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.



MacLane coherence theorem for "monoidal" category without 1
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.


