Relative extriangulated categories arising from half exact functors

TL;DR

This paper introduces a new method to construct closed subfunctors in extriangulated categories using half exact functors, unifying and extending previous approaches in various categorical contexts.

Contribution

It provides a general construction technique for closed subfunctors in extriangulated categories, encompassing existing methods and characterizing all such subfunctors when enough projectives are present.

Findings

New construction method for closed subfunctors from half exact functors

Unification of existing constructions across different categories

Complete characterization of closed subfunctors in categories with enough projectives

Abstract

Relative theories(=closed subfunctors) are considered in exact, triangulated and extriangulated categories by Dr\"{a}xler-Reiten-Smal{\o}-Solberg-Keller, Beligiannis and Herschend-Liu-Nakaoka, respectively. We give a construction method of closed subfunctors from given half exact functors which contains existing constructions. Moreover, if an extriangulated category has enough projective objects, then every closed subfunctor is obtained by this construction.