**Title**: Characterizing Cofree Cartesian Differential Categories

**Abstract**: Cartesian differential categories come equipped with a differential operator which formalizes the derivative from multivariable calculus. There has recently been renewed interest in cofree Cartesian differential categories. For any Cartesian left additive category X there exists a cofree Cartesian differential category Faa(X) over it, which satisfies the expected couniversal property, and this construction is known as the Faa di Bruno construction. A natural question to ask is whether well-known examples of Cartesian differential categories?

Therefore, we would like to answer the following: starting with only an arbitrary Cartesian differential category, how can we check if it is a cofree Cartesian differential category without knowing the base Cartesian left additive category?

In this talk, we will provide a characterization of cofree Cartesian differential categories using only internal structure, that is, as categories enriched over complete ultrametric spaces (where the metric is similar to that of power series) and whose base Cartesian left additive category is induced by maps whose derivative is zero. A consequence of this result is that the induced cofree Cartesian differential category comonad is of effective descent type. Furthermore, we also explain how many well-known Cartesian differential categories are NOT cofree.

This talk should be accessible to everyone! Even those unfamiliar with differential categories.