Category of categories

Simpson's lemma

Let be isomorphic categories such that

  1. is a category of categories;
  2. is a category of categories; and
  3. there exist categories isomorphic to and respectively.1

Then every category is isomorphic to a category and vice versa.

Proof

Since functors are precisely morphisms and functors determine composition, it follows that the isomorphism class of (as a category) is determined by the isomorphism class of . The same goes in the opposite direction.

A corollary is that any pseudoautistic category of categories containing categories isomorphic to and is autistic.2

#state/tidy | #lang/en | #SemBr

Footnotes

  1. The walking morphism and composition respectively. ↩

  2. 1999. FOM: Russell paradox for naive category theory ↩