Geodesic
The Structure Theorem for Weak Module Coalgebras
Matematičeskie zametki, Tome 88 (2010) no. 1, pp. 3-17.

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Let $H$ be a weak Hopf algebra, let $C$ be a weak right $H$-module coalgebra, and let $\overline C=C/C\cdot \operatorname{Ker}\operatorname{\sqcap}^{L}$. We prove a structure theorem for weak module coalgebras, namely, $C$ is isomorphic as a weak right $H$-module coalgebra to a weak smash coproduct $\overline C\times H$ defined on a $k$-space $$ \{\Sigma c_{(0)}\otimes h_2\varepsilon(c_{(-1)}h_1)\mid c\in C,\,h\in H\} $$ if there exists a weak right $H$-module coalgebra map $\phi\colon C\to H$.
Keywords: weak Hopf algebra, weak Hopf bicomodule, weak comodule coalgebra, weak smash coproduct, weak module coalgebra. 
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Yu. Wang; L. Yu. Jang. The Structure Theorem for Weak Module Coalgebras. Matematičeskie zametki, Tome 88 (2010) no. 1, pp. 3-17. http://geodesic.mathdoc.fr/item/MZM_2010_88_1_a0/

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