Mac Lane coherence theorem explained

In category theory, a branch of mathematics, Mac Lane's coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. But regarding a result about certain commutative diagrams, Kelly is states as follows: "no longer beseen as constituting the essence of a coherence theorem". More precisely (cf.

1. Counter-example

), it states every formal diagram commutes, where "formal diagram" is an analog of well-formed formulae and terms in proof theory.

The theorem can be stated as a strictification result; namely, every monoidal category is monoidally equivalent to a strict monoidal category.

Counter-example

It is not reasonable to expect we can show literally every diagram commutes, due to the following example of Isbell.

Let

Set _{0} \subset Set be a skeleton of the category of sets and D a unique countable set in it; note D x D = D by uniqueness. Let p : D = D x D \to D be the projection onto the first factor. For any functions f, g: D \to D , we have f \circ p = p \circ (f x g) . Now, suppose the natural isomorphisms \alpha: X x (Y x Z) \simeq (X x Y) x Z are the identity; in particular, that is the case for X = Y = Z = D . Then for any f, g, h: D \to D , since \alpha is the identity and is natural,f \circ p = p \circ (f x (g x h)) = p \circ \alpha \circ (f x (g x h)) = p \circ ((f x g) x h) \circ \alpha = (f x g) \circ p .Since p is an epimorphism, this implies f = f x g . Similarly, using the projection onto the second factor, we get g = f x g and so f = g , which is absurd.

Proof

Coherence condition

In monoidal category

C , the following two conditions are called coherence conditions:⊗ \colon C x C\to C called the tensor product, a natural isomorphism \alpha _{A,B,C} , called the associator:\alpha _{A,B,C} \colon (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C)

• Also, let

I an identity object and I has a left identity, a natural isomorphism λ _{A} called the left unitor:λ _{A} : I ⊗ A → A

as well as, let

I has a right identity, a natural isomorphism \rho _{A} called the right unitor:\rho _{A} : A ⊗ I → A .

Pentagon identity and triangle identity

See also

• Coherency (homotopy theory)
• Monoidal category
• Symmetric monoidal category
• Coherence condition

References

• 10.1017/S0960129508007184 . On traced monoidal closed categories . 2009 . Hasegawa . Masahito . Mathematical Structures in Computer Science . 19 . 2 . 217–244 .
• 10.1006/aima.1993.1055 . free . Braided Tensor Categories . 1993 . Joyal . A. . André Joyal . Street . R. . Ross Street . . 102 . 1 . 20–78 .
• Natural Associativity and Commutativity . 1911/62865 . October 1963 . MacLane . Saunders . Rice Institute Pamphlet - Rice University Studies .
• 10.1090/S0002-9904-1965-11234-4 . Categorical algebra . 1965 . MacLane . Saunders . Bulletin of the American Mathematical Society . 71 . 1 . 40–106 . free .
• Book: Mac Lane, Saunders . [{{Google books|MXboNPdTv7QC|page=165|plainurl=yes}} Categories for the working mathematician ]. Springer . New York . 1998 . 0-387-98403-8 . 37928530.
• Section 5 of Saunders Mac Lane, 10.1090/S0002-9904-1976-13928-6 . Topology and logic as a source of algebra . 1976 . Mac Lane . Saunders . Bulletin of the American Mathematical Society . 82 . 1 . 1–40 . free .
• Turning monoidal categories into strict ones . The New York Journal of Mathematics [Electronic Only] . 2001 . 7 . 257–265 . Schauenburg . Peter. 1076-9803.

Further reading

• Book: 10.1007/BFb0063106 . Coherence theorems for lax algebras and for distributive laws . Category Seminar . Lecture Notes in Mathematics . 1974 . Kelly . G. M. . 420 . 281–375 . 978-3-540-06966-9 .

External links

• Web site: Armstrong . John . Mac Lane's Coherence Theorem . The Unapologetic Mathematician. 29 June 2007 .
• Web site: Etingof . Pavel . Gelaki . Shlomo . Nikshych . Dmitri . Ostrik . Victor . 18.769, Spring 2009, Graduate Topics in Lie Theory: Tensor Categories §.Lecture 3 . MIT Open Course Ware.
• Web site: coherence theorem for monoidal categories . ncatlab.org.
• Web site: Mac Lane's proof of the coherence theorem for monoidal categories . ncatlab.org.
• Web site: coherence and strictification. ncatlab.org.
• Web site: coherence and strictification for monoidal categories. ncatlab.org.