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. ------_=_NextPart_001_01CB5C66.6607A3FA Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2//EN"> <HTML> <HEAD> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <META NAME="Generator" CONTENT="MS Exchange Server version 6.5.7654.12"> <TITLE>Not invariant but good</TITLE> </HEAD> <BODY> <!-- Converted from text/plain format --> <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> </BODY> </HTML> ------_=_NextPart_001_01CB5C66.6607A3FA-- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]