  We define generalized bialgebras and Hopf algebras and on this basis we
introduce quantum categories and quantum groupoids. The quantization of the
category of linear (super)spaces is constructed. We establish a criterion for
the classical value of the dimension of the polynomial function algebra on the
full subcategory of this quantum category specified by Sudbery type commutation
relations for quantum vector spaces (the ``Poincare -- Birkhoff -- Witt
property''). The criterion is the equality of the ``quantum constants''
$c_\a=c_\b^{\pm 1}$ of the quantum vector spaces. (These constants appeared in
earlier works on PBW for the multiparameter quantization of the general linear
group.) Links with the Yang-Baxter equation and the Yang -- Baxter structures
of quantum linear spaces are established. The role of categories as a
generalization of groups is discussed.
