Implication Details
Claim: If a category has biproducts and has filtered colimits and has filtered-colimit-stable monomorphisms and has products, then it satisfies CIP.
Proof: Let be a family of objects. For every finite subset the canonical morphism is a (split) monomorphism. Hence, their colimit is also a monomorphism, which is the canonical morphism .
This implication has a dual.
Show 22 categories using this implication
- trivial category
- category of abelian groups
- category of large families of abelian groups
- category of commutative monoids
- category of cochain complexes of abelian groups
- category of filtered vector spaces
- category of groups
- category of monoids
- category of left modules over a ring
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of rngs
- category of sequences of abelian groups
- category of abelian sheaves
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of long transfinite sequences of abelian groups
- category of vector spaces
- category of large families of vector spaces which are mostly zero
- category of graded abelian groups
- category of graded modules over a graded ring
- walking isomorphism