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

Abstract

Let $T$ be a monad on a category $\mathcal{C}$. 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 $E{M}_T$ 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 $E{M}_T$ is higher differential, and show that this holds if and only if every higher derivation on a module $M\in E{M}_T$ extends uniquely to its module of quotients $Q_{\tau }(M)$.