Nonabelian algebraic topology is a program developed by Ronnie Brown and his collaborators. The idea is to redo and enhance algebraic topology by making use of the tool of strict omega-groupoids and in particular the crossed complexes equivalent to them, which generalize the complexes of abelian groups traditionally used in algebraic topology.
I hope it is helpful to relate my (Ronnie Brown's) experiences from the 1960s and later with nonabelian cohomology.
In writing my book on topology in the 1960s, I got offended by having to make a detour to get the fundamental group of the circle, and then was attracted by Paul Olum?’s paper referenced below. I extended Olum’s work to a Mayer–Vietoris type sequence? in the second paper below, and this enabled one to compute the fundamental group of, for example, a wedge of circles.
(I use an MV sequence in Topology and Groupoids in connection with pullbacks of covering spaces.)
So I decided to use this account for the book, thus giving students the advantage, it seemed, of an introduction to cohomological ideas.
The problem was that the account when written in detail came to 30 pages (or maybe 40) and when looked at in the cold light of day seemed incredibly boring (a full account is different from Olum’s research account).
I was at the time looking for exercises and came across Philip Higgins’ paper on presentation?s of groupoids, which used free products with amalgamation of groupoids. So I decided to give an exercise on the fundamental groupoid of a union. Then I felt I ought to write out a solution. When I had done this, it seemed streets ahead in exposition of all that nonabelian cohomology stuff and moreover, when souped up to the fundamental groupoid on a set of base points, gave results not reachable by the MV sequence; for example you could not with the MV sequence deduce the precise calculation of the fundamental group of a union of two open sets whose intersection had say 150 path components. (This anomaly is also significant, in illustrating the limitations of exact sequences.)
So I decided to switch to an exposition of groupoids in 1-dimensional homotopy theory (also spurred by a meeting with George Mackey? in 1967 where he told me of his work on ergodic groupoids, which is now seen as a preliminary to Noncommutative Geometry).
It occurred to me that if one could come to the groupoid idea from two distinct directions, then there was likely to be more in this than met the eye.
At the same time, an examination of the proof of the van Kampen theorem for groupoids, suggested that the theorem should have an extension to all dimensions, if one could define homotopy gadgets with the right properties. Another stimulus was the proof (used in the book) by Frank Adams? (circulated in handwritten lecture notes) of the cellular approximation theorem, which had analogies to parts of the van Kampen proof, but failed to get algebraic results because, apparently, of the lack of an appropriate algebraic gadget in dimension .
It took 9 years to find such a gadget in dimension , and another 3 to get them in all dimensions, in work with Philip Higgins.
It seemed to me unfortunate that this work aroused the opposition, for reasons never explained to me, of Frank Adams, who told people the whole programme was “ridiculous”. His opinion became the opposite only when I told him (1985?) of the extension to the non simply connected case of the Blakers-Massey description of of a triad, using the nonabelian tensor product (work with Jean-Louis Loday?).
The higher order van Kampen theorems, and the often nonabelian calculations which result, have not been obtained by cohomological methods, but only by working directly with structures appropriate to the geometry of higher homotopies, i.e. forms of strict multiple groupoids. This confirms the comment of Philip Hall?, Philip Higgins’ supervisor, that one should not try to force the geometry into a given algebraic mode, but search for the algebra which models the geometry. So it seems to me that algebraic topology has been mainly restricted to, or not got out of, the single base point and “group”, not “groupoid”, mode, nor appreciated the possibilities of colimit type theorems in algebraic (and geometric?) topology – no algebraic or geometric topology text (except mine!) mentions the higher order van Kampen work with Philip Higgins.
You can also see this restriction in the contrast between the unsymmetrical, choice laden, definition of the second relative homotopy group, with its compositions in one direction (recall the limitations of “Lineland” described in “Flatland”) and the definition of the fundamental double groupoid of a pointed pair of spaces , with its compositions in directions. This contrast gets more significant in higher dimensions.
For all these reasons, my inclination is to look for the applications of the “appropriate” (whatever that is!) structures rather than cohomology with coefficients in such structures, where lots of detail is likely to get lost. Also, in making calculations it is convenient to work with strict algebraic structures, where the notion of colimit is more comprehensible. Even there, it has been a problem to make say colimit calculations with crossed modules into a symbolic computer algebra format. See the work by Chris Wensley? listed below.
These results could not have been obtained without the intuitions on multiple compositions easily allowed by a cubical approach.
One of the key observations for this programme was that one could define a strict homotopy double groupoid for a pointed pair of spaces, and that this was closely related to the well known fundamental crossed module of a pair of spaces, first considered by J.H.C. Whitehead?. His paper listed below was a key source of ideas.
The natural extension of this observation is to construct a strict cubical ω-groupoid? of a filtered space , and find its relation to the quite classical homotopically defined fundamental crossed complex functor . The proofs here are non trivial. By proving using a colimit theorem for one can shortcut singular homology?, and obtain old and new results in algebraic topology, including some explicit calculations of homotopy groups, even as modules over the fundamental group. This working with filtered space is not unreasonable since they abound. For example, classifying spaces often come with convenient filtrations, as do geometric realisation?s of simplicial or cubical sets. These ideas generalise of course to multifiltered space?s or -cubes of spaces. It is not so clear that one must work with a kind of bare topological space, and so have little handle on which to construct invariants, except say by first taking a singular complex, or using multipaths.
The main idea of the Higher Homotopy van Kampen Theorems is to model algebraically the gluing of homotopy types, or limited models of such.
An indication of a beginnings of a Čech type approach to nonabelian cohomology using groupoids and crossed complexes is given in the new book, Chapter 12. This has not been developed in terms of sheaf theory.
Another big gap in comparison with traditional algebraic topology is intersection theory? and Poincare duality?, although the (quite complicated) machinery of tensor products is available in the crossed complex context.
An obvious gap is also that of extending Grothendieck’s work on the fundamental group!
Nonabelian algebraic topology in particular provides a context for and makes some use of nonabelian cohomology.
One of the main motivations for the development of Nonabelian algebraic topology was the observation that the Seifert-van Kampen theorem is most naturally understood as being not about homotopy groups, but about the fundamental ∞-groupoid of a space. Since the assignment of fundamental -groupoids to spaces is an -category-valued co-presheaf, one can understand this in the context of nonabelian cosheaf homotopy. Remarks on this are in the blog entry Codescent and the van Kampen theorem.
Ronnie Brown has published a long series of articles over the years developing the ideas of nonabelian algebraic topology.
Currently a comprehensive monograph is in preparation on the subject of the algebraic topology of filtered spaces using crossed complexes and cubical omega-groupoids with connections. A current version of this monograph is available on the web at
near the bottom of the page.
The use of crossed complexes allows the notion of free crossed resolution, and so clear links with standard homological algebra. The place of strict -fold groupoids in nonabelian homological algebra needs much further work. One key is that the higher homotopy van Kampen theorem allows some algebraic information on gluing homotopy types, giving some colimit information on nonabelian structures. The traditional invariants, e.g. homotopy groups, -invariants, …, then have to be extracted from this larger structure.
Olum, P., Non-abelian cohomology and van Kampen’s theorem, Ann. Math. 68 (1958) 658–667.
Brown, R., On a method of P. Olum, J. London Math. Soc. 40 (1965) 303–304.
Brown, R., Elements of Modern Topology, McGraw Hill, Maidenhead, 1968.
Brown, R., Topology and Groupoids, Booksurge, 2006.
Higgins, P.J., Presentations of groupoids, with applications to groups, Proc. Camb. Phil. Soc., 60 (1964) 7–20.
Brown, R. and Higgins, P.J., On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc.(3) 36 (1978) 193–212.
Brown, R. and Higgins, P.J., Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981) 11–41.
Whitehead, J.H.C., Combinatorial Homotopy II, Bull. Amer. Math. Soc., 55 (1949), 453–496.
Brown, R. Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems, in Handbook of Algebra 6, Edited M. Hazewinkel, Elsevier, 2009.
Wensley, C.D. and Alp, M., XMOD, a GAP share package for computation with crossed modules, GAP Manual, (1997), 1355–1420.