Cartesian differential categories revisited

TL;DR

This paper refines the concept of Cartesian differential categories, demonstrating their broader applicability, their relationship with comonads, and their behavior under restriction structures, thus advancing the theoretical framework of differential categories.

Contribution

It introduces a more general definition of Cartesian differential categories and explores their properties, including their comonadic nature and behavior with restriction structures.

Findings

Generalized Cartesian differential categories are comonadic over Cartesian categories.

Every Cartesian category has an associated cofree differential category.

Categories with restriction structures are closed under splitting restriction idempotents.

Abstract

We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over Cartesian categories, so that every Cartesian category has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.