Phantom ideals and cotorsion pairs in extriangulated categories (Q1722072)

The approximation theory of an exact category was developed by \textit{X. H. Fu} et al. [Adv. Math. 244, 750--790 (2013; Zbl 1408.18022)] as a generalization of the classical approximation theory for subcategories. In the paper under review, the authors establish the approximation theory in an additive category equipped with an additive bifunctor, and consider it in extriangulated categories. It is proved in the paper that if \((\mathcal{C}, \mathbb{E}, \mathfrak{s})\) is an extriangulated category with enough injective objects and projective objects then there exists a bijective correspondence between any two of the following classes: (1) special precovering ideals of \(\mathcal{C}\); (2) special preenveloping ideals of \(\mathcal{C}\); (3) additive subfunctors of \(\mathbb{E}\) having enough special injective morphisms; and (4) additive subfunctors of \(\mathbb{E}\) having enough special projective morphisms. Moreover, it is proved that if \((\mathcal{C}, \mathbb{E}, \mathfrak{s})\) is an extriangulated category with enough injective objects and projective morphisms then there exists a bijective correspondence between the following two classes: (1) all object-special precovering ideals of \(\mathcal{C}\) ; (2) all additive subfunctors of \(\mathbb{E}\) having enough special injective objects.