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 Interval category and [[Ordinal category|]] 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

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Footnotes

1. The walking morphism and composition respectively. ↩

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