Algebraic Homotopy: crossed and simplicial algebraic homotopy and non-Abelian cohomology

 

1. Algebraic models for homotopy types

  The theory of Cat^n-groups, crossed complexes and other related notions has been developed in an attempt to integrate the action of the fundamental group of a space into algebraic models of aspects of its homotopy type. Although the theory is relatively young, the foundations of it were laid by Whitehead as long ago as 1950. Its successes to date include wide ranging generalisations of the van Kampen theorem (Brown-Higgins, Brown-Loday) that allow calculations to be made of invariants of large classes of spaces, the discovery of a new non-abelian tensor product of groups (Brown-Loday) related to problems in algebraic K-theory and the homology of groups, and a complete algebraic description of n-types of connected spaces (Loday) generalising that given for 2-types by MacLane and Whitehead.
 
  The main problem solved  in this area has been that of giving descriptions of functors allowing direct passage from homotopy types to  Cat^n-groups, crossed complexes and the development of a new  integrated model (n-crossed complexes) encompassing both Cat^n-groups, and crossed complexes.  These developments allow a new, complete   and algebraic proof of Loday's result on n-types (paper 65 of the list of publications)and an ``easy''  categorical and unified proof of restricted forms of the Brown-Higgins and Brown-Loday  generalised van Kampen Theorem to be given.  Work with P.J.Ehlers  (ex-research student), shows that there may be a van Kampen `metatheorem' from which the known forms of this type of result may be derived. (Several papers are in preparation; see the list of publications)), which includes links to preprints.)
 
  One of the eventual aims of this research is to find the ``correct''  interpretation of these new invariants along the lines suggested by  Grothendieck in his notes ``Pursuing Stacks''. This area will involve the  interplay between these structures and homotopy coherence.
 

2. Equivariant homotopy theory,

(joint with Ronnie Brown, Marek Golasinski and Andy Tonks)
One of the aspects we have looked at it is the situation where there is a group acting on the space being modelled. We have used methods adapted from  homotopy coherence to analyse the analogues of crossed module and crossed complex techniques in this context.  This has resulted in several papers (66,73 and 83 of the list of publications).  The links with the Grothendieck programme for Pursuing Stacks are strong as the category of G-sets is an important example in that general area.
 

3. Pursuing Stacks, or `Grothendieck's dream'

(joint with R.Brown, K.H.Kamps (Hagen), and others.) This grew out of correspondence in 1982-3 between Brown and Porter in Bangor, and Grothendieck, then at Montpellier. That correspondence became linked to Grothendieck s 650 page manuscript ´A la poursuite des champs . It revolves around an attempt to extend the relationship (Galois-Poincar´e correspondence) between the fundamental groupoid of a space and the category of covering spaces, to higher homotopy invariants (n-types). There is considerable evidence that higher order versions of this classical theorem should exist, linking representations of homotopy n-types with categories of stacks of (n-1)-types. The precise relationship is still elusive however. This area is one of the main motivations for the development of higher dimensional category theory and the corresponding algebra.
 

4. Linear Representation of algebraic models for n-types.

Homotopy 1-types are modelled by groups, homotopy 2-types by crossed modules or cat1groups, and so on. There is a rich linear representation theory for groups, but as yet no fully developed corresponding theory for the models of higher homotopy n-types. For n = 2, two embryonic theories exist. One due to my ex-student, M. Forrester-Barker, is based on length 1 chain complexes of free modules, the other uses so called 2-vector spaces as considered by Kapranov and Voevodsky. The relationships between the two approaches are as yet unknown and the chain complex based theory needs further work to optimise the results found so far.
 

5. Weak n-categorical models for homotopy n-types.

Strict n-groupoids do not model all homotopy n-types. Loday s catn 1-groups do model all connected homotopy n-types, but in so doing repeat information enormously. Smaller models do exist for n = 2, but for n ¸ 3 there is as yet no detailed knowledge of how to produce a small model. The hope is that methods based on Segal categories and similar constructions, which do give weak models for all n, can be adapted to allow calculations.
 

6. The Crossed Menagerie

These are a set of notes initially prepared for a course of 4 lectures at the XVI Encuentro Rioplatense de \'{A}lgebra y Geometr\'{\i}a Algebraica, in Buenos Aires, 12-15 December 2006, and then extended for an MSC course in Ottawa the following Summer. They set out to introduce some of the family of crossed algebraic gadgetry that have their origins in combinatorial group theory in the 1930s and `40s, then were pushed much further by Henry Whitehead in the papers on Combinatorial Homotopy. Since about 1970, more information and more examples have come to light, initially in the work of Ronnie Brown and Phil Higgins in which crossed complexes were studied in depth. Explorations of crossed squares by Loday and Guin-Valery and, from about 1980 onwards, indicated their relevance to many problems in algebra and algebraic geometry, as well as to algebraic topology have become clear. More recently in the guise of 2-groups, they have been appearing in parts of differential geometry, and have, via work of Breen and others, been of central importance for non-Abelian cohomology. This connection between the crossed menagerie and non-Abelian cohomology is almost as old as the crossed gadgetry itself, dating back to Dedecker's work in the 1960s. Yet the basic message of what they are, why they work, how they relate to other structures, and how the crossed menagerie works, still need repeating, especially in that setting of non-Abelian cohomology in all its bewildering beauty.

Of course, there is a strong link with `Grothendieck's dream'.

A copy of the first 7 chapters of the notes can be found here.

This page was last modified on 21-03-2008. T.P.