  The subject of this article are cross product bialgebras without co-cycles.
We establish a theory characterizing cross product bialgebras universally in
terms of projections and injections. Especially all known types of biproduct,
double cross product and bicross product bialgebras can be described by this
theory. Furthermore the theory provides new families of (co-cycle free) cross
product bialgebras. Besides the universal characterization we find an
equivalent (co-)modular description of certain types of cross product
bialgebras in terms of so-called Hopf data. With the help of Hopf data
construction we recover again all known cross product bialgebras as well as new
and more general types of cross product bialgebras. We are working in the
general setting of braided monoidal categories which allows us to apply our
results in particular to the braided category of Hopf bimodules over a Hopf
algebra. Majid's double biproduct is seen to be a twisting of a certain tensor
product bialgebra in this category. This resembles the case of the Drinfel'd
double which can be constructed as a twist of a specific cross product.
