zbMATH — the first resource for mathematics

$$\mathcal K$$-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.

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)
Full Text: