Fonctorial Construction of Frobenius Categories

TL;DR

This paper develops a functorial approach to constructing Frobenius and stable categories from exact and triangulated categories using an exact or triangulated functor, generalizing existing concepts and linking different notions of stability.

Contribution

It introduces a functorial method to build Frobenius categories and stable categories from exact and triangulated categories via an exact or triangulated functor, unifying various constructions.

Findings

Constructs Frobenius categories from exact categories using functorial direct factors.

Defines $M$-stable categories for exact and triangulated categories through quotienting.

Links different notions of $M$-stable categories via a theorem of Keller and Vossieck.

Abstract

Let $\Ascr,\Bscr$ be exact categories with $\Ascr$ karoubian and $M$ be an exact functor. Under suitable adjonction hypotheses for $M$, we are able to show that the direct factors of the objects of $\Ascr$ of the form $MY$ with $Y \in \Bscr$ make up a Frobenius category which allow us to define an $M$-stable category for $\Ascr$ only by quotienting. In addition, we propose a construction of an $M$-stable category for $\Ascr,\Bscr$ triangulated categories and $M$ a triangulated functor. We illustrate this notion with a theorem of Keller and Vossieck which links the two notions of $M$-stable category.