CatDat

Implication Details

Claim: If a category has ℵ₁-filtered colimits, then it is Cauchy complete.

Proof: More generally, if $\kappa$ is any infinite regular cardinal, a category with $\kappa$-filtered colimits must be Cauchy cocomplete. This is because the walking idempotent is $\kappa$-filtered. See also Makkai-Pare, Prop. 2.2.1.