Re: Historical notes in Freyd's Abelian Categories

[porst <porst@uni-bremen.de>]

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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

On Sun, Mar 3, 2024 at 3:02 AM David Roberts <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>> wrote:
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:

Rédaction n°188. Ensembles. Chapitre IV. Structures (état 8 ?). Dieudonné, Jean<https://url.au.m.mimecastprotect.com/s/o-upCvl1g2SmrqN5UQiRQU?domain=archives-bourbaki.ahp-numerique.fr>, R188_nbr091, accès le 3/03/2024, http://archives-bourbaki.ahp-numerique.fr/items/show/602<https://url.au.m.mimecastprotect.com/s/Eng1CwV1jpSMgk8mUq_wZB?domain=archives-bourbaki.ahp-numerique.fr>

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:

Rédaction n°137. Ensembles. Chapitre III. Structures (état 5). Chevalley, Claude, R137_nbr040, accès le 3/03/2024, http://archives-bourbaki.ahp-numerique.fr/items/show/546<https://url.au.m.mimecastprotect.com/s/ZpKNCxngGkf6PDMVUY6vWm?domain=archives-bourbaki.ahp-numerique.fr>

The April 1953 draft has a corresponding heading, but the pages are missing.

David


On Sun, 3 Mar 2024, 2:03 pm David Roberts, <droberts.65537@gmail.com<mailto:droberts.65537@gmail.com>> wrote:
You can see the publication history of Chapter 4 of Théorie des Ensembles here:

http://archives-bourbaki.ahp-numerique.fr/elements-mathematique#elements-math1<https://url.au.m.mimecastprotect.com/s/01vRCyoj8PuvAKXwHRWYSK?domain=archives-bourbaki.ahp-numerique.fr>

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

http://archives-bourbaki.ahp-numerique.fr/items/browse?search=&type=5&sort_field=Dublin+Core%2CTitle&advanced[0][element_id]=99&advanced[0][type]=is+exactly&advanced[0][terms]=Th%C3%A9orie+des+ensembles<https://url.au.m.mimecastprotect.com/s/EYHOCzvkmpflOEgZHopFvB?domain=archives-bourbaki.ahp-numerique.fr>

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

On Sun, 3 Mar 2024, 1:09 pm Nikita Danilov, <danilov@gmail.com<mailto:danilov@gmail.com>> wrote:
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.


On Sat, 2 Mar 2024 at 19:55, Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> wrote:
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


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