n-category = (n,n)-category
n-groupoid = (n,0)-category
While associativity and uniticity of composition of k-morphisms holds only up to choices of higher morphisms, coherence is the demand that the collection of these choices forms a contractible ∞-groupoid.
It is the fact that this condition makes recourse among all (n,r)-categories to (∞,0)-categories = ∞-groupoids that it is easier to define and handle an (∞,r)-category than an (n,r)-category for finite : in the former case one just needs that certain spaces are contractible without necessarily being equal to the point, while in the latter case one demands some of these to be exactly equal to the point, which is a condition much harder to get under control. Accordingly, a coherence law in an (n,n)-category such as a bicategory () or a tricategory () or a tetracategory () is in degree a complicated equation that asserts that a certain contractible space of higher morphisms is exactly equal to the point.
We start the list of examples with a warning on how not to misunderstand the above definition.
The demand for contractible spaces of choices of associators and unitors is not to be confused with asserting contractible hom-spaces in general (which would make the theory of -categories trivial, anyqay!)
For instance in a braided monoidal category there is, in general, a non-contractible space of morphisms for any two objects , because the double braiding will not in general be equal to the identity morphism. But the braiding morphisms also is not the kind of structural morphism that the above definition refers to. In the definition of braided monoidal category it may look as if it is on par with the associator, but this is in fact not so:
in the context of -categories we may use the periodic table of higher categories to identify the braided monoidal category with a one-object-one-morphism 3-category. In this, the non-trivial double braiding is a nontrivial 3-morphism that represents a nontrivial element of the 3rd homotopy group of the 3-category. Demanding this to be trivial would be demanding the 3-caztegory to be trivial!
But the role of the associator, which now is a 3-morphism witnessing the non-associativitiy of 2-morphisms under horizontal composition is quite different. This is best seen by looking at a geometric definition of higher categories, such as an (∞,n)-category for : here the associator is a choice of horn-filler, and this choice by construction happens in a contractible space and by construction cannot contain nontrivial cells like the double braiding. This is also amplified by the following example
The coherence theorem for monoidal categories asserts that the with the standar definition of monoidal category, there is a unique composite of associators that re-bracket any sequence of tensor products.
sSet(\Delta, C) \to sSet(\Lambda^1, C)
is an acyclic Kan fibration. This means that its fibers are contractible ∞-groupoids. But these fibers are exactly the spaces whose points are choices of composition rules, whose morphisms are comparison maps between these, and so on.
Therefore the contractibility of these fibers is the coherence of the associators for the quasi-category.
In a Trimble n-category the space of choices of composing a sequence of morphisms is explicitly the topological space of surjections from the unit interval onto the length -interval . The coherence law of composition in a Trimble -category is the fact that these spaces are contractible.
One can consider coherence laws for algebraic structures other than -categories. See coherence theorem for more.