Concerning Ross Street's interesting remarks about history, I should clarify where the term "Beck condition" comes from. I would like to urge the readers of our categories list to study Jon's 1967 thesis which is now available via TAC Reprints. There one finds Jon's tripleability theorems which were used for example by Benabou and Roubaud in their 1970 paper on descent. Ross remembers that Jon arrived in Tulane in 1969 prepared to lecture on triples and descent. I heard Jon lecturing on that topic already in late 1967 at a meeting of the American Mathematical Society in Illinois. In particular, he explicitly stated the condition that I therefore called the "Beck condition" in my work on Hyperdoctrines (presented to an AMS meeting in NYC in early 1968). Later I saw this condition referred to as the Chevalley condition in a paper of J-L Verdier. I do not know whether Jon was familiar with that work of Chevalley. Some sort of "coequalizer in the base implies descent" property on a fibration F is of course true (in addition to F(X+Y) = F(X) x F(Y)) for those fibrations that exemplify a reasonable notion F of parameterized family. Tripleability provides a useful tool for analyzing these descents, but which are the tripleable (monadic) functors that could arise from descent in some fibration? Bill On Sat, 3 Nov 2007, Ross Street wrote:
History can be harder than mathematics. Yet, it is a worthy goal to get it right. This can require discussion and feedback. Here are some of my memories which I am quite happy for people to correct if they have a fuller picture.
Jean Benabou invented bicategories. In SLNM47 you will find the particular example of a bicategory Spn(E) whose morphisms are spans in a pullback-complete category E. You will also find the convention to refer to properties holding in the homs as local. I always thought it nice that the homs in Spn(E) were slice categories E / a x b, thereby unifying two uses of "local".
You will also find in that SLNM47 paper, the notion of morphism of bicategories and of homomorphism of bicategory. These a very useful concepts. They do compose in their own way. I believe there was no attempt to deny that the "indexed categories" of Pare-Schumacher are category-valued homomorphisms.
The 1969-70 academic year at Tulane University Math Dept was dedicated to Category Theory. Jack Duskin and I were there (doing some teaching as well as research) for the whole year. Saunders Mac Lane and Eduardo Dubuc were there for the first semester. Bernhard Banaschewski and Z. Hedrln were there for the second semester. However, we had a lot of visitors as well. In particular, Jean Benabou visited sometime in the first semester. In particular, I learnt from Benabou's lectures about the "Chevalley condition" for fibrations and how descent data were Eilenberg-Moore algebras. Jean gave me a copy of his Comptes Rendu article with Jacques Roubaud.
Very soon after Jean Benabou left, Jon Beck arrived. He asked me what the various visitors had talked about. When I told him about Benabou's lecture on descent, he said that that was what he had planned to talk about ("triples" and descent). I encouraged him to do so but he decided to change his topic. His topic by the way was also very interesting: using monads -- sorry, triples -- in homotopy theory and categorical coherence. This was before operads!
I wondered what would happen to Beck's work on descent. Category theorists were not prolific publishers. Then I found reference to the "Beck condition" in Bill Lawvere's papers of the time: it was what Benabou had called the "Chevalley condition". So, when I had need for a 2-categorical version of this involving comma objects instead of pullbacks, I called it the "Beck-Chevalley condition". This 2-categorical version expresses pointwiseness of Kan extensions and embodies Lawvere's formula for such extensions.
Also by the way, Lawvere's comma categories are generalized slice constructions so I proposed (not really wishing to introduce new notation but somewhat worried about using (f, g) as more than just the pair) using f/g for functors f and g into the same category.
Now, as much as I would love SIX bottles of GOOD champagne, I am not going to submit a suggestion for Jean's challenge. Composition of fibrations is a wonderful thing as is composition of homomorphisms of bicategories; but they do different jobs. It is hard enough to say fibrations are composable from the homomorphism viewpoint!
There is a thing about this that requires a mixture of the two views. Regard one fibration p : E --> A as a homomorphism E_ : A --> Cat. Keep the other q : A -- > B as a fibration. Then the homomorphism corresponding to the composite q p is a generalized left Kan extension of E_ along q.
Ross