# MacLane coherence theorem for “monoidal” category without 1

MacLane's coherence theorem for a monoidal category states that once the associators for 4-fold products are compatible (i.e., the pentagon axiom holds), it holds for n-fold products, so I can bracket n-fold products in any way I like.

What happens if I forget the unit, i.e. I consider "semigroup categories" as they are called in the book "Tensor Categories" by Etingof, Gelaki, Nikshych and Ostrik. Is it still true that the pentagon axiom implies that the bracketing of n-fold products does not matter?

The proof of the coherence theorem given in the book relies on the unit. Still, they say that "semigroup categories" categorify semigroups. Then they should better satisfy coherence in my opinion. But I don't know.

• Yes, it's Theorem 3.1 in MacLane's excellent paper "Natural associativity and commutativity". – Atsche Mar 5 '18 at 8:48
• You may wish to make your comment an actual answer, and, after the enforced delay, accept it. – David Roberts Mar 5 '18 at 8:50

## 2 Answers

Sorry, I noticed this an hour after posting: Yes, it's true, see Theorem 3.1 in MacLane's excellent paper "Natural associativity and commutativity".

It's a mystery to my why this is so scarcely mentioned.

• I assume for similar reasons to the lack of emphasis on rings without unit in more classical algebra. I've never once encountered a monoidal category without unit in nature. – Kevin Arlin Mar 6 '18 at 21:12
• Well, but "pedagogically" it's not so wise in my opinion with regards to the pentagon axiom. I find it much better, as in MacLane's paper, to first introduce multiplication, then talk about what coherence means, then prove that coherence is equivalent to pentagon axiom, then introduce what a unit is. Suddenly, all makes sense. – Atsche Mar 6 '18 at 21:37
• @KevinCarlson finite abelian groups, say? – მამუკა ჯიბლაძე Mar 6 '18 at 23:29
• @მამუკა ჯიბლაძე Haha, you got me-similarly, I've heard of the rng of even integers. Atsche, that's a nice point. I think the ratio of people who mention Maclane's then to people who have actually read his paper is probably unjustly small. – Kevin Arlin Mar 7 '18 at 0:49

The proof of Mac Lane's coherence theorem is fairly ad hoc. However, higher dimensional rewriting provides some general methods to prove coherence-like theorems. Using this, it is very easy to prove the coherence of "semigroup categories".

See for example "Coherence in monoidal track categories", by Y. Guiraud and P. Malbos.

• Hm, looking at this, I'm fine with MacLane's down to earth thing...:) – Atsche Mar 8 '18 at 7:08