Pivotal Weak Turaev $\pi$-Coalgebras

In this paper, we first introduce the notion of the $\pi$-pivotal elements in a weak Turaev $\pi$-coalgebra $H$ and show that the representation category of $H$ is a pivotal crossed category if and only if there is a $\pi$-pivotal element in $H$. Also we discuss the relation between $\pi$-pivotal elements and $\pi$-ribbon elements of a quasitriangular weak Turaev $\pi$-coalgebra. Finally, we obtain a generalized Deligne Type theorem.