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

[John Baez <john.baez@ucr.edu>]

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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




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