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Let π be a group. The main purpose of this paper is to provide further examples of crossed π-categories in the sense of Turaev. For this, we first introduce the notion of weak Hopf π-algebra as the dual notion of weak Hopf π-coalgebra and investigate the properties of weak Hopf π-algebra keeping close to weak Hopf algebra in sense of Böhm et al. It is shown that the category of the copresentations of weak Hopf π-algebra is braided crossed π-category. Finally, we shall consider the notion of weak Doi-Hopf group module in the weak Hopf π-algebra setting, and discuss the separability of a class of functors for the category of weak Doi-Hopf π-modules to the category of comodule over a suitable coalgebras. Also, the applications of our theories are presented.

Constructing New Crossed Group Categories Over Weak Hopf Group Algebras