Coherence condition

Template:Short description Template:Multiple issues In mathematics, and particularly category theory, 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.

An illustrative example: a monoidal category

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

${\alpha }_{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C)$

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

$((A_N\otimes A_{N-1})\otimes A_{N-2})\otimes \cdots \otimes A_1)\to (A_N\otimes (A_{N-1}\otimes \cdots \otimes (A_2\otimes A_1)).$

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

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 $A,B,C,D$, the following diagram commutes.

Any pair of morphisms from $((\cdots (A_N\otimes A_{N-1})\otimes \cdots )\otimes A_2)\otimes A_1)$ to $(A_N\otimes (A_{N-1}\otimes (\cdots \otimes (A_2\otimes A_1)\cdots ))$ constructed as compositions of various ${\alpha }_{A,B,C}$ 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 Template:Nowrap be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms Template:Nowrap and Template:Nowrap. By composing these with f, we construct two morphisms:

Template:Nowrap, and

Template:Nowrap.

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

Template:Nowrap.

Associativity of composition

Let Template:Nowrap, Template:Nowrap and Template:Nowrap 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:

Template:Nowrap, and

Template:Nowrap.

We have now the following coherence statement:

Template:Nowrap.

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

• Pseudoalgebra

References

• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite book
• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal

External links

• Template:Cite arXiv