2-cells but which is standardly presented concretely, where the interchange axiom becomes a not entirely trivial theorem. I think it's fair to say that operations automatically satisfy all the right coherence laws if you define them using universal properties. We study what happens when coherence fails. These groupoids are used in the construction of the free associative category. These groups are shown to be more than just free groups.
Some progress is made in classifying noncoherence of associative categories.
Each structure has interesting theorems that can be proven about it. If coherence theory is to be thought of as some type of higher-dimensional algebra, then there are many structures that are of interest.