Characterizations of higher derivations and higher differential torsion theories in Eilenberg-Moore categories of monads

TL;DR

This paper introduces and characterizes higher derivations on monads and their modules within Eilenberg-Moore categories, linking them to torsion theories and module quotients, with several illustrative examples.

Contribution

It defines higher derivations on monads and modules, providing their characterization and exploring conditions for higher differential torsion theories in Eilenberg-Moore categories.

Findings

Higher derivations on monads are characterized via ordinary derivations.

Higher derivations on modules extend uniquely to modules of quotients under certain conditions.

Torsion theory is higher differential if and only if derivations extend uniquely.

Abstract

Let $T$ be a monad on a category $\mathscr{C}$. In this paper, we introduce the notion of higher derivations on the monad $T$ and characterize them in terms of ordinary derivations on $T$. We also define higher derivations on modules over the monad $T$ in the Eilenberg-Moore category $EM_T$ and establish their characterization in a similar manner. We provide several examples that illustrate and support our results. Furthermore, we examine the conditions under which a torsion theory on $EM_T$ is higher differential, and show that this holds if and only if every higher derivation on a module $M \in EM_T$ extends uniquely to its module of quotients $Q_{\tau}(M)$.