Cartesian Differential Comonads and New Models of Cartesian Differential Categories

TL;DR

This paper introduces Cartesian differential comonads, generalizing existing constructions to produce new examples of Cartesian differential categories using algebraic structures like power series and Zinbiel algebras.

Contribution

It defines Cartesian differential comonads and demonstrates how they generate new Cartesian differential categories from various algebraic frameworks.

Findings

New examples of Cartesian differential categories from algebraic structures

Generalization of the coKleisli construction for differential categories

Broader applicability of Cartesian differential categories

Abstract

Cartesian differential categories come equipped with a differential combinator that formalizes the derivative from multi-variable differential calculus, and also provide the categorical semantics of the differential $\lambda$-calculus. An important source of examples of Cartesian differential categories are the coKleisli categories of the comonads of differential categories, where the latter concept provides the categorical semantics of differential linear logic. In this paper, we generalize this construction by introducing Cartesian differential comonads, which are precisely the comonads whose coKleisli categories are Cartesian differential categories, and thus allows for a wider variety of examples of Cartesian differential categories. As such, we construct new examples of Cartesian differential categories from Cartesian differential comonads based on power series, divided power algebras, and Zinbiel algebras.