We consider categories of generalized perverse sheaves, with
relaxed constructibility conditions, by means of the process of
gluing $t$-structures and we exhibit explicit abelian
categories defined in terms of standard sheaves categories which
are equivalent to the former ones. In particular, we are able to
realize perverse sheaves categories as non full abelian
subcategories of the usual bounded complexes of sheaves
categories. Our methods use induction on perversities. In this
paper, we restrict ourselves to the two-strata case, but our
results extend to the general case.