Bourbaki and category theory again

["George Janelidze" <janelg@telkomsa.net>]

The topic "Bourbaki and category theory" has several interesting aspects,
and I shall write at least one more message about that. At the moment, under
the permission of Francis Borceux, I would like to show one of his messages
to me written a few days ago:

"...I had a look at

Nicolas Bourbaki
Elements de mathematiques
Theorie des ensembles
Chapitre 4
Second edition
Hermann 1966

just to refresh my mind. The first edition of this book had been published
in 1957.

First of all an obvious remark.

The work of Bourbaki is by no means a "one shot work". It took many decades
to produce it. Many books and chapters have had successive editions,
sometimes very different from the previous ones. A new edition was generally
reflecting all the progresses and changes of point of view made since the
previous edition. And if generally a new edition is "Revised and augmented",
the second edition of the book mentioned above is at once presented as
"Revised and reduced...". I am sure that many people would be able to
comment infinitely on this peculiar aspect.

The point of Bourbaki in Section 1 of this Chapter 4 is to define what a
mathematical structure is ("une espece de structure"). This is done in a
very general setting, which includes most mathematical structures you can
think of, whatever their nature: algebraic, topological, ordered,
categorical, and so on. It is in this first section that the notion of
"isomorphism" is investigated.

Section 2 of Chapter 4 is devoted to the notion of "morphism" for such a
mathematical structure (this Section is called "Morphismes et structures
derivees" ... thus even the title of the section refers explicitly to the
notion of "morphism"). The notions of "initial structure" and "final
structure", so popular in categorical topology, are at once investigated.
The notion of product is presented as a special case of an initial
structure.

Section 3 is then devoted to the general "universal problem" when comparing
two mathematical structures. The "existence theorem" is stated and proved in
Subsection 3.2 in terms of the existence of products and a "solution set
condition".

So what?

Well, category theory intends to study objects and morphisms, without any
reference to a mathematical structure which would have given rise to them.
And this proved to be the very natural and efficient context where to
introduce the notions of limit and adjoint functors and prove the basic
theorems about them.

But indeed, the "adjoint functor theorem" is essentially present in
Bourbaki, even if the abstract notions of "category" and "functor" never
appear.? The theorem is proved in the case of "two mathematical structures"
in the very general sense defined by Bourbaki.

The great merit of Peter Freyd has been to put the "universal problem for
two structures" in the elegant context of categories and adjoint functors.
And to have given a corresponding elegant proof of the "adjoint functor
theorem". This has made the question fully transparent, in opposition to the
heavy technicalities found in Bourbaki.

Now was Peter Freyd aware of the result of Bourbaki? Probably, since in
those days Mac Lane and Eilenberg had regular contacts with the Bourbaki
group. But there is no shame at all -- just merit -- to generalize an
existing result, especially to put it in its "right context". But why to
care about these questions of priorities? Everybody knows the Fermat theorem
... but did he really prove it?

As a matter of comparison, also the "nine lemma" and the "snake lemma" did
exist before the invention of abelian categories. But abelian categories
provided a beautiful and natural context where to study these lemmas. And of
course, these lemmas have been further investigated in much more general
contexts than just abelian categories. Like for adjoint functors, further
studied in enriched, bi, 2, pseudo or lax contexts. To whom should we give
credit for such results? To the author of the very first result of that
kind? To the author of the more general result? To the author of the result
which "you" consider as most "natural". Really, I am not interested in
argueing on this. Did you already count the number of "Pythagoras theorems"
in mathematics ... or the number of "Galois theorems"?

Now, all right. As a "has been" category theorist, I consider abstract
categories as the most natural setting for studying the adjoint functor
theorem. But I am also aware that rapidly category theorists leave the
context of "abstract" categories for more specific "mathematical structures"
in order to prove more precise theorems. They study theories giving rise to
algebraic categories, accessible categories, topological categories,
classifying toposes, and so on. All these theories (Lawvere, sketches,
coherent theories, ...) fall under the scope of Bourbaki's "structures".
Thus Bourbaki did prove his "universal mapping theorem" in a general setting
which includes (probably) all concrete mathematical examples that you can
find in categorical books as applications of the more general Freyd adjoint
functor theorem. But nevertheless, as far as universal problems are
concerned, I consider Freyd's approach as much more elegant..."

End of message of Francis Borceux copied by George Janelidze




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