Coherence condition

In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".

The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra.

Coherent isomorphism

In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing canonical isomorphisms. But in some cases, such as prestacks, there can be several canonical isomorphisms and there might not be an obvious choice among them.

In practice, coherent isomorphisms arise by weakening equalities; e.g., strict associativity may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a weak 2-category from that of a strict 2-category.

Replacing coherent isomorphisms by equalities is usually called strictification or rectification.

Coherence theorem

Mac Lane's coherence theorem states, roughly, that if diagrams of certain types commute, then diagrams of all types commute.^[1] A simple proof of that theorem can be obtained using the permutoassociahedron, a polytope whose combinatorial structure appears implicitly in Mac Lane's proof.^[2]

There are several generalizations of Mac Lane's coherence theorem.^[3] Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".^[4]

Homotopy coherence

Coherence condition

A coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.

Part of the data of a monoidal category is a chosen morphism ${\displaystyle \alpha _{A,B,C}}$, called the associator:

${\displaystyle \alpha _{A,B,C}\colon (A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)}$

for each triple of objects ${\displaystyle A,B,C}$ in the category. Using compositions of these ${\displaystyle \alpha _{A,B,C}}$, one can construct a morphism

${\displaystyle ((A_{N}\otimes A_{N-1})\otimes A_{N-2})\otimes \cdots \otimes A_{1})\rightarrow (A_{N}\otimes (A_{N-1}\otimes \cdots \otimes (A_{2}\otimes A_{1})).}$

Actually, there are many ways to construct such a morphism as a composition of various ${\displaystyle \alpha _{A,B,C}}$. One coherence condition that is typically imposed is that these compositions are all equal.^[5]

Typically one proves a coherence condition using a coherence theorem, which states that one only needs to check a few equalities of compositions in order to show that the rest also hold. In the above example, one only needs to check that, for all quadruples of objects ${\displaystyle A,B,C,D}$, the following diagram commutes.

Any pair of morphisms from ${\displaystyle ((\cdots (A_{N}\otimes A_{N-1})\otimes \cdots )\otimes A_{2})\otimes A_{1})}$ to ${\displaystyle (A_{N}\otimes (A_{N-1}\otimes (\cdots \otimes (A_{2}\otimes A_{1})\cdots ))}$ constructed as compositions of various ${\displaystyle \alpha _{A,B,C}}$ are equal.

Further examples

Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.

Identity

Let f : A → B be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms 1_A : A → A and 1_B : B → B. By composing these with f, we construct two morphisms:

f o 1_A : A → B, and

1_B o f : A → B.

Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement:

f o 1_A = f  = 1_B o f.

Associativity of composition

Let f : A → B, g : B → C and h : C → D be morphisms of a category containing objects A, B, C and D. By repeated composition, we can construct a morphism from A to D in two ways:

(h o g) o f : A → D, and

h o (g o f) : A → D.

We have now the following coherence statement:

(h o g) o f = h o (g o f).

In these two particular examples, the coherence statements are theorems for the case of an abstract category, since they follow directly from the axioms; in fact, they are axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.

See also

• Coherence condition
• Canonical isomorphism
• 2-category
• Pseudoalgebra
• Tricategory