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

This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\mathbb{k}$-algebras $\Lambda$ and the category ${\mathcal{A}}_{\Lambda }^p$ associated with $\Lambda$. 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.