Simpson's lemma

Let $𝖢 1 ≅ 𝖢 2$ be isomorphic categories such that

$𝖢 1$ is a category of categories;
$𝖢 2$ is a category of categories; and
there exist categories $𝖬, 𝖭 \in 𝖢 1$ isomorphic to $2 ――$ and $3 ――$ respectively.¹

Then every category $𝖠 \in 𝖢 1$ is isomorphic to a category $𝖡 \in 𝖢 2$ and vice versa.

Proof

Since functors $𝖬 \to 𝖠$ are precisely morphisms and functors $𝖭 \to 𝖠$ determine composition, it follows that the isomorphism class of $𝖠$ (as a category) is determined by its isomorphism class in $𝖢 1$ . 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

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

The walking morphism and composition respectively. ↩
1999. FOM: Russell paradox for naive category theory ↩

Simpson's lemma

Footnotes