Coherence condition explained
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
, called the
associator:
\alphaA,B,C\colon(A ⊗ B) ⊗ C → A ⊗ (B ⊗ C)
in the category. Using compositions of these
, one can construct a morphism
((AN ⊗ AN-1) ⊗ AN-2) ⊗ … ⊗ A1) → (AN ⊗ (AN-1 ⊗ … ⊗ (A2 ⊗ A1)).
Actually, there are many ways to construct such a morphism as a composition of various
. 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
, the following diagram commutes.
Any pair of morphisms from
(( … (AN ⊗ AN-1) ⊗ … ) ⊗ A2) ⊗ A1)
to
(AN ⊗ (AN-1 ⊗ ( … ⊗ (A2 ⊗ A1) … ))
constructed as compositions of various
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 be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms and . By composing these with f, we construct two morphisms:
, and
.Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement:
.
Associativity of composition
Let, and 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:
, and
.We have now the following coherence statement:
.
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
References
- 10.1016/0021-8693(64)90018-3 . On MacLane's conditions for coherence of natural associativities, commutativities, etc . 1964 . Kelly . G.M . Journal of Algebra . 1 . 4 . 397–402 .
- Book: Kelly . G. M. . Laplaza . M. . Lewis . G. . Mac Lane . Saunders. 10.1007/BFb0059553 . Coherence in Categories . Lecture Notes in Mathematics . 1972 . 281 . 978-3-540-05963-9 .
- 10.1016/0022-4049(86)90005-8 . A universal property of the convolution monoidal structure . 1986 . Im . Geun Bin . Kelly . G.M. . Journal of Pure and Applied Algebra . 43 . 75–88 .
- Book: 10.1007/978-1-4612-0783-2_11 . Tensor Categories . Quantum Groups . Graduate Texts in Mathematics . 1995 . Kassel . Christian . 155 . 275–293 . 978-1-4612-6900-7 .
- Book: 10.1007/BFb0059555 . Coherence for distributivity . Coherence in Categories . Lecture Notes in Mathematics . 1972 . Laplaza . Miguel L. . 281 . 29–65 . 978-3-540-05963-9 .
- 10.1006/aima.1999.1881 . free . A Coherent Approach to Pseudomonads . 2000 . Lack . Stephen . Advances in Mathematics . 152 . 2 . 179–202 .
- Natural Associativity and Commutativity . 1911/62865 . October 1963 . MacLane . Saunders. Rice Institute Pamphlet - Rice University Studies .
- Book: Mac Lane, Saunders . Saunders Mac Lane . 1971 . Categories for the working mathematician . Graduate texts in mathematics . 4 . Springer . 7. Monoids §2 Coherence . 161–165 . Categories for the Working Mathematician . 10.1007/978-1-4612-9839-7_8 . 9781461298397 . https://link.springer.com/chapter/10.1007/978-1-4612-9839-7_8.
- 10.1016/0022-4049(85)90087-8 . Coherence for bicategories and indexed categories . 1985 . MacLane . Saunders . Paré . Robert . Journal of Pure and Applied Algebra . 37 . 59–80 .
- 10.1016/0022-4049(89)90113-8 . A general coherence result . 1989 . Power . A.J. . Journal of Pure and Applied Algebra . 57 . 2 . 165–173 .
- The syntax of coherence . Cahiers de Topologie et Géométrie Différentielle Catégoriques . 2000 . 41 . 4 . 255–304 . Yanofsky . Noson S. .
External links
- 2109.01249 . Malkiewich . Cary . Ponto . Kate . Coherence for bicategories, lax functors, and shadows . 2021 . math.CT .
