nLab coherence law

In the definition of $(n,r)$-categories

In the definition of (n,r)-categories in higher category theory the coherence laws assert that:

Semi-formal Definition

(Coherence)

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.

A coherence theorem is an assertion that with a given definition of n-category-structure, coherence is satisfied.

Remark

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 $n$: 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 ($n = 2$) or a tricategory ($n=3$) or a tetracategory ($n = 4$) is in degree $n+1$ a complicated equation that asserts that a certain contractible space of higher morphisms is exactly equal to the point.

Examples

We start the list of examples with a warning on how not to misunderstand the above definition.

Warning

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 $(n,r)$-categories trivial, anyway!)

For instance in a braided monoidal category there is, in general, a non-contractible space of morphisms $x \otimes y \to x \otimes y$ for any two objects, because the double braiding $B_{y \otimes x}\circ B_{x \otimes y}$ 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 $(n,r)$-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 $B_{y \otimes x}\circ B_{x \otimes y}$ 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-category to be trivial!

But the role of the associator, which now is a 3-morphism witnessing the non-associativity 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 $n = 3$: 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

Monoidal categories

Example

(monoidal categories)

The coherence theorem for monoidal categories asserts that with the standard definition of monoidal category, there is a unique composite of associators that re-bracket any sequence of tensor products.

Quasi-categories

Example

(quasi-categories)

For $C$ a simplicial set that is a quasi-category we have (as discussed there) that the canonical morphism

$$sSet(\Delta[2], C) \to sSet(\Lambda^1[2], C)$$

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.

Trimble $\omega$-categories

Example

(Trimble $\omega$-categories)

In a Trimble n-category the space of choices of composing a sequence of $n$ morphisms is explicitly the topological space $Top_{0,1}(I, I^{\vee n})$ of surjections from the unit interval $[0,1]$ onto the length $n$-interval $[0,n]$. The coherence law of composition in a Trimble $n$-category is the fact that these spaces are contractible.

$n$-Groupoids

Example

In an n-groupoid modeled as a $(n+1)$-coskeletal Kan complex, the coherence law is the condition that every $(n+1)$-sphere $\partial \Delta^{n+1}$ has a unique filler. This says that the corresponding space of choices is a point.

More generally, for a homotopy n-type modeled as a Kan complex, the coherence law is just that all these sphere fillers exist, which only says that there is a contractible space of choices.

For other algebraic structures

One can consider coherence laws for algebraic structures other than $(n,r)$-categories. See coherence theorem for more.

Related concepts

• rectification

References

The coherence law of associativity is stated in

• Hermann Grassmann, §3 of Ausdehnungslehre, 1844