Multivariate functorial difference

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}.

💡 Research Summary

The paper “Multivariate Functorial Difference” develops a categorical theory of partial difference operators for a broad class of functors between presheaf categories, extending the author’s earlier single‑variable construction (Paré 2024) to the multivariable setting. The central technical device is the Jacobian profunctor, a profunctor‑valued analogue of the classical Jacobian matrix, which aggregates all partial difference operators of a functor into a single object. This Jacobian enables a lax chain rule (Theorem 4.2): for functors (F\colon\mathsf{Set}^{A}\to\mathsf{Set}^{B}) and (G\colon\mathsf{Set}^{B}\to\mathsf{Set}^{C}) there is a natural transformation \