Abstract
Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $\lambda$-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation of a more discrete nature such as the calculus of finite differences. On the other hand, change action models have been shown to capture these examples as well as more "exotic" examples of differentiation. But change action models are very general and do not share the nice properties of Cartesian differential categories. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.
Highlights
In the early 2000’s, Ehrhard and Regnier introduced the differential λ-calculus [ER03], an extension of the λ-calculus equipped with a differential combinator capable of taking the derivative of arbitrary higher-order functions
We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories
The relevance of Cartesian differential categories lies in their ability to model both “classical” differential calculus and the differential λ-calculus
Summary
In the early 2000’s, Ehrhard and Regnier introduced the differential λ-calculus [ER03], an extension of the λ-calculus equipped with a differential combinator capable of taking the derivative of arbitrary higher-order functions. While Cartesian differential categories have proven to be an immensely successful formalism, they have, by design, some limitations They cannot account for certain “exotic” notions of derivative, such as the difference operator from the calculus of finite differences [Ric54]. We introduce Cartesian difference categories (Section 4.2), whose key ingredients are an infinitesimal extension operator and a difference combinator, whose axioms are a generalization of the differential combinator axioms of a Cartesian differential category. In this version, we correct the Cartesian difference structure of the Kleisli category of the tangent bundle monad. The proposed infinitesimal extension and difference combinator for said Kleisli category in the conference paper [APL20] was based on a result from another paper [APO19].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
