Dear Jim and all, This is reply to Jim Stasheff's question below. I hope it is helpful to relate my own experiences from the 1960s and later wrt nonabelian cohomology. In writing my book on topology, I got offended by having to make a detour to get the fundamental group of the circle, and then was attracted by Olum's paper {O}lum, P. \newblock \enquote{Non-abelian cohomology and {van} {Kampen}'s theorem}. {Ann. Math.} \textbf{68} (1958) 658--667. I extended his result to a Mayer-Vietoris (MV) type exact sequence in ``On a method of P. Olum'', {\em J. London Math. Soc.} 40 (1965), 303-304. 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 of an introduction to cohomology 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 presentations 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). 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 (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 n>1. It took 9 years to find such a gadget in dimension 2, and another 3 to get them it 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 \pi_3 of a triad, using the nonabelian tensor product (work with Loday). I did make some attempt to get the groupoid van Kampen result from nonabelian cohomology, see 40. (with P.R. HEATH and H. KAMPS), ``Groupoids and the Mayer-Vietoris sequence'', {\em J. Pure Appl. Alg.} 30 (1983) 109-129. and perhaps the earlier 11. ``Groupoids as coefficients'', {\em Proc. London Math Soc.} (3) 25 (1972) 413-426. But 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 mentions the 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 \rho_2(X,A), with its compositions in 2 directions: see 25. (with P.J. HIGGINS), ``On the connection between the second relative homotopy groups of some related spaces'', {\em Proc. London Math. Soc.} (3) 36 (1978) 193-212. 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. These results could not have been obtained without the intuitions on multiple compositions easily allowed by a cubical approach. In any case, it is useful to analyse successes and failures, in deciding how to proceed. Hence my above account of some failures in nonabelian cohomology! What if anything have I missed? See also http://www.bangor.ac.uk/r.brown/einst.html How can one apply these (strict, for my preference!!) multiple groupoids in other areas???? (recall Grothendieck's work on the fundamental group!!) Good luck Ronnie jim stasheff wrote:

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Jim asks: ____________________________________

How does

`Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids'.

relate to the current rage for `nonabelian cohomology'?

if it's answered in the above paper/book, more precise coordinates please very helpful except, like Urs Schreiber, you answer my question only

RONALD BROWN wrote: partially

do you *address* how yours relates to the current rage in alg geom, math phys, cat theory or what ever?

Ronnie, you already know but others may not Paul Olum @ 1950 already refers to `nonabelian cohomology'!!

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With regard to the book, that is dealt with in Chapter 12, "Nonabelian cohomology of spaces and of groups". The idea is that cohomology requires coefficients, and those dealt with in this book are `crossed complexes', which include groups, groupoids, and crossed modules, as well as algebraic models of spaces with homotopy groups in only two dimensions 1 and n > 1 . Crossed complexes form a linear model of homotopy types, i.e. no Whitehead product. So they are limited, but nonetheless linear models, as in linear algebra, have uses. They can be nonabelian in dimensions 1 and 2, and give complete information on 2-types (which is a step up from groups).

The use of crossed modules as coefficients in nonabelian cohomology was pioneered by Dedecker, first for cohomology of groups, and later, in two papers in Canad. J. Math., of spaces.

Crossed complexes as coefficients for cohomology appear in a Brown-Higgins paper on the classifying space of a crossed complex referred to earlier.

Wrt filtered spaces: the theme of the book is that the functor \Pi: (filtered spaces) \to (crossed complexes) defined using fundamental groupoid \pi_1(X_1,X_0) and relative homotopy groups \pi_n(X_n,X_{n-1},x), x\in X_0, n \geq 2, can serve as a new foundation for algebraic topology because \Pi preserves certain colimits and certain tensor products, proofs of which (not so easy!) do not require singular homology or simplicial approximation. (The proofs do require the cubical higher homotopy groupoid \rho of a filtered space.) Many spaces, e.g. CW-complexes, classifying spaces of groups, or of higher categories, come with a filtration, or several filtrations, so this starting point is not unreasonable. If you are just given a space, you can get a filtered space by taking the (or a) singular complex and realising that as a CW-complex.

The properties of BC, the classifying space of the crossed complex, require knowledge of the homotopy properties of simplicial or cubical sets.

One reason \rho is useful is that it allows compositions in n directions in dimension n, whereas relative homotopy groups are limited to a 1-dimensional composition; and their definition is unaesthetic since it involves choices. They still form an essential part of the theory, and are useful for calculation rather than conjecture or proof, and because they are easier to relate to standard algebraic topology.

There is a way of getting a Cech theory.

I do not know the uses of crossed complexes in analogues of Poincar\'e duality.

More general nonabelian cohomology requires more general coefficients, for which there are candidates, either strict models, e.g. crossed n-cubes of groups, or there is a lot of play with weak higher categorical structures. With the crossed complex model, you can do some calculations, particularly of 2-types.

It seems reasonable to start by developing the linear theory, and, as I tried to communicate, Henry Whitehead's Combinatorial Homotopy II was a very powerful pointer to the structures required and the possible applications. There is a great string of work also by Hans Baues, in several books, going further in many ways, but from a different standpoint.

Hope that helps

Ronnie

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