Something more can be said about the relations between Bourbaki, Samuel’s 1948 paper, and Freyd’s GAFT: It certainly is true that what Bourbaki writes is influenced (if not written) by Samuel. From the categorical perspective there is a certain irony between
the formulations in Bourbaki and the Samuel-paper: While Samuel’s original paper is surprisingly categorical in nature (though he apparently didn’t know the 1945 Eilenberg-Mac Lane - paper and, hence, didn’t use the concepts of category and functor), this
has been lost in the Bourbaki text.
In some detail:
1. The setting Samuel considers (T-sets and T-mappings) and describes by a set of axioms express the following (in categorical language): T-sets and T-mappings form a category T which is equipped with a faithful functor |-|: T —> Set and this reflects
isomorphisms and lifts products, equalizers, and intersections. His S-T-mappings between S-sets X and T-sets Y then are most naturally to be interpreted as S-mappings X —> EY, where E:T —> S is a functor commuting with the underlying functors |-| (and, hence,
preserves limits).
2. The only non-standard axiom he introduces, he uses (only) to show that (in categorical language) the functor E satisfies the solution set condition.
His „universal mapping problem“ then becomes: Show that for every X in T there exists an E-universal morphism! And his proof can be read as follows: "The claim is true since T has and E preserves limits and E satisfies the solution set condition“ and is
done essentially as in Freyd’s book — except for the language. For more details see my arXiv-posting 2310.19528 (October 2023).
Hans-E. Porst
Am 03.03.2024 um 11:24 schrieb William Messing <messing@math.umn.edu>:
The 1958 edition of Bourbaki Théorie des Ensembles, Chapitre IV, Structures, has the appendix discussing at length and defining in a precise sense the word "canonique". Why this was suppressed in all subsequent editions has seemed both idiotic
and inexplicable to me.
William Messing
Well, I got a chance to look as I'm not unfamiliar with the Bourbaki Archives.
The cited result from Nikita (CST22 in Chapter IV, 3.2) was already included in:
dating to September 1953. See the attachment.
The paper
Pierre Samuel, "On universal mappings and free topological groups", Bulletin of the American Mathematical Society, 54, juin 1948, p. 591-598.
was a big influence on this section (and note that P. Samuel was also a Bourbaki collaborator).
The following 1950 draft does not have the cited result:
The April 1953 draft has a corresponding heading, but the pages are missing.
David
You can see the publication history of Chapter 4 of Théorie des Ensembles here:
http://archives-bourbaki.ahp-numerique.fr/elements-mathematique#elements-math1
If people have access to older copies they might check the 1957 and 1966 editions of the standalone ch4, and the 1970 edition of the full book. More patient people m8ght like to dig through the drafts at
Eilenberg was active in Bourbaki, don't forget, and was writing drafts on category theory. I can't recall when he ceased working with them offhand.
David
CAUTION: External email. Only click on links or open attachments from trusted senders.
This reminds me of a (probably trivial) question that occurred to me some time ago. N. Bourbaki's Theory of Sets has a result very similar to the general adjoint functor theorem (CST22 in Chapter IV, 3.2). This volume was printed about 4 years
after Abelian Categories.
Is the history behind the Bourbaki's version known?
Thank you,
Nikita.
Peter has called my attention to the existence of some historical notes in the preface to the TAC reprint (TR-3) of Abelian categories. In particular, he had already essentially discovered the general adjoint functor, at least for reflective subcategories)
in his undergrad honors thesis, even though adjoints had not yet been defined. The preface in the TAC reprint includes things not in any other published version of the book.
Michael