  We introduce the notion of a categorical join, which can be thought of as a
categorification of the classical join of two projective varieties. This notion
is in the spirit of homological projective duality, which categorifies
classical projective duality. Our main theorem says that the homological
projective dual category of the categorical join is naturally equivalent to the
categorical join of the homological projective dual categories. This
categorifies the classical version of this assertion and has many applications,
including a nonlinear version of the main theorem of homological projective
duality.
