Category of categories

Simpson’s lemma

Let 𝖢1 ≅𝖢2 be isomorphic categories such that

1. 𝖢1 is a category of categories;
2. 𝖢2 is a category of categories; and
3. there exist categories 𝖬,𝖭 ∈𝖢1 isomorphic to Interval category and [[Ordinal category|3――]] respectively.¹

Then every category 𝖠 ∈𝖢1 is isomorphic to a category 𝖡 ∈𝖢2 and vice versa.

Proof

Since functors 𝖬 →𝖠 are precisely morphisms and functors 𝖭 →𝖠 determine composition, it follows that the isomorphism class of 𝖠 (as a category) is determined by the isomorphism class of 𝖢1. The same goes in the opposite direction.

A corollary is that any pseudoautistic category of categories containing categories isomorphic to 𝟤 and 𝟥 is autistic.²

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Footnotes

1. The walking morphism and composition respectively. ↩

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