Groupoids generalise groups
Double groupoids and multiple compositions
"If at first, the idea is not absurd, then there is no hope for it. Albert EinsteinWhat might seem "absurd" is that these two simple ideas are the start of a new approach to basic algebraic topology and homotopy theory, unmentioned in standard texts.
Context for Higher Dimensional Group Theory
This is a one page (pdf) diagram of the historical development of notions of higher dimensional algebra, from the background to group theory (Galois, Gauss, ...) via the van Kampen Theorems and structured categories to Pursuing Stacks. (October 19, 2006)
Comments on Higher Dimensional Group Theory These are regularly (or irregularly!) updated notes on the area.
`Out of Line' Link to pdf and html files of a new version of a presentation as a Friday Evening Discourse to the Royal Institution of Great Britain in May, 1992. The title refers both to the area being non traditional and to the idea of higher dimensional algebra.
Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids This 703 page text was published August, 2011, as EMS Tracts in Mathematics vol 15. It gives a full account with history and background intuitions of this new approach to basic algebraic topology at the border between homology and homotopy, using cubical higher homotopy groupoids defined on filtered spaces as a key tool in proving a central Higher Order Seifert-van Kampen Theorem for the fundamental crossed complex of a filtered space. It may be ordered from European Mathematical Society Publishing House A downloadable pdf is available from my web page Nonabelian algebraic topology .
Applications of Higher Homotopy van Kampen Theorems This is a brief list of papers stating or applying such theorems.Topology and Groupoids Link to details of this book. The generalisation of the Seifert-van Kampen Theorem from a theorem for the fundamental group on a space with base point to a theorem on the fundamental groupoid on a space with a set of base points opened the door to new ideas on higher homotopy groupoids. Here is a link to a recent paper on this. Readers may like to try try their hand at rewriting traditional iterated loop space theory but on a space with several base points! Note also that this book is the only text on topology to deal with orbit spaces and orbit groupoids.
Bibliography on Higher Dimensional Group Theory. This area is now storming ahead as a web search on "Higher Dimensional Algebra" shows.
Groupoids in Mathematics It was through finding in 1965 the use of groupoids in 1-dimensional homotopy theory that the idea of using groupoids in higher homotopy theory emerged. This brief survey with links gives an account of various uses of groupoids.
Here is a link to a discussion on mathoverflow on generalized categories for higher homotopy groupoids.
Link to Preprint page with several beamer presentations.
Link to Publication list, with many downloadable.
(NOTE: The following was written in 1996. I have now decided to put later comments in the separate page Comments .)
I thought of the idea of Higher Dimensional Group Theory in the mid 1960s and saw it as a method for obtaining new homotopical information by generalising to higher dimensions the fundamental group of a space with base point.
Such a programme was put forward by Dehn, and by Cech in 1932, but the idea was thought to have been killed by the discovery that a natural analogue in higher dimensions of the fundamental group, the higher homotopy groups, were commutative. It was shown that a set with two group structures satisfying a natural compatability condition known as the interchange law was in fact simply a commutative group, and the two group structures coincided. Thus the higher homotopy groups are analogous to but do not generalise the fundamental group. It was on this ground that Cech's paper for the 1932 Zurich ICM was famously rejected by Alexandroff and Hopf, so that only a small paragraph by Cech appeared in the Proceedings. No method was seen at that time of bringing non commutative structures successfully into higher dimensional homotopy theory.
Eventually, this situation changed with the notion of groupoid as a generalisation of that of group. It became clear that a set with two groupoid structures satisfying the analogous interchange law was a much more complicated object than just a group, and could be regarded as `more non abelian' than a group.
I came into this story with the realisation, expressed in my book on Topology published in 1968, that all of the basic theory of the fundamental group obtained a better expression, with more powerful theorems and more elegant proofs, if expressed in terms of the fundamental groupoid of a space X on a set A of base points, rather than restricting to the fundamental group on a single base point. The immediate problem was to calculate the fundamental group of a space which was the union of two subsets with a non connected intersection W. Since W had more than one component, it was not clear in which component to put the base point. A reasonable solution turned out to be to use one point in each component, i.e. to use a set of base points. The stimulus for this work was a 1964 paper by Philip Higgins on Presentations of groupoids with applications to groups, and this was very helpful for turning a theorem into a computational method. This direction was strengthened by a conversation with G.W. Mackey in 1967, when he told me of his work on ergodic groupoids. It seemed that if the idea of groupoids occurred in such widely differing subjects, then it must have a lot going for it. I was not then aware of the origin of the concept of groupoids in number theory, namely the generalisation to the quaternary case of work of Gauss on the composition of binary quadratic forms, which is a major part of his Disquitiones Arithmeticae.
As Grothendieck wrote in 1985:
- "The idea of making systematic use of groupoids (notably fundamental groupoids of spaces, based on a given set of base points), however evident as it may look today, is to be seen as a significant conceptual advance, which has spread into the most manifold areas of mathematics. ... In my own work in algebraic geometry, I have made extensive use of groupoids - the first one being the theory of the passage to quotient by a ``pre-equivalence relation'' (which may be viewed as being no more, no less than a groupoid in the category one is working in, the category of schemes say), which at once led me to the notion (nowadays quite popular) of the nerve of a category. The last time has been in my work on the Teichmüller tower, where working with a ``Teichmüller groupoid'' (rather than a ``Teichmüller group'') is a must, and part of the very crux of the matter ...."
In view of the success of groupoids in 1-dimensional homotopy theory, and since I had some training in the higher dimensional theory, the question of the applicability of the groupoid notion in higher dimensional homotopy theory was unavoidable. There were two possible extreme answers:
- Groupoids were useful in dimension 1 but, as for non commutative groups, can become only of little use in higher dimensions.
- Groupoids can become increasingly useful in higher dimensions.
One pointer was that the result referred to above which made higher homotopy groups commutative did not apply to the groupoid case. The same method of proof yielded only that a set with two compatible groupoid structures contained a family of commutative groups. This crucial difference seemed not to have been investigated and was worth pursuing from the point of view of curiosity alone.
A further starting point was the intuition that the proof of the groupoid version of the Van Kampen Theorem should generalise to higher dimensions, provided the right functors with the right properties were available. That is, there was an idea of a proof in search of a theorem. It was for this reason that a reference to such a higher dimensional theorem appeared in the Introduction to this last paper.
Part of a method of attack was to go through standard treatments of homotopy theory attempting to replace the word 'group' by `groupoid' and to see whether it was possible by appropriate modifications to arrive at a sensible theory which did more than the standard one. This turned out to be a non trivial task, which is ongoing.
There were two simple intuitions involved. One was the notion of an
That is, we know how to cut things up, but do we have available an algebraic control over the way we put them together again? For formulae on a line, we have standard conventions for well formed formulae in group or groupoid theory such as
but we have only a few instances of "2-dimensional formulae".
The other intuition was that one needed a generalisation to higher dimensions of the idea of expressing one side of a square as a composite of the other three sides, or their inverses. It is interesting that here the step from a linear statement to a 2-dimensional statement should need a lot of apparatus; it took us a long time to find an appropriate formulation.
There are two important, related and relevant differences between groupoids and groups. One is that groupoids have a partial multiplication, and that the condition for two elements to be composable is a geometric one (namely the end point of one is the starting point of the other). This partial multiplication allows for groupoids to be thought of as "groups with many identities". The other is that the geometry underlying groupoids is that of directed graphs, whereas the geometry underlying groups is that of based sets, i.e. sets with a chosen base point. It is clear that graphs are more interesting than sets, and can reflect more geometry. Hence people find in practice that groupoids can reflect more geometry than can groups alone.
An argument usually made for groups is that they give the mathematics of reversible processes, and hence have a strong connection with symmetry. This argument applies even more strongly for groupoids. For groups, the processes all start and return to the same position. For groupoids, the processes may start at one point and finish at another. This is clearly a more flexible situation.
Indeed, to analyse a reversible process you need to look at the intermediate processes. A convenient way of doing this is by using groupoids. This confirms a basic intuition that in dimension 1 groupoids are more convenient than groups for writing down an 'algebraic inverse to subdivision'.
There were however several arguments for continuing an exploration of these higher dimensional ideas, although it was of course unclear what exactly could be the applications, or how `central' these would be.
There was a gut feeling that there ought to be some mathematics which enabled
one properly to express in algebraic terms the general notion of "algebraic
inverse to subdivision", and that this would be important for a class of
problems central in mathematics, namely the local-to-global problems. As
an example of the problem, suppose a triangle is subdivided into pieces as
shown in the following picture:
How does one describe algebraically the way it should be put together again?
This very general kind of cutting up is difficult to handle, and the easiest way of attack was to start with cubical methods, since the algebra of compositions of a subdivided cube is relatively easy to handle. Attacks on the more general case are in David Jones' thesis (for the groupoid case only) and in work of Richard Steiner (for the categorical case).
- In view of the importance of group theory in mathematics, a higher dimensional theory should be of great significance, and it would be foolish to turn away from a chance to investigate it. The potential contribution to science and mathematics was far greater than any possible loss of time. Not many such opportunities may occur in one lifetime! In any creative endeavour, one has to catch ideas as they fly past.
- The idea of using groupoids rather than groups had not been tried before, as far as one could tell, and some simple definitions were easy to make.
- Ehresmann had already in his difficult book "Catégories et structure" (Dunod, Paris, 1964) defined the notion of multiple groupoid, and given some simple examples, so that one had something to start with. So these objects could be simply defined: it was not so clear if they were the 'right' ones.
- The idea would not go away!
- As noticed earlier, one aspect is simply that of curiosity. If it did not work, then the reasons for that could also be interesting.
- Perhaps the really serious argument is not the above but that the hypothesis that higher dimensional group(oid) theory will become of great importance is much more interesting and more fun than the opposite hypothesis. If one pursues exciting hypotheses, then some of them might work out well, and the obstructions to the validity of the hypotheses will in any case be of considerable interest. In general, it is a good idea to disregard the many arguments for not trying a new approach.
It is now over 30 years later, and it is reasonable to assess whether or not the programme has been successful.
Certainly, the hypothesis that there is a "many variable group(oid) theory" which can be seen to bear to ordinary group theory a relation similar to that of many variable to 1-variable calculus, is hardly known, let alone accepted. Even the notion of groupoids as a more flexible tool than groups in some situations is only beginning to be widely appreciated. It is not so easy to find a book on group theory or algebraic topology which even mentions groupoids.
Perhaps the most significant of the books which use the notion seriously is Connes "Non commutative geometry". He states that Heisenberg discovered quantum mechanics by considering the groupoid of transitions for the hydrogen spectrum, rather than the usually considered group of symmetry of an individual state. This fits with the previously expounded philosophy. The main examples of groupoids in his book are equivalence relations and holonomy goupoids of foliations.
This programme is however relevant to a programme of Grothendieck on `anabelian algebraic geometry', for which you can see some indication in an extract from his `Esquisse d'un Programme'.
Criteria for success
Tests for a theory which is successful in a mathematical and scientific rather than sociological sense could be the following. A successful theory would be expected to yield:
- a range of new algebraic structures, with new applications and new results in traditional areas;
- new viewpoints on classical material;
- better understanding, from a higher dimensional viewpoint, of some phenomena in group theory;
- new computations with these objects, and hence also in the areas in which they apply;
- new algebraic understanding of the structure of certain geometric situations;
- a stimulus to new ideas in related areas;
- a range of unexplored ideas and potential applications;
- the solution of some classical famous problems.
Interestingly, all these criteria, except the last, seem to be well satisfied.
New algebraic structures
Here are some structures which came out of this theory: omega groupoids; crossed differential algebras; the non abelian tensor product of groups which act on each other, and, under similar circumstances, a non abelian exterior product; induced crossed modules; coproducts of crossed P-modules; non abelian tensor products for various kinds of algebras, such as Lie algebras, associative algebras, commutative algebras; crossed n-cubes of groups (and of Lie algebras, associative algebras, commutative algebras, ...); simplicial T-complexes; infinity groupoids; tensor products of crossed complexes.
New applications of these structures
New computations of certain homotopy groups, and even homotopy types; applications to Poisson structures; new notions of isologisms for p-groups; new estimates for the size of the Schur multiplier; new computations of identities among relations.
New viewpoints on classical material
One of the comforting features of working in this area has been that classical ideas kept on appearing, so that one felt sure that the work was not hurtling off into totally isolated territory.
One of the early examples of this was the link between double groupoids and crossed modules. These had been defined by J.H.C. Whitehead in 1946, as a part of what we can see as his programme of extending to higher dimensions the methods of combinatorial group theory of the 1930s. In one direction, the analysis of Tietze transformations, this led to simple homotopy theory. In another, it led to crossed modules, Whitehead products, and the much more neglected theory of `homotopy systems', now called crossed complexes. The latter are analogues of chain complexes, but with non abelian information in dimensions 1 and 2. It was important to us to find that these crossed complexes were equivalent to forms of infinity groupoids.
One of the deepest examples of these links is that between multiple groupoids and commutator calculus, as shown in work of Ellis and Steiner giving an equivalence between catn-groups and crossed n-cubes of groups.
Understanding of some phenomena in group theory
There are several areas I would emphasise here: one is the notion of symmetry of groups, and of higher order cohomology. It has been well known for many years that the symmetry of a group G can be considered as part of a crossed module G -> Aut(G). There has recently been more understanding of the next level, namely that the symmetries of a crossed module form part of a still higher level structure, namely a 2-crossed module, or a related structure called a crossed square, and that this higher order symmetry is related to the study of homotopy 3-types. These ideas have been put into very general situations by L. Breen.
A special case of the notion of induced crossed module replaces the normal closure NQ(P) of a subgroup P of a group Q by a morphism d: C -> NQ(P) whose kernel is of topological significance (it is the second homotopy group of the mapping cone of BP -> BQ), but no other algebraic descriptions has been given.
The classical Schreier theory of non abelian extensions of groups can be described in terms of free crossed resolutions. This makes the theory more analogous to the usual abelian extension theory, and also, through the choice of small free crossed resolutions, allows for direct computations.
I refer again to the page on non abelian tensor products , where this new construction gives a finer analysis of the commutator subgroup of a group, and so yields new results in group theory.
New computations of these objects, and in their applications
At the start of the theory in 1966, few examples of double groupoids were known. One of the earliest results of Brown and Spencer was an equivalence between certain kinds of structured double groupoids and crossed modules. Since lots of examples of crossed modules were known, this immediately gave rise to many examples of double groupoids.
Recently, Brown and Wensley gave many theoretical and computational results on what are called 'induced crossed modules', which arise in topological examples such as the computation of the homotopy 2-type of mapping cones. These results yielded a number of surprising examples of crossed modules, and led to work by Wensley and Alp on a GAP share package for computation with crossed modules, and for listing crossed modules. There have also been computations by a number of authors on the non abelian tensor product, and these led to the calculation of certain homotopy groups and homotopy 3-types, using the Generalised Van Kampen Theorem. The fact that one can compute some homotopy types directly is a strong argument for this method, and shows that the algebra captures the geometry in a form accessible to computation.
It is interesting that simplicial groups have been extensively used in homotopy theory, and could be considered as a form of `higher dimensional group'. However, it is difficult to use them directly for describing `algebraic inverses to subdivision'. There are important relations between simplicial groups and other structures which have arisen in this area.
New algebraic understanding of geometry
The chief examples here to my mind are the applications of the Generalised Van Kampen Theorems (GVKTs) proved by Brown and Higgins in 1978, 1981, and by Brown and Loday in 1987. These theorems explain some phenomena in homotopy theory and give new computations which seem unreachable by other methods.
In this area, there seems considerably more potential than has so far been achieved. Multiple groupoids are complicated objects, as is shown by the fact that n-fold groupoids model homotopy n-types. A major part of the problem is to use this complication successfully in a geometric situation, and for this one has to construct a non trivial n-fold groupoid arising from the geometry. Work mentioned above indicates that this can be done in areas of differential topology and geometry, and that we have so far only scratched the surface.
In the work on the Van Kampen theorems, a major part of the difficulty in getting started was to construct higher homotopy groupoids, let alone use them in a computational way. The first successful example in dimension 2 was found by Brown and Higgins in 1974, nine years after the idea was first mooted. This may seem a long time working on the topic, but it should be remembered that this was 42 years after Cech defined higher homotopy groups, and that many distinguished mathematicians had been working in homotopy theory and algebraic topology, but had failed to use groupoids in any computational manner and had not conceived of this idea, or had failed to obtain a successful conclusion. Once a good definition of homotopy double groupoid was found, a proof of the 2-dimensional Van Kampen Theorem came along very satisfactorily, embedding exactly the original intuitions.
I have met people who had thought for six weeks or so of the idea of using squares in an algebraic fashion in homotopy theory, or for asking if time could be 2-dimensional, and had concluded sadly that it did not work, or that it was not clear how to make real progress. We thought of it for nine years, and in the end concluded that it worked rather well, when one thought if it in the right kind of way. Chris Spencer remarked at one point that it was a peculiar subject, because when you finally saw what was going on, then the "obvious" and "natural" method actually worked in the way it clearly ought to have done from the start! The problem was to throw away lots of gangue in one's thinking to expose the real ore.
The subtle equivalences between certain kinds of multiple groupoids and other complex algebraic structures, for example crossed n-cubes of groups, suggest that if one could find say a triple groupoid in a differential geometric situation, then its very existence should provide an algebraic control of the situation, from which one could immediately read of geometric results.
For example, Loday published his work on "Spaces with finitely many non trivial homotopy groups" in 1982, which gave a construction of the fundamental cat-n-group of an n-cube of spaces. Some further details of this non trivial construction were given by Gilbert, and its existence, particularly when combined with the equivalence of categories proved by Ellis and Steiner between catn-groups and crossed n-cubes of groups, immediately yields non trivial results, namely families of laws governing generalized Whitehead products. This equivalence of categories and the use of cat-n-groups in a GVKT also strongly suggests that this family of laws is exactly right.
In the work of Brown and Loday, the new determination of the third homotopy group of the suspension of a K(G,1) as the kernel of a 'commutator morphism' from the non abelian tensor square of G, is in fact exercise 1 in the new theory. One would like to have similar 'easy results' in some differential geometric situation, but the appropriate multiple groupoids have not so far emerged, except for the double groupoids of the theory of Poisson structures.
The GVKTs are examples of local-to-global theorems involving non abelian structures. The fact that these structures enable the expression of the GVKTs is a point in favour of these particular structures, and suggest that they should have applications in other local-to-global problems.
It is clear that the GVKTs give results in homotopy theory which are so far unobtainable by other methods. In view of the difficulty of homotopy theory, this shows that the theory works. Indeed, it seems one needs these methods to calculate even just the second homotopy group of a mapping cone C(f) of a map f: BP -> BQ of classifying spaces of groups induced by a morphism of groups.
As another example, a paper by Goodwillie gives a general position argument in order to prove over a number of pages a connectivity result for n-ads, due originally to Barratt and Whitehead in Proc. London Math. Soc. (3) 6 (1956) 417-439. The latter paper also determines the critical group, assuming the intersection of the subspaces is 1-connected. The Van Kampen Theorem for n-cubes of spaces, together with the results of Ellis and Steiner, give a determination of this critical group assuming only the intersection is connected. The more general algebra is needed even to describe the critical group, and of course this description implies the connectivity result.
A stimulus to new ideas in related areas
The notion of a simplicial complex which is a Kan complex but in which the `fillers' are given canonically, the so called `unique thin fillers', arose in Keith Dakin's thesis , and he stated a beautifully clear and simple set of three axioms for a simplicial T-complex. He was though able to prove the equivalence with crossed complexes only in the rank 2 case, when one gets crossed modules.
The notion of thin structure became a key feature of the work of Brown and Higgins both in proving an equivalence between $omega$-groupoids and crossed complexes, and in the proof of the Generalised Van Kampen Theorem (GVKT) for $omega$-groupoids, since it enabled a higher dimensional version of the the multiple compositions of Homotopy Addition Formulae required for this proof of the GVKT. This work was published in 1981.
This work influenced, and was influenced by, the work of Nick Ashley in 1978, which completed Keith Dakin's work in proving an equivalence between crossed complexes and simplicial T-complexes.
Street used the term `hollow' for a related notion in dealing with parity complexes . However the terminology has now settled on `thin'. Dominic Verity has obtained an equivalence between `globular' $omega$-categories and certain 'T-structured' simplicial sets (though this has yet to be published). There is a reference in a paper of Street. Interestingly, the Australian work on higher order categories ran parallel to and independently of the work at Bangor on higher order groupoids for many years, though the higher order groupoid theory, and its relation to simplicial T-complexes, was available in full detail by 1981 (see [A, B]) and one part of this was even generalised in [C]. A definition of cubical T-complex was given, and the relation with crossed complexes was stated, in 1977, see also [D].
A strong impetus to the work of Ross Street came from discussions on higher order cocycle conditions with the theoretical physicist John Roberts in 1975. The work at Bangor was of course largely motivated by the development of GVKTs and their applications, but non abelian cohomology was always at the back of one's mind since an initial stimulus was work of Olum on non abelian cohomology and Van Kampen's theorem for the fundamental group, dated from 1963.
The success of higher dimensional groupoid theory, and the very term 'higher dimensional algebra', has stimulated work on applications of infinity categories in theoretical physics. A notable exponent here is John Baez. It is felt strongly that a unified field theory including gravitation has to involve a mathematics in which the notion of equality is weakened. This inevitably leads to a search for a complete definition of 'weak infinity category', whose possibility was advertised in "Pursuing stacks".
A range of new possibilities and unexplored ideas
There are several situations which would make splendid proofs of success but which have hardly been explored:
- Explain some scattering phenomenon for fundamental particles using higher
order symmetry and higher order groupoids.
The arguments for this are several.
- One is the general argument for what Wigner called "The unreasonable effectiveness of mathematics in the physical sciences". This might be thought too general an argument, since it could be applied to almost any new kind of mathematics. However, we are not talking about any kind of mathematics but a new kind which reflects some basic intuitions not previously mathematicised. It is this kind of mathematics which has a possibility of "unreasonable effectiveness".
- The more particular argument is that a particle itself is given because of a certain symmetry. Thus the symmetries of particles, and of families of particles, need a higher order symmetry. In particular, this higher order symmetry should be necessary for the analysis of scattering properties of families of particles.
- In view of the work of Heisenberg referred to above, one should perhaps here be asking for a mixture of higher order symmetries and transitions. An appropriate algebra for this could be provided by higher order groupoids.
- Another argument is that the interactions of complex objects such as fundamental particles need an algebra which is concerned with complex gluings governed by geometric conditions. The existence of the GVKTs makes the algebra in which these theorems reside appropriate candidates for consideration in other situations.
- On the other hand, strict multiple groupoids need not be the 'correct' structures for the subtle considerations of higher order symmetry. It seems that one needs 'lax structures', in which laws such as associativity hold only up to some kind of homotopy, where this homotopy is given in a controlled fashion. This topic is known as coherence theory, and is the subject of much current work.
- A futher problem is that whereas functional analysis has been strongly linked with work on groupoids and non commutative geometry, there has been no such link with multiple groupoids.
- Use higher order groupoids in algebraic geometry and algebraic number theory.
- Again the argument here is the importance of local-to-global considerations in this area.
- Another aspect is the general use of homological algebra in these areas.
- Grothendieck long argued for non abelian cohomology in algebraic number theory, but the work that was done by him and his students (for example Giraud), was, though expressed in general terms such as topos theory, still only a beginning in low dimensions. The long manuscript by Grothendieck on "Pursuing stacks" (600pp, 1983) had a considerable part of its theme that of the possibilities for non abelian cohomology. When I told him in 1985 of the result of Loday that n-fold groupoids determine completely homotopy n-types, he said this was "Absolutely beautiful." It is a pity that he has abandoned his work on the "Long march to Galois theory", of which "Pursuing Stacks" was to be volume 1. However, this is being pursued by others.
- Extensive work by L. Breen has gone a long way to setting up algebraic foundations for such applications, continuing a programme suggested to him by Grothendieck in 1975.
Use notions of higher order algebra in discussing the mathematics of the
brain or of other organs.
- This is a nicely wild speculation (but possibly mild in comparison with some of the speculations on the relations between mathematics and the notion of consciousness). The point is to suggest that it is very unlikely that a purely 'linear' mathematics can describe the enormous complexities of interactions which take place in the brain. As was illustrated a long time ago in Abbott's fable 'Flatland', the beings who lived on a line had very limited possibilities for interactions!
- The mathematics of the brain has surely to consider the evolution of structures, their growth and change. Methods of category theory have been applied by Andrée Ehresmann and her collaborators in the development of such ideas.
- It would clearly be naive to suppose that current and recently developed ideas of higher dimensional algebra could immediately and successfully apply in this area. Nonetheless, the existence of such structures does at least allow for experimental possibilities.
We leave the readers to enjoy their own further speculations.
The solution of some classical famous problems
I have put this in for discussions sake. I rate it low in the rank of evaluation criteria, though this view is controversial.
Clearly, if a new theory does solve a classical problem, then that is a sufficient indication of the success of the theory. It is not however necessary, though many would like to argue that it is.
New theories do not emerge like Venus Anadyomene, fully formed from the sea. New viewpoints need to be encouraged, and their influence can be properly assessed only over many years. It is also a much more exciting prospect for students if they can see that there is a new range of possibilities, in which even elementary aspects are currently unexplored.
The point of this investigation was and is to investigate what is actually there, not to solve other peoples' problems, not to go like Columbus with the intention of seeking gold, and failing to notice the richness of the flora and fauna of the new lands. The value of the new lands cannot be evaluated until they are investigated.
When we started in the mid 1960s, very little was known, and the seeking and finding of this rich theory has been based on just a few clues. The actual technical and conceptual difficulties of making progress is still a problem, and there are many problems which one feels should get solved and somehow do not.
On the other hand, we may well, for many reasons, be missing out on some good applications.
The step from the theory in dimension 1 to higher dimensions has proved a difficult one to realise, but it has now been well established since the mid 1970s, and new applications are being found regularly. The notion of `higher dimensional algebra' has occurred in manifold areas, from differential geometry to computer science. It is closely related to low dimensional topology.
A general aspect of higher dimensional algebra is the contrast between the 1-dimensional viewpoint and the higher dimensional viewpoint. At least in the groupoid case, there is a kind of translation system between the two, manifesting itself in a number of complicated equivalences of categories of algebraically defined structures. One then finds that some statements which are tautologous in the higher dimensional viewpoint become complex statements in the linear viewpoint. An example is the case of a cube: from the higher dimensional viewpoint, all one can say is that a cube has faces. >From the linear viewpoint, we have the so called `homotopy addition lemma', which states intuitively that the boundary of a cube is the alternating sum of its faces, but a precise formulation is not so easy to state or to prove. Yet this idea is at the foundation of algebraic topology.
These contrasting viewpoints, namely on a line and higher dimensional, have proved crucial in the theory and applications. For example, both viewpoints have been necessary even to state what it means for an n-cube to have commuting boundary. For a 2-cube, i.e. a square, this is no problem: we have formulae of the usual type ab = cd. But even for a 3-cube, we are already in a new kind of mathematics. These basic ideas were central to the formulation and proofs of the first versions of the Generalised Van Kampen Theorems.
Ross Street has used the term "post modern algebra" for the area described here and including also the study of higher order algebraic structures up to coherence. These algebraic structures have proved important in many areas - the prime example is the use of monoidal categories and related structures such as braided categories in knot theory, operator algebras, and many other areas.
Some may be surprised that such a philosophical approach as explained above should have led us to new structures which underly certain kinds of geometry, and that these have led to specific calculations which are not obtainable by other methods. To me, it seems difficult to know of another method of procedure in this case. The article "The methodology of mathematics" gives a start on a full discussion of the practice of mathematical research. One method in this practice is that of attempting to conceive of and construct the kind of mathematics which embodies an intuition and would seem to be appropriate for a class of problems. This is `top down' rather than `bottom up'. One can hope that the two procedures will meet somewhere.
In this case, the start was a particular example of the general problem of describing algebraically how big things are built out of little things, and calculating and computing the resulting behaviour of the big things. Such problems are of course of wide occurrence, and come under the title of local-to-global problems. We need lots more new ideas to handle such problems, and many methods of developing the appropriate mathematics.
The Generalised Van Kampen Theorems give some new information in algebraic topology, and hence in group theory and in low dimensional topology, using notions of multiple groupoids. In fact, in the range in which they work, they can give a lot of new information. Outside this range, we are still usually in the dark, whereas other methods have yielded many results.
As mentioned above, looser structures called monoidal categories have been strongly used in knot theory, operator algebras, algebraic geometry, theoretical computer science and theoretical physics, and new notions of "weak infinity categories" are being developed and exploited. How are all these ideas related? What other structures may be appropriately used?
The fact that related ideas have occurred in so many apparently different areas is a strong argument for their importance. As mentioned earlier, there was a similar argument for considering the prospective importance of groupoids. Nowadays, groupoids are used in so many areas such as symmetry or even projective geometry , sometimes as a matter of routine, there really should be no doubt that they are an often convenient and necessary extension of the notion of group.
Since relatively few people have worked on these ideas of higher dimensional
group theory, compared say with group theory itself, it is likely that there
is a lot more country to be explored, and important discoveries to be made.
It is certainly of interest to be told of areas in which it could be thought
that such a theory worthy of the title should be able to contribute, but
in which it at present does not. It is such areas which would be uncharted
territory for these methods.
Acknowledgements: This work would not have succeeded without the happy collaborations with Philip Higgins and with Jean-Louis Loday, who have shown the strength of being able to cut through thickets and the insight to make the ways straight; with Chris Spencer, with whom the algebra first got off the ground; and also of many other collaborators and research students.
I would also like to thank two who gave us crucial hints: Graeme Segal mentioned to me in 1971 work of Deligne on the classification, in terms of 3-dimensional cohomology, of group objects in the category of groupoids. This eventually resolved itself into the equivalence between group-groupoids and crossed modules, which was found by Verdier in the mid 1960s, although when he was told about it in 1976 he claimed to have forgotten it! Matters of this kind were certainly under consideration in the Grothendieck school in the mid 1960s.
Terry Wall in a conversation in 1975 gave me the geometric insight that the collapsing arguments I needed to show that the construction of higher homotopy groupoids in our cubical context would actually work.
It is also a pleasure to acknowledge the stimulation of much correspondence
with Alexander Grothendieck in the years 1982-85, with many ideas which still
need assimilation. I refer to his `Esquisse
d'un programme' (1983) for his views on the correspondence.
Research work at Bangor on groupoids, higher order groupoids, and related areas
Publications from Bangor: Ronnie Brown, Lew Hardy, Peter Stefan, Tim Porter, Nick Ashley, David Jones, Mark Lawson, Andy Tonks, John Shrimpton, Chris Wensley.
Collaborators on joint papers: Chris Spencer, Lew Hardy, Philip Higgins, Jean-Louis Loday, Sid Morris, Phil Heath, Johannes Huebschmann, Graham Ellis, Heiner Kamps, Nick Gilbert, Jean-Marc Cordier, David Johnson, Edmund Robertson, Hans Baues, Razak Salleh, Kirill Mackenzie, Marek Golasinski, Mohammed Aof, Rafael Sivera, Osman Mucuk, George Janelidze, Ilhan Icen, James Glazebrook.
Research students at Bangor (with date of completion): Lew Hardy (1974), Tony Seda (1974), A. Razak Salleh (1975), Keith Dakin (1976), Nick Ashley (1978), David Jones (1984), Graham Ellis (1984), Fahmi Korkes (1985), Ghafar H. Mosa (1987), Mohammed Aof (1988), Fahd Al-Agl (1988), Osman Mucuk (1993), Andy Tonks (1993), Ilhan Icen (1996), Phil Ehlers (1994), J. Shrimpton (1990), Zaki Arvasi (1995), Murat Alp (1997), Ali Mutlu (1998), Anne Heyworth (1998), Emma Moore (2001).
Updated December 4, 2010
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