Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler's Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodule-monad and a algebra-comonoid over it. As an application we obtain a generalization of another classical theorem from the Hopf algebra literature, due to Schneider, which itself is an extension of Sweedler's result (to the setting of Hopf Galois extensions).
Keywords: monad, comonad, bimonad, Beck's theorem, Hopf module, Doi-Koppinen Hopf module, Hopf Galois, Sweedler's Fundamental Theorem, Schneider's Structure Theorem, Hilbert's Theorem 90
2010 MSC: 16T05, 16T15, 18A40, 18C15, 18D10, 18D35
Theory and Applications of Categories, Vol. 27, 2013, No. 13, pp 263-326.
Revised 2013-04-10. Original version at: