The Category of Hypergroups: seeking for a generalized version of Abelian Categories

TL;DR

This paper explores the categorical structure of abelian hypergroups to develop a generalized framework for homological algebra applicable to hyper-algebras, aiming to extend classical algebraic concepts.

Contribution

It introduces the notions of hyper (almost) preadditive and hyper (almost) abelian categories as a foundation for a broader homological algebra theory for hyper-algebras.

Findings

Defined hyper (almost) preadditive and hyper (almost) abelian categories

Established initial categorical properties of abelian hypergroups

Outlined a pathway for developing homological algebra for hyper-algebras

Abstract

In this paper we study categorical properties of the category of abelian hypergroups that leads to the notion of hyper (almost) preadditive and hyper (almost) abelian categories. Our goal is to create a path towards a general theory of homological algebra for hyper-algebras. This is a first attempt to achieve this goal. We hope to improve the definitions and results, and provide more examples soon.