Conceptual differential calculus part ii: Cubic higher order calculus

Abstract

Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The "local linear algebra" defined in Part I is generalized by "higher order local linear algebra". The underlying combinatorial object of such higher algebra is the natural n-dimensional hypercube, and so we qualify this calculus as "cubic". More precisely, we define two versions of conceptual cubic calculus: "full" and "symmetric cubic". The theory thus initiated sheds new light on several foundational issues.