Mac Lane coherence theorem

In category theory, a branch of mathematics, Mac Lane coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. 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.

Counter-example

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

Let ${\mathsf{S}\mathsf{e}\mathsf{t}}_0\subset \mathsf{S}\mathsf{e}\mathsf{t}$ be a skeleton of the category of sets and D a unique countable set in it; note $D\times D=D$ by uniqueness. Let $p:D=D\times D\to D$ be the projection onto the first factor. For any functions $f,g:D\to D$, we have $f\circ p=p\circ (f\times g)$. Now, suppose the natural isomorphisms $\alpha :X\times (Y\times Z)\simeq (X\times Y)\times Z$ 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,

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

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

References

• Mac Lane, Saunders (1998). Categories for the working mathematician. New York: Springer. ISBN 0-387-98403-8. OCLC 37928530. 
• 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.

External links

• 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/