For $n=2$, turns out that all 2-$categories$ are strictifiable, i.e., the notions of weak and strict bicategories coincide. However for $n \geq 3$, the situation becomes wild. In particular, there exists a counterexample sufficiently intuitive (well, as much intuitive as something can be in that context :)) from topology, and a discrepancy between weak and strict 3-groupoids shows up. In general the narrative goes as follows
"It follows that any non-symmetric braided monoidal category is a weak 3- category not equivalent to a strict one ... " (Tom Leinster - Higher Operads and Higher Categories, pg. xxii.)
Now I am going back at the beginning of this post, and particularly to the definition of bicategories. We know that there is a certain axiom showing up, the pentagon axiom, when someone writes down the definition of such a category. For someone with an idea of what a monoidal category is, isn't a big surprise or an ambiguity. Moreover, someone can observe that this pentagon is the same thing as the "associahedron" $K_4$. My question is, what happens with the diagrams in an arbitrary $n$-category (choose $n=3$ if you like), and how we generalise this pentagon axiom? Can we make explicit usage of the three dimensional associahedron $K_5$ to describe the generalisation of it in a 3-category for instance? If so, can something be said accordingly for arbitrary $n$ and $K_{n+2}$?
P.S.
I know that writing out diagrams even for 3-cateegories is a difficult task to do, therefore any suggestions for relevant papers or texts along with comments how can someone think of the generalisation of the pentagon axiom might be helpful.