\(\mathcal K\)-purity and orthogonality.

*(English)*Zbl 1060.18001Let \(\mathcal C\) a locally \(\lambda\)-presentable category, \(\mathcal K\) a full subcategory of \(\mathcal C\), and \(f: A\to B\) an arbitrary morphism in \(\mathcal C\). The morphism \(f\) is called \(\lambda\)-presentable if it is a \(\lambda\)-presentable object in the coslice category \(A/\mathcal C\). This property is proved to be equivalent to the fact that \(f\) is a pushout of some morphism between \(\lambda\)-presentable objects in \(\mathcal C\). The morphism \(f\) is said to be a \(\mathcal K\)-epi if, for any pair of morphisms \((g, h): B\rightrightarrows K\) with \(K\in\mathcal K\), we have: \(gf= hf\Rightarrow g= h\). The morphism \(f\) is called \(\mathcal K_\lambda\)-pure if, for any \(\mathcal K\)-epi \(h: C\to D\) in \(\mathcal C\), any morphism \(m: C\to A\) and \(n: D\to B\) such that \(fm =nh\), there exists some morphism \(d: D\to A\) such that \(dh= m\). The morphism \(f\) is called strongly \(\mathcal K_\lambda\)-pure if for any factorization \(f= nh\) of \(f\), we have: \(h\) is a \(\lambda\)-presentable \(\mathcal K\)-epi \(\Rightarrow h\) is a split mono. It is proved that these two notions coincide whenever \(\mathcal K\) is closed in \(\mathcal C\) under products and \(\lambda\)-directed colimits. The author uses the strong notion of purity to obtain a characterization of classes of objects defined by orthogonality with respect to \(\lambda\)-presentable morphisms. Those classes are natural examples of reflective subcategories defined by proper classes of morphisms.

Reviewer: Y. Diers (Faches-Thumesnil)

##### MSC:

18A20 | Epimorphisms, monomorphisms, special classes of morphisms, null morphisms |

18C35 | Accessible and locally presentable categories |

03C60 | Model-theoretic algebra |

18G05 | Projectives and injectives (category-theoretic aspects) |