Implication Details
Claim: If a category has filtered colimits, then it has ℵ₁-filtered colimits.
Proof: Every -filtered category is also -filtered, i.e. filtered. Therefore, every -filtered diagram is also a filtered diagram, hence has a colimit by assumption.
Show 26 categories using this implication
- category of abelian sheaves
- category of compact Hausdorff spaces
- category of finite sets and bijections
- category of free abelian groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of metric spaces with continuous maps
- category of pointed topological spaces
- category of pseudo-metric spaces with non-expansive maps
- category of sets
- category of sets and relations
- category of sheaves
- category of topological spaces
- category of Z-functors
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- delooping of an infinite uncountable group
- partially ordered collection of ordinal numbers
- trivial category
- walking fork
- walking idempotent
- walking isomorphism
- walking parallel pair