  Classically, the projective duality between joins of varieties and the
intersections of varieties only holds in good cases. In this paper, we show
that categorically, the duality between joins and intersections holds in the
framework of homological projective duality (HPD) [K07], as long as the
intersections have expected dimensions. This result together with its various
applications provide further evidences for the proposal of homological
projective geometry of Kuznetsov and Perry [KP18]. When the varieties are
inside disjoint linear subspaces, our approach also provides a new proof of the
main result "formation of categorical joins commutes with HPD" of [KP18]. We
also introduce the concept of an $n$-HPD category, and study its properties and
connections with joins and HPDs.
