  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.
