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, skew multicategories, as well as ordinary cartesian and symmetric multicategories and monoidal categories, 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.
翻译:我们定义和研究LNL的多元类别,它们抽象了古典线性逻辑的判断结构,并带有指数。许多现有结构可以作为LNL的多元类别,包括LNL的辅助类别、线性指数共毛形、LNL的多元类别、ILL的指数类别、储存的线性分配类别、交点和坚固的月经、CBPV-结构、极化钙模型、Skew的多类,以及普通的碳酸盐和对称多类和单子类别、多类和线性分配和*自主类别。为了统一研究这类结构类别,我们定义了LNL的理论概念,这样可以将这些结构的每一种类别与某些此类学说的代数加以识别。我们表明,LNL理论的免费代数值可由后代计算法提出,而每一种理论的形态都引出其2类的正值。