  We introduce the notion of a categorical cone, which provides a
categorification of the classical cone over a projective variety, and use our
work on categorical joins to describe its behavior under homological projective
duality. In particular, our construction provides well-behaved categorical
resolutions of singular quadrics, which we use to obtain an explicit quadratic
version of the main theorem of homological projective duality. As applications,
we prove the duality conjecture for Gushel-Mukai varieties, and produce
interesting examples of conifold transitions between noncommutative and honest
Calabi-Yau threefolds.
