From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4573 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki and Categories Date: Sat, 13 Sep 2008 16:31:19 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020035 13904 80.91.229.2 (29 Apr 2009 15:47:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:15 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sat Sep 13 19:13:26 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 13 Sep 2008 19:13:26 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KedHZ-0005iJ-Oh for categories-list@mta.ca; Sat, 13 Sep 2008 19:07:57 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 43 Original-Lines: 56 Xref: news.gmane.org gmane.science.mathematics.categories:4573 Archived-At: Dear Colleagues, I think the first things to say about "Bourbaki and Categories" are: (a) It is very obvious that the invention of category theory was by far the greatest discovery of 20th century mathematics. (b) Bourbaki Tractate is another great event, of a very different kind of course, which will be a treasure for the Historians of next centuries. It shows how the members of a very leading group of a leading mathematical country were thinking in the middle of the same century (well, up to their internal disagreements; after all, Eilenberg and Grothendieck were also there at some point...). (c) Accordingly, Bourbaki Tractate is the best evidence showing how hard it was to understand (even and especially for such brilliant mathematicians!) that there is something even better that Cantor paradise. (d) Defining structures, Bourbaki makes very clear that morphisms are important (and some form of universal properties are important). But morphisms are NOT defined in general: it is simply a class of maps between structures of a given type closed under composition and having isomorphisms (which ARE defined) as its invertible members. And... every interested student will ask: if so, why not defining a category? Let me also add what is less important but still comes to my mind: (e) Bourbaki approach to structures has a hidden very primitive form of what was later discovered by topos theorists: in order to define a structure they need a 'scales of sets', which is build using finite products and power sets (no unions and no colimits of any kind!). (f) According to Walter Tholen's message, Karl Heinrich Hofmann says: "...it is truly surprising that the theory of categories (S. Eilenberg and S. Mac Lane, 1946) was almost demonstrably ignored as the mother of all structure theories. This was hardly sustainable in commutative algebra anymore...". Very true (except 1946), but it is much-much-much worse in homological algebra, where the absence of categories and functors (having a section called "Functoriality" though) in Bourbaki's presentation is most amazing. (g) A few days ago Tom Leinster has explained to us that "disinformation is *deliberate* false information, false information *intended* to mislead". Fine, but sometimes false information is created by ignorance so badly, that it sounds right to call it disinformation (Don't you agree, Tom?). And... look at http://en.wikipedia.org/wiki/Bourbaki : There is a section called "Criticism of the Bourbaki perspective", which, among other things, says: "The following is a list of some of the criticisms commonly made of the Bourbaki approach:^[13]..." (where [13] is a book of Pierre Cartier; I have not seen that book, and so I am not making any conclusions about it). The list has seven items with no category theory in it! George Janelidze