Braided crossed category over crossed group-cograded weak Hopf quasigroups

TL;DR

This paper extends the theory of weak Hopf quasigroups by defining group-cograded structures and introduces a new category of Yetter-Drinfeld weak quasimodules, establishing their categorical properties.

Contribution

It generalizes previous results to weak Hopf coquasigroups and constructs new categories with braided and crossed structures.

Findings

Yetter-Drinfeld weak quasimodule category is a crossed category

Yetter-Drinfeld module subcategory is braided crossed

Generalization to weak Hopf coquasigroups

Abstract

In this paper, we generalizing the main result in Liu[10] to weak Hopf coquasigroups case. We first define and study group-cograded weak Hopf quasigroups, which generalize both group-cograded Hopf quasigroups and weak Hopf group-coalgebras. Then we introduce the notion of p-Yetter-Drinfeld weak quasimodule over group-cograded weak Hopf quasigroups H. If the antipode of H is bijective, we show that the category YDWQ(H) of Yetter-Drinfeld weak quasimodules over H is a crossed category, and the subcategory YD(H) of Yetter-Drinfeld modules is a braided crossed category.