`Double modules', double categories and groupoids, and a new homotopical double groupoid Ronald Brown* School of Computer Science Bangor University Gwynedd LL57 1UT, U.K. March 19, 2009 Bangor Math Preprint 09.01 Abstract We give a rather general construction of double categories and so, under* * further conditions, double groupoids, from a structure we call a `double module'. We also give a homotopical construction of a double groupoid from a tria* *d consisting of a space, two subspaces, and a set of base points, under a condition which also impl* *ies that this double groupoid contains two second relative homotopy groups.1 Introduction Double categories were introduced by Ehresmann [Ehr65] as an example of structu* *red category. Ex- amples of double groupoids were shown to arise from crossed modules in [BS76b ,* * BS76a] and this result was applied in [BH78 ] to give a 2-dimensional version of the Van Kampen* * Theorem for the fundamental group, namely a colimit theorem for the fundamental crossed module * *of a based pair. These results were generalised to crossed modules of groupoids, and to higher d* *imensions, in [BH81a ]. A more general version of the construction of double groupoids was given in [BM* *92 ] in terms of a core diagram. Another construction of double groupoids due to the author was ta* *ken up by Lu and Weinstein in [LW89 ] for purposes of Poisson groupoids. The first aim of this note is to give a generalisation of this last construc* *tion. We do not obtain an equivalence of categories, and so in effect the construction shows that double * *groupoids can be quite complex objects. Perhaps they should be considered among the basic structures i* *n mathematics. A classification of double groupoids is given in [AN ]. The second aim (Section 2) is to give a new construction of a double groupoi* *d for a topological space X with three subspaces A, B, C such that C A\B. Here C is thought of as* * a set of base points, and the double groupoid is well defined if the two induced morphisms ss2(A, c) * *! ss2(X, c), ss2(B, c) ! ss2(X, c) have the same image. Under this condition, the double groupoid also * *contains the two relative homotopy groups ss2(X, A, c), ss2(X, B, c) for all c 2 C. This is an * *extension of results of [BH78 ]. Thus this construction has the advantage of generality, symmetry, and * *multiple compositions in either directions, advantages not available for the traditional relative hom* *otopy groups. The relation between the two constructions is unclear, and it is hoped that * *this paper will encourage further study of the area. The bibliography gives other uses of double categories and groupoids, [Spe77* *, BM99 , DP93, Ehr63b, BJ04, Mac99], but is not intended to be exhaustive. *_____________________________________ email: r.brown@bangor.ac.uk. This research was partially supported by INTAS * *grant 93-436 ext `Algebraic K-theory, groups and categories', and a Leverhulme Emeritus Fellowship (2002-2004). 1AMSCLass2000: 18B40,18D05,55E30 KEYWORDS: double categories, double groupoids, crossed modules 1 1 Double modules and double categories We are trying to find a mathematical expression for the following diagram and a* *ssociated equations: ___a_____b___ | | | || | u||m v n |w| |__ __||____|_ ___// | c | d | 2 * * (1) | | | fflffl| x| p y q |z 1 | | | |_____|_____|_ e f We interpret each square as a 2-cell: thus m is thought of as av( uc : m This might be formalised as the equation av= ucm. So our four squares give us boundary equations av = ucm, bw= vdn, cy = xep, dz= yfq. We would like to `compose' such squares. The base point of each square is thoug* *ht of as the bottom right hand corner, so that is where the centre values m, n, p, q are `located'.* * So in order to `compose' m and n we need to translate m to the same corner as n. Thus we assume an actio* *n (m, d) 7! md satisfying md = dmd, and similarly my = ymy. (The data and axioms for this will* * be explained below: here we are concerned with formulae!) Thus we deduce from the above rules that abw = avdn avy = ucmy = ucmdn = ucymy = ucdmdn = uxepmy. We now construct from the above `data' a double category D. The squares of D* * will be quintuples (m : u acv) such that av = ucm (compare [Ehr63a, Spe77, BM99 ]). The horizontal and vertical compositions are given by: (m : u acv)O2(n : v bdw)= (mdn : u abcdw), (m : u acv)O1(p : x cey)= (p my : ux aevy). In virtue of the above calculations, given the data of diagram (1), these gi* *ve compositions with the correct boundary equations. We also would like the interchange law, namely that the two possible ways of* * composing the four squares in diagram (1)give the same answer. A direct calculation shows that thi* *s is equivalent to the rule myfq = qmdz, * * (2) again given the boundary equation dz = yfq, and this makes geometric sense in t* *erms of the choices of `transporting' m to the bottom right hand corner of the square involving q. Now we need to give a structure in which the above calculations make sense. Our notation for categories will be that we write s, t : C ! Ob C for the so* *urce and target maps of the category C so that an arrow c of C is an arrow c : sc ! tc, and composit* *ion cd is defined if and only if tc = sd. 2 Definition 1.1A double P -module consists of three morphisms of categories all * *over the identity on objects: M B H BB~B | BB |OE B__Bfflffl| V ____//_P such that M is totally intransitive, i.e. is a union of monoids, so that s = t * *on M. We write M(x) for M(x, x). Further, there are given right actions of both H and V on M. This mean* *s that if m 2 M(x) and d 2 H(x, y) then there is defined md 2 M(y) and the usual axioms for an act* *ion are satisfied, namely (mm1)h = mhmh1, mhk = (mh)k, whenever these make sense, and 1h = 1, m1 =* * m, and similarly for the action of V on M. We do not suppose these actions commute, but nonetheless we agree to write * *mdzwhen tm = sd and td = sz. However dz is here interpreted formally, or, if you like, as an ar* *row of the free product category H * V over the same set of objects as H, V, M. Thus tmdz= tz. In order to write our axioms in a way which agrees with the above calculati* *ons, we agree `evaluate in P ' means apply the morphisms ~, OE, _ to the given equation to give an equa* *tion in P . Thus the equation `ucm = av evaluated in P ' means not only that (_u)(OEc)(~m) = (_a)(OEv) but also that the equation makes sense in that sa = su, ta = sv, tu = sc, tc = tvtm. Similarly, md = dmd evaluated in P means that tm = sd and (~m)(OEd) = (OEd)(~md). With this agreed, the axioms are: if m, q 2 M, d, f 2 H, y, z 2 V (i)then md = dmd, my = ymy, evaluated in P; (ii)if also yfq = dz in P , then in M, myfq = qmdz. Now we have our main result, which follows from what was written above: Theorem 1.2 Given a double P -module as above, then the compositions O1, O2 gi* *ve the structure of double category, which is a double groupoid if all of M, H, V, P are groupoids. Some special cases are of interest. Example 1.3 If M consists only of identities, then D is the double groupoid of* * squares from H and V which "commute in P ". This is used in [LW89 ]. Example 1.4 Suppose that V = P and _ is the identity. Then we obtain a diagram ~ OE M ____//_Poo__H together with actions of P and H on M. The axioms now imply that ~ is a crossed* * module. 3 Example 1.5 Let H and V be subgroups of the group P and let M be a subgroup of* * P which is normal in both H and V . Then the inclusions, and the conjugation actions of H * *and V on M give the structure of a double P -module, from which we can obtain a double groupoid. Example 1.6 A semicore diagram is defined in [BM92 ] to consist of a commutati* *ve diagram of morphisms of groupoids j M _____//BH BBB | ~BBB__BOEfflffl|| P and an action of P on M such that ~ is a crossed module, j is an inclusion of a* * totally intransitive subgroupoid, and if m 2 M, h 2 H and h-1mh is defined, then h-1mh = mOEh. It fo* *llows that M is normal in H. Conversely, given such a morphism of crossed modules, we obtain a * *double P -module as in 1.1 with V = P . Notice also that if N = Ker OE, then N operates triviall* *y on M. Remark 1.7 This double category also has a thin structure in the sense that th* *e set of quintuples of the form (1 : u acv) form a subdouble category of the main double category. Remark 1.8 It is shown in [BS76a] that from a double groupoid one can recover * *two crossed modules, a kind of horizontal one and a vertical one. However the groupoid conditions ar* *e needed to recover the actions, and we know no way to do this in the category case. From a double cate* *gory one can recover two 2-categories, by restricting to the subdouble categories where either the h* *orizontal, or vertical, edge categories are discrete. 2 A new construction of a homotopical double groupoid Let X* = (X; A, B; C) be a space X with three subspaces A, B, C such that C A* * \ B. Let RX* be the space of maps f : I2 ! X which map the edges @1 I2 in direction 1 into B* *, the edges @2 I2 in direction 2 into A, and map all the vertices @@I2 into C. This is shown in the * *following diagram: C o__B____|o|C __/2/_ | | fflffl| A || X |A| 1 |o_____|_o C B C The boundary maps and degeneracies give the following geometric structure to* * RX*, in which RX1 is the set of maps (I, {0, 1}) ! (A, C) and RX2 is the set of maps (I, {0, * *1}) ! (B, C): ____//_ (RX*,OO1,OO2)__//_(RX2,oO2)o_OO ||| ||| ||| ||| |fflffl|fflffl|fflffl|fflffl|//_| (RX1, O1)________//_Coo_ Clearly the set RX* obtains two compositions O1, O2 from the usual composition * *of squares in the two directions, while RX2, RX1 have just one composition. These compositions a* *re of course not associative, nor do they have identities, but they do have reverses, -1, -2. Th* *ey do however satisfy the interchange law. Further the face and degeneracies respect the composition* *s. The following theorem generalises results from [BH78 ]. Theorem 2.1 Let aeX* be the quotient of RX* by the relation of homotopy rel ve* *rtices of I2 and through the elements of RX*. Then the compositions O1, O2 are inherited by aeX** * to give it the structure of double groupoid if the following condition holds: 4 (Con) For all c 2 C, the induced morphisms ss2(A, c) ! ss2(X, c), ss2(B, c) ! s* *s2(X, c) have the same image. Further, under this condition, the natural morphisms ss2(X, A; C) ! (aeX*, O1), ss2(X, B; C) ! (aeX*, O2) are injective. Proof The proof is a small elaboration of a similar proof for the case A = B * *in [BH78 ]. Details are also given in [Bro99]. The class in aeX* of an element ff of RX* is written <>. We develop only the horizontal case; the other follows by symmetry. So, le* *t us consider two elements <>, <> 2 D such that <<@+2ff>> = <<@-2fi>>, i.e. we have conti* *nuous maps ff, fi : (I2; @2 I2, @1 I2; @2I2) ! (X; A, B; C) and a homotopy h : (I, @(I)) x I ! (A, C) from ff|{1}xIto fi|{0}xIrel vertices, i.e. h(0xI) = y and h(1xI) = x. We define* * now the composition by <> +2<> = <> = <<[ff, h, fi]>>. This is given in a diagram by __B______c____B____ | | | | | | | | A ||ff A| h A| fi |A| |_____|_____|_____|_ B d B To prove this is independent of the choices made we chose two other represe* *ntatives ff02 <> and fi02 <> and a homotopy h0from ff0|{1}xIto fi0|{0}xI. Using them, we get __B______c____B____ | | | | | | | 0 | A ||ff0A| h0 A| fi |A| |_____|_____|_____|_ B d B which should give the same composition in aeX*. Let OE : ff ' ff0, _ : fi :' fi* *0 be homotopies of the required type. They with h, h0give rise to a diagram of the following kind. ________||____________||_ 3 | | | | ~ | | | | ____- | ff0 | h0 | fi0 | | 2 | | | | | |__ ___|__ __|___ __|_ |? OE _ 1 |_______|______________||_ | | | | | | | | | ff | h | fi | | | | | ________||_____________||_ Figure 1: Filling the hole in the middle We seem to have a hole in the middle. The key point is that all homotopies are * *rel vertices. So the bottom face of this hole may be filled by a constant homotopy. Then we can use * *a retraction to fill the hole, and this will give a cube in A, whose top face is a map (I2, @@I2) ! (A, * *c). By our assumption 5 (Con), this is deformable rel boundary and in X to a map (I2, @@I2) ! (B, c). T* *his homotopy is now added in direction 1 to the homotopy of the middle hole, and squashed down to g* *ive another filler of the hole; the composition in direction 2 of these three cubes is now a homotopy* * rel vertices through maps (I2; @2 I2, @1 I2; @2I2) ! (X; A, B; C) as required. The verification of the groupoid axioms is entirely analogous to the case o* *f the fundamental groupoid. We now verify the interchange law. Suppose given an array of composable elements of aeX*: ` ' <><> <><> This gives rise to a partially filled array 0 1 ff h fi @ k k0A fl h0 ffi However because of the rel vertices hypothesis on the homotopies h, h0, k, k0th* *e hole in the middle can be filled with a constant map. Reading the resulting matrix in two ways gives t* *he required interchange law. For the last part, we identify the second relative homotopy group ss1(X, B,* * c) with the homotopy classes of maps (I2, @-1, J2-,1) ! (X, B, c), with composition given by additio* *n in direction 2. So we have a morphism j : ss1(X, B, c) ! ae2(X; A, B; C) which we have to prove injec* *tive. Suppose then j(ff) is nulhomotopic in R2(X; A, B; C) and let H be such a nullhomotopy. We c* *onstruct another nulhomotopy H0 of ff which is a nullhomotopy of maps (I2, @-1, J2-,1) ! (X, B, * *c). To this end we use the connections i which are available in the cubical si* *ngular set S (X) of a space, as in [BH81b , AABS02 , GM03 ], for example. The main point is tha* *t a connection : S (X)n ! S (X)n+1, defined using the functions max, min, gives a kind of dege* *neracy in which_two__ adjacent faces of (f) coincide. It is convenient to represent these_symbolical* *ly_as |_, __|, |, |. The traditional cubical degeneracies are analogously represented by ||, __. In our* * current situation we surround the homotopy H by connections and constant homotopies, and also using * *the hypothesis (Conn) to obtain another homotopy from this time ff surrounded by constant maps* * or ||. In particular, the hypothesis (Conn) is used twice to obtain homotopies ,, ,0as part of the fo* *llowing picture of the new homotopy. To show this the following picture gives the picture at t 2 [0, 1* *], but the connections are actually applied on 2-dimensional faces in directions 1 and 2 of H. The wiggly * *lines denote constant homotopies. This also illustrates that one of the aims of bringing in connectio* *ns and 2-dimensional rewriting was to give a more algebraic method of constructing homotopies than p* *reviously available. For another application of such rewriting, to rotations, see [Bro82]. f|flOB__fflOB__|fflOB__||B__fflO|fflO2//_ f|flO fflO| fflO| |fflO |fflOfflffl| f|flO, fflO| ff,0lO||fflO|fflO1 f|flO tfflO|||fftlO|||f|flO|fflO f|flO__fflO|_fflO|_|fflO_|fflO___ |fflOA| B | A |fBflOfflO| |fflO | | |fflO fflO| |fflO|_AHt A __| |f|f|lfflOO| |fflO | | |fflO fflO| |fflO_|_/o/o|/o/o_|fflO_fflO|_/o/o/o/o__ |fflO B | B fflO| |fflO | fflO| |fflOfflO|_ B __| fflOfflO| |fflO | fflO| |_________________|_____|_/o/o/o/o/o/o/o/o/o/o/o/o/o/* *o/o/o/o/o/o Remark 2.2 Even in the case C is a singleton, the condition (Conn) is a non tr* *ivial condition needed to make aeX* = ss0(RX*) a double groupoid. Of course it is satisfied if A = B,* * giving the double groupoid used in [BH78 ] to prove a 2-dimensional van Kampen theorem. 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