Multivariate functorial difference

TL;DR

This work generalizes finite difference calculus to multivariable functors between presheaf categories by introducing profunctor-based Jacobians and lax chain rules. It develops tense (multivariable analytic) functors that preserve complemented subobjects, constructs a multivariable Newton-analytic framework, and establishes a Newton-series comonad whose convergence recovers soft analytic functors. The core contributions include the discrete Jacobian via $oldsymbol{ riangle}[F]$, a comprehensive set of limit/colimit and chain-rule laws, and a Kan-extension formulation of multivariable analytic functors. The results provide a robust, category-theoretic analogue of multivariate difference calculus with applications to recovering functors from iterated differences and characterizing fixed points as soft analytic functors. The framework connects to existing analytic functor theory and tangent/differential category ideas, offering a structured path for multivariate categorical differentiation and approximation.

Abstract

Partial difference operators for a large class of functors between presheaf categories are introduced, extending our difference operator from \cite{Par24} to the multivariable case. These combine into the Jacobian profunctor which provides the setting for a lax chain rule. We introduce a functorial version of multivariable Newton series whose aim is to recover a functor from its iterated differences. Not all functors are recovered but we get a best approximation in the form of a left adjoint, and the induced comonad is idempotent. Its fixed points are what we call soft analytic functors, a generalization of the multivariable analytic functors of Fiore et al.~\cite{FioGamHylWin08}.

Theorems & Definitions (122)

• Definition 1.1
• Proposition 1.1
• Definition 1.2
• Proposition 1.2
• proof
• Example 1.1
• Definition 2.1
• Proposition 2.1
• Proposition 2.2
• Definition 2.2