[[Sent on behalf of nathanael.arkor@gmail.com as it did get sent out/approved properly -- trying to fix this, apologies]] Dear all, While we are on the subject of provenance, and since it has been mentioned, it seems worth taking the opportunity to address the claim that Charles Ehresmann was responsible for the notion of 2-category. This assertion appears in Eilenberg and Kelly's 1965 La Jolla article "Closed Categories". However, the paper they cite (the 1963 "Catégories structurées") does not contain a definition of 2-category, nor the terminology. (It does introduce the notion of double category, but Ehresmann does not consider 2-categories as a special case.) It seems likely that Eilenberg and Kelly intended to cite Ehresmann's similarly named 1965 book "Catégories et Structure", in which 2-categories are mentioned. However, when I wrote to Andrée Ehresmann regarding the origin of the notion of 2-category, she wrote that Charles Ehresmann did not introduce 2-categories himself, but that they were introduced by his student Bénabou as a special kind of double category, and by different authors as categories enriched in Cat. As far as I am aware, 2-categories appeared in print for the first time in 1965, as instances of enriched categories in two different papers introducing the notion of enriched category: Bénabou's "Catégories relatives" (C. R. Acad. Sci. Paris 260), and Maranda's "Formal categories" (Canadian Journal of Mathematics 17). Consequently, my understanding is that it is appropriate to attribute the notion of 2-category jointly to Bénabou and Maranda. Best, Nathanael ________________________________ From: Ross Street Sent: 13 November 2023 1:58 PM To: j.adamek ; urs.schreiber Cc: JS Lemay ; Categories mailing list Subject: Re: The game of the name: Standard constructions, triples, monads, fundamental constructions Dear Urs, Jiri, and All It is not always clear from the publication record who thought of things first. Bénabou's Intro to bicategories was published by SLNM in 1967. This is before any Oberwolfach category conference. However, it is after the La Jolla 1965 conference where Eilenberg-Kelly presented closed categories with the type-set published paper appearing in 1966. By that time we had Mac Lane's coherence theorem for monoidal categories and Kelly's reduction of the axioms to two. Bénabou's multiplicative categories tried to incorporate coherence in the definition but there is a problem with his definition. In my opinion, Bénabou's contribution was to define bicategory (using the two axiom approach) as the several object version of monoidal category, thereby initiating weak higher category theory. This took courage. John Gray was already working with 2-categories, a concept of Charles Ehresmann. John told me in 1968-9 that he was convinced of the importance of bicategories because of the example of spans in a category with pullbacks. I was already convinced by Bénabou but felt the world might not be ready for papers using them. Also in the Eilenberg-Kelly paper, there were closed and monoidal functors. They agree in the closed monoidal case. These were of the lax rather than strong kind. I understand that Eilenberg pushed the presentation of their paper into emphasising closed over monoidal categories, but both are there. In Chapter IV Section 3 on examples, they point out that a closed functor from 1 to sets is a monoid and from I into abelian groups is a ring, etc.; they recognized that monoidal functors from 1 were monoids. Bénabou's morphisms of bicategories (lax functors) became the several object version. Jack Duskin told me in 1968 that Bénabou had the construction of a 2-category C' for each category C such that lax functors out of C amounted to 2-functors out of C'. I don't know how that fits historically with the result of Lawvere for C = 1, that André Joyal mentioned, which appeared in the ``Zurich triples book'' SLNM 80 (1969). Ross > On 12 Nov 2023, at 11:13 pm, Jirí Adámek wrote: > > Dear All, > > Bill Lawvere once told me that 'monad' had been the idea of Eilenberg. > Later I asked him by email about the details and he answered the following: > > I do not remember in which year it was. (Maybe 1968, judging from vague allusions in Springer Lecture Notes 80.) In any case it was in the common > room of the old castle at Oberwolfach when Sammy came out from behind the piano and announced the change. His informal speech emphasized that > the word would inflect well: 'monadic' etc. He also explicitly said > that nobody would ever confuse it with Leibnitzian monads. > > Jiri >