Rectangular torsion theories

In this paper we introduce and study \emph{rectangular torsion theories}, i.e.\ those torsion theories $(\text{C},\text{T},\text{F})$ with $\text{C}$ a pointed category, where the canonical functor $\text{C}\to \text{T}\times \text{F}$ is an equivalence of categories. In particular, we show that these are precisely the internal rectangular bands in the 2-category of pointed categories.