Mac Lane coherence theorem

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 be seen as constituting the essence of a coherence theorem". More precisely (cf. #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.

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

Let ${\displaystyle {\mathsf {Set}}_{0}\subset {\mathsf {Set}}}$be a skeleton of the category of sets and D a unique countable set in it; note ${\displaystyle D\times D=D}$by uniqueness. Let ${\displaystyle p:D=D\times D\to D}$be the projection onto the first factor. For any functions ${\displaystyle f,g:D\to D}$, we have ${\displaystyle f\circ p=p\circ (f\times g)}$. Now, suppose the natural isomorphisms ${\displaystyle \alpha :X\times (Y\times Z)\simeq (X\times Y)\times Z}$are the identity; in particular, that is the case for ${\displaystyle X=Y=Z=D}$. Then for any ${\displaystyle f,g,h:D\to D}$, since ${\displaystyle \alpha }$is the identity and is natural,

${\displaystyle f\circ p=p\circ (f\times (g\times h))=p\circ \alpha \circ (f\times (g\times h))=p\circ ((f\times g)\times h)\circ \alpha =(f\times g)\circ p}$.

Since ${\displaystyle p}$is an epimorphism, this implies ${\displaystyle f=f\times g}$. Similarly, using the projection onto the second factor, we get ${\displaystyle g=f\times g}$and so ${\displaystyle f=g}$, which is absurd.

In monoidal category ${\displaystyle C}$, the following two conditions are called coherence conditions:

• Let a bifunctor ${\displaystyle \otimes \colon \mathbf {C} \times \mathbf {C} \to \mathbf {C} }$called the tensor product, a natural isomorphism ${\displaystyle \alpha _{A,B,C}}$, called the associator:

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

• Also, let ${\displaystyle I}$an identity object and ${\displaystyle I}$has a left identity, a natural isomorphism ${\displaystyle \lambda _{A}}$called the left unitor:

${\displaystyle \lambda _{A}:I\otimes A\rightarrow A}$

as well as, let ${\displaystyle I}$has a right identity, a natural isomorphism ${\displaystyle \rho _{A}}$called the right unitor:

${\displaystyle \rho _{A}:A\otimes I\rightarrow A}$.

To satisfy the coherence condition, it is enough to prove just the pentagon and triangle identity, which is essentially the same as what is stated in Kelly's (1964) paper.

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

• Armstrong, John (29 June 2007). "Mac Lane's Coherence Theorem". The Unapologetic Mathematician.
• 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.
• "coherence theorem for monoidal categories". ncatlab.org.
• "Mac Lane's proof of the coherence theorem for monoidal categories". ncatlab.org.
• "coherence and strictification". ncatlab.org.
• "coherence and strictification for monoidal categories". ncatlab.org.
• "pentagon identity". ncatlab.org.