From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6224 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: Not invariant but good Date: Sat, 25 Sep 2010 00:01:57 -0400 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1285451633 15593 80.91.229.12 (25 Sep 2010 21:53:53 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 25 Sep 2010 21:53:53 +0000 (UTC) Cc: "Eduardo J. Dubuc" , , , To: "Categories list" Original-X-From: majordomo@mlist.mta.ca Sat Sep 25 23:53:51 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Ozcgn-00021C-4g for gsmc-categories@m.gmane.org; Sat, 25 Sep 2010 23:53:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39068) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Ozcfq-0004kt-97; Sat, 25 Sep 2010 18:52:50 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ozcfn-0000NW-Cl for categories-list@mlist.mta.ca; Sat, 25 Sep 2010 18:52:47 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6224 Archived-At: Dear all, Very briefly. Many good things in mathematics are depending on the choice=20 of a representation which is not invariant under equivalences, or under isomorphisms. Modern geometry would not exists=20 without coordinate systems. This is true also of algebra and category theory. Algebraic structures are often described by=20 generators and relations. Homological algebra is using non-canonical projective or injective resolutions. Choosing a base point may help=20 computing the fundamental group of a topological space. Choosing a triangulation may help computing the homology groups. Invariant notions are often constructed from notions which are not. For example, the Euler characteristic of a space is best explaned by using a triangulation. Another example from homotopy theory:=20 the notion of homotopy pullback square in a Quillen model category is=20 invariant under weak equivalences, but its definition depends on=20 the notion of pullback square which is not invariant under weak = equivalences! Part of the art of mathematics is in constructing invariant notions=20 from non-invariant ones. We should recognize the usefulness and importance of the latter. Please, let us not call them "evil"! Best, Andr=E9 PS: We should reserve the word "evil" to name things that really are. ------_=_NextPart_001_01CB5C66.6607A3FA Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Not invariant but good

Dear all,

Very briefly.

Many good things in mathematics are depending on the choice
of a representation which is not invariant under equivalences,
or under isomorphisms. Modern geometry would not exists
without coordinate systems. This is true also of algebra
and category theory. Algebraic structures are often described by
generators and relations. Homological algebra is using non-canonical
projective or injective resolutions. Choosing a base point may help
computing the fundamental group of a topological space.
Choosing a triangulation may help computing the homology groups.
Invariant notions are often constructed from notions which are not.
For example, the Euler characteristic of a space
is best explaned by using a triangulation.

Another example from homotopy theory:
the notion of homotopy pullback square in a Quillen model category = is
invariant under weak equivalences, but its definition depends on
the notion of pullback square which is not invariant under weak = equivalences!

Part of the art of mathematics is in constructing invariant notions
from non-invariant ones. We should recognize the usefulness and
importance of the latter. Please, let us not call them = "evil"!

Best,
Andr=E9

PS: We should reserve the word "evil" to name things that = really are.

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