Implication Details
Claim: If a category is finitely accessible, then it has filtered-colimit-stable monomorphisms.
Proof: Let be a finitely accessible category and let be a set of finitely presentable objects which generates under filtered colimits. Consider as a full subcategory and consider the restricted Yoneda embedding . It preserves filtered colimits (essentially by the definition of a finitely presentable object) and all limits, in particular monomorphisms. It also reflects monomorphisms since is a generating set. Therefore, since and hence the functor category has filtered-colimit-stable monomorphisms, this is also true for .
Show 22 categories using this implication
- category of Banach spaces with linear contractions
- category of compact Hausdorff spaces
- category of filtered vector spaces
- category of finite sets and bijections
- category of finite sets and surjections
- category of Hausdorff spaces
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of non-empty sets
- category of partially ordered sets
- category of preordered sets
- category of sets with a distinguished subset
- category of small categories
- category of torsion abelian groups
- category of torsion-free abelian groups
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- delooping of an infinite uncountable group
- discrete category on two objects
- empty category
- dual of the category of sets
- dual of the category of topological spaces