Ronnie Brown wrote:

Dear all,

André makes a proper and convincing case for the importance of pointed spaces in algebraic topology . Part of this importance is the difficulty of homotopy theory, which necessitates approximations, and these are sometimes adequate and convenient. Henry Whitehead introduced two methods to approximate homotopy theory: one was to describe special cases such as the category of polyhedra which were n-dimensional and r-connected, e.g. n small, or r near to n. or r about half n - cf the stable range what Schlessinger and I call `shallow' in rational homotopy theory, still manageable for r about 1/3 n The other was stabilisation. However, in introducing CW-complexes in ``Combinatorial Homotopy I'' he explains that he does not stick to the single vertex case since he wants to include covering spaces. There is often no canonical choice of base point in a covering space; there are also advantages in an account which does not require connectivity. .

e.g. an n-fold covering as a fibre bundle cf the Galois correspondence jim

I was told that one of Philip Hall's dictums was that you want the algebra to model the geometry, and not try to force it into a previously known format.

YES! one of the problems in math bio is to have the math model the biology and not try to force it into a previously known format.

Grothendieck wrote in part:

Choosing paths for connecting the base points natural to the situation to one among them, and reducing the groupoid to a single group, will then hopelessly destroy the structure and inner symmetries of the situation, and result in a mess of generators and relations no one dares to write down, because everyone feels they won't be of any use whatever, and just confuse the picture rather than clarify it. I have known such perplexity myself a long time ago, namely in Van Kampen type situations, whose only understandable formulation is in terms of (amalgamated sums of) groupoids.

... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]