Cotorsion pairs in comma categories

Abstract

Let \(\cal{A}\) and \(\cal{B}\) be abelian categories with enough projective and injective objects, and \(T \colon\cal{A}\rightarrow\cal{B}\) a left exact additive functor. Then one has a comma category (\(\mathopen{\cal{B} \downarrow T}\)). It is shown that if \(T \colon\cal{A}\rightarrow\cal{B}\) is \(\cal{X}\)-exact, then is a (hereditary) cotorsion pair in \(\cal{A}\) and is a (hereditary) cotorsion pair in \(\cal{B}\) if and only if is a (hereditary) cotorsion pair in (\(\mathopen{\cal{B}\downarrow T}\)) and \(\cal{X}\) and \(\cal{Y}\) are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories \(\cal{A}\) and \(\cal{B}\) can induce special preenveloping classes in (\(\mathopen{\cal{B}\downarrow T}\)).