A Topological Reimagining of Integration and Exterior Calculus

Abstract

A novel, highly general construction of integration, function calculus, and exterior calculus was achieved in this paper, allowing for integration of unital magma valued functions against (compactified) unital magma valued measures over arbitrary topological spaces. The Riemann integral, geometric product integral, and Lebesgue integral were all shown as special cases. Notions similar to chain complexes were developed to allow this general form of integration to define notions of exterior derivative for differential forms, and of derivatives of functions too. Fundamental realizations, some quite surprising, were achieved on the deepest natures of key concepts of analysis including integration, orientation, differentiation, and more. It's clear that further applications such as calculus on fractals, stochastic calculus, discrete calculus, and many other novel forms of analysis can all be achieved as special cases of this theory.