# Motivation/Intuition for the Pentagon Axiom

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?

• If we make a circular composition of the associators and arrive back to the same parenthesizing we started, then we should obtain the identity. The same follows somehow to possible parenthesizing of any number of tensored objects.. Mar 10, 2020 at 8:28
• @Berci Thank you for your comment! I have also found the nLab page enlightening (which makes me feel kind of stupid for not having seen that before). If you want to expand your comment a bit more to an answer, because I believe it might be useful to other people, feel free to. Otherwise you may copy past your comment to an answer and if there are not more detailed answers I will accept it, of course! Mar 10, 2020 at 8:42

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 $$A\otimes(B\otimes (C\otimes D)))$$.

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.