A new higher homotopy groupoid: the fundamental globular !-groupoid of a filtered space Ronald Brownyz February 23, 2007 Abstract We show that the graded set of filter homotopy classes rel vertices of * *maps from the n-globe to a filtered space may be given the structure of globular !-group* *oid. The proofs use an analogous fundamental cubical !-groupoid due to the author and Phil* *ip Higgins. This method also relates the construction to the fundamental crossed compl* *ex of a filtered space, and this relation allows the proof that the crossed complex associa* *ted to the free globular !-groupoid on one element of dimension n is the fundamental cross* *ed complex of the n-globe. Contents Introduction * * 2 1 Disks, globes, and cubes * * 6 2 The free globular !-groupoid on one generator * * 9 3 Closed monoidal structure * * 10 _____________________________________ MSC Classification:18D10, 18G30, 18G50, 20L05, 55N10, 55N25. KEY WORDS: fil* *tered space, higher homotopy van Kampen theorem, cubical singular complex, free globular groupoid. ySchool of Computer Science, University of Wales, Dean St., Bangor, Gwynedd,* * LL57 1UT, UK; email: r.brown@bangor.ac.uk zThe author was supported for part of this work by a Leverhulme Emeritus Fel* *lowship (2002-2004). 1 4 The higher homotopy van Kampen Theorem * * 11 5 Nerves and classifying spaces of globular !-groupoids * * 12 A The globular site * * 13 B The cubical site * * 13 References * * 17 Introduction By the n-globe Gn we mean the subspace of Euclidean n-space Rn of points x such* * that kxk 6 1 but with the cell structure for n > 1 Gn # e0 [ e1 [ [ en 1 [ en. * * (1) This structure will be given precisely in section 1. A filtered space is a compactly generated space X# and a sequence of subspac* *es X # X0 ` X1 ` ` Xn ` ` X# . * * (2) A map of filtered spaces f # Y ! X is a map f # Y# ! X# such that f#Yn# ` Xn * *for all n > 0. This gives the category FTop of filtered spaces. A filter homotopy ft # f0 ' f1* * is a continuous family of filtered maps ft # Y ! X for 0 6 t 6 1. The n-globe Gn has a skeletal filtration giving a filtered space Gn. If X i* *s a filtered space ij~`~'~ n then we have a globular singular complex R X which in dimension n is FTop#G ,* * X #. We ij~`~'~ will in appendix A explain the structure of R X as a globular set. We define ij~`~'~ ij~`~'~ ae X # #R X = j#, * * (3) * * ij~`~'~ where j is the relation of filter homotopy rel vertices. It will be clear that * *ae X inherits from ij~`~'~ R X the structure of globular set. Our main result is the following: 2 Theorem 0.1 (Main theorem) There are compositions ffii, 1 6 i 6 n in dimensio* *ns n > 1 ij~`~'~ giving the globular set ae X the structure of globular !-groupoid. ij~`~'~ We call ae X the fundamental globular higher homotopy groupoid of the filtere* *d space X . The proof of this theorem goes via the notion of cubical higher homotopy groupoid o* *f a filtered space, established in [BH81b ]. It should be useful therefore to put these resu* *lts in context. The overall aim of work on higher homotopy groupoids may be subsumed in the* * following diagram and its properties: __________________________// topologicalQdataoo_______________________algebraic data QQQ B nnnn QQQQ nnnn * * (4) QQQQ nnnn U QQQQ((QQ vvnnnnBn topological spaces The aim is to find suitable categories of topological data, algebraic data and * *functors as above, where U is the forgetful functor and B # U ffi B, with the following properties: (1) the functor is defined homotopically and satisfies a higher homotopy van * *Kampen theorem (HHvKT)1, in that it preserves certain colimits; (2) ffi B is naturally equivalent to 1; (3) there is a natural transformation 1 ! B ffi preserving some homotopical i* *nformation. The purpose of (1) is to allow some calculation of by gluing simple examples,* * such as convex subsets, following the use of the fundamental groupoid in [B06 ]. This c* *ondition (1) at present also rules out some widely used algebraic data, such as for example * *simplicial groups or groupoids, or differential graded algebras, since for those cases no * *such functor is known. (2) shows that the algebraic data faithfully captures some of the top* *ological data. The imprecise (3) gives further information on the algebraic modelling. The fun* *ctor B should be called a classifying space because it often generalises the classifying spac* *e of a group or groupoid. It has also been found useful in the homotopy classification of maps. Here is a table illustrating the possibilities. _____________________________________ 1Jim Stasheff has suggested this term to the author, instead of the previous* *ly used Generalised van Kampen Theorem, to make clear the higher homotopy information contained in theorems of* * this type. 3 ____________________________________________________ | Topological data | Algebraic data | |______________________________|_____________________ | | space with base point | groups | |_____________________________|_____________________| | space with set of base points | groupoids | |_______________________________|___________________ | | pointed pair of spaces | crossed modules | |______________________________|____________________ | | filtered space |crossed complexes | |______________________________|____________________ | | n-cube of pointed spaces | catn-groups | |______________________________|____________________ | | n-cube of pointed spaces cr|ossed n-cube of groups | |______________________________|____________________ | Strong results in the last two cases are shown in [BL87 , ES87]. In this paper we will deal only with the case of filtered spaces, which of * *course includes the first three cases. There are still several choices of algebraic data as sho* *wn in the following diagram of equivalent categories, which is taken from [Bro99 ]: cubical __#a#___//_oocubical_ T-complexes !-groupoids with connectionsRR#f#OO || RRR ||| RR))R || globular * * (5) poly-T-complexesOO #b#||| || kk!-groupoids55kk #e#| || kkkkkkkkkk |fflffl fflffl|uu#c#kkkkk|kkkkk simplicial ___________//_crossedoo_ T-complexes #d# complexes Each arrow here denotes an explicit functor which is an equivalence of categori* *es. The equiv- alences (a) and (b) are in [BH81a ]; (a) is an essential technical tool in the * *use of cubical !-Gpd s. The equivalence (c) is in [BH81c ], and this with (b) implies the equi* *valence (f); a direct form of this equivalence is given in the much harder category case in [A* *ABS02 ]. The equivalence (d) is due to Ashley in [Ash88 ]. The equivalence (e) is due to Jon* *es [Jon88 ]. The different forms of algebra reflect different geometries, those of disks, gl* *obes, simplices, cubes, as shown in dimension 2 in the following diagram. It is because the geometry of convex sets is so much more complicated in dimens* *ions > 1 than in dimension 1 that new complications emerge for the theories of higher or* *der group theory and of higher homotopy groupoids. 4 A classical homotopical functor on filtered spaces is the fundamental cross* *ed complex X of a filtered space, defined using relative homotopy groups (in the case X0 is * *a singleton) by Blakers, [Bla48]. Major achievements of the papers [BH81a , BH81b ] were o to define a homotopical functor, which here we call ae2, from filtered sp* *aces to cubical !-groupoids with connections (and hence also to cubical T-complexes), whi* *ch in di- mension n is the filter homotopy classes rel vertices of filtered maps In* * ! X (but see Remark 1.3); o to prove that this functor preserved certain colimits; o to relate ae2 with the classical functor from filtered spaces to crosse* *d complexes, and so to prove that preserves certain colimits. The proofs do not involve traditional techniques such as singular homology or s* *implicial approxi- mation. The results give nonabelian information in dimensions 6 2, and in highe* *r dimensions give information on the action of the fundamental group. Thus the Relative Hure* *wicz Theo- rem is a corollary of a HHvKT [BH81b , Example 6, p.34]. Analogous methods were* * used by Ashley in [Ash88 ] to define a functor ae from filtered spaces to simplicial T* *-complexes, and his ideas contributed to [BH81b ]. However there has been a lack of a directly defined homotopical functor fro* *m filtered spaces to globular !-groupoids, and this gap will be filled in this paper. The definition of classifying space is most convenient via well developed s* *implicial con- structions. In this way we get the classifying space of a crossed complex, [BH9* *1 ]. Its proper- ties are further exploited in, for example, [BGPT , Mar07 , MP06 ]. The equivalence of the category of globular !-groupoids with the category o* *f cubical !- groupoids with connection, and the monoidal closed structure on the latter cons* *tructed in [BH87 ], implies a monoidal closed structure on the category of globular !-grou* *poids. Fur- ther it is shown in [BH91 ] that the simple rule #f# #g# 7! #f g# gives a natur* *al transformation ae2 X ae2 Y ! ae2#X Y # for any filtered spaces X , Y , where X Y is the usual tensor product of fi* *ltered spaces given by [ #X Y #n # Xp Yq. p#q#n The induced transformation on crossed complexes is shown in [BB93 ] to be an is* *omorphism if X , Y are cofibred and connected. It follows from the above that there is a* * natural trans- formation ij~`~'~ ij~`~'~ ij~`~'~ ae X ae Y ! ae #X Y #. 5 This could be difficult to construct directly. This natural transformation may * *be used to enrich the category of filtered spaces over the monoidal closed category of globular !* *-groupoids. It should be apparent from the above that it is the cubical case which give* *s the possibility of formulating and of proving theorems; the basic reason is that cubical theory* * is handy for subdivision and its inverse, multiple compositions, and is also good for te* *nsor products. Many theorems can then, by equivalences of categories, be translated to the oth* *er cases. However the proofs for the cubical cases, particularly the properties of thin e* *lements and T- complexes, involve also the use of crossed complexes and the equivalence of cat* *egories (a,b) of diagram (5). Crossed complexes also have a well developed homotopy theory, * *[BG89 ], and have a clear relation with chain complexes with operators, [BH91 ]. The rel* *ation with simplicial theory is useful because of the wide development of simplicial theor* *y. Finally, the relation with the globular theory could be useful because of the wide familiari* *ty of uses of weak structures and lax functors and natural transformations: for example, com* *pare the discussion of Schreier theory using crossed complexes in [BH82 , BP96 ] with th* *e use of 2- groupoids in [BBF05 ]. Calculational applications are usually made using crosse* *d complexes. For example, the paper [BP96 ] uses the notion of small free crossed resolution* * to give small parametrisations of some nonabelian extensions of groups. 1 Disks, globes, and cubes Our results follow from an analysis of the relations between globes and cubes. * *These results are probably well known but need to be done carefully for our purposes. We give real space Rn the Euclidean norm kxk2 # x21# x22# # x2n. We embed* * Rn in Rn#1 as usual by x 7! #x, 0#. The n-cube In will be the subset of Rn of points * *x such that jxij 6 1 for all i. Thus I # I1 is identified with # 1, 1# and we also identify* * In with the n-fold product of I with itself. The n-disk is the subspace Dn of Rn of points x with kxk 6 1. The #n 1#-sp* *here Sn 1 is the subspace of Dn of points x with kxk # 1. We define the n-globe Gn to be Dn as a space, but with the cell structure Gn # e0 [ e1 [ [ en 1 [ en. Here for i < n the closed cell ~eiis the set of points x # #x1, . .,.xn# 2 Gn s* *uch that kxk # 1, xj # 0 for j < n i and xn i > 0. This convention is in keeping wit* *h the relationship with cubes which we find convenient. Note that the #n 1#-skeleto* *n of Gn is contained in Sn 1. For each of Q # , 2, ij~`~'~we have a singular complex SQ X of a topologica* *l space X, giving the well known simplicial and cubical singular complex, and also a `globular' s* *ingular complex consisting of maps Gn ! X. We will later describe this as a `globular set'. 6 Definition 1.1 We now define by induction maps OEn # In ! Gn, n > 1, with the f* *ollowing properties, for x # #x1, . .,.xn# 2 In: (i) OE1#x1# # x1; (ii)jxij # 1 for some i # 1, . .,.n if and only if kOEn#x#k # 1; (iii)jxij # 1 for some i # 1, . .,.n implies #OEn#x##j # 0 for j < i. We set for x # #t, y# 2 R Rn 1: p ______________ OEn#t, y# # #t 1 kOEn 1#y#k2, OEn 1#y##. * * (6) First note that if x # #t, y# then kOEn#x#k2 # t2 # #1 t2#kOEn 1#y#k2. This easily proves (ii) and (iii) by induction. * * 2 ij~`~'~2 The maps OEn # In ! Gn induce a map ~OE# S X ! S X. We define the globular subset flK of a cubical set K to agree with K in dime* *nsions 0, 1 and to be in dimension n > 2 the set of k such that @i k 2 Im "i11, i # 2, . .,.n. ij~`~'~2 * * 2 Proposition 1.2 The image of ~OE# S X ! S X is exactly the globular subset of* * S X. Proof We prove by induction from the formula for OEn that the image is globular* *. Let pi1# Rn ! Rn i be the projection omitting the first i coordinates. Suppose that OE* *n 1~@i # fn 1pi11. Then OEn@~i#1# f0n 1pi1where f0n 1#x# # #0, fn 1#x##. For the converse, we prove by induction that these are the only identificati* *ons that OEn makes. Suppose OEn#t, y# # OEn#t0, y0#. Then OEn 1#y# # OEn 1#y0# and p ______________ p _______________ t 1 kOEn 1#y#k2# t0 1 kOEn 1#y0#k2. Thus if kOEn 1#y#k 6# 1 then t # t0. But kOEn 1#y#k # 1 implies some jyij # 1, * *by the inductive hypothesis. * * 2 Let X be a filtered space. Then we obtain three filtered singular complexe* *s RQ X for Q # D, ij~`~'~, 2 defined as graded sets by #RQ X #n # FTop#Qn, X #. 7 There are also associated graded homotopy sets aeQ X which in dimension n are * *given by the quotient maps pQ # RQ X ! aeQ X # RQ X = j where j is the relation of homotopy rel vertices through filtered maps. In the cases Q # D, 2 it is known that these graded sets obtain additional * *structure giving us for Q # D the fundamental crossed complex X and for Q # 2 what is called * *in [BH81b ] the fundamental (cubical) !-groupoid (with connections) of X . However the proo* *f that the standard compositions on R2 X are inherited by ae2 X is non trivial, as is th* *e crucial result that p2 is a Kan fibration of cubical sets. Remark 1.3 In [BH81b ], the homotopies are not taken rel vertices and a condi* *tion J0 is imposed, that each map `I2! X0 may be extended to a map I2 ! X1. This conditio* *n is in many ways inconvenient. The filling processes used in the proofs can all be* * started by assuming instead that the homotopies are rel vertices so that the maps `I2! X0 * *required to be extended are in fact all constant. The details will be available in [BHS08 ]* *. 2 Our first main result is: Theorem 1.4 The induced map ij~`~'~ OE # ae X ! ae2 X is injective. ij~`~'~ * * 2 Proof Let #ff#, #fi# 2 #ae X #n be such that OE #ff# # OE #fi#, that is #ffOE#* * # #fiOE# in #ae X #n. Let H # ffOE j fiOE be such a homotopy. Then H is a map In#1 ! X such that writing * *In#1 # In I, each Ht # In ! X is a filtered map. We use a folding map # In ! In given by Definition 3.1 of [AABS02 ] (see* * Definition B.2) which has the property that factors through OE. We now define a new homotopy Kt # Ht # In ! X. Then Kt is a globular homot* *opy ffOE j fiOE. But, by assumption, ffOE, fiOE are already globular maps. So t* *he proof is completed with the following lemma. Lemma 1.5 If a # In ! X is a globular map, then a is globularly equivalent * *to a. Proof Since is a composition of the folding operations _i, it is sufficient * *to prove that _ia jij~`~'~a. We follow the proof of [AABS02 , Proposition 3.4]. By the defin* *ition of _i: _ia # #i@i#1a ffii#1a ffii#1 i @#i#1a. But @i#1a and @#i#1a By the laws (A2) we obtain, since a is globular, that i @i#1a 2 Im i "i # Im "2i# Im "1#1"i. 8 So standard contractions of the two cubes i @i#1a yield a homotopy of _ia jij* *~`~'~a through globular maps. * * 2 It now follows that ff, fi # Gn ! X are globularly equivalent. * * 2 This proof is a higher dimensional version of an argument in section 6 of * *[BHKP02 ]. ij~`~'~ Corollary 1.6 The compositions in ae2 X are inherited by ae X to give the la* *tter the structure of globular !-groupoid. ij~`~'~ We do not know how to prove directly that ae X may be given this structure of* * globular !-groupoid. 2 The free globular !-groupoid on one generator Let X be a filtered space. Then we have a diagram of maps of homotopy sets ij~`~'~j 2 # X #n #i! #ae X #n #! #ae X #n. * * (7) We know from [BH81b ] that the composition j ffi i is injective. We already kn* *ow that j is ij~* *`~'~ injective. It follows that i is injective. Thus the globular !-groupoid ae X * * contains the crossed complex X , and the results of [BH81c ] show that the latter generates* * the former as !-Gpd . We need below the following result. * * ij~`~'~ Theorem 2.1 If G is a globular !-groupoid, then there is a filtered space X s* *uch that ae X ,# G. Proof Let C be the crossed complex associated with the !-groupoid G under the e* *quivalence (c) of diagram (5). By Corollary 9.3 of [BH81b ], there is a filtered space X * *such that X ,# C. (Here X is the classifying space BC filtered by Xn # BC#n#where C#n#is the n* *th truncation ij~`~'~ of C.) It follows that ae X ,#G. * * 2 ij~`~'~n Theorem 2.2 The globular !-groupoid ae G is the free globular !-groupoid on t* *he class of the identity map, and its associated crossed complex is isomorphic to Gn. ij~`~'~n Proof Let ' # Gn ! Gn denote the identity map, and #'# its class in ae G . Let * *H be a globular ij* *~`~'~n !-groupoid and let x 2 Hn. We have to show there is a unique morphism ff # ae G* * ! H ij~`~'~ such that ff#'# # x. By Theorem 2.1 we may assume H is of the form ae X for so* *me filtered 9 ij~`~'~ space X . Then x has a representative g # Gn ! X . It follows that ae #g###'## * *# x. This proves existence of such a morphism. ij~`~'~n * * n Suppose fi # ae G ! H is another morphism such that fi##'## # x. Then fl#f* *f#, fl#fi# # G agree on the generating element cn 2 ssn#Gn, Gnn 1, 1# of that group. However* * Gn is generated as crossed complex by all elements dcn 2 ssr#Gnr, Gnr 1, 1# for all * *globular face operators d from dimension n to dimension r for 0 6 r 6 n. Since ff, fi are mor* *phisms of !-groupoids, ff# dcn# # dff cn # dfi cn # fi#d cn# . Therefore ff, fi agree on * * Gn. But ij~`~'~n the latter generates ae G as !-groupoid. So ff # fi. * * 2 The form of this crossed complex may be deduced from the cubical Homotopy * *Addition Lemma, [BH81a , Lemma 7.1]. 8 >< x#1 x2 # x1 # x#2 ifn # 2, ffix # x#3 #x2 #u2x x#1# #x3 #u3x# x#2# #x1 #u1xifn # 3, >:P n i # u x i#1# 1# fxi #xi # i g ifn > 4 (where ui # @#1@#2 b- @#n#1#. In the case when x is globular, this reduces to ffix # x#1# x1 ifn > 2. Notice that this is a groupoid formula if n # 2. 3 Closed monoidal structure The category of cubical !-groupoids with connection is monoidal closed, [BH87 ]* *. We recall from that paper how the tensor product is defined. For cubical !-Gpd s F, G, H, we define a bimorphism b # F, G ! H * * (8) to be a family of functions b # bp,q # Fp Gq ! Hp#q such that if x 2 Fp, y 2 * *Gq and p # q # n then: ( b#@ffix, y# if1 6 i 6 p, (i)@ffib#x, y# # b#x, @ffiyp# ifp # 1 6 i 6 n; ( b#"ix, y# if1 6 i 6 p # 1, (ii)"ib#x, y# # b#x, "i py# ifp # 1 6 i 6 n # 1; ( b# ix, y# if1 6 i 6 p, (iii) ib#x, y# # b#x, i py# ifp # 1 6 i 6 n; 10 (iv)b#x ffiix0, y# # b#x, y# ffiib#x0, y# if1 6 i 6 p and x ffiix0 is defined * *inF; (v)b#x, y ffijy0# # b#x, y# ffip#j b#x, y0# if1 6 j 6 q and y ffijy0 is defin* *ed inG; The tensor product of cubical !-groupoids F, G is given by the the universal* * bimorphism F, G ! F G: that is any bimorphism F, G ! H uniquely factors through a morph* *ism F G ! H. We next recall a result from [BH91 ]. Proposition 3.1 Let X , Y be filtered spaces. Then there is a natural transfor* *mation j # ae2 X ae2 Y ! ae2#X Y #. Proof This natural transformation is determined by the bimorphism ##f#, #g## 7! #f g# where f # Ip ! X , g # Iq ! Y . The proof that this is well defined and gives a* * bimorphism is routine, given the geometry of the cubes, that Ip Iq ,#Ip#q , and the well* * definedness of compositions on filter homotopy classes, as proved in [BH81b ]. * * 2 It is proved in [BH91 ], by considering the corresponding free crossed comp* *lexes, that this morphism is an isomorphism if X , Y are skeletal filtrations of CW-comple* *xes, and in [BB93 ] that this is an isomorphism if X , Y are connected and cofibred. Because the categories of cubical and of globular are equivalent, and the fo* *rmer has a monoidal closed structure, this is inherited by the latter. So we deduce from the above results: Theorem 3.2 Let X , Y be filtered spaces. Then there is a natural transformat* *ion ij~`~'~ ij~`~'~ ij~`~'~ j # ae X ae Y ! ae #X Y # which is an isomorphism if X , Y are connected and cofibred. 4 The higher homotopy van Kampen Theorem Suppose for the rest of this section that X is a filtered space. We suppose g* *iven a cover U # fU~g~2 of X such that the interiors of the sets of U cover X. For each i 2* * n we set Ui # Ui1" " Uin, Uii# Ui " Xi. Then Ui0` Ui1` is called the induced filtr* *ation Ui 11 ij~`~'~ of Ui. So the globular homotopy !-groupoids in the following ae -diagram of the* * cover are well defined: F ij~`~'~ia//_F ij~`~'~~c//ij~`~'~ i2 2 ae U__b_//_~2 ae U _____ae X * * (9) F Here denotes disjoint union (which is the same as coproduct in the category o* *f globular !-groupoids); a, b are determined by the inclusions ai # U~ " U~ ! U~, bi # U~ * *" U~ ! U~ for each i # #~, ~# 2 2; and c is determined by the inclusions c~ # U~ ! X. Definition 4.1A filtered space X is said to be connected if the following cond* *itions hold for each n > 0 # ffl If r > 0, the map ss0X0 ! ss0Xr, induced by inclusion, is surjective; i.e. * *X0 meets all path connected components of all stages of the filtration Xr. ffl (for n > 1): If r > n and x 2 X0, then ssn#Xr, Xn, x# # 0. * * 2 Theorem 4.2 Suppose that for every finite intersection Ui of elements of U, th* *e induced filtra- tion Ui is connected. Then (C) X is connected; ij~`~'~ (I) c in the above ae -diagram is the coequaliser of a, b in the category of g* *lobular !- groupoids. Proof This follows from Theorem B of [BH81b ], i.e. the analogous theorem for* * ae2, and the fact that the equivalence from the category of globular !-groupoids to that* * of cubical ij~`~'~ 2 !-groupoids with connections takes ae X to ae X . * * 2 5 Nerves and classifying spaces of globular !-groupoids Here we just show how to define a simplicial nerve N G of a globular !-groupoi* *d G, by the standard procedure: ij~`~'~ #N G#n # !-Gpd #ae n, G#. * *(10) The geometric realisation of this simplicial set then defines the classifying s* *pace BG of G. However it is not so easy to see how to exploit this. The classifying space of* * a crossed complex is applied in for example [BH91 , BGPT , Mar07 , MP06 ]. 12 A The globular site We now recall from [BH81c ] a definition which in [Str87], and later work, is t* *ermed that of a globular set. This is a sequence #Sn#n>0 of sets with two families of functio* *ns di # Sn! Si, i # 0, . .,.n 1, si # Si! Sn,i # 0, . .,.n 1, satisfying the following laws, where ff, fi # : (i)dffidfij# dffifori < j, ff, fi # ; (ii)sjsi # si fori < j; 8 >: si for j > i. A globular site GS is a small category such that globular sets can be identi* *fied with con- travariant functors GS ! Set. We want to identify such a site whose objects are* * the globes Gn of section 1. We therefore define maps ~di# Gi! Gn, ~si# Gn! Gi * *(11) p ________ x 7! #0n i, 1 kxk2, x#,#x1, . .,.xn#7! #x1, . .,.xi# * *(12) for i < n, where 0j # #0, . .,.0#. ____-z___" j B The cubical site Let K be a cubical set, that is, a family of sets fKn; n > 0g with face maps @f* *fi# Kn ! Kn 1 #i # 1, 2, . .,.n; ff # #, # and degeneracy maps "i # Kn 1 ! Kn #i # 1, 2, . .,.n# * *satisfying the usual cubical relations: @ffi@fij# @fij@1ffi #i < j#, (B.* *1)(i) "i"j# "j#1"i #i 6 j#, (B.1* *)(ii) 8 ><"j 1@ffi #i < j# @ffi"j# "j@ffi 1 #i > j# (B.1* *)(iii) >: id #i # j# 13 We say that K is a cubical set with connections if it has additional structure* * maps (called connections) #i, i # Kn 1 ! Kn #i # 1, 2, . .,.n 1# satisfying the relation* *s: ffi fij# fij#1 ffi #i < j# (B.* *2)(i) ffi ffi# ffi#1 ffi (B.* *2)(ii) ( "j 1 ffi #i < j# ffi"j# (B.2* *)(iii) "j ffi 1 #i > j# ffj"j# "2j# "j#1"j, (B.2* *)(iv) ( fij@1ffi #i < j# @ffi fij# fi (B.2* *)(v) j @ffi 1 #i > j # 1#, @ffj ffj# @ffj#1 ffj# id, (B.2* *)(vi) @ffj jff# @ffj#1 jff# "j@ffj. (B.2* *)(vii) The connections are to be thought of as extra `degeneracies'. (A degenerate cub* *e of type "jx has a pair of opposite faces equal and all other faces degenerate. A cube of ty* *pe ffix has a pair of adjacent faces equal and all other faces of type ffjy or "jy .) The prime example of a cubical set with connections is the singular cubical* * complex K # S2 X of a space X. Here Kn is the set of singular n-cubes in X (i.e. continuous* * maps In ! X). The face maps are induced as usual by maps ~@i# In 1 ! In and the degeneracies * *by the projections pi # In ! In 1. The connections ffi# Kn 1 ! Kn are 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 complex S2 X has some further relevant structure, namely the compositio* *n of n- cubes in the n different directions. Accordingly, we define a cubical set with* * connections and compositions to be a cubical set K with connections in which each Kn has n * *partial compositions ffij #j # 1, 2, . .,.n# satisfying the following axioms. If a, b 2 Kn, then a ffijb is defined if and only if @j b # @#ja , and then ( ( @j #a ffijb# # @j a ff @ffja ffij 1@ffib#i < j# @i#a ffijb# # * *(B.3) @#j#a ffijb# # @#jb @ffia ffij@ffib#i > j#, The interchange laws. If i 6# j then #a ffiib# ffij#c ffiid# # #a ffijc# ffii#b ffijd# * *(B.4) 14 whenever both sides are defined. (The diagram ~ ~ __//_ a b |ffliffl c d j will be used to indicate that both sides of the above equation are defined and * *also to denote the unique composite of the four elements.) If i 6# j then ( "ia ffij#1"ib#i 6 j# "i#a ffijb## * *(B.5) "ia ffij"ib #i > j# ( ffia ffij#1 ffib#i < j# ffi#a ffijb## (B.* *6)(i) ffia ffij ffib#i > j# ~ # ~ ___// j a "ja j #j#a ffijb## # fflffl| (B.* *6)(ii) "j#1a j b j#1 ~ ~ ___// j a "j#1b j j #a ffijb## fflffl| (B.6* *)(iii) "jb j b j#1 These last two equations are the transport laws2. It is easily verified that the singular cubical complex S2 X of a space X s* *atisfies these axioms if ffij is defined by ( a#t1, . .,.tj 1, 2tj, tj#1, . .,.tn##tj* * 6 1_2# #a ffijb##t1, t2, . .,.tn# # * * 1 b#t1, . .,.tj 1, 2tj 1, tj#1, . .,.tn#* *#tj > _2# whenever @j b # @#ja. * * ij~`~'~ We will now describe two graded subsets of a cubical set K. The globular su* *bset K consists in dimension n of the elements a such that @ffia 2 Im "i11, i # 1, . .,.n. The * *diskal subset KD consists in dimension n of the elements a such that @ffia 2 Im "n11 for #ff, i#* * 6# # , 1#. Clearly ij~`~'~ KD ` K ` K. Proposition B.1 If K is a cubical set with compositions, then the compositions * *ffii are inherited ij~`~'~ ff ij~`~'~ij~`~'~ ffi ij~`~'~ by K so that if di # Kn ! Kn i is defined by a 7! #@1# #a#, then K becomes a * *globular * *ij~`~'~ set with compositions. If further K is a cubical !-category (-groupoid), then K* * is a globular !-category (-groupoid). _____________________________________ 2Recall from [BS76] that the term connection was chosen because of an analog* *y with path-connections in differential geometry. In particular, the transport law is a variation or speci* *al case of the transport law for a path-connection. 15 It is proved in [BH81a ] that if K is a cubical !-groupoid then KD inherits* * the structure ij~`~'~ of crossed complex, and in [BH81c ], see also [AABS02 ], that K inherits the * *structure of globular !-groupoid. A globular !-category is a globular set as above with category structures f* *fii on Sn 0 6 i 6 n 1 for each n > 0 such that ffii has Si as its set of objects and Di , D#i, * *Ei as its initial, final, and identity maps. These category structures must be compatible, that is: (i)if i > j and ff # then Dffi#x ffijy# # Dffix ffijDffiy, whenever the left hand side is defined; (ii)Ei#x ffijy# # Eix ffijEiy in Sn whenever the left hand side is defined; (iii)(The interchange law) if i 6# j then #x ffijy# ffii#z ffijw# # #x ffiiz# ffij#y ffijw# whenever both sides are defined. It is standard to write both sides of the interchange law (when defined) as ~ ~ ___/j/ x y fflffl| z w i Definition B.2Let K be a cubical set with connections and compositions. The fol* *ding opera- tions are the operations _i, r, m #Kn ! Kn defined for 1 6 i 6 n 1, 1 6 r 6 n and 0 6 m 6 n by _ix # #i@i#1x ffii#1x ffii#1 i @#i#1x, r # _r 1_r 2. ._.1, m # 1 2. . .m# _1#_2_1# . .#._m 1 . ._.1#. * * 2 Note in particular that 1, 0 and 1 are identity operations. Here is a picture of _1 # K2 ! K2: 16 ______|| | ____|||_ | |||_| |____|||| | | ___/1/ _1#x# # | | fflffl| | x | |____ | 2 |_||||| |__|||| |___| | |_____| Proposition B.3 Let K be a cubical set with connections and compositions. The `* *folding'operator n #Kn ! Kn satisfies @i nx 2 Im "i11 for 1 6 i 6 n and x 2 Kn. That is, Im * *is contained in the globular subset of K. This is part of Proposition 3.3(iii) of [AABS02 ]. Note that the compositions * *are needed to define n but this property of n does not require any axioms on the compositio* *ns, but only the properties (B1), (B2) giving the relations between cubical operations and c* *onnections. References [AABS02] Al-Agl, F. A., Brown, R. and Steiner, R. `Multiple categories: the eq* *uivalence of a globular and a cubical approach'. Adv. Math. 170 (1) (2002) 71-118. [Ash88] Ashley, N. `Simplicial T-complexes and crossed complexes: a nonabelia* *n version of a theorem of Dold and Kan'. Dissertationes Math. (Rozprawy Mat.) 2* *65 (1988) 1-61. With a preface by R. Brown. [BB93] Baues, H. J. and Brown, R. `On relative homotopy groups of the produc* *t filtration, the James construction, and a formula of Hopf'. J. Pure Appl. Algebr* *a 89 (1-2) (1993) 49-61. [Bla48] Blakers, A. L. `Some relations between homology and homotopy groups'.* * Ann. of Math. (2) 49 (1948) 428-461. [BBF05] Blanco, V., Bullejos, M. and Faro, E. `Categorical non-abelian cohomo* *logy and the Schreier theory of groupoids.' Math. Z. 251 (1) (2005) 41-59. [B06] Brown, R., Topology and groupoids, Booksurge LLC, (2006) xxv+512pp. (* *previous editions: McGraw Hill, 1968; Ellis Horwood, 1988.) [BG89] Brown, R. and Golasinski, M., `A model structure for the homotopy th* *eory of crossed complexes', Cah. Top. G'eom. Diff. Cat. 30 (1989) 61-82. 17 [BGPT] Brown, R., Golasinski, M., Porter, T. and Tonks, A. `Spaces of maps * *into classifying spaces for equivariant crossed complexes II: the general topological* * group case', K-theory 23 (2001)129-155. [BH81a] Brown, R. and Higgins, P. J. `The algebra of cubes'. J. Pure Appl. A* *lg. 21 (1981) 233-260. [BH81b] Brown, R. and Higgins, P. J. `Colimit theorems for relative homotopy* * groups'. J. Pure Appl. Algebra 22 (1) (1981) 11-41. [BH81c] Brown, R. and Higgins, P. J. `The equivalence of #-groupoids and cro* *ssed com- plexes'. Cahiers Topologie G'eom. Diff'erentielle 22 (4) (1981) 371-* *386. [BH82] Brown, R. and Higgins, P. J. `Crossed complexes and non-abelian ext* *ensions', Category theory proceedings, Gummersbach, 1981, Lecture Notes in Mat* *h. 962 (ed. K.H. Kamps et al, Springer, Berlin, 1982), pp. 39-50. [BH87] Brown, R. and Higgins, P. J. `Tensor products and homotopies for !-* *groupoids and crossed complexes'. J. Pure Appl. Algebra 47 (1) (1987) 1-33. [BH91] Brown, R. and Higgins, P. J., `Crossed complexes and chain complexes* * with oper- ators', Math. Proc. Camb. Phil. Soc. 107 (1990) 33-57. [BH91] Brown, R. and Higgins, P. 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