Abstract
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 \({R=\{R_{\alpha, \beta} \in H_{\alpha} \otimes H_\beta\}_{\alpha, \beta\in\pi}}\) 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.
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