objects such that commutes with certain colimits
A pure subobject is a monomorphism – hence a subobject of some object in some category – which is a pure morphism: such that any sufficiently small system of equations involving constants in that admits a solution in also admits a solution in . This generalises the classical notions of ‘pure group’ and ‘pure submodule’.
Given any morphisms , , in , if both and are -compact and ,
then there exists a (not necessarily unique) morphism in such that . (We do not assert any compatibility with , however.)
A -pure subobject is a -pure monomorphism.
A retract is a -pure subobject in any category, for any .
Conversely, any -pure subobject in Set is a retract.
If is an injective module and is any module containing as a submodule, then the inclusion is -pure. (This can be checked directly without recourse to the fact that any injective submodule is a retract!)
In any category:
The class of -pure morphisms is closed under composition.
If is a -pure morphism, then so is .
If , then any -pure morphism is also -pure.
In a -accessible category, any -pure morphism is necessarily monic.
This is LPAC, Prop. 2.29.
If is a -accessible category with pushouts, then any -pure subobject in is a -filtered colimit in of retracts in .
This is LPAC, Prop. 2.30.
In a -accessible category, every -pure morphism is a regular monomorphism.
This is LPAC, Prop. 2.31.
Let be a -accessible category, and let be a full subcategory of that is closed under -filtered colimits for some regular cardinal . Then, is a -accessible category for some regular cardinal sharply larger than if and only if is closed under -pure subobjects in .
In particular, a category is accessible if and only if there is a fully faithful functor where is small, creates colimits for all -filtered diagrams, and is closed under -pure subobjects in .
This is LPAC, Cor. 2.36.