[Paper Review] Cartesian Differential Comonads and New Models of Cartesian Differential Categories

Sacha Ikonicoff, Jean-Simon Pacaud Lemay|arXiv (Cornell University)|Aug 9, 2021

Mathematical Control Systems and Analysis4 citations

TL;DR

This paper introduces Cartesian differential comonads—comonads whose coKleisli categories are Cartesian differential categories—thereby generalizing the construction of Cartesian differential categories from differential category comonads. It constructs new examples using power series, divided power algebras, and Zinbiel algebras, extending the categorical semantics of differentiation beyond traditional 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.

Motivation & Objective

• To generalize the construction of Cartesian differential categories by identifying comonads whose coKleisli categories are Cartesian differential categories.
• To provide a broader class of examples of Cartesian differential categories beyond those derived from differential categories.
• To explore new algebraic structures—power series, divided power algebras, and Zinbiel algebras—as sources of Cartesian differential comonads.
• To lay the foundation for future work on integration, antiderivatives, and Eilenberg-Moore categories in the context of Cartesian differential comonads.

Proposed method

• Define Cartesian differential comonads as comonads for which the coKleisli category inherits a Cartesian differential structure.
• Construct the differential combinator on the coKleisli category using the deriving transformation and comonad structure maps.
• Demonstrate that the coKleisli category of a Cartesian differential comonad satisfies the seven axioms of Cartesian differential categories.
• Provide explicit constructions of Cartesian differential comonads on categories of reduced power series, divided power algebras, and Zinbiel algebras.
• Verify that the derived differential combinator satisfies the standard identities of multivariable calculus, such as the chain rule and symmetry of partial derivatives.
• Use the coKleisli construction to lift smooth maps (as coKleisli maps) and define their derivatives categorically.

Experimental results

Research questions

• RQ1 What conditions must a comonad satisfy to ensure its coKleisli category is a Cartesian differential category?
• RQ2 How can new examples of Cartesian differential categories be constructed from algebraic structures like power series and divided power algebras?
• RQ3 Can Zinbiel algebras support a Cartesian differential comonad structure, and what does the induced differential combinator look like?
• RQ4 What is the relationship between Cartesian differential comonads and the known embedding of Cartesian differential categories into coKleisli categories of differential (storage) categories?
• RQ5 How might integration and antiderivatives be generalized in the context of Cartesian differential comonads?

Key findings

• Cartesian differential comonads are precisely the comonads whose coKleisli categories are Cartesian differential categories, providing a generalization of the standard construction from differential categories.
• The coKleisli category of the reduced power series comonad on the category of R-modules forms a Cartesian differential category, with the derivative of a power series map defined via the standard multivariable derivative.
• The coKleisli category of the divided power algebra comonad on the category of R-modules is a Cartesian differential category, where the differential combinator corresponds to the derivative of a divided power polynomial.
• The coKleisli category of the Zinbiel algebra comonad on the category of R-modules is a Cartesian differential category, with the differential combinator given by the Zinbiel derivative: ∂ΓV(p)(x,y,x*,y*) = x*⊗y + y⊗x* + y*⊗x + x⊗y*.
• The paper identifies potential candidates for integral combinator transformations in the power series and Zinbiel algebra cases, suggesting a path toward integrating the theory of differentiation with integration.
• Future work suggests that the Eilenberg-Moore category of a Cartesian differential comonad may be a tangent category, generalizing known results from differential categories.