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John Baez (Jul 08 2023 at 17:22):Who introduced the term [[monoidal category]], and when?
I had thought it was Mac Lane in this paper:
But now that I'm looking through this paper, I don't see the word "monoidal" anywhere!
He seems to just state his coherence theorem, Theorem 5.2, without actually giving any name to the kind of categories to which it applies!
John Baez (Jul 08 2023 at 17:24):Even weirder: in section 6 he calls this sort of category a "bicategory". :surprise:
John Baez (Jul 08 2023 at 17:25):Yup, that's right: he writes down the definition of monoidal category and calls it a "bicategory".
He says that bicategories were introduced by Bénabou:
Bicategories have been introduced independently by several authors. They are in Bénabou [1], with a different but equivalent definition of "coherence," but without any finite list of conditions sufficient for the coherence.
John Baez (Jul 08 2023 at 17:27):... but [1] is not Bénabou's famous 1967 paper on bicategories! Instead it's this:
John Baez (Jul 08 2023 at 17:28):So Mac Lane seems to have made up the word "bicategory" as a name for "monoidal category". :dizzy:
John Baez (Jul 08 2023 at 17:31):(Or maybe Bénabou did???)
John Baez (Jul 08 2023 at 17:37):Google is really going downhill: I'm trying to get ahold of Bénabou's paper "Categories avec multiplication" and the top link is to a paper entitled "Tensor-centric warfare".
David Egolf (Jul 08 2023 at 18:58):In case you're still looking for that paper, I think I found it here: https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1965.item
It's on page 1887. (I found that link here: https://comptes--rendus-academie--sciences-fr.translate.goog/page/consulter-les-anciens-numeros-1835-1996_fr/?_x_tr_sl=fr&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=sc#1835 )
John Baez (Jul 08 2023 at 18:59):Thanks, I am!
John Baez (Jul 08 2023 at 19:09):This paper is short, and interesting even though I can't read French. It does not mention the words "bicategory" or "monoidal category". It talks about "categories with multiplication".
It states the coherence laws for what we'd now call a monoidal category in Axiom 1, which in modern jargon says roughly that "all diagrams built using associators and unitors commute" - thus sidestepping the need to list finitely many commutative diagrams that guarantee this. So Mac Lane's big contribution was to find finitely many diagrams that do the job, and prove the [[coherence theorem for monoidal categories]], which says that they do the job.
Mike Shulman (Jul 08 2023 at 19:50):The notes at the end of Chapter VII of "Categories for the working mathematician" say
Monoidal categories were first explicitly formulated by Bénabou [1963, 1964], who called them "catégories avec multiplication" and by Mac Lane [1963b], who called them "categories with multiplication"; the renaming is due to Eilenberg.
Bénabou [1963] is "Catégories avec multiplication", and Mac Lane [1963b] is "Natural associativity and commutativity". But there is no "Bénabou [1964]" in the bibliography, and he doesn't give any citation for Eilenberg. d-:
Mike Shulman (Jul 08 2023 at 20:08):I wonder if the first published use of "monoidal category" was in the Eilenberg-Kelly paper "Closed categories" (1966). In the introduction they write
In Chapter II we consider closed categories which possess a tensor product... These considerations lead us to the notion of a monoidal category, which is a catégorie avec multiplication in the terminology of Bénabou ([1], [2], [3]).
The citations are to "Catégories avec multiplication" (1963), "Algèbre élémentaire dans les catégories avec multiplication" (1964), and "Catégories relatives" (1965).
Mike Shulman (Jul 08 2023 at 20:08):I guess probably "Algèbre élémentaire dans les catégories avec multiplication" is what Mac Lane meant to cite with "Bénabou [1964]".
John Baez (Jul 08 2023 at 20:24):Great, thanks! So the term "monoidal category" is definitely due to Eilenberg, but Mac Lane doesn't say when his buddy called them that. Your guess sounds good to me.
John Baez (Jul 08 2023 at 20:38):I got into this when reading Sinh's thesis. She uses what we'd call monoidal categories, with the pentagon and other identities listed, but she calls them "catégories AU" and says she is following Neantro Saavedra-Rivano's thesis for her terminology. She gives a date of 1970 with a question mark for that thesis, but the Mathematical Geneaology says it's 1972. Saavedra-Rivano was another student of Grothendieck so she may have gotten the information before his thesis was done.
She talks about
"-catégories" which have a functor obeying no laws,
"catégories associative" which are -catégories with an associator obeying the pentagon identity
"catégories AU" which are monoidal categories
"catégories AC" which are -categories with an associator obeying the pentagon identity and also a symmetry obeying the hexagon identity, and
"catégories ACU" which are symmetric monoidal categories.
So, it's a bit baroque by modern standards but all the definitions of these things are just what a modern mathematician would give.
Patrick Nicodemus (Jul 09 2023 at 06:30):John Baez said:
Great, thanks! So the term "monoidal category" is definitely due to Eilenberg, but Mac Lane doesn't say when his buddy called them that. Your guess sounds good to me.
Reminds me of the story Borceux tells in his categorical algebra book about Mac Lane and Eilenberg arguing over the history of category theory at a conference
Patrick Nicodemus (Jul 09 2023 at 06:30):
John Baez (Jul 09 2023 at 10:51):Heh, that's funny. In his later years, Bénabou became quite abusive on the category theory mailing list, complaining that Johnstone, Street etc. did not give him enough credit for his work. So one problem with not attempting to properly attribute people's work is that you wind up offending them.
John Baez (Jul 09 2023 at 10:53):However, I'm curious about the early history of monoidal categories purely because it's interesting. I wrote up my findings so far here:
Nathanael Arkor (Jul 09 2023 at 12:17):Yup, that’s right: he writes down the definition of monoidal category and calls it a ‘bicategory’.
This is funny not just because Mac Lane used the term "bicategory" to mean monoidal category after Bénabou used the terminology to mean what we would now call a bicategory, but also because Mac Lane introduces a different concept called a "bicategory" in his earlier 1950 paper Duality for groups (there he uses the term to mean a kind of factorisation system). So Mac Lane seemed rather fond of overloading that term!
Nathanael Arkor (Jul 09 2023 at 12:25):I wonder if the first published use of “monoidal category” was in the Eilenberg–Kelly paper “Closed categories” (1966)
This is also the earliest reference I was able to find to the terminology. Given that it was only 3 years after the introduction, and Eilenberg was one of the authors, this seems likely.
Steve Awodey (Jul 09 2023 at 14:32):John Baez said:
Heh, that's funny. In his later years, Borceux became quite abusive on the category theory mailing list, complaining that Johnstone, Street etc. did not give him enough credit for his work. So one problem with not attempting to properly attribute people's work is that you wind up offending them.
I think @John Baez means Benabou, not Borceux - who made an appearance (as a monk!) at the CT in Louvain-la-Neuve just last week. :smile:
John Baez (Jul 09 2023 at 15:12):I meant Benabou.
How did Borceux "make an appearance as a monk"??
John Baez (Jul 09 2023 at 15:17):Nathanael Arkor said:
This is funny not just because Mac Lane used the term "bicategory" to mean monoidal category after Bénabou used the terminology to mean what we would now call a bicategory, [...]
Really? After? The latest I've seen Mac Lane saying "bicategory" to mean "monoidal category" is 1965, at the end of his paper "Categorical algebra". The earliest I've seen Bénabou using "bicategory" in its modern sense is 1967, in his paper "Bicategories".
but also because Mac Lane introduces a different concept called a "bicategory" in his earlier 1950 paper Duality for groups (there he uses the term to mean a kind of factorisation system). So Mac Lane seemed rather fond of overloading that term!
That's really interesting. I guess he just kept throwing it at the wall, seeing if it would stick.
Nathanael Arkor (Jul 09 2023 at 15:25):John Baez said:
Really? After? The latest I've seen Mac Lane saying "bicategory" to mean "monoidal category" is 1965, at the end of his paper "Categorical algebra".
Oh, in the n-Category Café blog post, it says that Natural associativity and commutativity was published in 1968, rather than 1963, which is what confused me. I guess that's a typo.
John Baez (Jul 09 2023 at 15:25):Whoops, it's 1963.
Steve Awodey (Jul 09 2023 at 16:09):John Baez said:
I meant Benabou.
How did Borceux "make an appearance as a monk"??
he gave a tour of a local monastery in costume.
Morgan Rogers (he/him) (Jul 09 2023 at 16:28):Photo credit @Jacques Darné (if I'm not mistaken).
5a49f2e7-755f-49b0-b424-9d0b05cb61af.jpg
Mateo Carmona (Jul 09 2023 at 19:17):John Baez said:
I got into this when reading Sinh's thesis. She uses what we'd call monoidal categories, with the pentagon and other identities listed, but she calls them "catégories AU" and says she is following Neantro Saavedra-Rivano's thesis for her terminology. She gives a date of 1970 with a question mark for that thesis, but the Mathematical Geneaology says it's 1972. Saavedra-Rivano was another student of Grothendieck so she may have gotten the information before his thesis was done.
She talks about
"-catégories" which have a functor obeying no laws,
"catégories associative" which are -catégories with an associator obeying the pentagon identity
"catégories AU" which are monoidal categories
"catégories AC" which are -categories with an associator obeying the pentagon identity and also a symmetry obeying the hexagon identity, and
"catégories ACU" which are symmetric monoidal categories.
So, it's a bit baroque by modern standards but all the definitions of these things are just what a modern mathematician would give.
Grothendieck began employing the term "-catégories" as early as late 1964/early 1965 when he embarked on developing the main lines of the theory of motives (see for example https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/motives.pdf) within the framework of tannakian categories (a name introduced by Saavedra). He initially referred to them as "catégories tensorielles rigides" and later as "-catégories de Galois-Poincaré". It is worth noting that he not only had knowledge of Bénabou's work but also coincidentally attended the defense of his thesis, as mentioned in a letter dated 15.2.1983 (https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/agrb_web.pdf). The idea I would like to convey is that although Grothendieck did not introduce the concept himself, he was actively working with it in its early stages, well before his encounters with Sinh or Saavedra. It is interesting to mention that Manin uses the word "monoidal" in his 1968 paper "Correspondences, motifs, and monoidal transformations, Mat. Sb. (N.S.) 77, 119 (1968), 475–507," but unfortunately, I was unable to find a copy of it (and it is in Russian).
David Michael Roberts (Jul 10 2023 at 08:13):monoidal = моноидальные and here it is: https://gdz.sub.uni-goettingen.de/id/PPN510932592_0119?tify=%7B%22pages%22%3A%5B489%5D%2C%22view%22%3A%22info%22%7D
John Baez (Jul 10 2023 at 09:19):Thanks, @Mateo Carmona!
David Michael Roberts (Jul 10 2023 at 09:50):I don't think monoidal here is quite what we want: on the first page there the phrase (Google translated) : "a birational morphism decomposing into a sequence of monoidal transformations with non-singular centers". Later, for X (and I believe X') varieties, he talks about a monoidal transformation X'->X.
Mateo Carmona (Jul 10 2023 at 18:30):David Michael Roberts said:
I don't think monoidal here is quite what we want: on the first page there the phrase (Google translated) : "a birational morphism decomposing into a sequence of monoidal transformations with non-singular centers". Later, for X (and I believe X') varieties, he talks about a monoidal transformation X'->X.
My initial guess was that it was related but knowing it is something different is more interesting. I will see it later.
Mateo Carmona (Jul 10 2023 at 18:33):@John Baez Maybe you find interesting the following letter of 1974 that Grothendieck wrote to Deligne, Verdier, and Giraud advertising the work of Sinh: [see https://agrothendieck.github.io/divers/LGDVG23674scan.pdf ]
John Baez (Jul 11 2023 at 10:50):Thanks! Since I don't read French, I wonder if I can figure out what he's saying here. (I can slowly fight my way through it.)
John Baez (Jul 11 2023 at 11:15):This part is interesting because it sheds a bit of light on what happened after Sinh sent her hand-written thesis to Grothendieck:
I have recently received a copy of the manuscript in form (the making of which has gone through a thousand difficulties - it is doubtful whether it will be possible to have other copies in the near future, unless I make some according to this copy) and I sent an official report on the subject of this work to M. Ta Quang Buu, Minister of Higher and Technical Education of the RDV, report of which I am sending you a copy attached.
John Baez (Jul 11 2023 at 11:16):(Google translation; could easily be improved.)
Ralph Sarkis (Jul 11 2023 at 11:26):John Baez said:
(Google translation; could easily be improved.)
I've heard many people say deepl is better.
John Baez (Jul 11 2023 at 11:28):Here's what it produces:
I have received a copy of the manuscript in form (the making of which was done at the cost of a thousand difficulties - it is doubtful that it will be possible to have other copies in the near future, unless they are made from this copy) and I have sent an official report on the subject of this work to Quang, Minister of Higher and Technical Education of Lat RDV, a copy of which I enclose for your information.
John Baez (Jul 11 2023 at 11:29):"unless they are made from this copy" seems better than "unless I make some according to this copy".