Mac Lane coherence theorem

It is not reasonable to expect we can show literally every diagram commutes, due to the following example of Isbell.[2]

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.

• Saunders Mac Lane, Categories for the Working Mathematician, 2nd edition, Springer GTM, 1998.
• Section 5 of Saunders Mac Lane, Topology and Logic as a Source of Algebra (Retiring Presidential Address), Bulletin of the AMS 82:1, January 1976.

• https://ncatlab.org/nlab/show/coherence+theorem+for+monoidal+categories
• https://ncatlab.org/nlab/show/Mac+Lane%27s+proof+of+the+coherence+theorem+for+monoidal+categories
• https://unapologetic.wordpress.com/2007/06/29/mac-lanes-coherence-theorem/