Implication details
Claim: If a category has exact filtered colimits, then it has filtered-colimit-stable monomorphisms.
Proof: This is because is a monomorphism iff the diagram is a pullback, and if a functor preserves finite limits, it preserves pullbacks in particular.
This implication has a dual.
Show 23 categories using this implication
- category of large families of abelian groups
- category of Banach spaces with linear contractions
- category of compact Hausdorff spaces
- category of F(I)-sets
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of locally ringed spaces
- category of M-sets
- category of metric spaces with non-expansive maps
- category of extended metric spaces
- category of pseudo-metric spaces with non-expansive maps
- category of sets
- category of large families of sets
- category of large families of sets which are mostly singletons
- indiscrete category of sets
- category of sheaves
- category of abelian sheaves
- category of long transfinite sequences of abelian groups
- 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
- walking isomorphism