ .................................................................* *............................................................................@ This is in fact equivalent to: .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *...................................._.......................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *......................................................................... .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ .........................................................* *............................................................................@ The reader can check that in each case the two sides of the equation have t* *he same 2-shells (but not as subdivided). The second equation can also be written in matrix notation as: 2 __ 3 2 __ 3 | ff-1 || | 6 7 6 7 6 ff- ff+ 7 = 6 ff- ff+ 7. (H* *CL0) 4 3 2 5 4 2 3 5 __| || ff+1 __| __ The equivalence of (HCL) to (HCL0) can be seen by adding | | | , (resp* *ectively ____|||) to the top (resp. bottom) of the two sides of (HCL), using equatio* *ns (2) for further cancellation of connections, and then absorbing identities. We note that a morphism of double categories with connections preserves com* *mutativity of cubes. The following theorem will be crucial for the proof of the van Kampen theor* *em. Theorem 2.5 Let K be a double category with connections. Then any composition* * of comm- utative 3-shells is commutative. 7 The proof is left to section 5. Proposition 2.6 The functor tr : T ! D has a left adjoint_sk : D ! T which ass* *igns to a double groupoid with connections D the triple groupoid |_|D which agrees with D* * in dimensions 6 2 and in dimension 3 consists of the commutative 3-shells in D. Remark 2.7 (i) Our homotopy commutativity lemma has been called also the `hom* *otopy addition lemma', and has been formulated in different equivalent forms in certa* *in 2-dimensional groupoid type settings using inverses and reflections (cf. [5], Proposition 5, * *[15], Definition 3.8, [4], Proposition 5.5). Also [6], x5, 7, sets up, in the situation of cubical mu* *ltiple groupoids with connections, truncation and skeleton functors trn and skn and an n-dimensional * *homotopy addition lemma as a formula for the boundary of an n-cube. The definition and * *use of thin elements in dimension n is a key part of the proof of the generalised van Kampe* *n theorem in [7]. The homotopy addition lemma presented here applies to a general 2-dimension* *al category type setting. It is also a special case of results on cubical !-categories with* * connections in [1, Section 9]. (The fact that any composition of commutative shells is commutative* * is not stated in the last reference, but does follow easily from the results stated there.) (ii) Homotopy commutative cubes have been defined and investigated by Spenc* *er and Wong in the more special setting of what they call special double categories with co* *nnection ([15], Definition 1.4). By this they mean a double category (with connections), K, su* *ch that the horizontal subdouble category of squares in K of the form 1 ..........................................* *............................................................................@ .................................... .................................... ..........................................* *...... a...........................................* *.....................................................................b .................................... .................................... ..........................................* *............................................................................@ 1 is a groupoid under +2. Spencer and Wong show that this is a suitable setting * *to deal with (homotopy) pullbacks/pushouts and gluing/cogluing theorems in homotopy theory. It is not difficult to show that the definition of a homotopy commutative c* *ube as given in [15], Definition 3.8, is equivalent to our definition of commutative cube. As we mentioned before the fact that commutativity of cubes is preserved un* *der composition will be crucial for our purposes. Spencer and Wong have a similar result in the* *ir setting ([15], Proposition 3.11). The basic notions of relative homotopy carry over from topology to algebra.* * Let K be a double category with connections. A square u in K is called a relative homotop* *y if the two opposite edges @-2u, @+2u are identities. A relative homotopy between squares * *u, u0 in K is a commutative cube ff such that ff-1= u, ff+1= u0 and the remaining faces are relative homotopies. If, in addition, the remaining* * faces are thin, then u and u0are called thinly equivalent From (T0) and (T1), we conclude: 8 (T3) A thin square which is a relative homotopy is an identity. From Definition 2.2, making use of [8], 3.1(i), we read off the following * *lemma: Lemma 2.8 Squares which are thinly equivalent, coincide. This key lemma is used later to show that a construction is independent of the * *choices made. 3 The homotopy double groupoid, æ (X) This section is adapted from [4], and the reader should refer to that source fo* *r fuller details. 3.1 The singular cubical set of a topological space We shall be concerned with the low dimensional part (up to dimension 3) of the * *singular cubical set R (X) = (Rn(X), @-i, @+i, "i) of a topological space X. We recall the definition (cf. [6]). For n > 0 let Rn(X) = Top (In, X) denote the set of singular n-cubes in X, i.e. continuous maps In -! X, where I * *= [0, 1] is the unit interval of real numbers. We shall identify R0(X) with the set of points of X. For n = 1, 2, 3 a sing* *ular n-cube will be called a path, resp. square, resp. cube, in X. The face maps @-i, @+i: Rn(X) -! Rn-1(X) (i = 1, . .,.n) are given by inserting 0 resp. 1 at the ithcoordinate whereas the degeneracy ma* *ps "i: Rn-1(x) -! Rn(X) (i = 1, . .,.n) are given by omitting the ithcoordinate. The face and degeneracy maps satisfy t* *he usual cubical relations (cf. [6], x 1.1; [14] , x 5.1). A path a 2 R1(X) has initial point * *a(0) and endpoint a(1). We will use the notation a : a(0) ' a(1). If a, b are paths such that a(1* *) = b(0), then we denote by a + b : a(0) ' b(1) their concatenation, i.e. æ a(2s), 0 6 s 6 1_ (a + b)(s) = 1 2 b(2s - 1), _26 s 6 1 If x is a point of X, then "1(x) 2 R1(X), denoted ex, is the constant path at x* *, i.e. ex(s) = x for alls 2 I. If a : x ' y is a path in X, we denote by -a : y ' x the path reverse to a,* * i.e. (-a)(s) = a(1 - s) for s 2 I. In the set of squares R2(X) we have two partial composition* *s +1 (vertical composition) and +2 ( horizontal composition) given by concatenation in the fir* *st resp. second variable. 9 Similarly, in the set of cubes R3(X) we have three partial compositions +1,* * +2, +3. The standard properties of vertical and horizontal composition of squares a* *re listed in [6], x1. In particular we have the following interchange law. Let u, u0, w, w02 R2(X* *) be squares, then (u +2 w) +1 (u0+2 w0) = (u +1 u0) +2 (w +1 w0), whenever both sides are defined. More generally, we have an interchange law for* * rectangular decomposition of squares. In more detail, for positive integers m, n let 'm,n :* * I2 -! [0, m] x [0, n] be the homeomorphism (s, t) 7- ! (ms, nt). An m x n subdivision of a squ* *are u : I2 -! X is a factorization u = u0O'm,n; its parts are the squares uij: I2-! X defined by uij(s, t) = u0(s + i - 1, t + j - 1). We then say that u is the composite of the array of squares (uij), and we use m* *atrix notation u = [uij]. Note that as in x1, u +1 u0, u +2 w and the two sides of the interc* *hange law can be written respectively as ~ ~ ~ ~ u u w , [u w], 0 0 . u0 u w Finally, connections -, + : R1(X) -! R2(X) are defined as follows. If a 2 R1(X) is a path, a : x ' y, then let -(a)(s, t) = a(max (s, t)); +(a)(s, t) = a(min (s, t)). The full structure of R (X) as a cubical complex with connections and compositi* *ons has been exhibited in [1, 12]. 3.2 Thin squares In the setting of a geometrically defined double groupoid with connection, as i* *n [6], (resp [4]), there is an appropriate notion of geometrically thin square. It is proved in [* *6], Theorem 5.2 (resp. [4], Proposition 4), that in the cases given there, geometrically and a* *lgebraically thin squares coincide. In our context the explicit definition is as follows: Definition 3.1 (1) A square u : I2 - ! X in a topological space X is thin if th* *ere is a factorisation of u pu u : I2 -u! Ju -! X, where Ju is a tree and u is piecewise linear (PWL, see below) on the boundary * *@I2 of I2. Here, by a tree, we mean the underlying space |K| of a finite 1-connected 1* *-dimensional simplicial complex K. A map : |K| -! |L| where K and L are (finite) simplicial complexes is PWL* * (piecewise linear) if there exist subdivisions of K and L relative to which is simplicia* *l. (2) Let u be as above, then the homotopy class of u relative to the boundar* *y @I2 of I is called a double track. A double track is thin if it has a thin representative. 10 3.3 The homotopy double groupoid of a Hausdorff space The data for the homotopy double groupoid, æ (X), will be denoted by (æ 2(X), æ1(X), @-1, @+1, +1, "1), (æ 2(X), æ1(X), @-2, @+2, +2, * *"2) (æ 1(X), X, @- , @+ , +, "). Here æ1(X) denotes the path groupoid of X of [13]. We recall the definition. Th* *e objects of æ1 (X) are the points of X. The morphisms of æ1(X) are the equivalence classes * *of paths in X with respect to the following relation ~T. Definition 3.2 Let a, a0 : x ' y be paths in X. Then a is thinly equivalent t* *o a0, denoted a ~T a0, if there is a thin relative homotopy between a and a0. We note that ~T is an equivalence relation, see [4]. We use : x ' y to* * denote the ~T class of a path a : x ' y and call the semitrack of a. The groupoid str* *ucture of æ1 (X) is induced by concatenation, +, of paths. Here one makes use of the fac* *t that if a : x ' x0, a0 : x0 ' x00, a00: x00' x000are paths then there are canonical t* *hin relative homotopies (a + a0) + a00' a + (a0+ a00) : x ' x000(rescale) a + ex0' a : x ' x0; ex + a ' a : x ' x0(dilation) a + (-a) ' ex : x ' x (cancellation). The source and target maps of æ1(X) are given by @-1 = x, @+1 = y, if : x ' y is a semitrack. Identities and inverses are given by "(x) = resp.- = <-a>. In order to construct æ2(X), we define a relation of cubically thin homotopy on* * the set R2(X) of squares. Definition 3.3 Let u, u0be squares in X with common vertices. (1) A cubically t* *hin homotopy U : u T u0between u and u0is a cube U 2 R3(X) such that (i) U is a homotopy between u and u0, i.e. @-1(U) = u, @+1(U) = u0, (ii) U is rel. vertices of I2, i.e. @-2@-2(U), @-2@+2(U), @+2@-2(U), @+2@+2(U) are constant, (iii) the faces @ffi(U) are thin for ff = 1, i = 1, 2. (2) The square u is cubically T -equivalent to u0, denoted u T u0if there * *is a cubically thin homotopy between u and u0. 11 Proposition 3.4 The relation T is an equivalence relation on R2(X). Proof The reader is referred to [4] for a proof. If u 2 R2(X) we write {u}T, or simply {u}T, for the equivalence class of u * *with respect to T . We denote the set of equivalence classes R2(X) T by æ2(X). This inherits * *the operations and the geometrically defined connections from R2(X) and so becomes a double gr* *oupoid with connections. A proof of the final fine detail of the structure is given in [4]. Definition 3.5 An element of æ2 (X) is thin if it has a thin representative (in* * the sense of Definition 3.1). From the remark at the beginning of this subsection we infer: Lemma 3.6 Let f : æ (X) ! D be a morphism of double groupoids with connect* *ion. If ff 2 æ2(X) is thin, then f(ff) is thin. 3.4 The homotopy addition lemma Let u : I3 ! X be a singular cube in a Hausdorff space X. Then by restricting u* * to the faces of I3 and taking the corresponding elements in æ2 (X), we obtain a cube in æ (* *X) which is commutative by the homotopy addition lemma for æ (X) ([4], Proposition 5.5). Co* *nsequently, if f : æ (X) ! D is a morphism of double groupoids with connections, any singul* *ar cube in X determines a commutative 3-shell in D. 4 The van Kampen Theorem The general setting of the van Kampen theorem is that of a local-to global prob* *lem which can be explained as follows: Given an open covering U of X and knowledge of each æ (U) for U in U, gi* *ve a determination of æ (X). Of course we need also to know the values of æ on intersections U \ V and on t* *he inclusions from U \ V to U and V . F We first note that that the functor æ on Top preserves coproducts , sinc* *e these are just disjoint union in topological spaces and in double groupoids. It is an advantag* *e of the groupoid approach that the coproduct of such objects is so simple to describe. Suppose we are given a cover U of X. Then the homotopy double groupoids in * *the following æ-sequence of the cover are well-defined: G a G c æ (U \ V ) ' æ (U) -! æ (X). * * (5) (U,V )2U 2 b U2U The morphisms a, b are determined by the inclusions aUV : U \ V ! U, bUV : U \ V ! V for each (U, V ) 2 U 2and c is determined by the inclusion cU : U ! X for each * *U 2 U. 12 Theorem 4.1 (van Kampen theorem) If the interiors of the sets of U cover * *X, then in the above æ-sequence of the cover, c is the coequaliser of a, b in the category* * of double groupoids with connections. A special case of this result is when U has two elements. In this case the * *coequaliser reduces to a pushout. The proof of the theorem is a direct verification of the universal property* * for the coequaliser. So suppose D is a double groupoid, and G f : æ (U) ! D U2U is a morphism such that fa = fb. We require to construct uniquely a morphism F * *: æ (X) ! D such that F c = f. It is convenient to write fU for the restriction of f to æ (* *U). First, F is uniquely defined on objects, since if x 2 X then x 2 U for some* * U 2 U and so F (x) = F c(x) = fU (x), and the condition fa = fb ensures this value is indepe* *ndent of U. We next consider F on an element 2 æ1(X). By the Lebesgue covering lemm* *a, we can write = + . .+. where ui is a path in a set Ui of the cover, and so determines an element * * in æ1(Ui). The rule fa = fb implies that the elements fUi are composable in D, and, since* * F is a morphism, and F c = f, their sum must be F . A similar argument applies to an element 2 æ 2(X), where this time we * *choose a subdivision of ff as a multiple composition [ffij] with ffijlying in some Uij. * * We must have F = [fij]. Now we must prove that F can be well defined by choices of this kind. Indep* *endence of the subdivision chosen is easily verified by superimposing subdivisions. The hard p* *art is to show the effect of a homotopy. We start in dimension 1. Suppose h : u ~T v is a thin homotopy of paths u, v. Then there is a factor* *isation p h : I2 -! J -! X where J is a tree, and is PWL on the boundary @I2 of I2. Now p-1(U) covers J. Choose a subdivision of J so that the open star of eac* *h edge of the subdivision is contained in a set of p-1(U). By a grid subdivision of In we mean a subdivision determined by equally spa* *ced hyperplanes of the form xi = constant, i = 1, . .,.n. Choose a grid subdivision of I2 such* * that maps each little square s of the subdivision into some open star St(s) of an edge of* * J, and so (s) p-1U(s) for some U(s) 2 U. Assign extra vertices to J as of the vertices of the grid subdivision. Consider a non boundary edge e of the grid subdivision, with adjacent squar* *es s, s0. Deform on e, keeping its image in St(s) \ St(s0) so that is PWL on e. Doing this f* *or each edge of a square s gives a homotopy of the restriction | @s with image always in S* *t(s). The HEP allows this homotopy to be extended to a homotopy of | s still with values in* * St(s). These determine a homotopy ' 0rel @I2 such that if h0= p 0, then h0| s is a thin s* *quare lying in 13 U(s) 2 U for each s of the grid subdivision. The class in æ2(U(s)) is m* *apped by fU(s) to a thin square in D. The condition fa = fb implies these squares are composab* *le, and by the property (T2) of thin squares, their composite is a thin square. By (T3) of x2,* * this composite is an identity, and so F () = F (). Thus F is well defined. The proof that F in this dimension is a morphism of groupoids is immediate * *from the definition. We apply a similar argument in dimension 2. Suppose W : ff T fi is a cubically thin homotopy. Make a grid subdivision* * of I3 into subcubes c such that W maps c into a set U(c) 2 U. Then c determines a commutat* *ive 3-shell in æ (U(c)) which is mapped by fU(c)to a commutative 3-shell in D. Again by fa * *= fb but now in the next dimension, these commutative 3-shells are composable in D to give a* * commutative 3-shell C. But the faces of W not in direction 1 are given to be thin homotopie* *s. The argument as above shows that the faces of C not in direction 1 are thin squares in D. By* * Lemma 2.6, F () = F (), and so F is well defined. It is easy to check that F is a morphism of double groupoids with connecti* *ons. This completes the proof of our main result, apart from the proof of Theorem 2.5. 5 Proof of Theorem 2.5 The proof uses some 2-dimensional rewriting using connections of the type used * *since the 1970s. However there are some tricky points which we would like to emphasise. We often have to rearrange some block subdivision of a multiple composition* *. The gen- eral validity of this process is discussed in [10], and its application to doub* *le categories with connections in [8]. Here we point out the following. Let ff, fi be squares in a double category such that fl = ff fi= ff +2 fi is defined. Suppose further that ~ ~ ~ ~ ff1 fi1 ff = fi = . ff2 fi2 If ff1 +2 fi1, ff2 +2 fi2 are defined, then we can write ~ ~ ff1 fi1 fl = . ff2 fi2 However, if we rewrite ~ ~ ~ ~ ff01 fi01 ff = 0, fi = 0 ff2 fi2 then we cannot write ~ ~ ff01fi01 fl = 0 0 ff2 fi2 unless we are sure the compositions ff01+2 fi01, ff02+2 fi02are defined. Thus* * care is needed in 2-dimensional rewriting. 14 We now proceed with the proof that all compositions of commutative 3-cubes * *are comm- utative. We first do the case of direction 2. This requires the transport laws. Let ff, fi be 3-shells in K such that ff+2= fi-2. We assume that ff, fi ar* *e commutative, so that the HCL holds for each. Then ~ __ - ~ | (ff +2 fi)1__| @ odd(ff +2 fi)= - +_(by definition) (ff +2 fi)3(ff +2 fi)2_ ~ __ - - ~ | ff1 +1 fi1__| = - - + __ (by definition of +2 for 3-shells) ff3 +2 fi3 fi2 __ 2 __ __ 3 | __ ff-1__|| | __ - __ 7 = 64|| | fi1 __ __|5 (by the transport laws) __ __ ff-3fi-3 fi+2__ __ 2 __ __ 3 | __ ff-1__| || __ - __ +7 = 64ff-3_ fi2 __ fi35 (by HCL for fi) __ + __ || | fi1 __ __| 2 __ __ 3 | __ ff-1__| || __ + __ +7 - + = 64ff-3_ ff2 __ fi35 (since fi2 = ff2 ) __ + __ || | fi1 __ __| 2 __ __ 3 __ __ ff-2ff+3 fi+3 __ __ + 7 = 64| __ ff1 __| ||5 (by HCL for ff) __ + __ | | | fi1 __ __| ~__ - ~ + __ (ff +2 fi)2(ff +2 fi)3 = __ + (by transport laws and composition * *rules) | (ff +2 fi)1 __| = @even(ff +2 fi) (as required.) We now consider the case of +3 and for this it turns out to be convenient t* *o use (HCL0). We write the left hand side of this as @0oddand the right hand side as @0even. * * Suppose then 15 ff +3 fi is defined. Then 2 __ 3- | (ff +3 fi)1 @0odd(ff +3 fi)= 4(ff +3 fi)-3(ff +3 fi)+25 __| || 2 __ - -3 2 __ - -3 | ff1 +2 fi1 | ff1 fi1 = 4ff-3ff+2+2 fi+25 = 4ff-3ff+2 fi+25 __| || __| || || 2 __ -3 || | fi1 = 4ff-2ff+3 fi+25 (by HCL 0) ff+1 || || 2 __ -3 || | fi1 = 4ff-2fi-3 fi+25 (as ff+3= fi-3) ff+1 __| || 2 __ 3 || || | = 4ff-2fi-2 fi+35 (by HCL 0) ff+1fi+1 || 2 __ 3 || | = 4(ff +3 fi)-2(ff +3 fi)+35= @0even(ff +3 fi) (ff +3 fi)+1 || as required. The proof for +1 is similar to the last one, but using (HCL). We leave it t* *o the reader. To see the complications of these ideas in higher dimensions, see [1] and a* *lso p.361-362 of [11], which deal with the 4-cube. Note also that when applied to æ (X), these calculations imply the existenc* *e, even a con- struction, of certain homotopies which would otherwise be difficult to find. Th* *e possibility of calculating with such homotopies is indeed one of the aims of this theory. References [1] F.A. Al-Agl, R. Brown and R. Steiner, Multiple categories: the equivalence* * of a globular and a cubical approach, Advances in Maths. 170 (2002) 71-118. [2] R. Brown, Groupoids and van Kampen's theorem, Proc. London Math. Soc. (3) * *17 (1967) 385-401. [3] R. Brown, Topology: a geometric account of general topology, homotopy typ* *es, and the fundamental groupoid, Ellis Horwood, Chichester (1988). [4] R. Brown, K.A. Hardie, K.H. Kamps and T. Porter, A homotopy double groupoi* *d of a Hausdorff space, Theory and Applications of Categories 10 (2002) 71-93. [5] R. Brown and P.J. Higgins, On the connection between the second relative h* *omotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978), 193-* *212. 16 [6] R. Brown and P.J. Higgins, On the algebra of cubes, J. Pure Applied Algebr* *a 21 (1981), 233-260. [7] R. Brown and P.J. Higgins, Colimit theorems for relative homotopy groups',* * J. Pure Appl. Algebra 22 (1981) 11-41. [8] R. Brown and G.H. Mosa, Double categories, thin structures and connections* *, Theory and Applications of Categories 5 (1999), 163-175. [9] R. Brown, and C.B. Spencer, Double groupoids and crossed modules, Cahiers * *Top. G'eom. Diff. 17 (1976) 343-362. [10] R. Dawson and R. Par'e, `General associativity and general composition for* * double cate- gories', Cahiers Top. G'eom. Diff. Cat. 34 (1993) 57-79. [11] P. Gaucher, `Combinatorics of branchings in higher dimensional automata', * *Theory and Applications of Categories, 8 (2001) 324-376. [12] M. Grandis and L. Mauri, Cubical sets and their site, Theory and Applicati* *ons of Cate- gories 11 (2003) 185-211. [13] K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausd* *orff space, Applied Cat. Structures 8 (2000), 209-234. [14] K.H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory, Wo* *rld Scientific, 1997. [15] C.B. Spencer and Y.L. Wong, Pullback and pushout squares in a double categ* *ory with connection, Cah. Top. G'eom. Diff. 24 (1983), 161-192. [16] E.H. van Kampen, On the connection between the fundamental groups of some * *related spaces, American J. Math. 55 (1933) 261-267. 17