Tangent category theory comes in different flavours: Cockett and Cruttwell, after generalizing Rosický's notion of a tangent category, introduce restriction tangent categories and tangent fibrations, and Cockett, Lemay and Lucyshyn-Wright introduced tangent monads as monads which lift a tangent structure to the category of algebras. Can we formalize these notions? Is there a common framework to capture these constructions? How do these notions behave with other constructions of tangent category theory, such as vector fields, differential bundles, or connections?
In this talk, I introduce the formal notion of tangent objects and give some examples and applications of the formal approach. I discuss the formal construction of vector fields, show how to recover in the formal context the usual operations between vector fields, such as Lie bracket, and classify vector fields in some of the examples provided.