Differential restriction categories

TL;DR

This paper introduces a new algebraic structure combining cartesian differential categories and restriction categories to model smooth maps on open subsets of Euclidean space, with applications to partial linearity and algebraic properties.

Contribution

It develops a novel restriction differential category framework that captures smooth partial maps algebraically and explores its models and properties.

Findings

New algebraic framework for smooth partial maps

Models for the combined differential and restriction structure

Lifting of differential restriction structure through completion operations

Abstract

We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $\R^n$ in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.