Cartesian Difference Categories

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.

A Finite Axiomatisation of Finite-State Automata Using String Diagrams

“…It has been answered recently in [90] for copy-delete monoidal categories 14 , and in [65] for traced copy-delete categories. Finally, we point out the recent work [3], which studied rewriting for monoidal closed categories: classes of monoidal categories with a 'function space' which are relevant to the semantics of programming languages.…”

Cartesian Differential Kleisli Categories

“…In [21], Manzyuk studied the tangent bundle monad of categorical models of the differential λ-calculus. In [2,3], Alvarez-Picallo and the author studied the tangent bundle monad for Cartesian difference categories, a slight generalization of Cartesian differential categories by adding extra nilpotent structure. So for a Cartesian k-differential category X, its tangent bundle monad [3, Sec 6.1] is the monad T := (T, µ, η) which is defined as follows:…”

“…All the necessary requirements that need to be checked for the tangent bundle to be a Cartesian kdifferential monad can be found in [3,Lem 6.4]. In particular, in the term calculus, ( 6) is expressed as the equality:…”

“…Note that the derivative of a vector field is no longer a vector field. For more details on the Kleisli category of the tangent bundle monad, see [3,Sec 6.3].…”