Multivariate functorial difference

TL;DR

This paper develops a multivariable difference calculus for functors between presheaf categories, introducing Jacobian profunctors and a functorial Newton series to approximate and analyze these functors.

Contribution

It extends difference operators to multivariable functors, introduces Jacobian profunctors, and constructs a functorial Newton series for functor approximation.

Findings

Introduced multivariate partial difference operators for functors.

Defined the Jacobian profunctor and established a lax chain rule.

Developed a functorial Newton series with a left adjoint 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}.