LNL polycategories and doctrines of linear logic

We define and study LNL polycategories, which abstract the judgmentalstructure of classical linear logic with exponentials. Many existing structurescan be represented as LNL polycategories, including LNL adjunctions, linearexponential comonads, LNL multicategories, IL-indexed categories, linearlydistributive categories with storage, commutative and strong monads,CBPV-structures, models of polarized calculi, Freyd-categories, and skewmulticategories, as well as ordinary cartesian, symmetric, and planarmulticategories and monoidal categories, symmetric polycategories, and linearlydistributive and *-autonomous categories. To study such classes of structuresuniformly, we define a notion of LNL doctrine, such that each of these classesof structures can be identified with the algebras for some such doctrine. Weshow that free algebras for LNL doctrines can be presented by a sequentcalculus, and that every morphism of doctrines induces an adjunction betweentheir 2-categories of algebras