Mixing
How to prove “The maps factoring through an injective object are precisely the null-homotopic maps”
$begingroup$
Thanks for your attention,
I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras,
I cannot prove this sentence:
"The maps factoring through an injective object are precisely the null-homotopic maps."
Here, the object is in the complex category.
Please give a complete proof or some references which take it!
category-theory representation-theory
edited Jan 7 at 11:14
OmG
2,482722
asked Dec 20 '10 at 12:47
Zani LuoZani Luo
241
$endgroup$
add a comment |
$begingroup$
Thanks for your attention,
I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras,
I cannot prove this sentence:
"The maps factoring through an injective object are precisely the null-homotopic maps."
Here, the object is in the complex category.
Please give a complete proof or some references which take it!
category-theory representation-theory
edited Jan 7 at 11:14
OmG
2,482722
asked Dec 20 '10 at 12:47
Zani LuoZani Luo
241
$endgroup$
1
$begingroup$
You should describe the context in which that statement appears.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:46
2
$begingroup$
In any case, this follows from the very special form which the injective objects have: to see this, notice that you can describe precisely the indecomposable injectives.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:49
$begingroup$
It's not true. Where in Happel's book does this sentence appear? What is true is that the null-homotopic maps are the ones factoring through $mathcal{S}$-injective objects, where $mathcal{S}$ is the class of short exact sequences of complexes that are split in each degree.
$endgroup$
– Jeremy Rickard
Jan 7 at 11:38
add a comment |
$begingroup$
Thanks for your attention,
I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras,
I cannot prove this sentence:
"The maps factoring through an injective object are precisely the null-homotopic maps."
Here, the object is in the complex category.
Please give a complete proof or some references which take it!
category-theory representation-theory
edited Jan 7 at 11:14
OmG
2,482722
asked Dec 20 '10 at 12:47
Zani LuoZani Luo
241
$endgroup$
Thanks for your attention,
I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras,
I cannot prove this sentence:
"The maps factoring through an injective object are precisely the null-homotopic maps."
Here, the object is in the complex category.
Please give a complete proof or some references which take it!
category-theory representation-theory
category-theory representation-theory
edited Jan 7 at 11:14
OmG
2,482722
asked Dec 20 '10 at 12:47
Zani LuoZani Luo
241
edited Jan 7 at 11:14
OmG
2,482722
asked Dec 20 '10 at 12:47
Zani LuoZani Luo
241
edited Jan 7 at 11:14
OmG
2,482722
edited Jan 7 at 11:14
OmG
2,482722
edited Jan 7 at 11:14
OmG
2,482722
2,482722
asked Dec 20 '10 at 12:47
Zani LuoZani Luo
241
asked Dec 20 '10 at 12:47
Zani LuoZani Luo
241
asked Dec 20 '10 at 12:47
Zani LuoZani Luo
241
241
1
$begingroup$
You should describe the context in which that statement appears.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:46
2
$begingroup$
In any case, this follows from the very special form which the injective objects have: to see this, notice that you can describe precisely the indecomposable injectives.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:49
$begingroup$
It's not true. Where in Happel's book does this sentence appear? What is true is that the null-homotopic maps are the ones factoring through $mathcal{S}$-injective objects, where $mathcal{S}$ is the class of short exact sequences of complexes that are split in each degree.
$endgroup$
– Jeremy Rickard
Jan 7 at 11:38
add a comment |
1
$begingroup$
You should describe the context in which that statement appears.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:46
2
$begingroup$
In any case, this follows from the very special form which the injective objects have: to see this, notice that you can describe precisely the indecomposable injectives.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:49
$begingroup$
It's not true. Where in Happel's book does this sentence appear? What is true is that the null-homotopic maps are the ones factoring through $mathcal{S}$-injective objects, where $mathcal{S}$ is the class of short exact sequences of complexes that are split in each degree.
$endgroup$
– Jeremy Rickard
Jan 7 at 11:38
1
1
$begingroup$
You should describe the context in which that statement appears.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:46
$begingroup$
You should describe the context in which that statement appears.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:46
2
2
$begingroup$
In any case, this follows from the very special form which the injective objects have: to see this, notice that you can describe precisely the indecomposable injectives.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:49
$begingroup$
In any case, this follows from the very special form which the injective objects have: to see this, notice that you can describe precisely the indecomposable injectives.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:49
$begingroup$
It's not true. Where in Happel's book does this sentence appear? What is true is that the null-homotopic maps are the ones factoring through $mathcal{S}$-injective objects, where $mathcal{S}$ is the class of short exact sequences of complexes that are split in each degree.
$endgroup$
– Jeremy Rickard
Jan 7 at 11:38
$begingroup$
It's not true. Where in Happel's book does this sentence appear? What is true is that the null-homotopic maps are the ones factoring through $mathcal{S}$-injective objects, where $mathcal{S}$ is the class of short exact sequences of complexes that are split in each degree.
$endgroup$
– Jeremy Rickard
Jan 7 at 11:38
add a comment |
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1
$begingroup$
You should describe the context in which that statement appears.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:46
2
$begingroup$
In any case, this follows from the very special form which the injective objects have: to see this, notice that you can describe precisely the indecomposable injectives.
$endgroup$
– Mariano Suárez-Álvarez
Dec 20 '10 at 21:49
$begingroup$
It's not true. Where in Happel's book does this sentence appear? What is true is that the null-homotopic maps are the ones factoring through $mathcal{S}$-injective objects, where $mathcal{S}$ is the class of short exact sequences of complexes that are split in each degree.
$endgroup$
– Jeremy Rickard
Jan 7 at 11:38