From: kramsay@aol.commangled (Keith Ramsay) Subject: Re: Query about Category Theory. Date: 19 Jun 2000 03:33:25 GMT Newsgroups: sci.math Summary: [missing] In article <8ihn4u$6uf$2@cantuc.canterbury.ac.nz>, mathwft@math.canterbury.ac.nz (Bill Taylor) writes: |Does anyone know of any cases where Category Theory (morphisms, functors & |all that) has helped solve an unsolved problem? I believe it helped Grothendieck and Deligne arrive at some of their ideas, like the ones used in Deligne's proof of the Weil conjecture. Everything done using �tale cohomology has a bit of categorical machinery running in the background. It helped me with a result on a family of elliptic curves (my half of a joint paper with Rajiv Gupta at UBC). As I worked on the problem, I realized that various of my constructions were actually simply pullbacks. There was a group action as well. Suppose you have a family E_t of elliptic curves, parametrized by t. In our key cases, the curves E_t all had extra automorphisms. If you started with some E_t and "followed it around" as t circled one of the points where the family degenerates, it might come back with an automorphism having been applied to it. Then there's a set of points on each one, and following the points as t varies, you get a (set of) curves (complex curves, i.e. Riemann surfaces). The automorphism causes some points to come around so that they're on the same component as one of the "other" points. There are in general fewer components than points being tracked; these points are sections of curves which may have multiple branches over the base (t). It was important to know the irreducible components of this collection of curves. I'd done some concrete calculations, but I was still finding it easy to slip up. Considered in concrete terms, I was finding it confusing to see how the group action played itself out, but a categorical lemma about what happens when you have a pullback diagram with a group acting on all of the objects (in a compatible way) helped out. It was a case where the usual category theorists' claim of having a useful higher level of abstraction proved to be true. This one fact was easiest for me to understand as a general fact about this situation, rather than as some observation about how these Riemann surfaces wrap around the branch points. Functors between categories "& all that" didn't play an obvious role, although I think it was of psychological help to know that (most of) what I was doing could be placed in the birational category, where up to birational equivalence the singularities aren't a problem (algebraic varieties are birationally equivalent to nonsingular ones). I suppose some of it could be interpreted as relating the category of varieties over the rationals with the category of varieties over the rationals with a G action on them, and the group action can itself be thought of as a functor. This was the paper which was rejected at the first journal we sent it to, on the advice of a referee who said we should state early on that n was an integer. Paragraph 1 described the problem in general. Paragraph 2 started by saying that throughout the paper, n would be an integer. :-( Keith Ramsay ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Query about Category Theory Date: 21 Jun 2000 15:19:53 GMT Newsgroups: sci.math In article <8ihn4u$6uf$2@cantuc.canterbury.ac.nz>, Bill Taylor wrote: >Does anyone know of any cases where Category Theory (morphisms, functors & all >that) has helped solve an unsolved problem? > >That is, a problem in some other branch of math, posed without reference to >categorical ideas, and previously unsolved, that was first solved via >Category T. > >I realize that CT provides a unifying framework for many seemingly disparate >ideas in math, and that is a fine thing of course; but I was just wondering >if it had this problem-solving capability. > >TIA for any helpful responses. This question about categories was forwarded to the category theory mailing list ... I append a reply. Briefly, the answer is that category theory has been used for solving *many* problems. A famous early example is the use of topos theory by Grothendieck and Deligne for solving the Weil conjectures... but by now abelian categories and derived categories are part of the working toolkit of anyone in homological algebra, and model categories are similarly part of the homotopy theorist's repertoire... because they help solve concrete problems! In physics, the rigorous construction of 3d TQFTs using monoidal categories is a nice example. ......................................................................... Date: Wed, 21 Jun 2000 15:23:33 +0100 From: Ronnie Brown To: categories@mta.ca Subject: categories: Re: A question on: I have a paper with Loday on calculating the third homotopy group of a suspension of a K(G,1) 51. (with J.-L. LODAY), ``Van Kampen theorems for diagrams of spaces'', {\em Topology} 26 (1987) 311-334. in terms of a new tensor product of non abelian groups. It would not be a sensible task to rewrite the paper without category theory (considered as a unifying principle, as a mode for efficient calculation in certain algebraic structures, and as a supplier of new algebraic structures). To go further back (and higher, of course) Grothendieck's extension of the Riemann-Roch Theorem uses category theory explicitly. In the 1950's people were trying to give algebraic proofs of this theorem, when AG came up with an algebraic proof of a vast generalisation. Then there is the categorical background to the proof of the Weil conjectures .... But the question is misplaced - in the early part of the 20th century, I expect some would ask if set theory was really necessary for a confirmed problem solver! The history of maths shows that maths greatest contribution to science, culture and technology has been in terms of expressive power, to give a language for intuitions which enables exact description, calculation, deduction. (Exam question: discuss the last statement, with an emphasis on particular examples!) It also allows for the *formulation* of new problems, which perhaps cause old interests to lapse as people perceive there are more exciting things to do. It is a narrow view to regard `important maths' as necessarily that which solves well known problems, and so leave evaluation as akin to a sports league table: how old is the problem? who has worked on it? etc, etc. The progress of maths is much more complicated and interesting than that! It won't help the public image of maths if it is seen that mathematicians believe the most important aspect of their subject is a (to the public) bizarre set of problems which seem to interest no one else. However, it is important that these questions be asked, together with questions on modes for evaluating `good maths'. As AG remarked in a letter, maths was held back for centuries for lack of the `trivial' concept of zero! For more discussion, look at http://www.bangor.ac.uk/ma/CPM/cdbooklet/knots-m.html knots2.html So let the debate be broadened and continued! Ronnie Brown (will someone please forward this to the newsgroup?) PS I am trying to trace a combinatorics problem solved in the 1950s using categories of paths, and which was at the time held up as the sort category theory could not do! Maybe my memory is failing! But the question put on the newsgroup is an old war horse! -------- Original Message -------- Subject: categories: A question on: Date: Mon, 19 Jun 2000 17:13:05 +1000 From: Ross Street To: "categories@mta.ca" I thought the CatNet would be interested in the following question which appeared on . The letter was pointed out to me by a colleague at Macquarie. --Ross ************************************************* From: mathwft@math.canterbury.ac.nz (Bill Taylor) Newsgroups: sci.math Subject: Query about Category Theory. Date: 18 Jun 2000 05:36:30 GMT Organization: Department of Mathematics and Statistics, University of Canterbury, Christchurch, NewZealand Does anyone know of any cases where Category Theory (morphisms, functors & all that) has helped solve an unsolved problem? That is, a problem in some other branch of math, posed without reference to categorical ideas, and previously unsolved, that was first solved via Category T. I realize that CT provides a unifying framework for many seemingly disparate ideas in math, and that is a fine thing of course; but I was just wondering if it had this problem-solving capability. TIA for any helpful responses. ---------------------------------------------------------------------------- --- Bill Taylor W.Taylor@math.canterbury.ac.nz