# Fundamental theorems of Doi-Hopf modules in a nonassociative setting

## TL;DR

This paper extends the fundamental theorems of Doi-Hopf modules to a weak non-associative setting, establishing categorical equivalences that unify various algebraic structures like Hopf algebras and quasigroups.

## Contribution

It introduces weak non-associative Doi-Hopf modules and proves a fundamental theorem, unifying multiple algebraic frameworks through categorical equivalence.

## Key findings

- Established a fundamental theorem for weak non-associative Doi-Hopf modules.
- Proved categorical equivalences encompassing Hopf algebras and quasigroups.
- Unified various algebraic structures within a common theoretical framework.

## Abstract

In this paper we introduce the notion of weak non-asssociative Doi-Hopf module and give the Fundamental Theorem of Hopf modules in this setting. Also we prove that there exists a categorical equivalence that admits as particular instances the ones constructed in the literature for Hopf algebras, weak Hopf algebras, Hopf quasigroups, and weak Hopf quasigroups.