Dear Steve, My answer to your straight answer is "yes and no"; that is: First of all I suggest to avoid discussing the reasons why some mathematicians do not see the categorical definition of cartesian product as anything special: this discussion continues, stops, and comes back again for a half of century, and I don't think you and I can make any further progress in it. Next, the idea of topos as generalized space is very nice and important, but it is too far. Long before it one learns easier things, such as, say, adjoint functors - which do not have a non-categorical definition! And then, immediately, there are amazing applications, such as seeing the geometric realization of a simplicial set as an outcome of the universal property of Yoneda embedding (which I mentioned in my previous message). Category theory is not a religion, and if someone discovers tomorrow something better than category theory, I shall be happy to study it. But this has not happened yet, and at the moment category theory provides the unique way to unify mathematical theories - and if one has new ideas or constructions in algebra, geometry, or logic - they should be understood and presented categorically - not with "ad hoc justifications" (using your expression). Unfortunately the number of mathematicians that were/are either ignorant or careless about this, is much larger then the number of those who tried to clean things up. The result is a dangerously growing chaos in abstract mathematics - and we certainly do not want applied mathematics to contribute to this chaos by telling us science fiction stories about things like string theory mixed up with operads and higher-dimensional categories! A simple categorical theory of torsors (via monads and resulting cohomology), and Galois theory in general categories (I mean what I call Galois theory) do exist, and your example of C_2 can be described via any of them. I am one of many people who can explain this to you if you are really interested. With best regards, George ----- Original Message ----- From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk> To: <categories@mta.ca> Sent: Sunday, March 19, 2006 8:25 PM Subject: categories: Re: cracks and pots
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.
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