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\(\mathcal K\)-purity and orthogonality. (English) Zbl 1060.18001
Let \(\mathcal C\) 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.

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)
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