Motivation/Intuition for the Pentagon Axiom

Question

I have just started reading a bit on monoidal categories, and there is I just can't make much sense of: the Pentagon Axiom.

To provide some context, we have a category $\mathcal{C}$ together with a tensor product $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ and an associator or associativity constraint $a$, which is a natural isomorphism $$a \colon \otimes(\otimes \times \text{id}) \to \otimes(\text{id} \times \otimes) \, .$$

For $\mathcal{C}$ to be a conoidal category we need the associator to satisfy the commutativity of this diagram

for all $A, B, C, D \in \mathcal{C}$.

While the existence of an associator and the left and right units make perfect sense to me, I can't think of the Pentagon Axiom as something "natural" to impose the associator. Can somebody give me some motivation as to why one wants this property to be satisfied in a monoidal category?

Answer 1

Suppose that your diagram didn't commute. For example, suppose that you started at the top left corner, and traveled to the top right corner using the two different paths available to you, and you got different answers. Then there wouldn't be a unique path from $(((A\otimes B)\otimes C)\otimes D$ to $A\otimes(B\otimes (C\otimes D)))$.

Why would we care about this?

The simple answer is this: you want this diagram to commute so that you have a coherent way of discussing the tensored product of four different objects (hence the name coherence diagram, which it is usually referred to as). In addition, if this diagram didn't commute, this wouldn't really model a lot of categories which we would otherwise morally view as monoidal. For example, the category of tangles Tang is a monoidal category, with a monoidal product being horizontal stacking of tangles. While there are five different ways of stacking 4 different tangles, it wouldn't make sense for them to suddenly become unequal; these are physical objects, and the resulting tangle should be the same.

The better answer is this: this diagram is used to prove an extremely important theorem relevant to monoidal categories; namely, Mac Lane's Coherence Theorem for monoidal categories. The theorem basically states that a large class of diagrams in any monoidal category commute. What this then says is that, given a tensored object with $n$ (non-identity) objects (but possibly including arbitrary instances of identity objects), there is a unique, canonical, isomorphism to any other way you could possibly write that object. To prove this, he uses the pentagonal diagram, alongside the triangular diagrams (let me know if you don't know what I'm talking about) in his proof.

The proof is extremely clever, and is an important read because Mac Lane figures out a way to solve a problem which has a very nontrivial solution. If you plan on working with monoidal categories, I highly suggest studying it.