From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3958 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Help! Date: Sun, 7 Oct 2007 10:09:19 +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 1241019627 11087 80.91.229.2 (29 Apr 2009 15:40:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:27 +0000 (UTC) To: "Categories list" Original-X-From: rrosebru@mta.ca Sun Oct 7 10:53:14 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Oct 2007 10:53:14 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeWUo-0000iP-SA for categories-list@mta.ca; Sun, 07 Oct 2007 10:48:38 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 15 Original-Lines: 109 Xref: news.gmane.org gmane.science.mathematics.categories:3958 Archived-At: Dear Michael, The question is so impossibly big, and it was asked and answered in various form so many times, and no responsibility for the originality/completeness of the answer can be assumed... So, I am not afraid to begin with a few obvious remarks, looking forward to seeing many other remarks from others: 1. Many people believe that mathematics is about mathematical structures, but what is a mathematical structure in general? According to Bourbaki, one should begin with two finite sequences of sets, say, A, B, C,... and X, Y, Z,...; let us call them constant sets and variable sets respectively. Then build any scale, which is a finite sequence of sets obtained by taking finite products and power sets of the sets above. Then, (briefly and roughly!) call a structure (of a fixed type) an element of one of the sets in the scale satisfying certain conditions. For example: (a) a topological space (defined via open sets) has no constant sets, one variable set X, and its structure is an element t in PP(X) that is closed under finite intersections and arbitrary unions. (b) a vector space has one constant set A ("the set of scalars"), one variable set X ("the set of vectors"), and its structure can be defined as element (a,b,c,d) of P(AxAxA)xP(AxAxA)xP(AxXxX)P(XxXxX), where a, b, c, d are addition of scalars, multiplication of scalars, scalar (-on-vector) multiplication, and addition of vectors respectively; that (a,b,c,d) should satisfy familiar conditions of course. Then, according to Bourbaki again, structures are useless without morphisms - but what is a morphism? It turns out that only isomorphisms can be defined, and the class of morphisms should in each case be CHOSEN depending on the "experience" of working with a given class of structures in such a way, that it is closed under composition and contains all isomorphisms (or, better, also determine isomorphisms as invertible morphisms). Is it possible that the most fundamental concept of mathematics is described as such a monster? Is not it better to study abstract categories? And what is the problem of defining morphisms? To answer this question one should learn about functors, covariant and contravariant ones, and what do they preserve and what not. 2. Set theory is wonderful: it gives precise mathematical definitions to concepts that were only intuitively understood before. But... it often makes definitions complicated. And category theory very often solves this problem by using universal properties. For example the set N - better to say, the structure (N,0,s) - of natural numbers is "designed to count"; therefore N should have the first element 0 (unfortunately 0 is better than 1) and the successor function s from N to N - and be "the best such", i.e. initial such. Moreover, developing basic properties of this structure out of initiality is much easier than out of Peano axioms. In fact all classical number systems have simple elegant definitions via universal properties. Moreover, the axioms of set theory itself are much less elegant than their elementary-topos-theoretic counterparts. 3. We can say that set theory is more fundamental than arithmetics: e.g. children learn addition by counting the number of elements in the disjoint union. But category theory is more fundamental than set theory: e.g. it makes the disjoint union a more natural operation... but, more importantly (a) we all know that, say, a+b=b+a, ab=ba, PvQ<=>QvP,... for all natural numbers a and b and all logical formulas P and Q - but one needs category theory to see these as the same result (and we can add cartesian products, free product, intersection, union, and many other operations to it). (b) or, say, we all know that composites of injections are injections and composites of surjections are surjections - but again, one needs category theory to see these as the same result; (c) and even exponentiation and implication are the same... and categorical logic follows... 4. Linear algebra tells us that instead of working with linear transformations of finite-dimensional vector spaces we can work with matrices, but one cannot formulate this properly without using the concept of equivalence of categories (the category of finite-dimensional K-vector spaces is equivalent to the category of natural numbers with matrices with entries from K as morphisms). And there are so many other equivalences and dualities (that are not isomorphisms) playing fundamental roles in various branches of mathematics. Not to mention that the aforementioned matrices themselves arise from a categorical observation (finite products = finite coproducts). 5. Proper understanding of Eilenberg - Mac Lane work, and the work of their followers, friends, and not-quite-friends in category theory and proper understanding of what the 21st century mathematics would be without it obviously requires far better knowledge of mathematics than the 21st century students have. But may be we should at least say that our Fields Medal (Grothendieck) is not less than any other Fields Medal... George Janelidze ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Friday, October 05, 2007 2:52 PM Subject: categories: Help! > What would you say to an undergraduate math club about categories? I have > been thinking about it, but I am not sure what to say. Talk about > cohomology, which is what motivated E-M? I don't think so. Talk about > dual spaces of finite-dimensional vector spaces? Maybe, but then what? > > Michael