Nonabelian Algebraic Topology Ronald Brown* July 15, 2004 UWB Math Preprint 04.15 Abstract This is an extended account of a short presentation with this title giv* *en at the Min- neapolis IMA Workshop on `n-categories: foundations and applications', Ju* *ne 7-18, 2004, organised by John Baez and Peter May. Introduction This talk gave a sketch of a book with the title Nonabelian algebraic topology * *being written under support of a Leverhulme Emeritus Fellowship (2002-2004) by the speaker an* *d Rafael Sivera (Valencia) [6]. The aim is to give in one place a full account of work b* *y R. Brown and P.J. Higgins since the 1970s which defines and applies crossed complexes and cu* *bical higher homotopy groupoids. This leads to a distinctive account of that part of algebra* *ic topology which lies between homology theory and homotopy theory, and in which the fundamental * *group and its actions plays an essential role. The reason for an account at this Workshop* * on n-categories is that the higher homotopy groupoids defined are cubical forms of strict multi* *ple categories. Main applications are to higher dimensional nonabelian methods for local-to* *-global prob- lems, as exemplified by van Kampen type theorems. The potential wider implicati* *ons of the existence of such methods is one of the motivations of this programme. The aim is to proceed through the steps ` ' ` ' (geometry)____//_underlying____//_(algebra)_//_(algorithms)_//_ computer * * . processes implementation ___________________________________ *I am grateful to the IMA for support at this Workshop, and to the Leverhulm* *e Trust for general support. 1 Nonabelian Algebraic Topology * * 2 The ability to do specific calculations, if necessary using computers, is seen * *as a kind of test of the theory, and one which also leads to seeking of new results; such calcula* *tions seem at this stage of the subject to require strict algebraic models of homotopy types.* * We obtain some nonabelian calculations, and it is this methodology which we term nonabeli* *an algebraic topology. It is fortunate that higher categorical structures, and in particular* * higher groupoid structures, do give nonabelian algebraic models of homotopy types which allow s* *ome explicit calculation. They have also led to new algebraic constructions, such as a nona* *belian tensor product of groups, of Lie algebras, and of other algebraic structures, with rel* *ations to homology of these structures (see the references in [3]). 1 Background Topologists of the early 20th century dreamed of a generalisation to higher dim* *ensions of the nonabelian fundamental group, for applications to problems in geometry and anal* *ysis for which group theory had been successful. The dream seemed to be shattered by the disc* *overy that ~Cech's apparently natural 1932 generalisation of the fundamental group, the hi* *gher homotopy groups ßn(X, x), were abelian in dimensions > 2. So `higher dimensional groups'* * seemed to be just abelian groups, and the dream seemed to be a mirage. Despite this, the relative homotopy groups ßn(X, A, x) were found to be in * *general non- abelian for n = 2, and as a result J.H.C. Whitehead in the 1940s introduced the* * term `crossed module' for the properties of the boundary map @ : ß2(X, A, x) ! ß1(A, x) and the action of ß1(A, x) on ß2(X, A, x). In investigating `adding relations t* *o homotopy groups' he proved the subtle result (call it Theorem W) that if X is obtained from A by* * adding 2-cells, then ß2(X, A, x) is the free crossed ß1(A, x)-module on the characteristic maps* * of the 2-cells. The proof used transversality and knot theory. A potentially new approach to homotopy theory derived from the expositions * *in Brown's 1968 book [2] and Higgins' 1971 book [11], which in effect suggested that most * *of 1-dimensional homotopy theory can be better expressed in terms of groupoids rather than group* *s. This led to a search for the uses of groupoids in higher homotopy theory, and in par* *ticular for higher homotopy groupoids. The basic intuitive concept was generalising from th* *e usual partial compositions of homotopy classes of paths to partial compositions of homotopy c* *lasses (of some form) of cubes. One aim was a higher dimensional version of the van Kampen theo* *rem for the fundamental group. A search for such constructs proved abortive for some years * *from 1966. However in 1974 we observed that Theorem W gave a universal property for ho* *motopy in dimension 2, which was suggestive. It also seemed that if the putative highe* *r dimensional Nonabelian Algebraic Topology * * 3 groupoid theory was to be seen as a success it should at least recover Theorem * *W. But Theorem W is about relative homotopy groups! It therefore seemed a good idea for us to * *look at the relative situation in dimension 2, that is, to start from a based pair X* = (X,* * A, C), where C is a set of base points, and define a homotopy double groupoid æ(X*) using maps* * of squares. The simplest, and most symmetric, possibility seemed to be to consider R2(X*) a* *s the set of maps f : I2 ! X which take the edges of I2 to the subspace A and take the verti* *ces to the set C, and then to form æ2(X*) as a quotient by homotopy rel vertices of I2 and thr* *ough elements of R2. Amazingly, this gave a double groupoid, whose 1-dimensional part was the* * fundamental groupoid ß1(A, C) on the set C. The proof is not entirely trivial, and uses a f* *illing technique. Previous to this, work with Chris Spencer in 1971-2 had investigated the no* *tion of double groupoid, shown a clear relation to crossed modules, and introduced the notion * *of connection on a double groupoid so as to define a notion of commutative cube in a double g* *roupoid, generalising the notion of commutative square in a groupoid. Once the construct* *ion of æ2(X*) had been given, a proof of a 2-dimensional van Kampen theorem came fairly quick* *ly. The paper on this was submitted in 1975, and appeared in 1978 [5] (after a third re* *feree's report, the reports of the first two referees being withheld). The equivalence between * *double groupoids with connection and crossed modules established by Brown and Spencer [7], and s* *o relating the double groupoid æ2(X*) to the second relative homotopy groups, enabled new calc* *ulations of some nonabelian second relative homotopy groups in terms of the associated cros* *sed module, and so enabled calculation of some homotopy 2-types. Thus there is considerable evidence that the nonabelian fundamental group i* *s naturally generalised to various forms of crossed modules or double groupoids, and that t* *hese structures enable new understanding and calculations in dimension 2. The important point * *is that in dimension 2 we can easily define homotopical functors, that is, functors define* *d in terms of ho- motopy classes of maps, in a manner analogous to that of the fundamental group * *and groupoid, and then prove, but not so easily, theorems which enable us to calculate direct* *ly and exactly with these functors without using homological techniques. As an example, we give the following problem. Given a morphism ' : P ! Q o* *f groups, calculate the homotopy 2-type of the mapping cone X of the induced map of class* *ifying spaces B' : BP ! BQ. This 2-type can be described completely in terms of the crossed * *module @ : '*(P ) ! Q `induced' from the identity crossed module 1 : P ! P by the morp* *hism ' [8]. There are some general results on calculating induced crossed modules, whil* *e for some calculations, we have had to use a computer. The following table, taken from [8* *], contains the results of computer calculations, using the package XMOD [1] in the symbolic co* *mputation system GAP [9]. The examples are for Q = S4 and various subgroups P of Q. The c* *omputer has of course full information on the morphisms @ : '*(P ) ! S4 in terms of gen* *erators of the groups in the table. The third column gives ß2 = Ker @ = ß2(X). The numbers 128* *.2322 and 96.67 refer to groups in the GAP4 table of groups. Nonabelian Algebraic Topology * * 4 _________________________________ | | P | ' P | ß | | |_|_____________|__*______|__2__|_|_ | | <(1, 2)> GL(|2, 3) |C | | | | | | 2 | | | | S G|L(2, 3) |C | | | | 3 | | 2 | | | | <(1, 2), (3, 4)>S|C |C | | | | |42 | 2 | | | | D | S C | C | | | | 8 | 4 2 | 2 | | | | C C |SL(2, 3) |C | | | | 3 3| | 6 | | | | <(1, 2)(3, 4)>12|8.2322C |C3 | | | | | 4| 2 | | | | C |96.67 |C | | |_|_______4______|_________|_4__|_| In fact the pairs of equal groups in the second column give pairs of isomorphic* * crossed modules, though the proof of this needs some calculation. It is not claimed that these results are in themselves significant. But the* *y do show that: (i) there are feasible algorithms; (ii) the homotopy groups, even with their st* *ructure as modules over the fundamental group, represent but a pale shadow of the actual homotopy * *type; (iii) this homotopy type can sometimes be represented by convenient nonabelian structures;* * (iv) we can use strict structures for explicit calculations in homotopy theory; (v) the col* *imit formulation allows for some complete calculations, not just up to extension; (vi) there is * *still the problem of determining the triviality or not of the first k-invariant in H3(S4, ß2). It may be argued that the use of crossed modules has not yet been extended * *to the geometric and analytic problems which we have described above as motivating the need for * *a higher dimensional version of the fundamental group and groupoid. This argument shows * *that there is work to be done! Indeed, there are few books on algebraic topology other th* *an [2] which mention even the fundamental groupoid on a set of base points. Once the definitions and applications of the homotopy double groupoid of a * *based pair of spaces had been developed, it was easy to guess a formulation for a homotopy gr* *oupoid and GvKT in all dimensions, namely replace: the based pair by a filtered space; the crossed module functor by a crossed complex functor; and the homotopy double groupoid by a corresponding higher homotopy groupoid. However the proofs of the natural conjectures proved not easy, requiring develo* *pments in both the algebra of higher dimensional groupoids and in homotopy theory. 2 Main results Major features of the work over the years with Philip J. Higgins and others can* * be summarised in the properties of the following diagram of categories and functors: Nonabelian Algebraic Topology * * 5 Diagram 2.1 filtered spacesH rr88r dHHHdHH æ rrrrrrrrr HHHHHH rrrrr HHHHHH rrrrrr HHHHH xxrrr|r|rOrU*r B HHHH$$HH cubical fl //crossedH !-groupoids ______________________complexesoo_ with connections ~ Let us first say something about the categories FTop of filtered spaces, an* *d !-Gpds of cubical !-groupoids with connections. A filtered space consists of a (compactly generat* *ed) space X and an increasing sequence of subspaces X* : X0 X1 X2 . . .Xn . . .X. With the obvious morphisms, this yields the category FTop. One example of a filtered space is a manifold equipped with a Morse functio* *n. This example is studied using crossed complex techniques in [12, Chapter VI] (free crossed c* *omplexes are there called homotopy systems). Another, and more standard, example of a filtered space is a CW -complex wi* *th its skeletal filtration. In particular, the n-cube In, the product of n-copies of the unit i* *nterval I = [0, 1], with its standard cell structure, becomes a filtered space In*, and so for a fi* *ltered space X* we can define its filtered cubical singular complex R2(X*) which in dimension n* * is simply FTop (In*, X*). Thus R2(X*) is an example of a cubical set: it has for n > 1 and i = 1, . .* *,.n, face maps @i from dimension n to dimension (n - 1), and in the reverse direction degenera* *cy maps "i, all satisfying a standard set of rules. We also need in the same direction as the "* *i an additional degeneracy structure, called connections, i, i = 1, . .,.n, which are defined * *by the monoid structures max , min on I. There are also in dimension n and for i = 1, . .,.n * *partially defined compositions Oigiven by the usual gluing a Oib of singular n-cubes a, b such th* *at @+ia = @-ib. There is an equivalence relation on R2(X*)n given by homotopy through fil* *tered maps rel vertices of In. This gives a quotient map p : R2(X*) ! æ2(X*) = (R2(X*)= ). * * (*) It is easily seen that æ2(X*) inherits the structure of cubical set with connec* *tions. A major theorem is that the compositions Oi are also inherited, so that æ2(X*) becomes * *a cubical !- groupoid with connections. The proof uses techniques of collapsing on subcompl* *exes of the n-cube. A development of these techniques to chains of partial boxes proves th* *e surprising result, essential for the theory, that the quotient map p of (*) is a cubical K* *an fibration. This Nonabelian Algebraic Topology * * 6 can be seen as a rectification result, by which we mean a result using deformat* *ions to replace equations up to homotopy by strict equations. An advantage of the functor æ and the category !-Gpds is that in this conte* *xt we can prove a Generalised van Kampen Theorem (GvKT), that the functor æ preserves certain c* *olimits. This yields precise calculations in a way not obtainable with exact or spectral* * sequences. The proof is `elementary' in the sense that it does not involve homology or* * simplicial ap- proximation. However the proof is elaborate. It depends heavily on the algebrai* *c equivalence of the categories !-Gpds and Crsof crossed complexes, by functors fl, ~ in the * *Main Diagram. This equivalence enables the notion of commutative cube in an !-groupoid, and t* *he proof that any composition of commutative cubes is commutative. It is also shown that flæ * *is naturally equivalent to a functor defined in a classical manner using the fundamental g* *roupoid and relative homotopy groups, as detailed below. In detail we set (X*)1 = ß1(X1, X0), the fundamental groupoid of X1 on the* * set of base points X0, and for n > 2 we let (X*)n be the family of relative homotopy * *groups ßn(Xn, Xn-1, x), x 2 X0. Setting C0 = X0 and Cn = (X*)n we find that C obtain* *s the structure of crossed complex, Diagram 2.2 . ._.__//_Cnffin//_Cn-1_//_._._.//_C2ffi2//_C1 t|| |t| |t| s||t|| fflffl|fflffl| |fflfflffflffl|flffl| C0 C0 C0 C0 where the structure and axioms are those universally satisfied by this example * * (X*). If X* is the skeletal filtration of a reduced CW -complex X, then (X*) sho* *uld be regarded as a powerful replacement for the cellular chains of the universal cover of X, * *with operators from the fundamental group of X. However the use of crossed complexes rather than t* *he cellular chains allows for better realisation properties. Also the functor allows for * *many base points, so that groupoids are used in an essential way. The GvKT for æ immediately transfers to a GvKT for this classical functor * *. We again emphasise that the proof of the GvKT for is elaborate, and tightly knit, but * *elementary, in that it requires no background in homology, or simplicial approximation. It* * is a direct generalisation of the proof of the van Kampen Theorem for the fundamental group* *oid. Indeed it was the intuitions that such a generalisation should exist that gave in 1966* * the impetus to the present theory. With this GvKT we deduce in the first instance: o the usual vKT for the fundamental groupoid on a set of base points; Nonabelian Algebraic Topology * * 7 o the Brouwer degree theorem (Sn is (n - 1)-connected and ßnSn = Z); o the relative Hurewicz theorem; o Whitehead's theorem that ß2(A [ {e2~}~, A, x) is a free crossed ß1(A, x)-* *module; o a more general excision result on ßn(A [ B, A, x) as an induced module (c* *rossed module if n = 2) when (A, A \ B) is (n - 1)-connected; o computations of ßn(A [ B, A \ B, x) when (A, A \ B), (B, A \ B) are (n - * *1)-connected. Whitehead's theorem, and the last two results for n = 2, are nonabelian results* *, and hence not obtainable easily, if at all, by homological methods. The assumptions required of the reader are quite small, just some familiari* *ty with CW - complexes. This contrasts with some expositions of basic homotopy theory, where* * the proof of say the relative Hurewicz theorem requires knowledge of singular homology theor* *y. Of course it is surprising to get this last theorem without homology, but this is because* * it is seen as a statement on the morphism of relative homotopy groups ßn(X, A, x) ! ßn(X [ CA, CA, x) ~=ßn(X [ CA, x) and is obtained, like our proof of Whitehead's theorem , as a special case of a* *n excision result. The reason for this success is that we use algebraic structures which model the* * geometry and underlying processes more closely than those in common use. The cubical classifying space B2C construction for a crossed complex C is g* *iven by the geometric realisation of a cubical nerve N2 C defined by (N2 C)n = Crs( (In*), C). The crossed complex C has a filtration C* by successive truncations, and this g* *ives rise to the filtered space B(C) = B2C* for which there is a natural isomorphism B(C) ~* *=C. The fundamental groupoid and homotopy groups of B2C are just the fundamental groupo* *id and `homotopy groups' of C. In particular, if X* is the skeletal filtration of a CW* * -complex X, and C = (X*), then ß1(C) ~=ß1(X, X0) and for n > 2, ßn(C, x) ~=Hn(Xex), the homolo* *gy of the universal cover of X based at x. The category FTop has a monoidal closed structure with tensor product given* * by [ (X* Y*)n = Xp x Yq. p+q=n It is also fairly easy to define a monoidal closed structure on the category !-* *Gpds since this category is founded in the structure on cubes. The equivalence of this category* * with that of Nonabelian Algebraic Topology * * 8 crossed complexes, Crs, then yields a monoidal closed structure on the latter c* *ategory, with exponential law Crs(A B, C) ~=Crs(A, CRS(B, C)). The elements of CRS (B, C) are: in dimension 0 just morphisms B ! C; in dimensi* *on 1, are homotopies of morphisms; and in dimensions > 2 are forms of `higher homotopies'* *. There is a complicated formula for the tensor product A B, as generated in dimension n b* *y elements a b, a 2 Ap, b 2 Bq, p + q = n, with a set of relations and boundary formulae* * related to the cellular decomposition of a product of cells Ep x Eq, where Ep has for p > 1 ju* *st 3 cells of dimensions 0, p - 1, p respectively. Using the filtered 1-cube I1*, and the values on this of æ and , we have u* *nit interval objects in each of the categories FTop, !-Gpds, Crs. This, with the tensor product, gi* *ves a cylinder object and so gives a homotopy theory on these categories. Another difficult result is that the functor preserves certain tensor pro* *ducts, for example of CW -filtrations. This result is needed to deduce the homotopy addition lemma* * for a simplex by induction, and also to prove the homotopy classification results. A major result is that there is a natural bijection of homotopy classes [ (X*), C] ~=[X, B2C] for any crossed complex C and CW -complex X, with its skeletal filtration. This* * is a special case of a weak equivalence B2(CRS( (X*), C)) ! (B2C)X . In fact the published version of this is for the simplicial classifying space B* * C, and this has given also an equivariant version of these results. In this theory the homotopy addition lemma (HAL) for a simplex falls out by* * a simple inductive calculation, since the usual representation of the n-simplex n as a * *cone on n-1 is exactly modelled in the crossed complex case, where the cone is defined usin* *g the tensor product. The explicit formulae for the tensor product, and so for any cone cons* *truction, give an easy calculation of the HAL formula. There is an acyclic model theory for crossed complexes similar to that for * *chain complexes. So we obtain an Eilenberg-Zilber type theorem for crossed complexes. This has b* *een developed with explicit formulae by Tonks. 3 Conclusion The book that is being written is divided into three parts, of which Part I is * *now available on the web [6]. Part I deals with the case of dimensions 1 and 2, and the proof an* *d applications Nonabelian Algebraic Topology * * 9 of the GvKT in this dimension. The reasons for this are that the step from dim* *ension 1 to dimension 2 is a big one, and the reader needs to grasp the new ideas and techn* *iques. Also a gentle introduction is needed to calculating with the (nonabelian) crossed modu* *les. Part II should be regarded as a kind of handbook on crossed complexes. The * *main properties of crossed complexes and of the functor are stated. Applications and calculat* *ions are deduced. Thus crossed complexes are presented as a basic tool in algebraic topology. The proofs in Part III of these basic properties of crossed complexes, such* * as the monoidal closed structure, and the GvKT, require however the use of the category of cubi* *cal !-groupoids with connection. The theory of these is developed in Part III. Appendices give some basic theory needed, such as some category theory (adj* *oint functors, colimits, etc.). A question asked at the Workshop was: Why concentrate on filtered spaces? T* *he only an- swer I can give is that they provide a workable basis for a theory of higher ho* *motopy groupoids. A more general theory, in some ways, is given by n-cubes of spaces, and the ass* *ociated catn- groups, but this does not have such an intuitive exposition as can be provided * *for the filtered case. Currently, there is in the absolute case no theory which works in dimensi* *ons > 2, and even the known dimension 2 case does not yet provide a range of explicit calcul* *ations. Peter May pointed out that, in contrast to stable homotopy theory, this wor* *k brings the fundamental group fully into the algebraic topology setting, and that such inpu* *t is surely needed for applications in algebraic geometry. Note that Grothendieck's increas* *ingly influential manuscript `Pursuing Stacks' [10] was intended as an account of nonabelian homo* *logical algebra (private communication). But that subject should in principle need `nonabelian* * algebraic topology' as a precursor. The book makes no attempt to explain the work on catn-groups, which allows * *for calculations in homotopy theory which are more complex, and indeed more nonabelian, than tho* *se given in the planned book. That work thus continues the story of Nonabelian Algebraic To* *pology, and references to many authors and papers are given in [3]. The references which follow include some survey articles with wider referen* *ces. We conclude with a diagram of historical context for the current theory1. References [1] M. Alp and C. D. Wensley, XMod, Crossed modules and Cat1-groups: a GAP4 pac* *kage, (2004) (http://www.maths.bangor.ac.uk/chda/) ___________________________________ 1I am grateful to Aaron Lauda for rendering this diagram in xypic. Nonabelian Algebraic Topology * * 10 [2] R. Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968. secon* *d edition as Topology: a geometric account of general topology, homotopy types, and t* *he fundamental groupoid, Ellis Horwood, Chichester (1988) 460 pp. [3] R. Brown, `Higher dimensional group theory', http://www.bangor.ac.uk/~mas010/hdaweb2.htm [4] R. Brown, `Crossed complexes and homotopy groupoids as non commutative tool* *s for higher dimensional local-to-global problems', Proceedings of the Fields Institute * *Workshop on Cat- egorical Structures for Descent and Galois Theory, Hopf Algebras and Semiab* *elian Cate- gories, September 23-28, 2002, Contemp. Math. (2004). (to appear), UWB Math* * Preprint 02.26.pdf (30 pp.) [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. [6] R. Brown and R. Sivera, `Nonabelian algebraic topology', (in preparation) P* *art I is down- loadable from (http://www.bangor.ac.uk/~mas010/nonab-a-t.html) [7] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers T* *op. G'eom. Diff., 17 (1976) 343-362. [8] R. Brown and C. D. Wensley, `Computation and homotopical applications of in* *duced crossed modules', J. Symbolic Computation, 35 (2003) 59-72. [9] The GAP Group, 2004, GAP - Groups, Algorithms, and Programming, version 4.4* * , Tech- nical report, (http://www.gap-system.org) [10] A. Grothendieck, `Pursuing Stacks', 600p, 1983, distributed from Bangor. N* *ow being edited by G. Maltsiniotis for the SMF. [11] P. J. Higgins, 1971, Categories and Groupoids, Van Nostrand, New York. Rep* *rint Series, Theory and Appl. Categories (to appear). [12] V. Sharko, 1993, Functions on manifolds: algebraic and topological aspects* *, number 131 in Translations of Mathematical Monographs, American Mathematical Society. Professor Emeritus R. Brown, Department of Mathematics, University of Wales, Bangor, Dean St, Bangor, Gwynedd LL57 1UT, United Kingdom http://www.bangor.ac.uk/~mas010 email: r.brown@bangor.ac.uk Nonabelian Algebraic Topology * * 11 Some Context for Nonabelian Algebraic Topology ________________ _______________ ________ | Gauss' || || Brandt's |||| __________ ___________| |Galois|||||composition|of|||||composition|of|| symmetry||||amodulorithme|Theory|||____||tic||||||____||binary|quadratic* *||||||||M//^^^^^^^ MMMM |________|___|ssss hhhhh||h ||| | quaternary || ___________MMM | ss hhhhhh |____forms____|___||quadratic|forms|* *|llll celestial| M&&Mfflyysssffl|hhhhhh __________vvll|_____________|__* *|l | || _groups__sshh_______________|// | __________|| mechanics||||____||||____|___________EE**|groupoids|____||algebraic||||V* *VV00bbbbbbbbKKmm""""" ____________EEEE |van Kampen's|||bbbbbbbbbbbbbbKK|topology|||____|* *|___hhQQQKKK""""* monodromy||||___|_________"||_Theorem___||__||"__________Yffff33gggK|homolog* *y||KKK"""**jj ___________|,fundamental||||||,YY@@ ???``?"""??invariant||||||KKKK**|_______* *_|__|*ttjj __________ ||_group___||___||WWWW ???"""???|theory|||____||_________%%K~* *~yyttjjj |homology|||____|| 44 WWWWWWWW ""???"??ww? jj|categories|||_* *___|| !!! _____________ææ WWWWWWWWW???"""wwww??? jj | _____________ii!fundamental||||||__________WWWWWW____________WWWW???""öo``* *`````jj--www||??? || higher |||| ||groupoid_||___|||free"""|||???W|cohomology|||||++?"//^^^* *^^^^^^|???? |homotopy || ii OE00 """resolutions||||____||of|groups||???||??? | || ii OEOE00"" ffl ??|__________|__|_______ffl* *ffl|#??88 || groups ||| i OE 00"" __________ffl q????qq # | || | ii OEOE0"" identities|| qqq ???? ## | structured|| (~Cech,|1932)|||___|________|||ffiffi||OE00ii"""qqq????# | || ____________|øi|groupoidsiin|||among||||00___________|||??|?categories||0d* *ddd##??qqq ø i qqq rrdddd##?? ___________|||i|differential|||||relations||||____||double||||||(Ehresmann* *)||||____||????000iiffiffiøjj#???qqqqfi | relative||ii ||topology_||___||000"""qqr|categories|____|??????00qqqfifi homotopy||||||iiiT OO """qq00qqqGGGrrrrrG____________||????????,,fififi ||groups_||___||i_________ii__________|||,,"|2-groupoids|____||________ØØ?* *""""%?yyrrreeeeeeeeeyqqqq||TTTTTTT##fififi T))T __yy fifi %%% iii |crossed|||grdoubleoupoids||||||___||||00b|nonabelian||||* *|bbfifi ___________iii`modules||||____|||__|__|algebraic||||||uu`cohomology||||_* *___||%%jjj||Pfififi jjj PPPP| fi FF | |crossed|||||____________zzuuuuttj|f|K-theory|____|_______%%fiffi||ÆÆÆPP* *PP::uflflflfifiOO complexes||||____||cubical||||||,,YYYYY|ppaa||cat1-groups||||aaaa|PPPPPu* *uuuflfl|| 00 | PPPP lll| ffll | ___________~~0ii!-groupoids||| | PPllllP(Loday,|1982)|||___|f|lfl| ||higher ||||tt|__________|__|ii____________ii_____________|ii||vvl|lllPPP* *PPLLßflfl%%000| |homotopy||||00```````%%%``````|/catn-groups||||____||PPPPPßßßflflfl||/````* *``````````oo```````````````````````````|| |VVVVV00_____________|``kk UUUU|PPPPßß flfl | |groupoids|||____||Generalized|||||uukkk**VVVVV000UUPPPPP___________|||Pflf* *lflUUUUUU| 000 van|Kampen ||||oo```_nonabelian__|||UUU|'crossed|||||'**U| 000|Theorems| |||||TTT ||tensor ||||ppaaan-cubes|of|||||||a* *aaaaaaa ___|__________|___|__|||products|||||0FFFT||groups__||__|||T* *TTTTTT ____________ |quadratic|||||F|##F |_________|__|FFFTTTTTTPP!! || higher|order|||||complexes|||||[[[[[[[[[[[[[[[[___________|||[[""T||TTTTT* *TT |symmetry|_||___|||____|__|ZZ||ZZZZZZ[[[[[[[[[[|computing|||||--[[|))ZZZZ* *GRR || RR________ZZZZZZZ|ZZZZZZZGGG))R |homotopy || || | |gerbes|||____||_______|||fflffl|||_types__||__||ZZZ|ZZZZZZ* *ZZZ##GG | | RRR|)multiple)||___________||ZZZZZZZZZ__________fflf* *fl|ZZZZRR|""zz || || groupoids|in|||||nonabelian||||ZZZZPursuing||||,,* *ZZZ|| || || |differential|||||algebraic|||||||||Stacks|||____* *|| || || ||topology_||___||topology||||____|||| || | | | | | | fflffl| fflffl| fflffl| fflffl| fflffl| fflffl|