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charset="utf-8" Something more can be said about the relations between Bourbaki, Samuel=E2= =80=99s 1948 paper, and Freyd=E2=80=99s GAFT: It certainly is true that wha= t Bourbaki writes is influenced (if not written) by Samuel. From the catego= rical perspective there is a certain irony between the formulations in Bour= baki and the Samuel-paper: While Samuel=E2=80=99s original paper is surpris= ingly categorical in nature (though he apparently didn=E2=80=99t know the = 1945 Eilenberg-Mac Lane - paper and, hence, didn=E2=80=99t 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 =E2=80=94> Set and this reflects isomorphisms and lifts products, equaliz= ers, and intersections. His S-T-mappings between S-sets X and T-sets Y the= n are most naturally to be interpreted as S-mappings X =E2=80=94> EY, where= E:T =E2=80=94> 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 =E2=80=9Euniversal mapping problem=E2=80=9C then becomes: Show that for= every X in T there exists an E-universal morphism! And his proof can be re= ad as follows: "The claim is true since T has and E preserves limits and E = satisfies the solution set condition=E2=80=9C and is done essentially as in= Freyd=E2=80=99s book =E2=80=94 except for the language. For more details s= ee my arXiv-posting 2310.19528 (October 2023). Hans-E. Porst Am 03.03.2024 um 11:24 schrieb William Messing : The 1958 edition of Bourbaki Th=C3=A9orie des Ensembles, Chapitre IV, Struc= tures, has the appendix discussing at length and defining in a precise sens= e the word "canonique". Why this was suppressed in all subsequent edition= s has seemed both idiotic and inexplicable to me. William Messing On Sun, Mar 3, 2024 at 3:02=E2=80=AFAM David Roberts > wrote: Well, I got a chance to look as I'm not unfamiliar with the Bourbaki Archiv= es. The cited result from Nikita (CST22 in Chapter IV, 3.2) was already include= d in: R=C3=A9daction n=C2=B0188. Ensembles. Chapitre IV. Structures (=C3=A9tat 8 = ?). Dieudonn=C3=A9, Jean, R188_nbr091, acc= =C3=A8s le 3/03/2024, http://archives-bourbaki.ahp-numerique.fr/items/show/= 602 dating to September 1953. See the attachment. The paper Pierre Samuel, "On universal mappings and free topological groups", Bulleti= n 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 Bou= rbaki collaborator). The following 1950 draft does not have the cited result: R=C3=A9daction n=C2=B0137. Ensembles. Chapitre III. Structures (=C3=A9tat 5= ). Chevalley, Claude, R137_nbr040, acc=C3=A8s le 3/03/2024, http://archives= -bourbaki.ahp-numerique.fr/items/show/546 The April 1953 draft has a corresponding heading, but the pages are missing= . David On Sun, 3 Mar 2024, 2:03 pm David Roberts, > wrote: You can see the publication history of Chapter 4 of Th=C3=A9orie des Ensemb= les here: http://archives-bourbaki.ahp-numerique.fr/elements-mathematique#elements-ma= th1 If people have access to older copies they might check the 1957 and 1966 ed= itions of the standalone ch4, and the 1970 edition of the full book. More p= atient people m8ght like to dig through the drafts at http://archives-bourbaki.ahp-numerique.fr/items/browse?search=3D&type=3D5&s= ort_field=3DDublin+Core%2CTitle&advanced[0][element_id]=3D99&advanced[0][ty= pe]=3Dis+exactly&advanced[0][terms]=3DTh%C3%A9orie+des+ensembles Eilenberg was active in Bourbaki, don't forget, and was writing drafts on c= ategory theory. I can't recall when he ceased working with them offhand. David On Sun, 3 Mar 2024, 1:09 pm Nikita Danilov, > wrote: CAUTION: External email. Only click on links or open attachments from trust= ed senders. ________________________________ This reminds me of a (probably trivial) question that occurred to me some t= ime ago. N. Bourbaki's Theory of Sets has a result very similar to the gene= ral adjoint functor theorem (CST22 in Chapter IV, 3.2). This volume was pri= nted about 4 years after Abelian Categories. Is the history behind the Bour= baki's version known? Thank you, Nikita. On Sat, 2 Mar 2024 at 19:55, Michael Barr, Prof. > 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 lea= st for reflective subcategories) in his undergrad honors thesis, even thoug= h adjoints had not yet been defined. The preface in the TAC reprint includ= es things not in any other published version of the book. Michael You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group = | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group = | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group = | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --Apple-Mail=_AFE930EE-CDB6-4D5A-822D-4D65A55A10DE Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
Something more can be said about the relations between Bourbaki, Samuel=E2= =80=99s 1948 paper, and Freyd=E2=80=99s GAFT: It certainly is true that wha= t Bourbaki writes is influenced (if not written) by Samuel. From the catego= rical perspective there is a certain irony between the formulations in Bourbaki and the Samuel-paper: While Samuel=E2=80=99s = original paper is surprisingly categorical in nature (though he apparently = didn=E2=80=99t know the  1945 Eilenberg-Mac Lane - paper and, hence, d= idn=E2=80=99t 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 =E2=80=94> Set and this reflects isomorphisms and lifts products, equalizers, and intersections. His S-T-ma= ppings between S-sets X and T-sets Y  then are most naturally to be in= terpreted as S-mappings X =E2=80=94> EY, where E:T =E2=80=94> 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 t= hat (in categorical language) the functor E satisfies the solution set cond= ition.

His =E2=80=9Euniversal mapping problem=E2=80=9C then becomes: Show tha= t 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 lim= its and E satisfies the solution set condition=E2=80=9C and is done essentially as in Freyd=E2=80=99s book =E2=80=94 except for the langu= age. 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.ed= u>:

The 1958 edition of Bourbaki Th=C3=A9orie des Ensembles, C= hapitre IV, Structures, has the appendix discussing at length and defining = in a precise sense the word "canonique".   Why this was supp= ressed in all subsequent editions has seemed both idiotic and inexplicable to me.

William Messing

On Sun, Mar 3, 2024 at 3:02=E2=80=AFA= M David Roberts <droberts.65= 537@gmail.com> wrote:
Well, I got a chance to look as I'm not unfamiliar with the Bourbaki A= rchives.

The cited result from Nikita (CST22 in Chapter IV, 3.2) w= as already included in:

R=C3=A9daction n=C2=B0188. Ensembles. Chapitre IV. Struct= ures (=C3=A9tat 8 ?). Dieudonn=C3=A9, Jean, R188_nbr091, acc=C3=A8s le 3/03/2024, http://archives-bourbaki.ahp-numerique.fr/items/show/602

dating to September 1953. See the attachment. 

The paper

Pierre Samuel, "On universal mappings and free topol= ogical groups", Bulletin of the American Mathematical Society, 54, jui= n 1948, p. 591-598. 

was a big influence on this section (and note that P. Sam= uel was also a Bourbaki collaborator).

The following 1950 draft does not have the cited result:<= /div>

R=C3=A9daction n=C2=B0137. Ensembles. Chapitre III. Struc= tures (=C3=A9tat 5). Chevalley, Claude, R137_nbr040, acc=C3=A8s le 3/03/202= 4, http://archives-bourbaki.ahp-numerique.fr/items/show/546

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

David


On Sun, 3 Mar 2024, 2:03 pm David Rob= erts, <droberts.65537@gmail.com> wrote:
You can see the publication history of Chapter 4 of Th=C3= =A9orie des Ensembles here:

http://archives-bourbaki.ahp-numerique.fr/elements-mathemat= ique#elements-math1

If people have access to older copies they might check th= e 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 w= riting drafts on category theory. I can't recall when he ceased working wit= h them offhand.

David

On Sun, 3 Mar 2024, 1:09 pm Nikita Da= nilov, <danilov@gmail.com> wrote:
CAUTION: External ema= il. Only click on links or open attachments from trusted senders.

This reminds me of a (probably trivial) question that occu= rred to me some time ago. N. Bourbaki's Theory of Sets has a result very si= milar to the general adjoint functor theorem (CST22 in Chapter IV, 3.2). Th= is volume was printed about 4 years after Abelian Categories. Is the history behind the Bourbaki's v= ersion known?

Thank you,<= /div>
Nikita.


On Sat, 2 Mar 2024 at 19:55, Michael = Barr, Prof. <barr.michael@mcgill.ca> wr= ote:
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 parti= cular, he had already essentially discovered the general adjoint functor, a= t least for reflective subcategories) in his undergrad honors thesis, even though adjoints had not yet been defi= ned.  The preface in the TAC reprint includes things not in any other = published version of the book.

Michael
 
 
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