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Keywords:
comma category; cocompatible functor; cotorsion pair
comma category; cocompatible functor; cotorsion pair
Summary:
Let $\mathcal {A}$ and $\mathcal {B}$ be abelian categories with enough projective and injective objects, and $T \colon \mathcal {A}\rightarrow \mathcal {B}$ a left exact additive functor. Then one has a comma category $(\mathopen {\mathcal {B} \downarrow T})$. It is shown that if $T \colon \mathcal {A}\rightarrow \mathcal {B}$ is $\mathcal {X}$-exact, then $(^\bot \mathcal {X}, \mathcal {X})$ is a (hereditary) cotorsion pair in $\mathcal {A}$ and $(^\bot \mathcal {Y}, \mathcal {Y})$) is a (hereditary) cotorsion pair in $\mathcal {B}$ if and only if $\bigl (\binom {^\bot \mathcal {X}}{^\bot \mathcal {Y}} \bigr ), \langle {\bf h}(\mathcal {X}, \mathcal {Y})\rangle )$ is a (hereditary) cotorsion pair in $(\mathopen {\mathcal {B}\downarrow T})$ and $\mathcal {X}$ and $\mathcal {Y}$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $\mathcal {A}$ and $\mathcal {B}$ can induce special preenveloping classes in $(\mathopen {\mathcal {B}\downarrow T})$.
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