Jim Stasheff wrote:
John and anyone else who cares to weigh in, here are some comments from the purely topological or rather homotopy theory side:
For both bundles and fibrations (e.g. over a paracompact base), your last slogan is the oldest:
FIBRATIONS WITH FIBER F OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F)
`the same as' referring to homotopy classes.
It's certainly old, but I mentioned another that may be older: COVERING SPACES OVER B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS FROM THE FUNDAMENTAL GROUP OF B TO AUTOMORPHISMS OF F. although one usually sees this special case (which I didn't bother to mention): CONNECTED COVERING SPACES OVER B WITH FIBER F ARE "THE SAME" AS TRANSITIVE ACTIONS OF THE FUNDAMENTAL GROUP OF B ON F which is usually disguised as follows: CONNECTED COVERING SPACES OVER B ARE "THE SAME" AS SUBGROUPS OF THE FUNDAMENTAL GROUP OF B Anyway, I wasn't trying to present things in historical order. I was trying present them roughly in order of increasing "dimension", starting with extensions of groups, then going up to 2-groups, then expanding out to groupoids, then going up to n-groupoids, and finally omega-groupoids... which are the same as homotopy types! And here, as usual, the n-category theorists meet up with the topologists - and find that the topologists have already done everything there is to do with omega-groupoids ... but usually by thinking of them of them as *spaces*, rather than omega-groupoids! It's sort of like climbing a mountain, surmounting steep cliffs with the help of ropes and other equipment, and then finding a Holiday Inn on top and realizing there was a 4-lane highway going up the other side. So, as usual, the main point of calling homotopy types "omega-groupoids" instead of "spaces" is not to reinvent topology, but to see how ideas from topology generalize to n-category theory. Think of spaces as omega-groupoids but use those as a springboard for omega-categories - or at least n-categories, perhaps just for low values of n if one is feeling tired. In the case at hand, the omega-groupoidal slogan: FIBRATIONS OF OMEGA-GROUPOIDS WITH FIBER F AND BASE B ARE "THE SAME" AS WEAK OMEGA-FUNCTORS FROM B TO AUT(F) is just a reformulation of: FIBRATIONS WITH FIBER F OVER THE SPACE B ARE "THE SAME" AS MAPS FROM B TO THE CLASSIFYING SPACE OF AUT(F) but it suggests a grandiose generalization: FIBRATIONS OF OMEGA-CATEGORIES WITH BASE B ARE "THE SAME" AS WEAK OMEGA-FUNCTORS FROM B^{op} TO THE OMEGA-CATEGORY OF OMEGA-CATEGORIES! I guess we can thank Grothendieck for making precise and proving a version of this with omega replaced by n = 1: FIBRATIONS OF CATEGORIES WITH BASE B ARE "THE SAME" AS WEAK 2-FUNCTORS FROM B^{op} TO THE 2-CATEGORY OF CATEGORIES. More recently people have been thinking about the n = 2 case, especially Claudio Hermida: 22) Claudio Hermida, Descent on 2-fibrations and strongly 2-regular 2-categories, Applied Categorical Structures, 12 (2004), 427-459. Also available at http://maggie.cs.queensu.ca/chermida/papers/2-descent.pdf He states something that hints at this: FIBRATIONS OF 2-CATEGORIES WITH BASE B ARE "THE SAME" AS WEAK 3-FUNCTORS FROM B^{op} TO THE WEAK 3-CATEGORY OF 2-CATEGORIES. where I'm using B^{op} to mean B with the directions of both 1-morphisms and 2-morphisms reversed. (Hermida follows tradition and calls this B^{coop} - "op" for reversing 1-morphisms and "co" for reversing 2-morphisms. But, it looks like we'll be needing to reverse all kinds of morphisms in n-category case, so we'll need a short name for that.) Best, jb