From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3141 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: cracks and pots Date: Sun, 19 Mar 2006 18:25:08 +0000 Message-ID: <9CC67FE4-AAAD-4ABD-90BB-06D5BA30CAF2@cs.bham.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v746.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019117 7410 80.91.229.2 (29 Apr 2009 15:31:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:31:57 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Wed Mar 22 23:00:45 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Mar 2006 23:00:45 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMG3K-0004vk-I5 for categories-list@mta.ca; Wed, 22 Mar 2006 22:59:58 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 87 Original-Lines: 83 Xref: news.gmane.org gmane.science.mathematics.categories:3141 Archived-At: On 17 Mar 2006, at 09:36, George Janelidze wrote: > ... I think if we really care about > relations between category theory and "other foundational > disciplines", we > should begin by explaining that category theory is not just a language > allowing one to call homology a functor, but that category theory has > beautiful constructions and results (some already from 1940s and 50s!) > making enormous simplifications/applications/illuminations in > neighbour > areas of pure mathematics, such as abstract algebra, geometry, and > logic. Dear George, I think the straight answer is that it is genuinely difficult. Even for elementary applications it is not easy. Try asking non- categorical topologists how they explain the product topology to students. Many will say, "This definition may look odd, but it turns out to work best." Others will produce various ad hoc justifications, such as "It's the definition that makes Tychonoff's theorem true." (Though that may be at least historically correct.) You point out that the product topology is the unique one such that projections are continuous and tupling preserves continuity, but they still don't see that as anything special. But with regard to certain advanced applications, there are pictures in the minds of the category theorists that do not translate at all easily to paper. Even the master expositors find it hard. I'm thinking for example of the idea of topos as generalized space. I have been working seriously with toposes (usually as generalized spaces) for about 15 years now and in some respects my understanding of them is quite deep. Yet there is still a huge gap in my understanding when it comes to their applications in algebraic geometry, Galois theory and algebraic topology, the kind of fields that gave rise to toposes in the first place. Somehow when I read the accounts I see a mass of machinery but no clear intuitions for what it is doing. This surprises me. A characteristic strength of category theory is that it is particularly good at explaining the underlying meaning of constructions, with its notion of universal properties, and with some beautiful tricks of categorical logic. So is it possible to explain, or illuminate, those particular categorical applications to someone like me? (Perhaps the challenge has already been met, and I've just missed the right book; and of course I eagerly await vol. 3 of the Elephant.) Here's a sample question where my categorical understanding falls short of the applications. If A is an Abelian group, then the space ^A of A-torsors is also a group (modulo canonical isomorphisms - the equational laws of group theory do not necessarily hold up to equality). The identity element is the regular representation of A on itself, group multiplication is "tensor product" of torsors, and inverses are got by inverting the A- action. It follows that if X is any space, then the collection of continuous maps from X to ^A is also a group, and this construction is self- evidently contravariant in X. Obviously it takes topos theory to formalize this, but already we can paint a picture. For example, suppose A is the cyclic group C_2 of order 2 and X is the circle. Then there are (classically) two isomorphism classes of maps T: X -> ^A, essentially because in going once right round the circle the variable torsor T(x) can come back either just how it started or with an automorphism swapping its two elements. The corresponding group is C_2. This looks like some kind of cohomology, so is it already part of the standard theory? I've never managed to follow all the machinery through. All the best, Steve Vickers