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

[Richard Garner <richard.garner@mq.edu.au>]

Earlier on Mac Lane did also use "bicategory" to mean more or less what
we would nowadays call "category with a proper factorisation system".
(Starting already in 1950 in "Duality for groups".) Isbell used this
terminology a lot and as late as 1964.

Richard

Ross Street <ross.street@mq.edu.au> writes:

> Dear John
>
> Well that is important for me to know/remember. 
> It must have been **that** use of the term ``bicategory'' that Jean was seeking approval
> from Mac Lane to use for the several object form.
> I do think the terms closed and monoidal category are due to the Eilenberg-Kelly
> however there would have been discussion of terminology at the LaJolla conference.
> Very tricky!
>
> Ross
>
>> On 14 Nov 2023, at 9:14 am, John Baez <john.baez@ucr.edu> wrote:
>> 
>> Hi -
>> 
>> It is not very important, but I was amused to discover recently that
>> Mac Lane's famous 1963 paper on monoidal categories, "Natural
>> associativity and commutativity," does not mention "monoidal
>> categories". Instead he called them "bicategories"!
>> 
>> Later in this paper he writes
>> 
>> "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."
>> 
>> This is not Bénabou's famous paper on bicategories: instead it's
>> "Catégories avec multiplication", where Bénabou introduces a
>> preliminary concept of monoidal category, which he called "catégorie
>> avec multiplication".
>> 
>> Furthermore, it's now recognized that Bénabou's formulation of
>> coherence for monoidal categories is not quite right. Benabou's
>> version is along the lines of "all diagrams formed by associators
>> and unitors commute", and he does not state this in a way that rules
>> out problematic cases caused by coincidental equations between
>> objects.
>> 
>> It seems the history of mathematics is endlessly tricky.
>> 
>> Best,
>> John Baez
>> 
>>