Change Actions: Models of Generalised Differentiation

TL;DR

This paper introduces change action models as a categorical framework for generalized differentiation, extending previous structures to arbitrary cartesian categories and connecting to various geometric and computational models.

Contribution

It generalizes change structures to arbitrary cartesian categories and develops change action models as a unifying categorical framework for higher-order differentiation.

Findings

Change action models arise naturally in geometric and computational settings.

Canonical constructions of change action models are possible on any cartesian category.

The framework connects to cartesian differential categories, group models, and Kleene algebra.

Abstract

Cai et al. have recently proposed change structures as a semantic framework for incremental computation. We generalise change structures to arbitrary cartesian categories and propose the notion of change action model as a categorical model for (higher-order) generalised differentiation. Change action models naturally arise from many geometric and computational settings, such as (generalised) cartesian differential categories, group models of discrete calculus, and Kleene algebra of regular expressions. We show how to build canonical change action models on arbitrary cartesian categories, reminiscent of the F\`aa di Bruno construction.