Announcement of paper

Announcement of paper

[Michael Makkai]

The paper "Computads and 2 dimensional pasting diagrams" has been posted
on my website http://www.math.mcgill.ca/makkai/. The paper is in several
pdf files.

With greetings: Michael Makkai

multiple compositions

[Ronnie Brown]

Dear Michael,

This email is suggested by your announcement at (
http://www.math.mcgill.ca/makkai/) of  papers on pasting and computads.

First, I hope it is useful to direct people to an early paper with a
definition of strict omega-categories, there called \infty-categories:
(with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids
and crossed  complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981)
371-386.
www.bangor.ac.uk/r.brown/pdffiles/x-comp.pdf
The main emphasis of this paper is  the equivalence in the title. Because
crossed complexes C have a classifying space BC which can also be
represented as a fibration over B\pi_1 C with fibre a topological abelian
group (in fact the classifying space of a chain complex) this implies that
the homotopy type of spaces represented by \infty-groupoids is limited. This
observation suggested to Grothendieck in 1982 the need to move to weak
\infty-categories (or groupoids) for dealing with matters of nonabelian
cohomology, which for him was a long standing aim. It was not till I met him
in 1986 that I  convinced him that strict n-fold groupoids really did model
all weak homotopy n-types, (Loday), at which he exclaimed `That is
absolutely beautiful!' There is still work to do on the connections with
nonabelian cohomology! And crossed complexes, though limited,  are certainly
useful for this, because of their close relation to chain complexes with
operators. (for a survey on crossed complexes, see  math.AT/0212274).

Second, I would like to raise some general questions on multiple
compositions and what is or should be the mathematics to deal with these.
For 2-categories, this seems to be pasting schemes. However the thrust of my
work since the 1970s has been to Higher Homotopy van Kampen theorems,
(HHvKTs) based on the question of the possible use of groupoids in higher
homotopy theory, given their success in 1-dimensional homotopy theory. The
key aim was to use cubical methods, because these gave a convenient
`algebraic inverse to subdivision', through the use of multiple
compositions, modelling steps in the proof of the usual vKT for groupoids.

Such HHvKTs were proved with Philip Higgins in dimension 2 in 1978, in all
dimensions (for crossed complexes) in 1981, and with Jean-Louis Loday in
1987 (for cat^n groups and so crossed n-cubes of group). All these theorems
have algebraic implications for homotopy types which seem unobtainable by
other means. The theorems with PJH use directly `algebraic inverse to
subdivision', while the proof with Loday uses some sophisticated algebraic
topology and simplicial methods (Waldhausen, Zisman, Puppe and some new
results). The work  obtains to a limited  extent a vision of Grothendieck of
what he termed `integration of homotopy types'; there are strong
connectivity assumptions so the theorems do not allow calculation of
everything, e.g. homotopy groups of spheres, and so some have said `the
theory has not fullfilled its promise' (report on a failed research
proposal). On the other hand, the theory does come within the scope of
`higher dimensional nonabelian methods for local-to-global problems', and
the new explicit calculations enabled and relations with combinatorial group
theory (e.g. the nonabelian tensor product of groups, bibiliography now of
90 items, http://www.bangor.ac.uk/~mas010/nonabtens.html) are pointers to
its success.

Possibly relevant to this is that I have never been able to write down a
proof of even the 2-dimensional HHvKT using 2-groupoids; and it would not
have been, or at least was not,  even conjectured in those terms.

My preference is for algebraic models of homotopy types which lead to some
explicit algebraic computations (hence the HHvKTs),  to new theorems, and
new relations with other areas.

My overall questions are therefore:
(i) to what extent can and should cubical methods be used in weak category
theory?
(ii) is there some operadic or other method from current ideas on higher
category theory which allows the use of `algebraic inverses to subdivision'
in all dimensions?
(iii) to what extent are these operadic methods generally useful in higher
dimensional nonabelian methods for local-to-global problems? (I first heard
of the term local-to-global problems from Dick Swan, when we worked on his
lecture notes on the Theory of Sheaves, Oxford, 1958.)

Of course subdivision allows the passage from global to local. The problem
is the converse, a key problem in maths and science (even biology and
engineering!). Anything which helps in this seems to me a Good Thing!

Greetings and Good Luck,

Ronnie Brown