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Wikipedia seems to have an answer

"The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was discovered in 1968 by Jean Bénabou."

https://en.wikipedia.org/wiki/Strict_2-category

But this answer seems to me a little bit odd.

SGA 4 of Grothendieck was held between 1963-1964. There we find the following remarks

"C’est le fait que les 𝒰-topos (éléments d’un univers 𝒱) forment une 2-catégorie, et non plus seulement une catégorie ordinaire comme les espaces topologiques ordinaires, qui constitue du point de vue technique la différence la plus importante entre la théorie des topos et celle des espaces topologiques. Ce fait est la source de certaines complications techniques auxquelles on a déjà fait allusion, mais aussi de faits essentiellement nouveaux par rapport à la topologie traditionnelle."

and yes, I know that SGA 4 was edited and published then in 1972 when the notion was already known. But there is another remark

"Le fait que les 𝒰-topos (éléments d’un univers 𝒱) forment une 2-catégorie permet en particulier de définir la notion d’équivalence de deux 𝒰-topos 𝐸, 𝐸′."

And it is very likely that a notion of equivalence of topoi was already in the years 63, 64 in the original seminars (this is a speculative remark).

SGA4 cites the thesis of Monike Hakim as a former reference but that thesis was published later.

Do we have some conclusive information to add to the hypothesis that was Grothendieck who introduced the notion of 2-categories?

We can also say a few words about the use of Cat, already in use in SGA 1 and suggested even in the Tohoku paper. Grothendieck wrote a paper related to Quillen in 1968 where it is already present the theory of n-categories.

If you think Wikipedia is right, can you please let me know how Grothendieck enters in this picture?

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    $\begingroup$ In Closed categories, Eilenberg and Kelly cite Ehresmann's 1963 Catégories structurées for the definition of 2-category, though I don't see which definition to which they're referring. However, Wikipedia is citing the 1965 Catégories et structures, of which I'm struggling to find a copy. $\endgroup$
    – varkor
    Oct 7 at 6:39
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    $\begingroup$ Bénabou in Catégories relatives cites his own "forthcoming" Algèbre élémentaire dans les catégories and Ehresmann's 1963 paper for the definition of 2-category. $\endgroup$
    – varkor
    Oct 7 at 7:10
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    $\begingroup$ @varkor - you are spot on - "Catégories structurées" is the right reference, it seems. I added an answer about it giving some extra context. $\endgroup$ Oct 7 at 7:26
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    $\begingroup$ I should clarify that it is definitely clear that Ehresmann defined double categories in the 1963 paper, which generalise 2-categories, but this is separate from the question of who defined 2-categories explicitly. Both Eilenberg–Kelly and Bénabou use the terminology "2-category", which suggests the meaning is known, but this term does not appear in the 1963 paper as far as I can tell. $\endgroup$
    – varkor
    Oct 7 at 8:14
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    $\begingroup$ Here are the links: Catégories avec multiplications and Algèbre élémentaire dans les catégories avec multipliplication. 2-categories appear in neither. $\endgroup$
    – varkor
    Oct 18 at 10:25

1 Answer 1

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Following the comment of varkor, I re-opened Catégories structurées of Charles Ehresmann (published 1963), and I believe that Section 4 "Catégories doubles" and Section 5 "Catégories $n$-uples" give the necessary definitions.

There is also an article of Guitart Sur les contributions de Charles Ehresmann à la théorie des catégories where it is suggested that Ehresmann gave that definition in 1961. It is quite possible that some of this was rediscovered independently, Guitart writes:

Ainsi, pour lui une catégorie est d’abord un graphe orienté, plus une loi de composition des paires de flèches consécutives : vers 1965 cela fait une différence entre lui et les autres catégoriciens, différence conceptuelle, à laquelle s'ajoute des différences de notations : il emploie "$(F,\underline{f},E)$" pour "$f\colon E\to F$", "$\alpha$ et $\beta$" pour "dom et cod", "$e', C, e$" pour "$\mathrm{Hom}_C(e,e')$". Avec parfois le choix d'une terminologie "provisoire" <...>, cela a certainement contribué à une mauvaise diffusion des travaux catégoriques d'Ehresmann jusqu'en 1970 environ.

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    $\begingroup$ Could you be explicit about where in section 5 the definition of 2-category is given? Probably my French is not good enough, but I only saw what looked like the definition of an n-fold category. $\endgroup$
    – varkor
    Oct 7 at 8:10
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    $\begingroup$ Guitart writes that Ehresmann gave the definition of double category in 1961, not of 2-category. $\endgroup$
    – varkor
    Oct 7 at 8:12
  • $\begingroup$ @varkor : You are right, I was too fast. It seems true that the definition of a 2-category is only implicit in the paper (perhaps he considered it less important?), specifically, if one looks at Section 6 "Structures d'ordre sur une catégorie" at the notions of "catégorie ordonnée" and "catégorie fonctoriellement ordonnée", one finds something sufficiently close (as far as I am concerned). $\endgroup$ Oct 7 at 11:46
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    $\begingroup$ I personally would like to see a precise definition, particularly as we know from other sources that "2-catégorie" was established terminology by 1965. $\endgroup$
    – varkor
    Oct 7 at 12:58
  • $\begingroup$ Many thanks for collecting this up, but I second @varkor to know who introduced 2-categories. $\endgroup$ Oct 7 at 15:51

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