  Given an abelian k-linear rigid monoidal category V, where k is a perfect
field, we define squared coalgebras as objects of cocompleted V tensor V
(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 V is the category of k-vector spaces, squared (co)algebras coincide with
conventional ones. If V is braided, a braided Hopf algebra can be obtained from
a squared one.
  Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras
in V and corresponding fibre functors to V (which is not the case with other
definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to
the problem of defining quantum groups in braided categories.
