Re: terminology

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

Dear André

There seems to me to be a tremendous amount of great work going on 
higher category theory, but when you write

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One lies in the fact that equivalent categories are considered to be the 
"same", 

even if [or] when they are not isomorphic.
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this seems to go against the grain of what I have been doing in groupoids  since I decided they were valuable in about 1965! It sounds like the old  canard `groupoids reduce to groups', so there must be some confusion in my mind on what you are saying. 

One thing that took me a while to realise was that it was not enough to study the fundamental groupoid or a fundamental group but one needed to consider intermediate cases, namely the fundamental groupoid on a set of base points chosen according to the geometry at hand. (`Algebraic topology'  has not understood this it seems.) The vertices of a groupoid give a spatial component to group theory, a kind of geography, and sometimes, even often, that is needed to model the geometry.  So for example it is useful  to replace the trefoil group which has 2 generators x,y and one relation  x^2=y^3 by the trefoil groupoid which is the double mapping cylinder (homotopy pushout) in groupoids of the two maps Z \to Z, given by squaring  and cubing. So we add an extra groupoid generator iota on different vertices which turns x^2 into y^3. This corresponds to the double mapping construction to give a CW-complex. 

So groupoids give the strict algebra of keeping the information which makes things the same. 

In higher dimensions we want not just commutative diagrams but control of  the ways of filling these diagrams. If the diagram is a pentagon (as we all know does happen) I would want a pentagon as part of the geometry, and the only question is how to deal with multiple compositions of various such objects, and that was the aim of David Jones thesis on Polyhedral T-complexes. The point is that the pieces to be composable have to be all faces but one of a general poyhedral `horn', the process of composing them  is the filler of the horn, and the composite of the pieces  is the remaining face of the filler. (It was not attempted to do this in category rather than groupoid terms, and that is still a mystery!) So you can see I have long been very sympathetic to using the Kan condition for describing algebraic or structural objects, but find the simplicial approach too awkward (for me, of course; I found the way Nick Ashley coped with that was amazing). 

I do not want to consider equivalent groupoids the same, as I may want to  use the spatial components to describe how they might be glued together.  It is partly the old tag of not throwing away information till the last possible moment. 

On the other hand, some computations are best done at the strict level, rather than the weak one. I mention here the rotations in my paper:

``Higher dimensional group theory'', in {\em Low dimensional topology},  London Math Soc. Lecture Note Series 48 (ed. R. Brown and T.L.  Thickstun, Cambridge University Press, 1982), pp. 215-238.

(see also a fuller exposition in the new book on Nonabelian algebraic topology), which would seem to be more difficult to write out at the lax level. The fact that the strict calculations imply the existence of certain homotopies is part of the interest. 

So in the work with Higgins a Kan fibration - from the singular filtered complex of a filtered space to the quotient to give a strict structure - ties in the lax and the strict in a necessary way for the theory and calculations. 

I am really searching for points of agreement. 

Best regards

Ronnie


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