Monoidal Category (Coherence Conditions) In the definition of a **monoidal category** below, can someone please explain the idea behind the coherence conditions, especially the pentagon diagram's construction. Why do we need four elements A,B,C, and D in category **C**? Is the idea to try and cover all possible associative groupings for all 4 objects in **C**? ![monoidal category](

The thing we want the structure of a monoidal category to have is that there is a unique coherence isomorphism between any two different ways of associating a product of arbitrary length, as well as introducing and eliminating an arbitrary number of unit factors.

For ordinary monoids, we know that general associativity follows from the basic $(ab)c = a(bc)$ form. (and similarly for unit introduction/elimination)

This is basically enough to prove that you can construct an arbitrary coherence morphism using just associators $\alpha_{ABC} : (A \otimes B) \otimes C \to A \otimes (B \otimes C)$ along with the two unitors, the monoidal product, and composition. This allows the monoidal structure to be defined using just these three natural isomorphisms.

So what remains is a way to ensure that coherence morphisms are unique. Over time, it has been shown that requiring just the pentagon and triangle laws is enough to guarantee uniqueness.