Relatively divisible and relatively flat objects in exact categories

TL;DR

This paper explores the concepts of relatively divisible and relatively flat objects within exact categories, establishing their relationships with cotorsion pairs and various exact structures, and providing new characterizations and properties.

Contribution

It introduces the notions of relatively divisible and flat objects in exact categories and links them to cotorsion pairs and different exact structures, offering new theoretical insights.

Findings

Relatively flat objects correspond to cotorsion pairs with projectively generated structures.

Relatively divisible objects correspond to cotorsion pairs with injectively generated structures.

Established Galois connections and closure properties among these concepts.

Abstract

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides with the class of relatively flat objects of $\mathcal{C}$ for some relative projectively generated exact structure, while $\mathcal{B}$ coincides with the class of relatively divisible objects of $\mathcal{C}$ for some relative injectively generated exact structure. We exhibit Galois connections between relative cotorsion pairs, relative projectively generated exact structures and relative injectively generated exact structures in additive categories. We establish closure properties and characterizations in terms of approximation theory.