Algebraic structures in comodule categories over weak bialgebras

Abstract

For a bialgebra L coacting on a -algebra A, a classical result states A is a right L-comodule algebra if and only if A is an algebra in the category of right L-comodules. We generalize this to the setting of weak bialgebras H. We prove that there is an isomorphism between the categories of right H-comodule algebras and the category of algebras in . We also recall and introduce the formulaic notion of H coacting on a -coalgebra and on a Frobenius -algebra, respectively, and prove analogous category isomorphisms. Many examples are provided throughout.