# Extensions of Operators, Liftings of Monads and Distributive Laws

## TL;DR

This paper generalizes the algebraic and categorical framework of the First Fundamental Theorem of Calculus, establishing equivalences between extensions of operators, liftings of monads and comonads, and distributive laws under new constraints.

## Contribution

It introduces a generalized class of constraints linking differential and Rota-Baxter operators and proves the equivalence of their extensions, liftings, and distributive laws.

## Key findings

- Extensions of operators are equivalent under new constraints.
- Liftings of monads and comonads correspond to these operator extensions.
- Distributive laws are characterized by the same constraints.

## Abstract

In a previous study, the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC) is shown to allow extensions of differential and Rota-Baxter operators on the one hand, and to give rise to categorical explanations using the ideas of liftings of monads and comonads, and mixed distributive laws on the other. Generalizing the FFTC, we consider in this paper a class of constraints between a differential operator and a Rota-Baxter operator. For a given constraint, we show that the existences of extensions of differential and Rota-Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are equivalent.