Re: Defense: Problems with "canonical"

On Oct 14, 2:45 pm, "Christian Reinbothe" <Christian.Reinbo...@t-
online.de> wrote:
----- Original Message -----
From: "Christian Reinbothe" <Christian.Reinbo...@xxxxxxxxxxx>

1. I cannot define the term "canonical base of Rx...xR".
I need a mathematical correct definition of the term "canonical".

2. In the book "Kategorien" of S. MacLane (ISBN 3-540-05634-3)
can be found several definitions of "canonical ...", but not a
definition of "canonical base of Rx..xR".
(As far as I understand it.)


There *is* no canonical basis of Rx..xR in the technical sense.
Intuitively, as kea@xxxxxxxxxxx mentioned in reply to your earlier
post, there's a *standard* basis, but the standard basis isn't
preserved by isomorphisms and therefore isn't distinguished by the
vector space structure.

3. In further literature there is the term "__THE__ canonical
isomorphism V -> V^**". In 4.3. of the above mentioned
preprints the term is not usable, because you get
__TWO__ "canonical" isomorphisms V -> V^**.


I don't have MacLane at hand, and can't comment on the specific
passage you cite. Certainly there exist "non-canonical" isomorphisms V
-> V^**. However, for each finite-dimensional V, there exists a
"distinguished" isomorphism V -> V^** ("map each element v to its
evaluation functional on V^*"). This family of isomorphisms (one for
each finite-dimensional vector space) makes a certain family of
diagrams commute, and that's the definition of "canonical".
Intuitively, these isomorphisms do not depend on a choice of basis,
but are distinguished by the vector space structure. (The "canonical
isomorphism" maps the *category* of finite-dimensional spaces to
itself, but one abuses language and calls specific isomorphisms
"canonical".)

By the way, in your manuscript (assuming it hasn't changed since last
week) you attempt to show the neutral element of a group isn't
"canonical" by defining a different group structure on the set G of
elements underlying a group. In the process you change the group under
consideration: The new binary operation is not generally the original
operation of G, and the new group has a different neutral element. As
kea points out, the neutral element of a group is *uniquely defined*
by the group structure.

Hope that's helpful,

Andrew D. Hwang
Dept of Mathematics and CS
College of the Holy Cross
Worcester, MA, 01610-2395, USA
(Posting address is invalid)

.