In category theory, a branch of mathematics, Mac Lane coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”.[1] 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.[2]

Let be a skeleton of the category of sets and D a unique countable set in it; note by uniqueness. Let be the projection onto the first factor. For any functions , we have . Now, suppose the natural isomorphisms are the identity; in particular, that is the case for . Then for any , since is the identity and is natural,

.

Since is an epimorphism, this implies . Similarly, using the projection onto the second factor, we get and so , which is absurd.

Proof

Notes

  1. Mac Lane 1998, Ch VII, § 2.
  2. Mac Lane 1998, Ch VII. the end of § 1.

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.


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