Squared Hopf algebras and reconstruction theorems

Given an abelian -linear rigid monoidal category , where is a perfect field, we define squared coalgebras as objects of cocompleted ⨂ (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If is the category of -vector spaces, squared (co)algebras coincide with conventional ones. If is braided, a braided Hopf algebra can be obtained from a squared one. Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras in and corresponding fibre functors to (which is not the case with the usual definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to the problem of defining quantum groups in braided categories.