🤖 AI Summary
This work addresses the problem of generalizing Joyal’s differential calculus for analytic functors in one variable to the multivariate setting between presheaf categories. To this end, the authors develop a monadic bicategory framework at the bicategorical level, introducing linear exponential comonoidal structure and codereliction transformations—thereby establishing, for the first time, a higher-dimensional categorical semantics for differential linear logic. The approach unifies semantic models of linear logic with combinatorial differential calculus, structurally extending analytic functors from the single-variable to the multivariate presheaf-categorical setting. Key contributions include: (i) the first bicategorical model of differential linear logic; (ii) a systematic lift of analytic functor calculus to a multivariate, intercategorical level; and (iii) novel formal tools for higher-order programming language semantics and algebraic combinatorics. The framework provides a principled, compositional foundation for differentiation in categorical logic and combinatorial species theory.
📝 Abstract
We develop further the theory of monoidal bicategories by introducing and studying bicategorical counterparts of the notions of a linear explonential comonad, as considered in the study of linear logic, and of a codereliction transformation, introduced to study differential linear logic via differential categories. As an application, we extend the differential calculus of Joyal's analytic functors to analytic functors between presheaf categories, just as ordinary calculus extends from a single variable to many variables.