CatDat

Implication Details

Assumptions: ℵ₁-filtered colimits

Conclusions: 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.