Mac Lane coherence theorem explained

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.

1. 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

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.

References

• Book: Mac Lane, Saunders . Categories for the working mathematician . Springer . New York . 1998 . 0-387-98403-8 . 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/