Cotorsion pairs in comma categories

Yuan, Yuan, He, Jian, and Wu, Dejun. "Cotorsion pairs in comma categories." Czechoslovak Mathematical Journal 74.3 (2024): 715-734. <http://eudml.org/doc/299303>.

@article{Yuan2024,
abstract = {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 $(\mathopen \{\mathcal \{B\}\downarrow T\})$.},
author = {Yuan, Yuan, He, Jian, Wu, Dejun},
journal = {Czechoslovak Mathematical Journal},
keywords = {comma category; cocompatible functor; cotorsion pair},
language = {eng},
number = {3},
pages = {715-734},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cotorsion pairs in comma categories},
url = {http://eudml.org/doc/299303},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Yuan, Yuan
AU - He, Jian
AU - Wu, Dejun
TI - Cotorsion pairs in comma categories
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 3
SP - 715
EP - 734
AB - 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 $(\mathopen {\mathcal {B}\downarrow T})$.
LA - eng
KW - comma category; cocompatible functor; cotorsion pair
UR - http://eudml.org/doc/299303
ER -