When is the map between cokernels a monomorphism, when is it an epimorphism?
$require{AMScd}$
Let us denote by $text{Arr}(mathcal{C})$ the arrow category of $mathcal{C}$ which has objects given by morphisms $f:Ato B$ in $mathcal{C}$, and has morphisms given from $(f:Ato B)to (g:Cto D)$ by commuting squares:
$$begin{CD}
A@>>> C\
@V{f}VV @V{g}VV\
B @>>> D
end{CD}$$
Say $mathcal{C}$ is abelian. Then $text{coker}$ is an endofunctor on the arrow category:
$$begin{CD}A@>>> C\@VfVV @VgVV\B@>>>D\@V{c_1}VV @Vc_2VV\text{coker}(f)@>>>text{coker}(g)end{CD}$$
(I.e. it takes $f:Ato B$ to $c_1:Bto text{coker}(f)$.)
When is $text{coker}(f)to text{coker}(g)$ a monomorphism, when is it an epimorphism?
I am hoping one can say something like "coker" preserves monomorphisms or epimorphisms.
abstract-algebra category-theory homological-algebra
asked Dec 10 '18 at 14:11
user616128
405
add a comment |
$require{AMScd}$
Let us denote by $text{Arr}(mathcal{C})$ the arrow category of $mathcal{C}$ which has objects given by morphisms $f:Ato B$ in $mathcal{C}$, and has morphisms given from $(f:Ato B)to (g:Cto D)$ by commuting squares:
$$begin{CD}
A@>>> C\
@V{f}VV @V{g}VV\
B @>>> D
end{CD}$$
Say $mathcal{C}$ is abelian. Then $text{coker}$ is an endofunctor on the arrow category:
$$begin{CD}A@>>> C\@VfVV @VgVV\B@>>>D\@V{c_1}VV @Vc_2VV\text{coker}(f)@>>>text{coker}(g)end{CD}$$
(I.e. it takes $f:Ato B$ to $c_1:Bto text{coker}(f)$.)
When is $text{coker}(f)to text{coker}(g)$ a monomorphism, when is it an epimorphism?
I am hoping one can say something like "coker" preserves monomorphisms or epimorphisms.
abstract-algebra category-theory homological-algebra
asked Dec 10 '18 at 14:11
user616128
405
1
It's an easy exercise to see that if $Bto D$ is epi, then so is $text{coker}(f)to text{coker}(g)$; it's an instructive one to find counterexamples to the converse. It's very easy also to find examples where monos aren't preserved.
– Max
Dec 10 '18 at 19:17
@Max Monomorphisms in the arrow category (which are pairs of monomorphisms in the base category I think) aren't preserved?
– user616128
Dec 11 '18 at 13:03
No, it's easy to find counterexamples.
– Max
Dec 11 '18 at 13:13
add a comment |
$require{AMScd}$
Let us denote by $text{Arr}(mathcal{C})$ the arrow category of $mathcal{C}$ which has objects given by morphisms $f:Ato B$ in $mathcal{C}$, and has morphisms given from $(f:Ato B)to (g:Cto D)$ by commuting squares:
$$begin{CD}
A@>>> C\
@V{f}VV @V{g}VV\
B @>>> D
end{CD}$$
Say $mathcal{C}$ is abelian. Then $text{coker}$ is an endofunctor on the arrow category:
$$begin{CD}A@>>> C\@VfVV @VgVV\B@>>>D\@V{c_1}VV @Vc_2VV\text{coker}(f)@>>>text{coker}(g)end{CD}$$
(I.e. it takes $f:Ato B$ to $c_1:Bto text{coker}(f)$.)
When is $text{coker}(f)to text{coker}(g)$ a monomorphism, when is it an epimorphism?
I am hoping one can say something like "coker" preserves monomorphisms or epimorphisms.
abstract-algebra category-theory homological-algebra
asked Dec 10 '18 at 14:11
user616128
405
$require{AMScd}$
Let us denote by $text{Arr}(mathcal{C})$ the arrow category of $mathcal{C}$ which has objects given by morphisms $f:Ato B$ in $mathcal{C}$, and has morphisms given from $(f:Ato B)to (g:Cto D)$ by commuting squares:
$$begin{CD}
A@>>> C\
@V{f}VV @V{g}VV\
B @>>> D
end{CD}$$
Say $mathcal{C}$ is abelian. Then $text{coker}$ is an endofunctor on the arrow category:
$$begin{CD}A@>>> C\@VfVV @VgVV\B@>>>D\@V{c_1}VV @Vc_2VV\text{coker}(f)@>>>text{coker}(g)end{CD}$$
(I.e. it takes $f:Ato B$ to $c_1:Bto text{coker}(f)$.)
When is $text{coker}(f)to text{coker}(g)$ a monomorphism, when is it an epimorphism?
I am hoping one can say something like "coker" preserves monomorphisms or epimorphisms.
abstract-algebra category-theory homological-algebra
abstract-algebra category-theory homological-algebra
asked Dec 10 '18 at 14:11
user616128
405
asked Dec 10 '18 at 14:11
user616128
405
edited Dec 10 '18 at 18:40
asked Dec 10 '18 at 14:11
user616128
405
asked Dec 10 '18 at 14:11
user616128
405
asked Dec 10 '18 at 14:11
user616128
405
405
1
It's an easy exercise to see that if $Bto D$ is epi, then so is $text{coker}(f)to text{coker}(g)$; it's an instructive one to find counterexamples to the converse. It's very easy also to find examples where monos aren't preserved.
– Max
Dec 10 '18 at 19:17
@Max Monomorphisms in the arrow category (which are pairs of monomorphisms in the base category I think) aren't preserved?
– user616128
Dec 11 '18 at 13:03
No, it's easy to find counterexamples.
– Max
Dec 11 '18 at 13:13
add a comment |
1
It's an easy exercise to see that if $Bto D$ is epi, then so is $text{coker}(f)to text{coker}(g)$; it's an instructive one to find counterexamples to the converse. It's very easy also to find examples where monos aren't preserved.
– Max
Dec 10 '18 at 19:17
@Max Monomorphisms in the arrow category (which are pairs of monomorphisms in the base category I think) aren't preserved?
– user616128
Dec 11 '18 at 13:03
No, it's easy to find counterexamples.
– Max
Dec 11 '18 at 13:13
1
1
It's an easy exercise to see that if $Bto D$ is epi, then so is $text{coker}(f)to text{coker}(g)$; it's an instructive one to find counterexamples to the converse. It's very easy also to find examples where monos aren't preserved.
– Max
Dec 10 '18 at 19:17
It's an easy exercise to see that if $Bto D$ is epi, then so is $text{coker}(f)to text{coker}(g)$; it's an instructive one to find counterexamples to the converse. It's very easy also to find examples where monos aren't preserved.
– Max
Dec 10 '18 at 19:17
@Max Monomorphisms in the arrow category (which are pairs of monomorphisms in the base category I think) aren't preserved?
– user616128
Dec 11 '18 at 13:03
@Max Monomorphisms in the arrow category (which are pairs of monomorphisms in the base category I think) aren't preserved?
– user616128
Dec 11 '18 at 13:03
No, it's easy to find counterexamples.
– Max
Dec 11 '18 at 13:13
No, it's easy to find counterexamples.
– Max
Dec 11 '18 at 13:13
add a comment |
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1
It's an easy exercise to see that if $Bto D$ is epi, then so is $text{coker}(f)to text{coker}(g)$; it's an instructive one to find counterexamples to the converse. It's very easy also to find examples where monos aren't preserved.
– Max
Dec 10 '18 at 19:17
@Max Monomorphisms in the arrow category (which are pairs of monomorphisms in the base category I think) aren't preserved?
– user616128
Dec 11 '18 at 13:03
No, it's easy to find counterexamples.
– Max
Dec 11 '18 at 13:13