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.
This implication has a dual.
Show 50 categories using this implication
- empty category
- trivial category
- discrete category on two objects
- category of large families of abelian groups
- category of finite sets and bijections
- delooping of a group
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of an infinite uncountable group
- category of compact Hausdorff spaces
- category of coproducts of Euclidean spaces
- category of F(I)-sets
- 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 free abelian groups
- category of finitely generated free modules over Z x Z
- category of finitely generated 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
- partially ordered collection of ordinal numbers
- category of pseudo-metric spaces with non-expansive maps
- category of partially ordered sets without isolated points
- category of sets and relations
- category of sets
- category of sets and bijections
- discrete category of sets
- category of large families of sets
- category of large families of sets and bijections
- category of large families of sets which are mostly empty
- category of large families of sets which are mostly singletons
- indiscrete category of sets
- category of sheaves
- category of abelian sheaves
- category of topological spaces
- category of pointed topological spaces
- category of long transfinite sequences of abelian groups
- category of uniform spaces
- category of large families of vector spaces which are mostly zero
- category of large vector spaces over a large field with a small basis
- category of Z-functors
- category of Z-sets
- cocompletion of a discrete–pair join
- category of ω-sets
- walking idempotent
- walking isomorphism