in which and are -presentable objects, the morphism factors through , i.e. there is some with .
Notice that the above definition does not requrie that also the morphism is factored, hence it does not express a lifting property.
In a -accessible category every -pure morphism is monic, hence exhibits a pure subobject. In a locally -presentable category -pure morphisms are, moreover, regular monomorphisms, and in fact coincide with the -directed colimits of split monomorphisms in the category of arrows ; more generally this characterization holds in all -accessible categories admiting pushouts.
(Adámek, Hub, Tholen).
with any right -module (from the left) yields an exact sequence of abelian groups.
Grothendieck has proved that faithfully flat morphisms of commutative schemes are of effective descent for the categories of quasicoherent -modules. But this was not entirely optimal, as there is in fact a more general class than faithfully flat morphisms which satisfy the effective descent. For a local case of commutative rings, Joyal and Tierney have then proved (unpublished) that the effective descent morphisms for modules are precisely the pure morphisms of rings (or dually of affine schemes). Janelidze and Tholen have reproved the theorem as a corollary of a result for noncommutative rings obtained using the Beck’s comonadicity theorem.
Bachuki Mesablishvili, Pure morphisms of commutative rings are effective descent morphisms for modules – a new proof, Theory and Appl. of Categories 7, 2000, No. 3, 38-42, tac
W.W. Crawley-Boevey, Locally finitely presented additive categories, Communications in Algebra 22(5)(1994), 1641-1674.
Christian U. Jensen, Helmut Lenzing, Model theoretic algebra: with particular emphasis on fields, rings, modules, Algebra, Logic and Applications 2, Gordon and Breach 1989.
Ivo Herzog, Pure-injective envelopes, pdf Journal of Algebra and Its Applications 2(4) (2003), 397-402.