Title: Cartesian differential categories and coCartesian differential monads
Cartesian Differential Categories (CDCs) are defined to introduce and study the notion of differential from calculus in a category theory point of view. In a CDC, morphisms between objects can be « derived », and this differentiation operation must satisfy a list of properties, including a version of the chain rule.
With JS Lemay, we introduced the notion of a coCartesian Differential Monad (cCDM) on a Category with finite biproducts to give the lightest apparatus on a monad which allows us to define a CDC structure on its Kleisli category.
In this introductory talk, we will motivate the definition for CDCs using the example of multivariable calculus on Euclidian spaces. We will define cCDMs, recall the construction of the Kleisli category for a monad, and show how to build a CDC structure on the Kleisli category of a cCDM, along with motivating examples. This joint work with JS Lemay is available on the ArXiv: https://arxiv.org/abs/2108.04304