Detection of filtered-colimit-stable monomorphisms
Claim
Let be a category with filtered colimits. Assume that is faithful functor which preserves monomorphisms and filtered colimits. If monomorphisms in are stable under filtered colimits, then the same is true for .For the record, here is the dual statement: Let be a category with cofiltered limits. Assume that is faithful functor which preserves epimorphisms and cofiltered limits. If epimorphisms in are stable under cofiltered limits, then the same is true for .
Proof
Since is faithful, it reflects monomorphisms. From here the proof is straight forward.Usage
This lemma is referenced in the following categories:
- category of pairs of sets
- category of pointed sets
- category of non-empty sets
- category of groups
- category of rings
- category of algebras
- category of rngs
- category of monoids
- category of M-sets
- category of small categories
- category of measurable spaces
- category of metric spaces with ∞ allowed
- category of topological spaces
- category of pointed topological spaces
- category of Hausdorff spaces
- category of sheaves
- category of Z-functors
- category of prosets