Implication Details
Claim: If a category has ℵ₁-filtered colimits, then it is Cauchy complete.
Proof: More generally, if is any infinite regular cardinal, a category with -filtered colimits must be Cauchy cocomplete. This is because the walking idempotent is -filtered. See also Makkai-Pare, Prop. 2.2.1.
Show 8 categories using this implication
- category of coproducts of Euclidean spaces
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of finite sets of cardinality a power of 3
- category of finitely generated free modules over Z x Z
- category of partially ordered sets without isolated points
- category of sets and relations
- walking idempotent