Properties and Characterisations of Cofree Cartesian Differential Categories

Cartesian differential categories come equipped with a differential operator which formalises the total derivative from multivariable calculus. Cofree Cartesian differential categories always exist over a specified base category, where the general construction is based on Fa\`a di Bruno's formula. A natural question to ask is, when given an arbitrary Cartesian differential category, how can one check if it is cofree without knowing the base category? In this paper, we provide characterisations of cofree Cartesian differential categories without specifying a base category. The key to these characterisations is, surprisingly, maps whose derivatives are zero, which we call differential constants. One characterisation is in terms of the homsets being complete ultrametric spaces, where the ultrametric is induced by differential constants, which is similar to the metric for power series. Another characterisation is as algebras of a monad. In either characterisation, the base category is the category of differential constants. We also discuss other basic properties of cofree Cartesian differential categories, such as the linear maps, and explain how many well-known Cartesian differential categories (such as polynomial or smooth functions) are not cofree.