Nel's category theory based differential and integral Calculus, or did Newton know category theory ?

TL;DR

This paper demonstrates how classical differential and integral calculus operations can be derived from category theory, providing a new foundational perspective applicable to complex functions and PDEs.

Contribution

It introduces a categorical framework that reconstructs classical calculus operations without generalized derivatives, extending to infinite-dimensional and non-convex domain functions.

Findings

Classical derivatives and integrals are obtainable via simple categorical constructions.

The framework applies to functions on non-convex domains with empty interior.

Supports advanced analysis in partial differential equations.

Abstract

In a series of publications in the early 1990s, L D Nel set up a study of non-normable topological vector spaces based on methods in category theory. One of the important results showed that the classical operations of derivative and integral in Calculus can in fact be obtained by a rather simple construction in categories. Here we present this result in a concise form. It is important to note that the respective differentiation does not lead to any so called generalized derivatives, for instance, in the sense of distributions, hyperfunctions, etc., but it simply corresponds to the classical one in Calculus. Based on that categorial construction, Nel set up an infinite dimensional calculus which can be applied to functions defined on non-convex domains with empty interior, a situation of great importance in the solution of partial differential equations