From: "Joyal, André" <joyal.andre@uqam.ca>
To: "Categories list" <categories@mta.ca>
Cc: "Eduardo J. Dubuc" <edubuc@dm.uba.ar>, <hoffnung@uottawa.ca>,
<dyetter@math.ksu.edu>, <baez@math.ucr.edu>
Subject: Not invariant but good
Date: Sat, 25 Sep 2010 00:01:57 -0400 [thread overview]
Message-ID: <E1Ozcfn-0000NW-Cl@mlist.mta.ca> (raw)
In-Reply-To: <E1OzIa7-0002CB-1L@mlist.mta.ca>
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,
André
PS: We should reserve the word "evil" to name things that really are.
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<P><FONT SIZE=2>Dear all,<BR>
<BR>
Very briefly.<BR>
<BR>
Many good things in mathematics are depending on the choice<BR>
of a representation which is not invariant under equivalences,<BR>
or under isomorphisms. Modern geometry would not exists<BR>
without coordinate systems. This is true also of algebra<BR>
and category theory. Algebraic structures are often described by<BR>
generators and relations. Homological algebra is using non-canonical<BR>
projective or injective resolutions. Choosing a base point may help<BR>
computing the fundamental group of a topological space.<BR>
Choosing a triangulation may help computing the homology groups.<BR>
Invariant notions are often constructed from notions which are not.<BR>
For example, the Euler characteristic of a space<BR>
is best explaned by using a triangulation.<BR>
<BR>
Another example from homotopy theory:<BR>
the notion of homotopy pullback square in a Quillen model category is<BR>
invariant under weak equivalences, but its definition depends on<BR>
the notion of pullback square which is not invariant under weak equivalences!<BR>
<BR>
Part of the art of mathematics is in constructing invariant notions<BR>
from non-invariant ones. We should recognize the usefulness and<BR>
importance of the latter. Please, let us not call them "evil"!<BR>
<BR>
Best,<BR>
André<BR>
<BR>
PS: We should reserve the word "evil" to name things that really are.<BR>
<BR>
<BR>
<BR>
<BR>
<BR>
</FONT>
</P>
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</HTML>
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[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next prev parent reply other threads:[~2010-09-25 4:01 UTC|newest]
Thread overview: 23+ messages / expand[flat|nested] mbox.gz Atom feed top
2010-09-24 15:44 subculture Eduardo J. Dubuc
2010-09-25 0:38 ` subculture Ruadhai
2010-09-25 23:10 ` RE : categories: subculture Joyal, André
2010-09-26 2:43 ` subculture David Leduc
2010-09-26 3:19 ` subculture Fred Linton
[not found] ` <AANLkTikJoHkO2M_3hnrQqqFq2_N2T9i6KF2DRFbHTujP@mail.gmail.com>
2010-09-26 3:43 ` subculture Eduardo J. Dubuc
2010-09-25 4:01 ` Joyal, André [this message]
[not found] ` <B3C24EA955FF0C4EA14658997CD3E25E370F59BE@CAHIER.gst.uqam.ca>
2010-09-26 3:29 ` Not invariant but good John Baez
2010-09-27 2:54 ` Peter Selinger
2010-09-27 15:55 ` RE : categories: " Joyal, André
2010-09-28 2:10 ` RE : " John Baez
2010-09-29 18:05 ` no joke Joyal, André
2010-09-30 2:53 ` John Baez
2010-09-28 10:18 ` RE : categories: Re: Not invariant but good Thomas Streicher
2010-09-29 21:25 ` Michael Shulman
2010-09-30 3:07 ` Richard Garner
2010-09-30 11:11 ` Thomas Streicher
2010-09-30 19:39 ` Michael Shulman
2010-09-30 11:34 ` Thomas Streicher
[not found] ` <20101001092434.GA9359@mathematik.tu-darmstadt.de>
2010-10-03 22:10 ` Michael Shulman
2010-09-27 5:36 John Baez
2010-09-28 23:11 ` Michael Shulman
2010-10-01 12:36 Thomas Streicher
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