Dear Bill, You ask: In the applications of algebraic topology to topology, where do the ‘basepoints’ originate? I'd like to give a different type of answer to that suggested in your list. (yet another is : whoopee, we have a group!) After I first thought of the van Kampen theorem for the whole fundamental groupoid I realised one wanted *computation*, and the whole fundamental groupoid of a space was too big for that. The fundamental group at a base point was too small in may cases, such as the circle. The solution that seemed right to me, and which took a while to find, was the fundamental groupoid on a set of base points chosen conveniently according to the geometry of a given situation. Eldon Dyer said I ought to take a hard line on this: if the connected space X is the union of 127 open sets whose intersections have 3,272 components, you do not want to take a single base point! Such situations (even with infinitely many components) occur in group theory applications, and analogously in topology in connected covering spaces over a union of 2 open sets. I advocated the fundamental groupoid on a set of base points in my 1968 book `Elements of Modern Topology', but this concept has I think not been mentioned in any later date algebraic topology text in English by other authors. Also group theorists are happy with graphs and free groups, but are very wary of the free groupoid on a graph! The only new result from that book that has been taken up is the gluing theorem for homotopy equivalences, but its origin is usually unacknowledged. Looking at the way this groupoid van Kampen theorem could be used, it seemed amazing to me that one could obtain *complete* information on a fundamental group by deducing that from knowledge of a larger structure, for which one had colimit information, whereas my tries with nonabelian cohomology gave only exact sequences. It seems that groupoids have the advantages of structure in dimensions 0 and 1, and that this is needed for what Grothendieck later called `integration of homotopy types'. Could one find analogous objects with structure in dimensions 0, 1, ...,n? This question led (after many years, and with fortunate collaborations) to higher dimensional van Kampen theorems, which gave quite new information on homotopy invariants, some of them nonabelian, e.g. second relative homotopy groups, triad homotopy groups, n-adic Hurewicz Theorems. Such results were published in 1978, 1981 (with Higgins), 1987 (with Loday). These theorems are, I think, not even mentioned in any texts on algebraic topology (except mine). Some current writers (Faria Martins, Kauffman, Ellis, Mikhailov,...) are using these techniques. So for me the question is: why are people unwilling to throw off the shackles of a single base point? I would welcome enlightenment. It may be that it is found just too hard to fit this idea into what is currently considered the `real world', and so to obtain new results. For example, what happens to iterated loop space theory, with more than one base point? Those interested in the sociology of science might like an excerpt from a lecture by Alan MacKay on icosahedral symmetry at the LMS in 1985. He said the reaction went through three phases: Phase 1) It is false. Phase 2) It is true, but unimportant. Phase 3) It is true, it is very important; and we have known it for years! All the best Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]