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August 3, 2009
Have added to my preprint list a link to a scan of the thesis of G.H. Mosa on `Higher dimensional algebroids and crossed domplexes', and above a link to the 2009 Askloster seminar.
January 25, 2009
Relevant to this discussion is the new web site nLab, which is a kind of wiki on n-categories.
September 25, 2008
Have just added to my preprint list a paper on `Algebraic colimit calculations in homotopy theory using fibred and cofibred categories'. This is an exposition to appear in the book on `Nonabelian algebraic topology', since it has the aim of systemising results relating groupoids, crossed modules over groupoids, and modules over groupoids. A secondary aim is to advertise that some colimit calculations of homotopical invariants have been known as possible since a publication in 1978 on a van Kampen type theorem for second relative homotopy groups, i.e. for 30 years, but are still not regarded as possible by topologists. Yet the proof is readable by a graduate student! I do not understand what has been going on!
September 22, 2008
There has been a recent very interesting discussion on the
category theory discussion
list (this is surely the most interesting of the discussion lists since
there is an eagerness to discuss matters such as value, direction and exposition)
with regard to expositions of category theory, which could result in more
availablity of expositions on generally available sites. John Baez
in this discussion (September 19) suggests that
"I can imagine a new Bourbaki who tries to explain all of mathematics in the language of categories. But I can also imagine a new Bourbaki who tries to explain all of mathematics in the language of infinity-categories. It may be a bit too late for the first one, and a bit too early for the second one." The question here might be: what should be an infinity-category? Workers in this field tend at the moment to concentrate on the globular approach, with little attention to the various cubical cases. For a discussion of this point see my paper `A new higher homotopy groupoid: the fundamental globular É-groupoid of a filtered space', Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343.
June 29, 2008
What are the boundaries of what one might expect, both high expectations and low expectations, from a proposed extension of a theory?
What one would like is that an extension from group theory to higher dimensional group theory would be analogous to the extension from one varable to many variable real analysis! The grounds for this analogy are the basic diagrams of subdivision of paths and of squares and cubes, on which the intuitions for Higher Homotopy van Kampen Theorems are based. I would be interested to have comments by email on this analogy.
This is related to the following question, also given below, with which this work started in 1965 or so:
Given that 1-dimensional homotopy theory is better described in most respects in terms of groupoids rather than groups, then how useful can groupoids be in higher homotopy theory?
The two extremes are that
The first possibility is boring and suggests the best strategy is: do nothing (in any research programme, it is always easy to find reasons for doing nothing!). The second possibility is exciting, in view of the wide range of uses and appplications of group theory. It is of interest to investigate obstructions to this possibility. This is a `no lose strategy' since if obstructions appear, that will be interesting, and if they are gradually overcome, that will be even more interesting! Following up this line of thought also engages with the question of :`What if?', which is known to those interested in innovation and imagination as a question of major importance!
Here are some current links to programmes in higher category theory:
IMA Minneapolis 2004
CRCG Göttingen, higher order structures in mathematics , 2006
Fields Institute 2007
Higher Structures in Geometry and Physics, January 15 - 19, 2007 IHP-Paris, France
Categories in geometry, Hungary, September, 2007
HOMCAT, Barcelona 2007-8
Trimester Program on Geometry and Physics, HIM, Bonn, May - August 2008
higher categorical structures in algebraic geometry, Pisa, September, 2008
and the continuing discussion on the Higher Dimensional Mathematics and Physics Weblog, the n-category cafe.
March 31, 2008
It is interesting that contributors to the n-category cafe use freely the term n-group, so it seems that my thesis of traditional group theory as level 1 of a theory extending in all dimensions, and which saw the light of day in the title of my paper `Higher dimensional group theory ' published in 1982, is quite widely accepted. Of course, there is the question of `What is and what should be a higher dimensional group?', and this is the heart of the matter. I have a recent paper to appear in Homotopy, Homology and Applications, on `A new higher homotopy groupoid: the fundamental globular omega-groupoid of a filtered space' (pdf file), which discusses this, from my point of view. I emphasise the cubical viewpoint to multiple groupoids, because in that context I have been able to formulate and prove theorems, which proves difficult in the globular approach. For speculation on various ideas, see the preprint on whiskered categories available, with others, from my preprint page.
The basic question in all this has been:
If groupoids are much more successful than groups in giving an account of 1-dimensional homotopy theory, then can groupoids be useful in higher dimensional homotopy theory?
My criterion for success of the theory was that it gave more powerful theorems with more natural and convenient proofs. As an indication of the opposition to this idea, a 2001 survey of `Mathematics in the 20th century' by Sir Michael Atiyah includes neither of the words `category' or `groupoid', and few recent books on algebraic topology or its history include the word groupoid. Yet there is vast evidence for the view that groupoids give a more flexible approach to symmetry than do groups. For more discussion, see the following link.
My first motivation throughout this work, whose basic intuitions go back to about 1966, was in terms of higher order van Kampen theorems. This local-to-global tool seems generally unexploited. I get the impression that people find it difficult to get their heads around these theorems, even in dimension 2. I wish I knew why! Perhaps it is because the tools for the proofs are quite unlike traditional tools in algebraic topology, and indeed need a lot of setting up in all dimensions. However the dimension 2 proof makes a nice lecture. It does need the concept of homotopy double groupoid of a based pair, which I find a great idea! The details are in my survey article, the first article in the electronic journal HHA. As I have said elsewhere, the basic question which arose from the 1968 edition of my book, now republished as Topology and Groupoids, was:
November 23, 2007. I have altered my November 4 posting below, in order to omit a name. On a separate page I state the theorem proved by Loday and I, which I like to call a Higher Homotopy van Kampen Theorem. The conclusion (C) implies the Barratt-Whitehead Connectivity Theorem, by applyng it to the n-cube of spaces derived by intersections from an (n+1)-ad. The exciting part though is (I), which allows some calculation. However the proof of the theorem is an induction involving both (C) and (I). It requires knowledge of a number of techniques in algebraic topology, and familiarity with crossed modules and related areas. I do not know how to summarise it. Note that to get an algebraic result you have to have the `right' algebra, and this theorem and its applications justify the algebra. Even the existence of the fundamental cat-n-group of an (n+1)-ad, and its associated crossed n-cube of groups, is important since it contains information on all the homotopy groups of sub r-ads and all the generalised Whitehead products. So it is a remarkable gadget. Of course the idea of dealing with colimits rather than exact sequences is very surprising, but very powerful, when the theorem applies.
November 5, 2007: I have not attended an algebraic topology meeting for a number of years, since the friendly atmosphere, wide range of topics, and interest in the range of my work have been more apparent at category theory meetings, and the tenor of discourse better reflects my own taste in mathematics and the way it develops. See for example the views of the Stanford Encyclopedia of Philosophy. I hope that at some stage some algebraic topologists will pursue the methods which I have developed with colleagues and students, but I agree that these methods are orthogonal to many current trends in algebraic topology, which is highly focussed on stable methods, i.e. as far away from the fundamental group as possible. Nonetheless, work of Faria Martins and Porter on Witten invariants, and of Porter and Turaev on HQFTs, are indications of possibilities. Perhaps the new Courant Centre with its focus on `Higher order structures in mathematics and physics' will give a lead. I have 67 of my published papers on essentially that higher order theme, but also lately including biology areas.
I should say that some papers referring to `van Kampen theorems for diagrams of spaces', have some errors due to the author(s) of those papers not ensuring that the required connectivity conditions wer satisfied! An erratum is available from Galway. The papers concerned are E1, E2, E3. Nonetheless, these papers have stimulated a lot of work on, for example, generalisations of Hopf's formula for the Schur multiplier. Since that is a homological result, it is not too surprising that other methods have been found to obtain these formulae.
It is also interesting that Jie Wu in `On combinatorial descriptions of the homotopy groups of certain spaces', Math. Proc. Camb. Phil. Soc. 130 (2001), 489-513, gives descriptions of all homotopy groups of the suspension of a K(G,1), but is not, as he says, able to recover the explicit calculations of the third homotopy group given in BL1 nor the extra homotopical information (e.g. Whitehead products) given in my paper in the Adams Memorial volume.
November 4, 2007: The last year has been busy and I had health problems from September which I won't bore people with, but it means not as much has got done as I would have hoped. Hence also this set of comments has been untouched for over a year.
In July 2006 I tried to convince readers of the algebraic topology discussion list (see `comments on Blakers-Massey') that Loday and I really had proved an algebraic version of the Blakers-Massey and Barratt-Whitehead connectivity theorems, but seem to have failed to have convinced some. This is a pity as there are many relations between the ideas outlined by Goodwillie in this correspondence and those in the Brown-Loday papers. The reason our papers go so much further than previous (and, it seems to me, current!) results in this area is that they were based on a thorough exploitation of crossed gadgetry (crossed modules and related structures), simplicial theory, and an intuition, based on previous work with Philip Higgins, that colimit theorems could be obtained, though with proofs of a different character. It is curious that the book by Hog-Angelloni/Metzler/Sieradski/ describes some consequences of the 2-dimensional van Kampen theorem proved by Higgins and I in 1975 (published 1978), such as Whitehead's classical theorem on free crossed modules, and some new results on homology, but fails to mention this 2-dvKT, which has other consequences, and was a model for later work. When asked, Frank Adams was unable, or refused, to explain what were his (very strong!) objections to my first paper with Philip Higgins, yet, strangely, supported the subsequent publication of a paper on Whitehead's theorem. Frank Adams was finally bowled over by the use of the nonabelian tensor product in the Blakers-Massey theorem, and became friendly and supportive. .When you think about it, that use `has' to be correct, since it is so natural a result and fits with so many things!
When I explained to Jean-Louis Loday in Strasbourg in 1981 how the relative Hurewicz theorem followed from a van Kampen type theorem, it soon occurred to us that the algebraic conjecture he was looking to prove for squares of spaces should be a consequence of a triadic Hurewicz theorem, which itself, by analogy, should follow from a triadic van Kampen theorem, and, as Jean-Louis remarked, the more general theorem may be easier to prove, as was the case here. The proof uses a wide range of techniques in algebraic topology, and the background to the paper was ably supported by Michel Zisman and Dieter Puppe. One aspect of these results, the non abelian tensor product of groups which act on each other, now has a bibliography of 90 papers, but none it seems by other algebraic topologists. Perhaps this is `post-modern homotopy theory'!
Alexander Grothendieck quickly observed that `the Bangor group were prepared to give time to foundations', and this led to `Pursuing Stacks' and our long correspondence. That manuscript grows in influence.
Two preprints have been prepared, and the one on globular groupoids has been submitted. Drafts with Rafael Sivera on fibrations of categories, and on the cubical classifying space of a crossed complex, are in preparation. The latter represents a decision to give a cubical exposition of this area in the book in preparation, fitting in with the other cubical work necessary for the main results. There has been a lot of work with Ion Baianu and Jim Glazebrook on the application of categorical and higher dimensional ideas in ontology and related areas, and this work is soon to appear in Axiomathes. As you may see in my publication list, I have also been concerned to give appropriate tributes to John Robinson, who died in April, 2007.
More widely, the subject of higher categorical structures is widening and surely running away from me. There was a big conference in Paris in January, 2007, and the scientific focus of the new Courant Centre at Gottingen in 'Higher Order Structures in Mathematics' is directed towards the construction and exploration of universal, highly organized structures in pure mathematics and mathematical physics.
Additional note: October 6, 2006 You can read extracts of two letters of Grothendieck to Ronnie Brown: one on speculation , and one on his later reaction to geometry at Bangor. Pursuing Stacks is due to be published in Documents Mathematiques in 2008, and G. Maltsiniotis' book `La theorie de l'homotopie de Grothendieck', Asterisque, 301, 2005, develops the ideas in Pursuing Stacks.
May 26, 2006
Two further quotes from the AG/RB correspondence are in the article `Analogy, concepts, ...', by Ronnie Brown and Tim Porter, one on zero, and another on working towards understanding, without worrying about its impact.
Additional note: (March 23, 2006)
I asked Mikhail Kapranov about the influence of Pursuing Stacks and Esquisse d'un Programme on Soviet mathematics in the 1980s. He replied:
March 21, 2006
From what I remember, Gelfand advocated reading both Esquisse and Pursuing stacks. Voevodsky was very interested in both anabelian geometry and higher stacks. Drinfeld was influenced by Esquisse in his paper on "Drinfeld associator" and a version of the Grothendieck-Teichmueller group appearing in the theory of quasi-Hopf algebras. This is probably the most serious influence on Soviet mathematics of the period.
Comments March 6, 2006
1. BOOK NOW AVAILABLE
Follow link Topology and Groupoids for more information.
This is a revised and extended version of my old topology book, previously published in 1968 and 1988, and long out of print. The revised version is xxv+512 rather than xvi+460 pages, and the material has been reorganised to emphasise the topology/groupoids translation. For ordering, see order page.
2. Rather than list my new preprints here, I refer you to the Bangor Mathematics Preprint List for 2005, 2006. Some recent work relevant to ours is a classification of double groupoids on http://arxiv.org/abs/math.CT/0602497 by Nicolas Andruskiewitsch and Sonia Natale, and work by Richard Steiner on thin fillers in cubical nerves at arXiv:math.CT/0601386 .
Comments February 16, 2006
A revised new edition of my 1968, 1988 book on `Topology' (McGraw Hill, Ellis Horwood) will be published hopefully within 3 weeks, at a reasonable price, with a purchasable e-version with colour and hyperref. It is retitled `Topology and groupoids', to reflect its character. The text has been revised and extended. For a view of the new cover, as pdf file, see here (1.78MB) Watch this space!
Work continues on following through the implications of higher dimensional algebra and category theory in biological areas. A paper presented at a conference on Neurobiology in Delhi in 2003 discusses the idea of an email analogy for a colimit. This was first presented to a Fields Institute Workshop in 2002. This and related ideas are developed in a paper Baianu et al.
Work continues on an EPSRC project `Higher dimensional algebra and differential geometry' which supported a Visiting Fellowship for Jim Glazebrook (Eastern Illinois) for 2003-2005. Three papers are in preparation, jointly with Tim Porter. This deals with crossed complexes and homotopy groupoids in relation to topics such as gerbes.
Some implications of category theory are considered in papers with Tim Porter on the notion of analogy, and with R. Paton on biology and hierarchical systems.
Comments: July 2, 2005
This is a brief update. The field of higher dimensional algebra is burgeoning, as a web search shows, and the influence of Pursuing Stacks is increasing, though its origins are not usually noticed. This was one reason for writing a web article on this. We are currently working (RB,TP, Jim Glazebrook) on applications of crossed complexes to areas related to stacks, gerbes, ...with EPSRC support for Jim as VF.
The increasing interest in higher groupoids is illustrated by the visit of Sonia Natale, for discussions on double groupoids in relation to quantum groups and groupoids, and so to Hopf algebras.
Work on Part II and IIIof the book on Nonabelian algebraic topology is progressing, though some technical difficulties emerged (for example the normalisation theorem for the crossed complex of a simplicial set) holding up the progress. The available material is helpful for other projects.
Some time was spent on helping Philip Higgins with the reprint of his book, on his paper on thin elements and commutative shells, and on the joint paper on the van Kampen theorem for the homotopy double groupoid of a Hausdorff space.
Comments: 23 August, 2004
The planned book on `Nonabelian algebraic topology' has made considerable progress under the support of the Leverhulme Emeritus Fellowship, and a general description and downloads may be obtained by clicking on: Preface and Part I . A paper based on the presentation on the book to the IMA, Minneapolis, Workshop on `n-categories: foundations and applications' (June 7-18, 2004, organised by John Baez and Peter May) may be found at UWB Math Preprint 04.15. The IMA Workshop showed the range of new ideas and concepts that are developing in Higher Dimensional Algebra.
I found a quotation from The Tempest, by William Shakespeare, on the role of the poet, which might also provoke thought on the role of mathematics.
It is more than time to make some additional comments for these pages, as much has happened in the period since 1999. So much, indeed, that I cannot envisage in the time available a full update of the bibliography. What I can hope to do is point out some trends and give some indications of more recent literature.
First of all, the title of this web page has not gained much currency, but a web search on "Higher dimensional algebra" shows that this more general term has caught on. It was introduced in my survey article [B:87] in the words: `A naive viewpoint is that n-dimensional geometry needs n-dimensional algebra'. John Baez has many articles and writings putting over this point of view, more than the three listed in the bibliography here. His web page `This week's find in mathematical physics' has many items in this area, and you could start at week 53 from 1995. There is much discussion in the Newsgroup: sci.physics.research.
A lot of research in Higher Dimensional Algebra is concerned, for good reasons, with notions of weak infinity category. A good survey of definitions on this is by Tom Leinster, published in TAC. Work on this was stimulated by the 1983 manuscript of Alexander Grothendieck, Pursuing Stacks, which is now being edited for publication by the SMF by G. Maltsiniotis . His `Travaux en course' available from that page are highly relevant to this area.
My own work in this area has taken a different line, since the basic motivation came from the search for higher homotopy groupoids and analogues of the van Kampen Theorem for the fundamental group. That theorem is a classical example of a non commutative local-to-global theorem. Now Atiyah in his paper `Mathematics in the 20th century' has given major trends as local to global, from commutative to non commutative, and increase in dimensions. All these trends are combined and realised in the Higher Order van Kampen Theorems for higher homotopy groupoids, and its applications, together with links to homology and K-theory, to which Atiyah also refers. Higher categorical structures give a vast range of essentially nonabelian algebraic structures with which to model geometric processes, and the range of applications will surely continue to increase in the 21st century.
My own view of `Higher dimensional algebra' is that it deals with algebraic structures where the domains of definitions of the operations are defined under geometric conditions. A general treatment of `partial algebraic structures' is given in a paper of Higgins' published in 1963.
For a survey on work on Crossed complexes and homotopy groupoids, see the preprint [Fields]. This contains an analogy for how a colimit works in terms of emails! In an email, the message that is supposed to be transferred is partitioned into bits, each bit is labelled, and then sent by some route to its destination, where all the bits are assembled into the transferred message. Of course we require that the assembled message should be independent of all the choices made on the way! This is how the proofs in the papers with Philip Higgins on the van Kampen theorem actually work. The algebra which makes this proof possible is surely of independent interest.
This explains the cubical approach which has dominated our work, since that approach is suitable for algebraic inverses to subdivision. The relation between the cubical and globular approach has become clearer with the paper [AABS:2002], which gives an equivalence of categories betweeen strict globular omega categories and cubical omega categories with connections. The latter objects form a monoidal closed category, as is not difficult to see, and so this structure transfers to the globular case. As far as I know, nobody has pursued the notion of weak cubical omega category.
There are subtleties on the homotopy theory of cubical sets, and a recent work on this is by Jardine developing ideas of Cisinski . A web search on `cubical' shows a resurgence of interest in cubical methods, but there are still open problems.
One of the aims and tests of the higher order van Kampen theory has been to obtain specific calculations, for example of homotopy 2-types. A recent example of the use of symbolic computation in this area is a paper with Chris Wensley, published in the J. Symbolic Computation, 2003. It is seen clearly in this paper how the second homotopy group, even as a module over the fundamental group, is but a pale shadow of the full homotopy 2-type, which itself can be calculated in some instances if seen as a crossed module. This aim of calculation is related to the concentration on strict omega categories and groupoids. However there is work using the monoidal closed structure on crossed complexes, by Brown and Gilbert, and Baues and Conduché, Baues and Brown, Baues and Tonks.
Grothendieck in his Esquisse d'un Programme (1984) refers to his aim over many years to develop Nonabelian homological algebra (see the extract). Yet one of the roots of homological algebra is algebraic topology, and so it is reasonable to consider what should be Nonabelian algebraic topology. A book is in preparation with this title by R. Brown and R. Sivera, planned to be published in the series of A. Bak for Kluwer. The aim of this book is mainly to give an exposition of the work on crossed complexes and cubical homotopy groupoids with Philip Higgins, with intuitions and explanations. More detail is given in [Lever].
The rather philosophical discussions at the end of this paper, and in the page on groupoids, are pursued in a wider context in the book `Towards a philosophy of real mathematics', by David Corfield.
Applications of crossed complexes to the study of Morse functions on manifolds are given in the book by Sharko.
The speculations on the relevance of higher dimensional algebra to the study of the brain are pursued in two papers [Longo, NBRCI].
For recent work of mine, see my preprint page.
For a picture of the Context of Higher Dimensional Group Theory, see the picture Context, kindly prepared for me by Aaron Lauda.
The following was l
The term Higher Dimensional Group Theory was introduced in 1982. The aim of this paper is to give a broad account of the background to the idea without much technical detail, so that it can be read by a wide audience.
The technical side is contained in the references given throughout, and the subject can be explored in the accompanying bibliography on 'Publications on groupoids and non linear homological and homotopical algebra'. This was initially built up for a research grant application, and then was directed at publications which referred to or were related to work at Bangor in these areas. It is now a suitable time to extend it, and suggestions would be welcomed.
I am developing a page on one aspect, the non abelian tensor product of groups. It is interesting that this construction, which can be seen as basic in group theory because of its relation to commutator theory, is also but a fragment of the general approach in terms of higher dimensional group theory.
In some places I have been speculative on the potential for this theory, on the grounds that this has to be done to make big conceptual gains or leaps, and to decide in what direction the theory might go next. These speculations also give some idea of the character of the theory.
These ideas arose for me in algebraic topology, and more specifically in homotopy theory, and the hope was that they would give new information. There was of course no certainty or even likelihood that the new theory would solve problems in algebraic topology which were currently occupying centre stage, though there was an expectation that the theory would give new answers to basic questions such as computing topological invariants, and in other areas where there was a combination of a paucity of work and a paucity of method. However, the form of the problems to which such a theory would successfully apply could not be determined till some portion of the theory was complete, with definitions, examples, theorems and proofs.
The aim of the theory was always more ambitious than that of providing new tools in homotopy theory. Problems in homotopy theory did however provide a suitable test-bed because of their notorious difficulty. Methods which provide new information there demonstrate a working mechanism and clearly have some merit.
A further advantage of homotopy theory is the generality and importance of the notion of deformation as an aid to and as a part of classification. This suggests the potential importance of new methods which might be applied not just to the homotopy theory of spaces but more generally.
The main aim was of course exploration: what did this potential new world contain? In attempting to investigate ideas of higher dimensional group theory it was necessary to step out of line. In order to convince you of the reality of these travels and the interest of the journey, I have to describe the journey, give some of the characteristics of the new world, and show some articles brought back from the journey which are I hope sufficiently strange to be convincing as evidence of a new country.
This article is also relevant to a discussion of our research methodology. There is some discussion of this in two articles [A, B] on my teaching and popularisation pages. Here it may be useful to give two quotations. The first is from P. Dirac, in one of his last addresses, and the second from Einstein (1916) (full quotation here).
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