Late Jon Beck has made some profound contributions to category theory, descent theory and homological algebra, mainly via studying monads, monadic cohomology and monadicity. Unfortunately some of his early manuscripts were lost and some of his results might have been lost in their original form and reappeared later from folklore or being rediscovered. Some times ago at category list there was a discussion (which got heated at the moment) about the results of Jean Beneabou’s school of descent theory in comparison with the Beck’s results. In Canada and US, the Beck’s environment, indexed categories of R. Paré and D. Schumacher, which are close to the approach via pseudofunctors from the original FGA of Grothendieck, were used. French school followed Grothendieck-Gabriel’s fibered categories instead. The comparison of the descent in bifibered categories with monadic descent has been first done in Benabou-Roubaud’s paper; moreover certain Beck-Chevalley property has been utilized. In category list discussion, it seems that some contibutors prefer to attribute all to Beck’s unpublished works only, and not taking seriously that fibered category formalism is nontrivially related and not the same generality as the monadic aprpoach of Beck. Here is my response to category list which has been disgarded by the moderator because the discussion has been already closed when I posted the comment:
Prof. Benabou wrote:
Sorry Ross, the “indexed categories” are not Pare-Schumacher, and not Lawvere. They are, as I mentioned to Barr, also due to the “fertile brain” of “you know who”.
Later he quotes 1962 as the year and mentiones SGA1.6 (what corresponds to the seminars 1960-61).
However, Grothendieck introduced pseudofunctors into descent theory a bit earlier.
The first published Grothendieck’s text dealing with subject has only pseudofunctors and NOT the fibered categories:
it is in those Bourbaki seminars, which are LATER collected as FGA (Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957--1962.])
The descent chapters of Bourbaki seminars are from 1959 and involve pseudofunctors (indexed category point of view).
Later, in SGA1 (corresponding to 1960-1961), the chapter 6 on fibrations and descent has been written by Pierre/Peter Gabriel and introduces fibered categories (proper) as the basic setup for the first time. As far as I heard from algebaric geometers indeed there was in reality some
input from Gabriel, not only Grothendieck, when rephrasing in a better language of fibered categories what is just partly supported by the footnote to Ch. 6 that the chapter is written down by Gabriel (the clarity of Gabriel’s style, parellel to the style in Gabriel-Zisman book, is quite recognizable!).
Thus in any case, a variant of indexed categories I mean pseudofunctors were under a different name introduced by Grothendieck, not by Pare and Schumacher: and for fibered categories Grothendieck was the one who introduced
the philosophy there of trading structure to property, but some (unclear in extent) part in making the transition
was played by Gabriel as well. This is the story known to most of people adhering to Grothendieck school and
I am surprised that the category list has that much confusion over it (including never mentioning Gabriel in this context).
As far as Benabou-Roubaud’s paper I indeed enjoyed the posts by Bunge and Benabou explaining the (sometimes subtle) differences from the known work of Beck (which fit into my earlier impressions).
I have retyped few years ago the C.R. paper into LaTeX, translated into English; though I should still recheck for typoses. I intended to post it to the arXiv as it is short and historically and pedagogically important while unavailable online and I hereto ask Prof. Benabou for permission (after rechecking the file together for correctness).
[I should remark that SGA1 is also on the arXiv, since 2002.]
It would be interesting if the discussion on Beck’s theorem and on Benabou-Roubaud’s theorem in this list would extend here to the case of higher categories. Lurie’s second volume
Derived algebraic geometry II: noncommutative algebra arXiv:0702299 has the Barr-Beck theorem for (infty,1)-categories. I recall discussing with A. Rosenberg and M. Kontsevich in 2004 extensively on the need for a Beck theorem in the setup of A-infty categories,
and tried to work on it but insufficiently to obtain it. My motivation was to explain certain version of noncommutative flag variety and they had another version long before needing
similar methods. In both cases Cohn universal localization was involved and we took a look of Neeman-Ranicki papers on usage of localization theory in algebraic K-theory implicitly involving higher Massey products in understanding the derived picture coming from Cohn’s localization. Roughly speaking, they understand Cohn’s localization, as H^0 following certain Bousfield localization at the level of chain complexes (earlier found by Vogel); extending this to comonads coming from a cover by Cohn localizations would give a comonad in the triangulated setup, andone wants to understand the quasicoherent sheaves for our examples obtained by gluing Cohn localizations by means of some Beck’s theorem in that setup). On July 16, 2004 MK and AR told me that they have just the day before
solved the problem, with a remark that they did not need to discuss “covers”, but I have not seen the A-infty version ever in print (some applications were given in a talk by MK at a conference in honour of van den Bergh’s birthday though; including getting usual formal schemes as noncommutative schemes via gluing along derived epimorphisms).
But Sasha Rosenberg has posted his lectures on the Max Planck site having Beck’s theorem in triangulated setup,
A. L. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57 (be careful: 105 pages)
page 36-37 (triangulated Beck’s theorem)
which contains a version of the Kontsevich-Rosenberg result in the triangulated setup: the proof uses the Verdier’s abelianization functor. It would be interesting to compare the result to Lurie’s result somehow, as well as to try to see somehow the Street’s orientals in the picture explicitly.
Side remark: has anybody ever stated exactly or even proved that any sensible notion of pseudo-n-limit for n greater than 2 would be represented by “n-categories of n-descent data” where the latter are constructed via Street’s orientals? in other words, I do not consider that
“n-descent object can be represented as the n-category of n-descent data”
is a tautology, but rather a conjecture worth working on
(even n=2 case of the assertion can not be found in print);
second in the Lurie’s framework can one formulate (similar to the descent object) the universal property of the appropriate Eilenberg-Moore category in a manner
parallel to the universal property of the descent object in 1-categorical situation??). In other words, one should see how for general n the orientals help to satisfy some
n-categorical universal property (the latter subject for n greater than 2 being to my inspection almost void in the literature anyway). At least for strict n-categories…
[[ Furthermore, what to do with descent data, orientals, nonabelian cocycles in other (usual) categories -- I mean, the combinatorics of nonabelian cocycles in Hopf algebra theory for example (e.g. Drinfeld twist, associator: the factor systems for cleft extensions of Hopf algebras etc.) seems to some extent superimposable to the one in group cohomology but still different and I do not know if orientals explain such things (cf. the bialgebra nonabelian cohomology in
Shahn Majid: Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras. Generalized symmetries in physics (Clausthal, 1993), 13--41, World Sci. Publ., River Edge, NJ, 1994. arXiv: hep-th/9311184 ]]
In 2-categorical situation, the 2-fibered categories are defined in Gray’s work and then again introduced and discussed at length by Claudio Hermida, who has good ideas on higher n (and I will be trilled to hear once that he found the time to return to the topic and give us good answers). An appendix in
Claudio Herminda: Descent on 2-fibrations and strongly 2-regular 2-categories. (English summary) Appl. Categ. Structures 12 (2004), no. 5-6, 427–459.
is discussing a 2-categorical version of Chevalley condition (truly, the subtle historical and terminological differences between B-R vs Beck contributions for 1d case are not made there). Beck’s theorem (proper) for pseudomonads is in the paper
Le Creurer, I. J.; Marmolejo, F.; Vitale, E. M.
Beck’s theorem for pseudo-monads.
J. Pure Appl. Algebra 173 (2002), no. 3, 293–313.