Filtered objects in extriangulated categories

Let R be an Artin ring and Θ={Θ(1),Θ(2),…,Θ(n)} be a family of objects in an Artin extriangulated R-category (C,E,s) such that E(Θ(j),Θ(i))=0 for all j≥i. In this article, we show that the class P(Θ) of the Θ-projective objects is a precovering class and the class I(Θ) of the Θ-injective objects is a pre-enveloping one in C. Furthermore, if C has enough projectives and enough injectives, we show that the subcategory F(Θ) of Θ-filtered objects is functorially finite in C. As an application, this generalizes the works by Ringel in a module category case and Mendoza–Santiago in a triangulated category case.