It may be helpful to say how I got into groupoids.

In the 1960s, I was writing a topology text and wanted to do the fundamental group of a cell complex, which required the van Kampen Theorem (I have now been persuaded to call this the Seifert-van Kampen theorem, as on wikipedia, so I call it SvKT). I was kind of irritated that this did not as then formulated give the fundamental group of the circle, so one had to make a detour and do all or a piece of covering space theory.

Then I found a paper by Olum on nonabelian cohomology and van Kampen's theorem which I extended to a Mayer-Vietoris type sequence which did give the fundamental group of the circle. Unfortunately, when written out in full, it was rather boring! I then came across a paper of Philip Higgins which included the notion of free product with amalgamation of groupoids. So I decided to put in an exercise using this notion for the fundamental groupoid of a space. Then I wrote out a solution for this, and it was so much nicer than the nonabelian cohomology stuff that I decided to make the account in terms of groupoids. It still needed the key notion of the fundamental groupoid on a set $C$ of base points, written $\pi_1(X,C)$. For the circle, this needed $C$ to have 2 elements. This result appeared in the first 1968 edition, and in subsequent ones, of the book on topology, but in no other topology text in English since then.

In 1967 I met George Mackey who told me of his work on ergodic groupoids. This persuaded me that the idea of groupoid was, or might be, more important than met the eye.

On writing out the proof of the SvKT for groupoids maybe 5 times, it occurred to me in 1965 that the proof should generalise to higher dimensions if one had the `right' gadget generalising $\pi_1(X,C)$. This was finally found with Philip Higgins in 1974 as the fundamental double groupoid $\rho_2(X,A,C)$ of a space $X$ with subspace $A$ and set $C$ of base points. So we got a SvKT in dimension 2, published in 1978, and had extended this to all dimensions by 1979. Work with Chris Spencer in 1971-2 on double groupoids and crossed modules was essential as a basis for all this.

The point I am making is that the initial aim of an improved proof of the fundamental group of the circle was very modest, but based on an aesthetic feeling, and the aim would not have got many marks for a research proposal! But in the end it opened out a new area.

One main driving force for the higher dimensional work was the intuitions of subdividing a square into little squares, and getting the inverse to that, i.e. composing the little squares into a big one. Another problem was that of expressing the idea of commutative cubes.

Philip Higgins told me of a remark of Philip Hall that one should try to make the algebra model the geometry, and not force it into an already known mold. I think that is what people were doing in avoiding the groupoid concept, despite its obvious nature. Indeed the idea of `change of base point' for the fundamental group is a bit like giving a railway timetable in terms of return journeys and change of start-- i.e. is bizarre.

Perhaps the moral is that is good to look for ways of expressing intuitions in a rigorous mathematical form. And if that means building up some maths from scratch, previous to definitions, examples, theorems, proofs, as was needed in the higher dimensional work, then that is a lot of fun! (More fun than doing someone else's problem!) But it may take a long time, need lots of attempts, and searching for related ideas, and as it gets going, hard work, and in our case fruitful collaborations.

Research students liked the idea of a big plan (what is or might be `higher dimensional group theory'?) and the attempts to pick from this something that might be doable.

I'd better not go on about the opposition!

Does that help?