Descent for functors to come

December 7, 2010

I am resuming these days my work on descent theory. In August/September I was thinking on issues related to the descent for
equivariant functors between categories where each category is itself glued from pieces. Equivariant in the sense of coactions of comonads or a similar formalism. One of the motivations is study of principal bundles over noncommutative schemes. The Cech cocycles are a bit tricky here in full generality, there are phenomena which do not exist or do not matter in commutative context.

I will meet these days Gabi Bohm from Budapest to talk about such issues; she is well versed in comonads and related issues. This will be also excuse to post here some standard and not so standard background from selected parts of descent theory. This post will grow in few days. Keep tuned :)

books buying, selling, publishing

July 14, 2010

Talking about scientific publishing is a large topic, it includes the problems like the expensive and raising journal prices, voluntary copyright transfer to the publisher and our free service on editorial boards and as reviewers, quality of journals, low reliability of various impact factors and so on. But today I will not talk the journals but rather books. I will start with a shortened personal story but then go into more hot topic of some contemporary disorientation of publishers and bookstore in publishing and selling good books.

As a child I did not have much access to the books in science although I was interested in science; my first advanced math books I bought with my father at about age of 16 in some second-hand bookstore; quickly after I was reading Postnikov’s volumes on geometry and algebraic topology and started buying books massively; In a way I would make myself happier by indulging into choosing and buying the books; those books in Russian were cheap at the time but the problem was that you had available only what was published very recently. For example, I would get the second volume of Penrose-Rindler’s book on spinor geometry but not the first as the first was translated to Russian earlier and was not available hence any more. One of the secrets of the cheap Russian books was that the publisher did not make too many copies and finance their staying for years in the stocks. The whole stock would be sold very quickly. I was told by Russians that in Moscow people would go to exhibitions like Soviet Exhibition where they would get books which were not available often in bookstore; while cheap, most of the books were rare to find.

Of course, after years, it became a problem that I could not fit the books in my room and other places and had to stock them in boxes and so on. Then also travel came, my graduate school in Wisconsin and my interests involved, new books were coming, at much higher price and lower pace. And then I got in a way saturated. I had a nice personal collection (parts of it were lost though in travels, movings and so on), but was often accessing good libraries and kind of got used that I can have what I need most of the time, unlike in my early history. One of the reasons of the saturation is that I had to narrow my main interests to professional ones, and read less and less other subjects like linguistics, and in my own field there are few surprises unnoticed: we know in advance that somebody is writing a major book so coming to a bookstore will rarely raise great interest. I became a book-quality sceptic: books are either known to me, or bad or out of my interests.

Now after nearly 10 years of not buying books much, and even not being anymore fond of entering bookstores, I felt some revival of my book apetite in recent weeks, and made some spontaneous excursions into bookstores. But now I travel less and bookstores which I specially liked like the one of Cambridge University Press in Cambridge, like the former foreign bookstore in Zagreb in Gundulićeva street, the University Bookstore in Wisconsin, one impressive bookstore (I do not recall the name) in Barcelona and so on are far from my reach timewise or spacewise.

The new generation buys books online and does not bother browsing. the choice is bigger, and often there are online excerpts. But I really get the feeling whether I like the book mostly only if I browse it in my own hands. Online I often get wrong impression on proportions and feeling of the style and content. So I would still like to have good bookstores.

In Zagreb, now you have very little choice, the only reasonable collection of foreign titles is the Algoritam. Of course you can order anything but lets focus on the browsing feel and real competent choosing by sitting and browsing within the bookstore.

Well, the collection has few meters of math, physics and computer science titles at all levels mixed (“mixed” is here the bad thing, though I am sufficiently experienced to find my way through wrong targeted parts of the stock). But you know Zagreb is not a big market for scientific books and the bookstores should sell the books which are of sufficiently broad interest. For example, proceedings volume of a conference, or extremely specialized topics are unlikely to find a buyer. So what happens is that such books stay in the pool, and of course the bookstore can not afford spacewise and timewise to keep so many books unsold for long time, so once the crap takes over the shelves, it is hard to replace it by more reasonable titles. People tell me that a reason is that some people order books and then decide not to buy them, so the bad books enter the bookstore unplanned. But I see many bad choices in more than one copy, so the books were really ordered not by such an error but otherwise.

So if you look through the Algoritam bookstore, math section, you see that the they did not choose famous and widely sold books, and series but chosen some random books from random publishers. For example you have some expensive PDE textbook written by some local experts in Beijing but you do not have the most famous textbooks on PDEs of Gilbarg and Trudinger or the one from Evans. In fact there are no books published by American Mathematical Society which is a very good, modern and reasonably cheap publisher. There are some Indian reprints of books which I know are unlawful by Indian law to be sold out of India, but in Algoritam they say that they bought it from a regular supplier. Strange that they learned of a strange supplier of strange reprints in India but do not know that AMS publishes good many quality books in the field. The Springer’s Yellow series is underrepresented and Algoritam does not seem to maintain the action of the Yellow sale, which is traditionally quite an event for math book fans.

I was also disappointed into finding so many new editions of outdated books. For example some sort of Oxford companion of philosophy of mathematics, talking so much of 19th century and earlier metaphysical thinking of nominalism and alike notions, while have no hint of the thrills brought by modern foundational, semantic and other developments from schools of Lawvere, or Grothendieck and so on, who changed mathematics so much.

There are of course, also the dedicated reprint series. Like the Dover. Well, I think that youngsters should be warned that Dover series is outdated in large. Not in the way it was before. There are still great books there, like Goldblatt’s Topoi, or Abrikosov, Gоr’коv, Dzyaloshinskii Quantum field theoretical methods in statistical physics , ever quoted Weyl’s Classical groups and so on. But Dover also published reprints of many minor authors; some books of historical type without copyright which can be found online for extremely rare users (for example Chandrasekhar’s Mathematical theory of black holes is a masterpiece, but not good for a contemporary student; it is choice of exact and detailed calculations which a very rare specialist will look apart from a rare consultation in a library). But you know, a student sees a book which is reasonably well written about a subject which is not known to her/him. And after reading few nicely written paragraphs will decide to buy an obsolete book. I mean something what looks readable may be suboptimal from today’s point of view, notation, conventions, language, knowledge and after many new discovered powerful shortcuts or stronger results which are now standard. So, the student may subjectively feel that he gained a great insight, while the learnt material is not as powerful as what modern introductions offer or is not as organized as a contemporary colleagues would expect.

quantization

March 19, 2010

Beck’s theorem vs. Benabou-Roubaud

January 7, 2010

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)
link
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
the assertion

“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.

starting

December 22, 2009

Hopefully, I will be starting soon. For now I just posted an entry containing my response to the historical discussion on the theorems of Beck and Benabou-Roubaud. For now I am more absorbed by nlab. My personal nlab pages can be found here and my web page at work is here.

Hello world!

August 31, 2009

Welcome to WordPress.com. This is your first post. Edit or delete it and start blogging!

Unfortunately, I dislike the very words blog and blogging.


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