  In this paper we prove that the inductive cocategory of a nilpotent
$CW$-complex of finite type $X$, $\indcocat X$, is bounded above by an
expression involving the inductive cocategory of the $p$-localisations of $X$.
Our arguments can be dualised to LS category improving previous results by
Cornea and Stanley. Finally, we show that the inductive cocategory is generic
for 1-connected $H_0$-spaces of finite type.
