Rectangular Torsion Theories

In this paper we introduce and study rectangular torsion theories , i.e. those torsion theories (C,T,F) ( C , T , F ) with C C a pointed category, where the canonical functor C→ T× F C → T × F is an equivalence of categories. We prove that rectangular torsion theories are pseudo-algebras for a suitable 2-monad on the 2-category of pointed categories. Interestingly, it turns out that this 2-monad is a 2-categorical counterpart of the monad for rectangular bands, i.e., non-empty semigroups satisfying $$xyz = xz$$ x y z = x z . So rectangular torsion theories can be viewed as internal rectangular bands in the 2-category of pointed categories. We also show that a triple (C,T,F) ( C , T , F ) is a rectangular torsion theory if and only if C C is a pointed category with full replete subcategories T,F T , F such that: (i) every morphism between an object in T T and an object in F F (in either direction) is a zero morphism; (ii) any object in C C decomposes as a biproduct of an object in T T and an object in F F , and, all biproducts of an object in T T with an object in F F exist.