  We define and study LNL polycategories, which abstract the judgmental
structure of classical linear logic with exponentials. Many existing structures
can be represented as LNL polycategories, including LNL adjunctions, linear
exponential comonads, LNL multicategories, IL-indexed categories, linearly
distributive categories with storage, commutative and strong monads,
CBPV-structures, models of polarized calculi, Freyd-categories, and skew
multicategories, as well as ordinary cartesian, symmetric, and planar
multicategories and monoidal categories, symmetric polycategories, and linearly
distributive and *-autonomous categories. To study such classes of structures
uniformly, we define a notion of LNL doctrine, such that each of these classes
of structures can be identified with the algebras for some such doctrine. We
show that free algebras for LNL doctrines can be presented by a sequent
calculus, and that every morphism of doctrines induces an adjunction between
their 2-categories of algebras.
