Let $j:U\to S$ be an (open) immesrion; let $P_U$ be a perverse (\'etale, though my question makes sense in the topological setting also) sheaf on $U$. Then I would like to say that the intermediate extension $j_{!*}P$ is 'based at' $U$. Note here that for any closed immersion $i:Z\to S$, $Z$ is disjoint from $U$, the zeroth perverse cohomology of both $Ri^*P_U$ and $Ri^!P_U$ is $0$.

In particular, if $P$ is a constant sheaf on a (connected smooth) variety $U$, I would like to say that $P$ is supported on any dense subvariety of $U$ (or at the generic point of $U$). Did anybody introduce something like this notion of 'base' for a perverse sheaf? The discussion Is there a 'classical' definition for the support of a perverse sheaves. made me think that the word 'support' is not appropriate here.