Coherence condition

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}\colon (A\otimes B)\otimes C\rightarrow 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})\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 $\alpha _{A,B,C}$ . One coherence condition that is typically imposed is that these compositions are all equal.^[1]

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 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.

Notes

^ (Kelly 1964, Introduction)

References

Kelly, G.M (1964). "On MacLane's conditions for coherence of natural associativities, commutativities, etc". Journal of Algebra. 1 (4): 397–402. doi:10.1016/0021-8693(64)90018-3.
Kelly, G. M.; Laplaza, M.; Lewis, G.; Mac Lane, Saunders (1972). Coherence in Categories. Lecture Notes in Mathematics. Vol. 281. doi:10.1007/BFb0059553. ISBN 978-3-540-05963-9.
Im, Geun Bin; Kelly, G.M. (1986). "A universal property of the convolution monoidal structure". Journal of Pure and Applied Algebra. 43: 75–88. doi:10.1016/0022-4049(86)90005-8.
Kassel, Christian (1995). "Tensor Categories". Quantum Groups. Graduate Texts in Mathematics. Vol. 155. pp. 275–293. doi:10.1007/978-1-4612-0783-2_11. ISBN 978-1-4612-6900-7.
Laplaza, Miguel L. (1972). "Coherence for distributivity". Coherence in Categories. Lecture Notes in Mathematics. Vol. 281. pp. 29–65. doi:10.1007/BFb0059555. ISBN 978-3-540-05963-9.
Lack, Stephen (2000). "A Coherent Approach to Pseudomonads". Advances in Mathematics. 152 (2): 179–202. doi:10.1006/aima.1999.1881.
MacLane, Saunders (October 1963). "Natural Associativity and Commutativity". Rice Institute Pamphlet - Rice University Studies. hdl:1911/62865.
Mac Lane, Saunders (1971). "7. Monoids §2 Coherence". Categories for the working mathematician. Graduate texts in mathematics. Vol. 4. Springer. pp. 161–165. doi:10.1007/978-1-4612-9839-7_8. ISBN 9781461298397.
MacLane, Saunders; Paré, Robert (1985). "Coherence for bicategories and indexed categories". Journal of Pure and Applied Algebra. 37: 59–80. doi:10.1016/0022-4049(85)90087-8.
Power, A.J. (1989). "A general coherence result". Journal of Pure and Applied Algebra. 57 (2): 165–173. doi:10.1016/0022-4049(89)90113-8.
Yanofsky, Noson S. (2000). "The syntax of coherence". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 41 (4): 255–304.

External links

Malkiewich, Cary; Ponto, Kate (2021). "Coherence for bicategories, lax functors, and shadows". arXiv:2109.01249 [math.CT].

[1] (Kelly 1964, Introduction)

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