The derived category is additive
Let $mathcal C$ be an abelian category. One way to see the derived category $D(mathcal C)$ is that it has
- the same objects as $operatorname{Ch}(mathcal C)$,
- roofs $Axleftarrow{simeq}Z_1rightarrow Z_2xleftarrow{simeq}Z_3rightarrowcdots xleftarrow{simeq}Z_nrightarrow B$ as morphisms.
To see that $D(mathcal C)$ is additive, it suffices to show that it contains finite biproducts, for then we can define the addition of morphisms in terms of $oplus$. So the goal is to find a biproduct of two objects $A, Bin D(mathcal C)$.
Clearly, for the object $Aoplus B$ from $operatorname{Ch}(mathcal C)$ there are inclusion morphisms $Ato Aoplus Bleftarrow B$. Let $T$ be an object with morphisms
$$Axleftarrow{simeq}C_1rightarrow T,quad
Bxleftarrow{simeq}C_2rightarrow T.$$
Note that it suffices to consider single-step roofs because the argument, once established, can be iterated for general roofs as above.
We see that there is a morphism $Aoplus Bxleftarrow{simeq} C_1oplus C_2to T$, making the diagram commute. However, I fail to show its uniqueness: Given another morphism $Aoplus Bxleftarrow{simeq} Zto T$, we have to show that both are equivalent, i.e., there is an object $Y$ with morphisms such that
$$begin{matrix}
&&Z\
&swarrow&uparrow&searrow\
A&leftarrow &Y&rightarrow &T\
&nwarrow&downarrow&nearrow\
&& C_1oplus C_2
end{matrix}$$
commutes, where $Yxrightarrow{simeq} A$.
Question: How to I find this object $Y$, showing uniqueness of the canonical morphism from $Aoplus B$ to $T$?
category-theory homological-algebra triangulated-categories derived-categories
asked Nov 28 at 8:55
Bubaya
398111
add a comment |
Let $mathcal C$ be an abelian category. One way to see the derived category $D(mathcal C)$ is that it has
- the same objects as $operatorname{Ch}(mathcal C)$,
- roofs $Axleftarrow{simeq}Z_1rightarrow Z_2xleftarrow{simeq}Z_3rightarrowcdots xleftarrow{simeq}Z_nrightarrow B$ as morphisms.
To see that $D(mathcal C)$ is additive, it suffices to show that it contains finite biproducts, for then we can define the addition of morphisms in terms of $oplus$. So the goal is to find a biproduct of two objects $A, Bin D(mathcal C)$.
Clearly, for the object $Aoplus B$ from $operatorname{Ch}(mathcal C)$ there are inclusion morphisms $Ato Aoplus Bleftarrow B$. Let $T$ be an object with morphisms
$$Axleftarrow{simeq}C_1rightarrow T,quad
Bxleftarrow{simeq}C_2rightarrow T.$$
Note that it suffices to consider single-step roofs because the argument, once established, can be iterated for general roofs as above.
We see that there is a morphism $Aoplus Bxleftarrow{simeq} C_1oplus C_2to T$, making the diagram commute. However, I fail to show its uniqueness: Given another morphism $Aoplus Bxleftarrow{simeq} Zto T$, we have to show that both are equivalent, i.e., there is an object $Y$ with morphisms such that
$$begin{matrix}
&&Z\
&swarrow&uparrow&searrow\
A&leftarrow &Y&rightarrow &T\
&nwarrow&downarrow&nearrow\
&& C_1oplus C_2
end{matrix}$$
commutes, where $Yxrightarrow{simeq} A$.
Question: How to I find this object $Y$, showing uniqueness of the canonical morphism from $Aoplus B$ to $T$?
category-theory homological-algebra triangulated-categories derived-categories
asked Nov 28 at 8:55
Bubaya
398111
the localization of an additive category is additive category.Thus you only need to check homotopy category is additive category.
– Sky
Dec 1 at 13:57
@Sky That $K(mathcal C)$ is additive is clear. However, it is not totally clear to me that the localisation of an additive category yields an additive one again.
– Bubaya
Dec 2 at 20:42
add a comment |
Let $mathcal C$ be an abelian category. One way to see the derived category $D(mathcal C)$ is that it has
- the same objects as $operatorname{Ch}(mathcal C)$,
- roofs $Axleftarrow{simeq}Z_1rightarrow Z_2xleftarrow{simeq}Z_3rightarrowcdots xleftarrow{simeq}Z_nrightarrow B$ as morphisms.
To see that $D(mathcal C)$ is additive, it suffices to show that it contains finite biproducts, for then we can define the addition of morphisms in terms of $oplus$. So the goal is to find a biproduct of two objects $A, Bin D(mathcal C)$.
Clearly, for the object $Aoplus B$ from $operatorname{Ch}(mathcal C)$ there are inclusion morphisms $Ato Aoplus Bleftarrow B$. Let $T$ be an object with morphisms
$$Axleftarrow{simeq}C_1rightarrow T,quad
Bxleftarrow{simeq}C_2rightarrow T.$$
Note that it suffices to consider single-step roofs because the argument, once established, can be iterated for general roofs as above.
We see that there is a morphism $Aoplus Bxleftarrow{simeq} C_1oplus C_2to T$, making the diagram commute. However, I fail to show its uniqueness: Given another morphism $Aoplus Bxleftarrow{simeq} Zto T$, we have to show that both are equivalent, i.e., there is an object $Y$ with morphisms such that
$$begin{matrix}
&&Z\
&swarrow&uparrow&searrow\
A&leftarrow &Y&rightarrow &T\
&nwarrow&downarrow&nearrow\
&& C_1oplus C_2
end{matrix}$$
commutes, where $Yxrightarrow{simeq} A$.
Question: How to I find this object $Y$, showing uniqueness of the canonical morphism from $Aoplus B$ to $T$?
category-theory homological-algebra triangulated-categories derived-categories
asked Nov 28 at 8:55
Bubaya
398111
Let $mathcal C$ be an abelian category. One way to see the derived category $D(mathcal C)$ is that it has
- the same objects as $operatorname{Ch}(mathcal C)$,
- roofs $Axleftarrow{simeq}Z_1rightarrow Z_2xleftarrow{simeq}Z_3rightarrowcdots xleftarrow{simeq}Z_nrightarrow B$ as morphisms.
To see that $D(mathcal C)$ is additive, it suffices to show that it contains finite biproducts, for then we can define the addition of morphisms in terms of $oplus$. So the goal is to find a biproduct of two objects $A, Bin D(mathcal C)$.
Clearly, for the object $Aoplus B$ from $operatorname{Ch}(mathcal C)$ there are inclusion morphisms $Ato Aoplus Bleftarrow B$. Let $T$ be an object with morphisms
$$Axleftarrow{simeq}C_1rightarrow T,quad
Bxleftarrow{simeq}C_2rightarrow T.$$
Note that it suffices to consider single-step roofs because the argument, once established, can be iterated for general roofs as above.
We see that there is a morphism $Aoplus Bxleftarrow{simeq} C_1oplus C_2to T$, making the diagram commute. However, I fail to show its uniqueness: Given another morphism $Aoplus Bxleftarrow{simeq} Zto T$, we have to show that both are equivalent, i.e., there is an object $Y$ with morphisms such that
$$begin{matrix}
&&Z\
&swarrow&uparrow&searrow\
A&leftarrow &Y&rightarrow &T\
&nwarrow&downarrow&nearrow\
&& C_1oplus C_2
end{matrix}$$
commutes, where $Yxrightarrow{simeq} A$.
Question: How to I find this object $Y$, showing uniqueness of the canonical morphism from $Aoplus B$ to $T$?
category-theory homological-algebra triangulated-categories derived-categories
category-theory homological-algebra triangulated-categories derived-categories
asked Nov 28 at 8:55
Bubaya
398111
asked Nov 28 at 8:55
Bubaya
398111
asked Nov 28 at 8:55
Bubaya
398111
asked Nov 28 at 8:55
Bubaya
398111
asked Nov 28 at 8:55
Bubaya
398111
398111
the localization of an additive category is additive category.Thus you only need to check homotopy category is additive category.
– Sky
Dec 1 at 13:57
@Sky That $K(mathcal C)$ is additive is clear. However, it is not totally clear to me that the localisation of an additive category yields an additive one again.
– Bubaya
Dec 2 at 20:42
add a comment |
the localization of an additive category is additive category.Thus you only need to check homotopy category is additive category.
– Sky
Dec 1 at 13:57
@Sky That $K(mathcal C)$ is additive is clear. However, it is not totally clear to me that the localisation of an additive category yields an additive one again.
– Bubaya
Dec 2 at 20:42
the localization of an additive category is additive category.Thus you only need to check homotopy category is additive category.
– Sky
Dec 1 at 13:57
the localization of an additive category is additive category.Thus you only need to check homotopy category is additive category.
– Sky
Dec 1 at 13:57
@Sky That $K(mathcal C)$ is additive is clear. However, it is not totally clear to me that the localisation of an additive category yields an additive one again.
– Bubaya
Dec 2 at 20:42
@Sky That $K(mathcal C)$ is additive is clear. However, it is not totally clear to me that the localisation of an additive category yields an additive one again.
– Bubaya
Dec 2 at 20:42
add a comment |
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the localization of an additive category is additive category.Thus you only need to check homotopy category is additive category.
– Sky
Dec 1 at 13:57
@Sky That $K(mathcal C)$ is additive is clear. However, it is not totally clear to me that the localisation of an additive category yields an additive one again.
– Bubaya
Dec 2 at 20:42