Re: The game of the name: Standard constructions, triples, monads, fundamental constructions

[Ross Street <ross.street@mq.edu.au>]

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 <j.adamek@tu-bs.de> 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
>



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