Filtered objects in extriangulated categories

TL;DR

This paper studies filtered objects in extriangulated categories, showing that certain classes of objects are precovering, preenveloping, and functorially finite, generalizing previous results in module and triangulated categories.

Contribution

It introduces the concept of filtered objects in extriangulated categories and proves their functorial finiteness, extending known results from module and triangulated categories.

Findings

$ ext{P}( heta)$ is a precovering class.

$ ext{I}( heta)$ is a preenveloping class.

$ ext{F}( heta)$ is functorially finite in $ ext{C}$.

Abstract

Let $R$ be an artin ring and $\Theta=\{\Theta(1),\Theta(2),\cdots,\Theta(n)\}$ be a family of objects in an artin extriangulated $R$-category $(\cal C,\mathbb{E},\mathfrak{s})$ such that $\mathbb{E}(\Theta(j),\Theta(i))=0$ for all $j\geq i$. In this paper, we show that the class $\cal P(\Theta)$ of the $\Theta$-projective objects is a precovering class and the class $\cal I(\Theta)$ of the $\Theta$-injective objects is a preenveloping one in $\cal C$. Furthermore, if $\cal C$ has enough projectives and enough injectives, we show that the subcategory $\cal F(\Theta)$ of $\Theta$-filtered objects is functorially finite in $\cal C$. As an appliacation, this generalizes the works by Ringel in a module category case and Mendoza-Santiago in a triangulated category case.