From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4566 Path: news.gmane.org!not-for-mail From: edubuc Newsgroups: gmane.science.mathematics.categories Subject: categories and disdain Date: Thu, 11 Sep 2008 19:01:57 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020032 13880 80.91.229.2 (29 Apr 2009 15:47:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:12 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Sep 12 11:44:39 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 11:44:39 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ke9nW-0005bE-Q3 for categories-list@mta.ca; Fri, 12 Sep 2008 11:38:58 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 36 Original-Lines: 73 Xref: news.gmane.org gmane.science.mathematics.categories:4566 Archived-At: this is about the recent thread "categories and functors" and "disdain for categories" Ten points: 1) It seems clear that E-M arrived to categories and functors by abstraction from the usual large categories of sets, groups, boolean algebras, modules, etc etc 2) It seems (less clearly) that Ehresman arrived to categories and functors by generalization from groupoids and morphisms of groupoids. 3) I agree with "It's been said before that the real insight of category theory --as something more general than groupoids, monoids, and posets-- is the notion of adjoint functors (including limits, etc)." Concerning this, I think that real breakthrough made by categories is the simple fact that they furnish the appropriate abstract structure to define the Bourbaki's concept of universal property. The fact that the singleton set is characterized by being a terminal object opens the way to characterize thousands of objects and constructions by being the terminal object in the appropriate category. Yoneda's lemma is the milestone. Everything is due to it. 4) I think the small-large business played no role at all in the rise of the concept of categories, neither in the rise of the disdain to them by many mathematicians. 5) "working" mathematicians were never afraid about paradoxes. In consequence, I think that phrases as "dare regarding these (perceived) monsters ...", "fear of the "illegitimately large" size", etc etc, are misleading and out of place. 6) Cantor did not "dare to think that there could be different levels of infinity", he discovered that they were different levels of infinity, and proceed to study this phenomena. This was not a bold action, he was just fascinated by the existence of different levels of infinity. He was not afraid of paradoxes either, he was very well aware of Russell paradox, but for him it was just another theorem. 7) It is often repeated that axiomatic set theory arise in order to eliminate paradoxes. False, axiomatic set theory arise in an attempt to understand Von Neumann accumulation process: Which were the axioms satisfied by the output of that process ? Answer: axiomatic set theory. 8) "In my experience, disdain for cat theory is due to papers with a very high density of unfamiliar names", I agree with this in the sense that this fact contributed to the rise of the disdain, but not as the single reason. I agree also with "Complicated sentences like this can be found in every fields. They are often the mark of a poor paper". It follows there must be other reasons (besides the abundance of poor papers in category theory, a fact that I found true) to explain the disdain. 9) A profound reason could be an instinctive opposition to real change in many people. The instinctive reaction against progress that may disrupt their own comfortable position. 10) The so self proclaimed "problem solvers" who disdain abstract theories often do not resolve any real problem. The just do "concrete nonsense". People who really solve true problems usually have a great respect for abstract theories. Of course, they are also many who just do "abstract nonsense" instead of contribute to the meaningful development of theories. eduardo dubuc