Who introduced the notion of 2-categories?

Question

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?

Answer 1

It appears that the definition of 2-category was introduced independently by two authors, both of whom independently introduced the modern notion of enriched category, for which 2-categories appeared as an example.

• Jean Bénabou gives 2-categories as example (5) in the 1965 paper Catégories relatives, under the name "2-catégories".
• Jean-Marie Maranda gives 2-categories as an example on the second page (p. 759) in the 1965 paper Formal categories, under the name "categories of the second type". The definition is spelled out in §2.

Both authors mention the connection to Ehresmann's double categories.

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Bénabou cites his own forthcoming thesis for the concept, which was eventually published under Structures algébriques dans les catégories (which is not precisely the name mentioned in the paper). He also points the reader to Ehresmann's 1963 Catégories structurées, but the definition of 2-category does not appear here; the reference is most likely due to the similar notion of double category, of which the notion of 2-category is a special case.

Other authors, such as Eilenberg and Kelly in the 1965 paper Closed categories, also cite Ehresmann's 1963 paper for the notion of 2-category. However, as it does not appear there, it is more likely they intended to cite Ehresmann's similarly named 1965 book Catégories et structures (which is the reference currently listed on Wikipedia), which does contain a discussion of 2-categories on page 324 in a historical note.

To settle the matter, I emailed Andrée Ehresmann, who wrote:

[Charles Ehresmann] did not introduced himself 2-catégories, which have been introduced soon after both by his student Jean Bénabou as special double categories, and by different other authors as categories 'enriched' in Cat

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The appearance in SGA 4 of 2-categories appears to have either been introduced during editing, or was based on word-of-mouth ideas. SGA 4, for instance, cites Monique Hakim's 1972 thesis, who introduced 2-categories by writing:

La notion de 2-catégorie est due a J. Benabou

Answer 2

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.