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. It is well known that differentiation can be turned into a compositional operation, using the tangent bundle construction, also called dual numbers in automated differentiation.
In this talk, we explain how Taylor expansion at any order can be similarly turned into a compositional operation, using an idea similar to that of jet bundles that appear in differential geometry.
We will then discuss how category theory is a nice framework to describe those ideas. Formally, Taylor expansion is captured as a functor, and the axioms of differential calculus boil down to naturality equations that turns this functor into a monad. This categorical structure is similar to the structure of higher order dual numbers that have found recent applications in automated differentiation.
No prior knowledge of category theory is required for most of this presentation.
