# New Braided $T$-Categories over Hopf (co)quasigroups

## TL;DR

This paper constructs a new braided T-category over Hopf (co)quasigroups, expanding the algebraic framework for automorphisms and categories related to Hopf quasigroups with bijective antipodes.

## Contribution

It introduces a novel category structure ${_{H} ext{YDQ}^{H}}( ext{α,β})$ and constructs a comprehensive braided T-category $ ext{YDQ}(H)$ incorporating these categories.

## Key findings

- Defines categories ${_{H} ext{YDQ}^{H}}( ext{α,β})$ for Hopf quasigroups.
- Constructs a braided T-category $ ext{YDQ}(H)$ with these categories as components.
- Provides a new algebraic framework for Hopf quasigroup automorphisms.

## Abstract

Let $H$ be a Hopf quasigroup with bijective antipode and let $Aut_{HQG}(H)$ be the set of all Hopf quasigroup automorphisms of $H$. We introduce a category ${_{H}\mathcal{YDQ}^{H}}(\alpha,\beta)$ with $\alpha,\beta\in Aut_{HQG}(H)$ and construct a braided $T$-category $\mathcal{YDQ}(H)$ having all the categories ${_{H}\mathcal{YDQ}^{H}}(\alpha,\beta)$ as components.