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.










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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

















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.










share|cite|improve this question






















  • 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















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.










share|cite|improve this question













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






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asked Nov 17 at 21:55









Qixiao

2,7651626




2,7651626












  • 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












  • @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

















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