Subject: categories: Juergen's question

From: Michael Barr <[email-masked]>
Date: Thu, 21 Jan 1999 15:20:59 -0500 (EST)

I want to comment on Juergen's question.  It is possible, I suppose,
that Grothendieck used the phrase "dualizing object" somewhere, but as
far as I am aware, he never did anything with them.  It is not, after
all, a difficult concept.  He certainly never talked about "*-autonomous
categories", although Grothendieck (and everyone else who ever gave it a
thought) was surely aware that finite dimensional vectors spaces and
finite abelian groups were such.  If he ever isolated the concept as an
interesting one, I am unaware of it and, in any case, I don't believe he
ever pursued it (someone would surely have let me know by now).  This
mad insistence on giving Grothendieck credit for every fleeting idea he
may (or even may not) have mentioned somewhere is a perfect example of
how the star system (no pun intended) has permeated our consciousness.
I am not, of course, blaming Grothendieck for any of this.  In another
instance, one of the best ideas I ever had has been named after Euler,
who never heard of cohomology groups.

Now, as I have said elsewhere, I was very much aware of the "pairs" of
vector spaces used by the topological vector space theorists when I
created the Chu construction.  I believe that the sources I had seen
stuck to the separated extensional case.  They did not mention that this
construction was originally due to Mackey, although I eventually (fairly
recently) tracked it down.  It was in his PhD thesis, in fact.  I
believe he did not stick to the separated extensional case.  But neither
he, nor anyone else I read talked about morphisms of pairs.  In the se
case, the morphisms are obvious.  In the general case a bit less so.
The duality was, of course, obvious and the main raison d'etre for the
pairs.  But, although it was obvious how to make the morphisms between
two pairs into a vector space, no one seems to have even raised the
question of making it into a pair.  That it is possible and even easly
struck me--and still strikes me--as an amazing bit of magic.

Michael

-------------------------------------------------------------------
History shows that the human mind, fed by constant accessions of
knowledge, periodically grows too large for its theoretical coverings, and
bursts them asunder to appear in new habiliments, as the feeding and
growing grub, at intervals, casts its too narrow skin and assumes
another... Truly the imago state of Man seems to be terribly distant, but
every moult is a step gained. 

- Charles Darwin, from "The Origin of Species"

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From: Dusko Pavlovic <[email-masked]>
Date: Mon, 25 Jan 1999 00:11:44 -0800

> how the star system (no pun intended) has permeated our consciousness.

I may have misunderstood, but I didn't think anyone even implied that
Grothendieck could be credited with *-autonomous categories. Juergen was just
asking about the history of the idea of dualizing object, which is just a part
of that structure, and certainly predates it.

The fact that an idea may have been in the air before it was captured in a
structure does not have to decrease the merit of capturing it; on the
contrary, it may also be thought of as a sign that it was an important idea,
or that capturing it wasn't easy. The fact that Wiles was drawing upon a rich
source of ideas does not devaluate his victory.

> who never heard of cohomology groups.

This is a remarkable phenomenon, isn't it? Cartesius also knew nothing of
Cartesian categories (or squares, or arrows...), and Frobenius could hardly
recognize the logical form of his reciprocity...

I think Etruscans had this religion, where they systematically attributed all
victories to the ancestors, so that the soldiers wouldn't take things too
personally.

With kind regards,
-- Dusko