CatDat

Implication Details

Claim: If a category is essentially countable and is thin and is ℵ₁-filtered, then it has a terminal object.

Proof: Let $\C$ be a thin, $\aleph_1$-filtered, and w.l.o.g. countable category. The identity diagram $\C \to \C$ admits a cocone. That is, there is an object $T$ with a morphism $A \to T$ for all $A \in \C$. Then $T$ is terminal.