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. 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. . 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. 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. Cases where the choice of a single base point is inconvenient are when there is a group action (as shown by Higgins and Taylor, 1981); in non-connected cases; in Nielsen fixed point theory (as shown by Philip Heath, and recently by Kate Proto); and for work in algebraic geometry by Zoonekynd. I like the description of the circle as obtained from the unit interval [0,1] by identifying 0 and 1 in the category of spaces; and of the integers as obtained from the `unit interval groupoid ' cal I (the indiscrete groupoid on {0,1}, and so finite) by identifying 0 and 1 in the category of groupoids. One gets analogous results for \pi_n as modules over \pi_1 in describing \pi_n(S^n \vee S^1). Of course this example can be done using covering spaces, but that seems a roundabout route. The Bass-Serre theory of graphs of groups often uses choice of base points and of trees; yet Higgins showed in 1976 that there was a nice normal form for the fundamental groupoid of a graph of groups. The action of Z_2 on S^1 by reflection has 2 fixed points, and is better described (look at quotients) by the action of Z_2 on the fundamental groupoid with these as base points rather than on the fundamental group on one of them (which one?). The success in 1967-8 (as it seemed to me) of groupoids in 1-dimensional homotopy theory led me ask for possible uses of groupoids in higher homotopy theory, and this has led with fortunate collaborations to the notions of `higher dimensional group theory' and `higher dimensional algebra', and various new algebraic structures with applications; an account of some aspects of these is in press with the EMS, and might be thought of as `towards nonabelian algebraic topology'. I confess not to have found the appropriate many base point version with applications of the fundamental cat^n-group of an n-cube of spaces for n> 1. So it is a case of horses for courses. Ronnie

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