Main Diagram of functors and monoidal closed categories which are exploited and need to be explained
by
Update: April 8, 2008
The papers which give this theory were developed over the period 1971-2001.
For a survey of the material to be contained in the book, see the article
`Crossed complexes and homotopy groupoids as non commutative tools
for higher dimensional local-to-global problems', Proceedings of the Fields
Institute Workshop on Categorical Structures for Descent and Galois Theory,
Hopf Algebras and Semiabelian Categories, September 23-28, Fields Institute
Communications 43 (2004) 101-130 (see
updated version to appear in Michiel Hazewinkel (ed), Handbook
of Algebra vol 6, Elsevier (2008/9)). It will be seen from this article, that the
structures which enable the full use of crossed complexes as a tool
in algebraic topology are substantial, intricate and interrrelated.
|
Main Diagram of functors and monoidal closed categories which are exploited and need to be explained |
To make it easier to see the intuitions behind this work we start in Part I (downloadable) with the 2-dimensional case, and the use of crossed modules and the homotopy double groupoid of a pair of spaces (X,A) with a set C of base points. In dimension 2 this consists of homotopy classes rel vertices of maps of a square into X which take the edges to A and the vertices to C. It is a kind of `symmetrical', many pointed version of relative homotopy groups, and has great advantages from the point of view of `algebraic inverses to subdivision', and constructing homotopies, which are necessary for our local-to-global theorem of the van Kampen type. This theorem enables the use of nonabelian colimit calculations in 2-dimensional homotopy theory, an ideas which is also relevant to geometric group theory, but is little referred to!
It should be emphasised that the functor Π from filtered spaces to crossed complexes is defined in terms of relative homotopy groups. Its major properties can be understood and are developed without resource to singular cohomology, or simplicial approximation. One application is a homotopical excision result, involving `induced modules and crossed modules', which has a corollary the Relative Hurewicz Theorem, and yields nonabelian results on second relative homotopy groups inaccessible to traditional methods.
A major result, generalising classical results of Eilenberg-Mac Lane, and using all the tools developed, is the homotopy classification of maps from a CW-complex to the classifying space BC of a crossed complex C. Recent work has further shown the utility of the classifying space for the homotopy classification of maps, relating the classical group theory of abstract kernels, and their obstructions to extensions, with mappings of an n-dimensional space into a space whose homotopy vanishes between 1 and n. Here the notion of fibration of crossed complexes and related exact sequences gives a view of basic obstruction theory.
The power behind these methods comes from the cubical ω-groupoids, which form a convenient category to obtain a monoidal closed structure, and to express `algebraic inverses to subdivision', essential for the local-to-global colimit results (Higher Homotopy van Kampen Theorem).
It is hoped that this exposition will help progress towards Grothendieck's vision of nonabelian cohomology.
Since the globular case is currently favoured for higher category theory,
it might be asked:
`Why are there not globular ω-groupoids in
the above diagram?'
The answer is that it was early proved with Higgins that they are
equivalent to crossed complexes, but we were
unable to do anything with them, either calculate, as we can with crossed
complexes, or conjecture and prove theorems, as we can with the cubical case.
Functors from simplicial sets or cubical sets to crossed complexes
are well studied and applied. In these cases, the notion of free crossed
complex is central to the applications, but hardly appears in the globular
case. The globular ω-groupoid on one free generator of dimension
n is analysed (probably for the first time!) in a recent paper to
appear in HHA on a higher homotopy globular ω-groupoid of
a filtered space (pdf).
To present an integrated view of the structures hinted at in the Main Diagram has required no essential change in the detailed proofs, but has needed reordering of the material, some extra clarification, and redrafting for consistency. Also there were some research problems to extend the theory. For example the theory of acyclic models presented problems with normalisation for the fundamental crossed complex of a simplicial set. This has now been resolved in a paper in JHRS. A not quite corrected version of the book has been submitted for review.
Recently, it has been decided that the simplicial classifying space theory is a bit of a diversion from the main cubical thrust, and so we are returning to the spirit of a previous cubical version of the classifying space, which dates back to 1982. An advantage of the cubical theory is that proofs are simpler, because the cubical Eilenberg-Zilber map is an isomorphism. Recent accounts of the homotopy theory of cubical sets are also helpful in this respect.
There are new expositions of calculating colimits in modules and crossed modules over groupoids, and calculating the tensor product of crossed complexes. So the final order of the book will not be as in the table of contents in Part I as downloaded.
It is interesting that there is still scepticism that a reworking of basic algebraic topology, returning to some intuitive roots in this way, is possible. Indeed the applications of groupoids in algebraic topology and combinatorial group theory are not widely accepted. The 2-dimensional van Kampen theorem is not referred to even in some books or papers using or proving some of its easy consequences, such as Whitehead's theorem on free crossed modules, for which methods not directly involved with the universal property of a free object seem preferred.
It is helpful that the preparatory book
`Topology
and
Groupoids'
is now available, for which here is a link to
a review. The work on the 1968 version
of this book convinced the author that all of
1-dimensional homotopy theory is better expresed by
consistently using groupoids, as this led to more powerful theorems
with simpler proofs (which seems OK to some of us). The natural question
arose:
Can the success of groupoids in 1-dimensional
homotopy theory be extended to higher homotopy theory?
This required development of the algebra of double groupoids and 11 years
work to develop the notion of higher homotopy
groupoid. As P.A. Dirac said in
one of his last addresses, but thinking primarily about physics: `.....one
must follow up a mathematical idea and see what its consequences are, even
though one gets led to a domain which is completely foreign to what one started
with.... ' . This contrasts with the often current emphasis on the
conformist notion of `the mainstream'.
A natural reply to those who taunt `Not mainstream!' is: `Not yet!'
Indeed, the head of the BBSRC, recently (THES, Oct 2008)
commented that there is a difference between `mainstream' and `cutting edge'.
We will communicate the further parts to those specially interested.
Philip Higgins' name is on the book in view of his large and inseparable input to the research, with his intuition, algebraic expertise and expository skills, though the responsibility for the correctness of the final version will still reside with the other two!
See below for downloading of Part I of three parts. This includes a self contained account many of the nonabelian results on second relative homotopy groups, as showing the intuitions for the higher dimensional case. We hope to attract comments and suggestions, which should be sent to either of
r.brown`symbolat'bangor.ac.uk
Rafael.Sivera`symbolat'uv.es
One overall theme of this book is the use in algebraic topology of some higher
categorical structures, which allow for the application of
higher dimensional nonabelian methods to certain
local-to-global problems.
Here is a link to an update
of the proposal (pdf) (or
html) for a Leverhulme Emeritus Fellowship
which supported this project.
From the Preface:
`Our aim for this book is to give a connected and we hope readable account of the main features of work on extending to higher dimensions the theory and applications of the fundamental group.'
`We describe algebraic structures in dimensions greater than 1 which develop the nonabelian character of the fundamental group: they are in some sense `more nonabelian than groups', and they reflect better the geometrical complications of higher dimensions than the known homology and homotopy groups. We show how these methods can be applied to determine homotopy invariants of spaces, and homotopy classification of maps, in cases which include some classical results, and allow results not available by classical methods.'
`In Part I we give some history of work on the fundamental group and groupoid, in particular explaining how the van Kampen theorem gives a method of computation of the fundamental group. We are then mainly concerned with the extension of this nonabelian work to dimension 2, using the key concept, due to J.H.C. Whitehead in 1946, of crossed module.'
`In Part II we extend the theory of crossed modules to crossed complexes, giving applications which include many basic results in homotopy theory, such as the relative Hurewicz theorem. This Part is intended as a kind of handbook of basic techniques in this border area between homology and homotopy theory.'
`However for the proofs of these results, particularly of the Higher Homotopy van Kampen theorem, and of the use of the tensor product and homotopy theory of crossed complexes, i.e. monoidal closed structures, we have to introduce in Part III another algebraic structure, that of cubical ω-groupoids with connections,' and to prove its equivalence with crossed complexes. This equivalence algebraicises some long standing geometric methods or intuitions in relative homotopy theory.
There are three possible downloads of pdf files. The small downloads contain the Preliminary material (Preface and Table of Contents), while the large one (190 pages) contains also all of Part I of the three planned Parts of the text, as well as the bibliography.
This is still a draft, and we expect details of these files to change.
The structures of Parts II and III may also change as the current material is worked on.
Downloads of pdf files:
small (Preface, TOC) (15 pages, 148KB)
larger (Preface, TOC, bibliography) (25 pages, 204KB)
full (with Part I) (190 pages, 1.01MB)
Link to `Higher dimensional group theory'.
date: October 14, 2008
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