### Who Invented Monoidal Categories?

#### Posted by John Baez

Most people would say Mac Lane invented monoidal categories. But the history is more complicated — as usual.

There are actually two questions here: who invented the *concept* of monoidal category, and who introduced the *term* ‘monoidal category’.

I had always assumed both were done by Mac Lane in this paper:

- Saunders Mac Lane, Natural associativity and commutativity,
*Rice University Studies***49**(4) (1963), 28-46.

But then I made the mistake of actually looking at this paper — and I didn’t see the word ‘monoidal’ anywhere! He states his coherence theorem, Theorem 5.2, without actually giving any name to the kind of categories to which it applies.

Even weirder: in the next section he calls this kind of category a ‘bicategory’.

Yup, that’s right: he writes down the definition of monoidal category and calls it a ‘bicategory’. Nowadays that means something else — something considerably more general.

Furthermore, Mac Lane says that bicategories were introduced by Bénabou:

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.

For a second that sounds reasonable. But wait: [1] is not Bénabou’s famous 1967 paper on bicategories — that hadn’t been written yet! Instead it’s this:

- Jean Bénabou, Catégories avec multiplication,
*Comptes Rendus des Séances de l’Académie des Sciences***256**(1963), 1887–1890.

This paper introduces what we’d now call monoidal categories! But it calls them ‘catégories avec multiplication’, meaning ‘categories with multiplication’. And as Mac Lane noted, it doesn’t list a finite set of coherence laws like the pentagon identity for the associator and triangle identities for the unitors: it basically just says all diagrams built from the associator and unitors commute.

So it’s all very confusing. In his famous paper on monoidal categories Mac Lane called them bicategories and attributed them to Bénabou — but he cites Bénabou’s paper on monoidal categories, not his famous paper on bicategories! 😵

The history of mathematics is so much simpler if you don’t actually read old papers.

So who invented the term ‘monoidal category’?

Mike Shulman seems to have figured it out. He wrote:

The notes at the end of Chapter VII of

Categories for the Working MathematiciansayMonoidal categories were first explicitly formulated by Bénabou [1963, 1964], who called them “catégories avec multiplication” and by Mac Lane [1963b], who called them “categories with multiplication”; the renaming is due to Eilenberg.

Bénabou [1963] is “Catégories avec multiplication”, and Mac Lane [1963b] is “Natural associativity and commutativity”. But there is no “Bénabou [1964]” in the bibliography, and he doesn’t give any citation for Eilenberg. d-:

I wonder if the first published use of “monoidal category” was in the Eilenberg–Kelly paper “Closed categories” (1966). In the introduction they write

In Chapter II we consider closed categories which possess a tensor product… These considerations lead us to the notion of a monoidal category, which is a

catégorie avec multiplicationin the terminology of Bénabou ([1], [2], [3]).The citations are to “Catégories avec multiplication” (1963), “Algébre élémentaire dans les catégories avec multiplication” (1964), and “Catégories relatives” (1965).

I guess probably “Algébre élémentaire dans les catégories avec multiplication” is what Mac Lane meant to cite with “Bénabou [1964]”.

So, ** Eilenberg invented the term ‘monoidal category’**. We’re just not completely sure of where and when.

And it seems that ** Bénabou invented the concept of monoidal category**. Even Mac Lane says the concept is in Bénabou’s paper, and was “introduced independently by several authors”.

(**Edit:** but read the comments! It turns out Bénabou’s definition was flawed. So now I’d say Mac Lane invented the (correct) concept of monoidal category.)

I got into these questions when writing an article about Hoàng Xuân Sính’s thesis, which she wrote under Grothendieck starting around 1967 and finishing in 1972. It’s called *Gr-catégories*, and it’s about monoidal categories where both the objects and morphisms have inverses. She made heavy use of what we would call monoidal categories, and she defines them in the modern way with the pentagon and other identities listed. But she calls them ‘catégories AU’, and she says she is following Neantro Saavedra-Rivano’s thesis *Catégories Tannakiennes* for her terminology. Saavedra-Rivano was another student of Grothendieck, and he finished his thesis around 1970. So both these theses give a good sense of how the French viewed monoidal categories in the late 1960’s.

Following Saavedra-Rivano, Hoàng Xuân Sính’s thesis discusses:

‘$\otimes$-catégories’, which have a $\otimes$ functor obeying no laws,

‘catégories associative’, which are $\otimes$-catégories with an associator obeying the pentagon identity

‘catégories AU’, which are what we’d call monoidal categories

‘catégories AC’, which are $\otimes$-categories with an associator obeying the pentagon identity and also a symmetry obeying the hexagon identity, and

‘catégories ACU’, which are what we’d call symmetric monoidal categories.

So, it’s a bit baroque by modern standards, but all the definitions of these things are just what a modern mathematician would expect!

By the way, Hoàng Xuân Sính does not cite Mac Lane’s famous 1963 paper “Natural associativity and commutativity”. Instead, she cites his later 1965 paper “Categorical algebra”. She does not cite Bénabou’s “Catégories avec multiplication”. Instead, she cites his *thesis* from 1966, which I have not read. She also cites the Eilenberg–Kelly paper “Closed categories” from 1966, so she would have had a chance to see the term ‘monoidal category’. But I don’t think she ever mentions this term.

And just in case you were wondering: Mac Lane’s 1965 expository paper “Categorical algebra” *also* does not use the term ‘monoidal category’. In the last section Mac Lane talks about things that we would call monoidal categories… *but again he calls them bicategories!*

## Re: Who Invented Monoidal Categories?

As this might not be obvious to all readers:

And thus not baroque at all, merely a naming of the requirements of each.