Title: Exponential Functions for Cartesian Differential Categories.

Abstract: We introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function from classical differential calculus. In particular, differential exponential maps can be defined without the need for limits, converging power series, multiplication, or unique solutions of certain differential equations — which most Cartesian differential categories do not necessarily have. Every differential exponential map induces a commutative rig, called a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, and the dual numbers exponential.