Author: ronniegpd Format: TextHere are some comments on the article http://ncatlab.org/nlab/show/strict+omega-groupoid and particularly the comment:
" Following work of J. H. C. Whitehead, in (Brown-Higgins) it is shown that the 1-category of strict ω-groupoids is equivalent to that of crossed complexes. This equivalence is a generalization of the Dold-Kan correspondence to which it reduces when restricted to crossed complexes whose fundamental group is abelian and acts trivially. More details in this are at Nonabelian Algebraic Topology."
The work on higher groupoids in topology was not following from Whitehead. It was all focused on two questions dating from say 1965: (i) could groupoids be useful in higher homotopy theory? (ii) how could one express a conjectured proof of a 2-dimensional van Kampen theorem? The latter seemed to need a form of fundamental homotopy double groupoid. The prospective algebra was developed with Chris Spencer in the early 1970s, and the relation with J.H.F. Whitehead's crossed mdoules clarified. (I have put our first version of this wok on my web site as /pdffiles/brown-spencerearly.pdf . (It was rejected for JPAA by Saunders on the basis of negative referees' reports.) In discussions with Philip Higgins in 1974 we argued 1. Whitehead's tricky theorem on free crossed modules was an example of a universal propery in 2-dimensional homotopy theory. 2. If
our conjectured 2-d vKT was any good it should have Whitehead's theorem as a Corollary. 3. Whitehead's theorem was about second relative homotopy groups. 4. We should therefore look for a homotopy double groupoid in a relative situation. This enabled us to write our paper, which was submitted in 1975 to the Proc. LMS, received with total disbelief, and no specific mathematical comment whatsoever, went to third referee, and in the end we agreed to cut it down by a third by omitting discussion, figures, and explanation, and it was published in 1978. By which time we had already dome the general case, for filtered spaces.
The generalisation of double groupoids to higher dimensions was considerably helped by PhD work of Keith Dakin, and Nick Ashley, on simplicial T-complexes. Two papers in JPAA in 1981 give the algebra and topology for the generalised Seifert-van Kampen theorem for the fundamental crossed complex of a filtered space. This a theorem yielding colimit theorems for relative homotopy groups, and specific calculations. Simplicial sets and simpliciual approximation are not used at all.
Two papers published in CTGD in 1981 kind or round off the story. One relates crossed complexes to infinity groupoids. now called strict globular omega-groupoids, and that paper has a definition of what is now called a globular set, as well as a definition of n-fold category. The second relates the cubical omega-groupoids with connectiosn to cubical T-complexes.
Discussion of this work in 1981 with Loday in Strasbourg suggested the possibility of a van Kampen theorem for his n-cat-groups, which we eventually relabelled cat^n-groups, to avoid the confusion with globular structures; i.e. we were dealing with n-fold categories rather than n-categories. This was eventually published in 1987 in Topology, while parallel work by Graham Ellis, eventually in collaboration with Richard Steiner, developed the algebra of crossed n-cubes of groups, and gave new applications of the Brown-Loday work.
This work enables new computations of n-ad homotopy groups and indeed of some n-types.
Curiously, the use of n-fold categories and of cubical methods does not seem to warrant a mention in the "cosmic cube".
I published a paper on a globular homotopy groupoid of a filtered space in HHA (2008), but it is interesting that the proof relies on the cubical results. It happens that neither globular nor simplicial methods were helpful for the local-to-global problems which I was pursuing, whereas the cubical methods enabled conjectures and proofs. Whitehead's use in CHII (1949) of what we now call crossed complexes, and which date back to Blakers in 1948, was certainly helpful, and this should be said. But Whithead in CHII was studying the homotopy classification of maps, and realisation problems. He was not thinking about higher categories; he naturally took a categorical viewwpoint, but focussed on specific topological problems, and particularly on usable models of certain homotopy types, under certain restrictive conditions, in order to make progress.
Here are some comments on the article http://ncatlab.org/nlab/show/strict+omega-groupoid and particularly the comment:
" Following work of J. H. C. Whitehead, in (Brown-Higgins) it is shown that the 1-category of strict ω-groupoids is equivalent to that of crossed complexes. This equivalence is a generalization of the Dold-Kan correspondence to which it reduces when restricted to crossed complexes whose fundamental group is abelian and acts trivially. More details in this are at Nonabelian Algebraic Topology."
The work on higher groupoids in topology was not following from Whitehead. It was all focused on two questions dating from say 1965: (i) could groupoids be useful in higher homotopy theory? (ii) how could one express a conjectured proof of a 2-dimensional van Kampen theorem? The latter seemed to need a form of fundamental homotopy double groupoid. The prospective algebra was developed with Chris Spencer in the early 1970s, and the relation with J.H.F. Whitehead's crossed mdoules clarified. (I have put our first version of this wok on my web site as /pdffiles/brown-spencerearly.pdf . (It was rejected for JPAA by Saunders on the basis of negative referees' reports.) In discussions with Philip Higgins in 1974 we argued 1. Whitehead's tricky theorem on free crossed modules was an example of a universal propery in 2-dimensional homotopy theory. 2. If our conjectured 2-d vKT was any good it should have Whitehead's theorem as a Corollary. 3. Whitehead's theorem was about second relative homotopy groups. 4. We should therefore look for a homotopy double groupoid in a relative situation. This enabled us to write our paper, which was submitted in 1975 to the Proc. LMS, received with total disbelief, and no specific mathematical comment whatsoever, went to third referee, and in the end we agreed to cut it down by a third by omitting discussion, figures, and explanation, and it was published in 1978. By which time we had already dome the general case, for filtered spaces.
The generalisation of double groupoids to higher dimensions was considerably helped by PhD work of Keith Dakin, and Nick Ashley, on simplicial T-complexes. Two papers in JPAA in 1981 give the algebra and topology for the generalised Seifert-van Kampen theorem for the fundamental crossed complex of a filtered space. This a theorem yielding colimit theorems for relative homotopy groups, and specific calculations. Simplicial sets and simpliciual approximation are not used at all.
Two papers published in CTGD in 1981 kind or round off the story. One relates crossed complexes to infinity groupoids. now called strict globular omega-groupoids, and that paper has a definition of what is now called a globular set, as well as a definition of n-fold category. The second relates the cubical omega-groupoids with connectiosn to cubical T-complexes.
Discussion of this work in 1981 with Loday in Strasbourg suggested the possibility of a van Kampen theorem for his n-cat-groups, which we eventually relabelled cat^n-groups, to avoid the confusion with globular structures; i.e. we were dealing with n-fold categories rather than n-categories. This was eventually published in 1987 in Topology, while parallel work by Graham Ellis, eventually in collaboration with Richard Steiner, developed the algebra of crossed n-cubes of groups, and gave new applications of the Brown-Loday work.
This work enables new computations of n-ad homotopy groups and indeed of some n-types.
Curiously, the use of n-fold categories and of cubical methods does not seem to warrant a mention in the "cosmic cube".
I published a paper on a globular homotopy groupoid of a filtered space in HHA (2008), but it is interesting that the proof relies on the cubical results. It happens that neither globular nor simplicial methods were helpful for the local-to-global problems which I was pursuing, whereas the cubical methods enabled conjectures and proofs. Whitehead's use in CHII (1949) of what we now call crossed complexes, and which date back to Blakers in 1948, was certainly helpful, and this should be said. But Whithead in CHII was studying the homotopy classification of maps, and realisation problems. He was not thinking about higher categories; he naturally took a categorical viewwpoint, but focussed on specific topological problems, and particularly on usable models of certain homotopy types, under certain restrictive conditions, in order to make progress.