Finite limits in derived categories
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Let $mathcal{A}$ be an abelian category with finite limits, let $D(mathcal{A})$ be its derived category.
(1)Does $K(mathcal{A})$ admit finite limits?
(2)Does $D(mathcal{A})$ admit finite limits?
(3) I think there are two notion of exactness for a functor $Fcolon D(mathcal{A})to D(mathcal{B})$, one of them is: it sends distinguished triangles to distinguished triangles. Another is the $F$ preserve finite limits(left exact) and colimits(right exact). I am just wondering if this two definitions are the same, or even if the question make sense.
derived-categories
asked Nov 17 at 21:55
Qixiao
2,7671627
add a comment |
up vote
1
down vote
favorite
Let $mathcal{A}$ be an abelian category with finite limits, let $D(mathcal{A})$ be its derived category.
(1)Does $K(mathcal{A})$ admit finite limits?
(2)Does $D(mathcal{A})$ admit finite limits?
(3) I think there are two notion of exactness for a functor $Fcolon D(mathcal{A})to D(mathcal{B})$, one of them is: it sends distinguished triangles to distinguished triangles. Another is the $F$ preserve finite limits(left exact) and colimits(right exact). I am just wondering if this two definitions are the same, or even if the question make sense.
derived-categories
asked Nov 17 at 21:55
Qixiao
2,7671627
Taking finite limits is not exact (in $A$), so is itself an operation that needs to be derived in the setting of derived categories: you need to consider what are called homotopy limits.
– Qiaochu Yuan
Nov 17 at 23:46
@QiaochuYuan Thanks for the comment, Apart from this I heard that derived category is not the correct setting to discuss construction of limits, etc, while $infty$-category are the natural setting where the construction work better, do you know some concrete statements I can read about? Or how can we upgrade derived category to $infty$-category? Thanks a lot!
– Qixiao
Nov 18 at 2:47
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $mathcal{A}$ be an abelian category with finite limits, let $D(mathcal{A})$ be its derived category.
(1)Does $K(mathcal{A})$ admit finite limits?
(2)Does $D(mathcal{A})$ admit finite limits?
(3) I think there are two notion of exactness for a functor $Fcolon D(mathcal{A})to D(mathcal{B})$, one of them is: it sends distinguished triangles to distinguished triangles. Another is the $F$ preserve finite limits(left exact) and colimits(right exact). I am just wondering if this two definitions are the same, or even if the question make sense.
derived-categories
asked Nov 17 at 21:55
Qixiao
2,7671627
Let $mathcal{A}$ be an abelian category with finite limits, let $D(mathcal{A})$ be its derived category.
(1)Does $K(mathcal{A})$ admit finite limits?
(2)Does $D(mathcal{A})$ admit finite limits?
(3) I think there are two notion of exactness for a functor $Fcolon D(mathcal{A})to D(mathcal{B})$, one of them is: it sends distinguished triangles to distinguished triangles. Another is the $F$ preserve finite limits(left exact) and colimits(right exact). I am just wondering if this two definitions are the same, or even if the question make sense.
derived-categories
derived-categories
asked Nov 17 at 21:55
Qixiao
2,7671627
asked Nov 17 at 21:55
Qixiao
2,7671627
asked Nov 17 at 21:55
Qixiao
2,7671627
asked Nov 17 at 21:55
Qixiao
2,7671627
asked Nov 17 at 21:55
Qixiao
2,7671627
2,7671627
Taking finite limits is not exact (in $A$), so is itself an operation that needs to be derived in the setting of derived categories: you need to consider what are called homotopy limits.
– Qiaochu Yuan
Nov 17 at 23:46
@QiaochuYuan Thanks for the comment, Apart from this I heard that derived category is not the correct setting to discuss construction of limits, etc, while $infty$-category are the natural setting where the construction work better, do you know some concrete statements I can read about? Or how can we upgrade derived category to $infty$-category? Thanks a lot!
– Qixiao
Nov 18 at 2:47
add a comment |
Taking finite limits is not exact (in $A$), so is itself an operation that needs to be derived in the setting of derived categories: you need to consider what are called homotopy limits.
– Qiaochu Yuan
Nov 17 at 23:46
@QiaochuYuan Thanks for the comment, Apart from this I heard that derived category is not the correct setting to discuss construction of limits, etc, while $infty$-category are the natural setting where the construction work better, do you know some concrete statements I can read about? Or how can we upgrade derived category to $infty$-category? Thanks a lot!
– Qixiao
Nov 18 at 2:47
Taking finite limits is not exact (in $A$), so is itself an operation that needs to be derived in the setting of derived categories: you need to consider what are called homotopy limits.
– Qiaochu Yuan
Nov 17 at 23:46
Taking finite limits is not exact (in $A$), so is itself an operation that needs to be derived in the setting of derived categories: you need to consider what are called homotopy limits.
– Qiaochu Yuan
Nov 17 at 23:46
@QiaochuYuan Thanks for the comment, Apart from this I heard that derived category is not the correct setting to discuss construction of limits, etc, while $infty$-category are the natural setting where the construction work better, do you know some concrete statements I can read about? Or how can we upgrade derived category to $infty$-category? Thanks a lot!
– Qixiao
Nov 18 at 2:47
@QiaochuYuan Thanks for the comment, Apart from this I heard that derived category is not the correct setting to discuss construction of limits, etc, while $infty$-category are the natural setting where the construction work better, do you know some concrete statements I can read about? Or how can we upgrade derived category to $infty$-category? Thanks a lot!
– Qixiao
Nov 18 at 2:47
add a comment |
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Taking finite limits is not exact (in $A$), so is itself an operation that needs to be derived in the setting of derived categories: you need to consider what are called homotopy limits.
– Qiaochu Yuan
Nov 17 at 23:46
@QiaochuYuan Thanks for the comment, Apart from this I heard that derived category is not the correct setting to discuss construction of limits, etc, while $infty$-category are the natural setting where the construction work better, do you know some concrete statements I can read about? Or how can we upgrade derived category to $infty$-category? Thanks a lot!
– Qixiao
Nov 18 at 2:47