Normed modules, integral sequences, and integrals with variable upper limits

Abstract

This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\mathbb{k}$-algebras $\hat{I}›$ and the category ${\mathcal{A}}_{\hat{I}›}^p$ associated with $\hat{I}›$. The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras.

28 pages; 2 figures