Taut functors and the difference operator

TL;DR

This paper develops a calculus of differences for taut endofunctors in the category of sets, introducing new rules and formulas that extend classical finite difference concepts to a categorical setting.

Contribution

It introduces a calculus of differences for taut endofunctors, including a lax chain rule and explicit formulas for various classes of functors, extending classical difference calculus to category theory.

Findings

Established a categorical calculus of differences for taut endofunctors.

Derived explicit difference formulas for polynomial, analytic, and reduced power functors.

Introduced covariant Dirichlet series and a Newton summation formula as an adjoint to the difference operator.

Abstract

We establish a calculus of differences for taut endofunctors of the category of sets, analogous to the classical calculus of finite differences for real valued functions. We study how the difference operator interacts with limits and colimits as categorical versions of the usual product and sum rules. The first main result is a lax chain rule which has no counterpart for mere functions. We also show that many important classes of functors (polynomials, analytic functors, reduced powers, ...) are taut, and calculate explicit formulas for their differences. Covariant Dirichlet series are introduced and studied. The second main result is a Newton summation formula expressed as an adjoint to the difference operator.