The combination of the theory of differential calculus with the theory of programming languages is an active field of research, with the development of automated differentiation and of the differential lambda calculus. This motivates the search for a compositional approach to differentiation. It is well known that differentiation can be tuned into a compositional operation, using a tangent bundle. It is less known however that the usual properties of the derivative can be described as a categorical structure on said tangent bundle.
Furthermore, we shall see that Taylor expansion at any degree can be described by a similar operator, reminiscent of jet bundles found in differential geometry and of higher order dual numbers found in automated differentiation. Surprisingly, this Taylor operation has the exact same categorical structure as the tangent bundle functor, even though its underlying combinatorics is much more complex.
This talk is the first part of a two talk series, but each talk can be followed independently.