Given a regular cardinal $\kappa$, a morphism $f: A\to B$ in a category $C$ is $\kappa$-pure if for every commutative square
in which $P$ and $Q$ are $\kappa$-presentable objects, the morphism $u:P\to A$ factors through $g$, i.e. there is some $h: Q\to A$ with $u = h\circ g$.
Notice that the above definition does not requrie that also the morphism $v$ is factored, hence it does not express a lifting property.
In a $\kappa$-accessible category $C$ every $\kappa$-pure morphism is monic, hence exhibits a pure subobject. In a locally $\kappa$-presentable category $\kappa$-pure morphisms are, moreover, regular monomorphisms, and in fact coincide with the $\kappa$-directed colimits of split monomorphisms in the category of arrows $C^2 = Arr(C)$; more generally this characterization holds in all $\kappa$-accessible categories admiting pushouts.
(Adámek, Hub, Tholen).
We work with unital, possibly commutative, rings and modules. Given a ring $R$, a morphism $f: M\to M'$ of left $R$-modules is pure if the tensoring the exact sequence of left $R$-modules
with any right $R$-module $N$ (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 $\mathcal{O}$-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 For a local case of commutative rings, Joyal and Tierney have then proved (unpublished) that the effective descent morphisms for modules are precisely the pure morphisms of rings (or dually of affine schemes).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.
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