Quasitriangular Hopf Group Coalgebras and Braided Monoidal Categories

Turaev introduced the notions of group coalgebras, Hopf group coalgebras and quasitriangular Hopf group coalgebras. Virelizier studied algebraic properties of Hopf π-coalgebras. In this paper, we give the definitions of a (crossed) left H-π-modules over a (crossed) Hopf π-coalgebra H, and show that the categories of (crossed) left H-π-modules are both monoidal categories. Finally, we show that a family \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R=\{R_{\alpha, \beta} \in H_{\alpha} \otimes H_\beta\}_{\alpha, \beta\in\pi}}$$\end{document} of elements is a quasitriangular structure of a crossed Hopf π-coalgebra H if and only if the category of crossed left H-π-modules over H is a braided monoidal category with braiding defined by R.