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

TL;DR

This paper introduces a new categorical framework for the Lebesgue integral with variable upper limits using normed modules over finite-dimensional algebras, leading to applications in categorification and algebraic dimension analysis.

Contribution

It provides a novel categorification of the Lebesgue integral with variable upper limits via normed modules and introduces an abstract integral with applications in function categorification and algebraic dimension.

Findings

Categorification of elementary functions including trigonometric and logarithmic functions.

A new approach to characterizing global dimensions of gentle algebras.

Redefinition of integration through an integral partially ordered set.

Abstract

This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\Bbbk$-algebras $\mathit{\Lambda}$ and the category $\mathscr{A}^p_{\mathit{\Lambda}}$ associated with $\mathit{\Lambda}$. 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.