Mixing
Can we “integrate†functors?
Clash Royale CLAN TAG#URR8PPP
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Let $F:mathcalCrightarrow mathcalC'$ be a functor between "nice" (e.g. abelian with enough injectives) categories. If F is not exact we can form the derived functors $F',F'',...$
Is it possible to reverse this process ("integrate" $F$) or stated reversely: Are there nice conditions on a functor $F$ to be a derived functor?
category-theory homological-algebra derived-functors
asked Jul 16 at 14:00
Takirion
661211
add a comment |Â
up vote
9
down vote
favorite
Let $F:mathcalCrightarrow mathcalC'$ be a functor between "nice" (e.g. abelian with enough injectives) categories. If F is not exact we can form the derived functors $F',F'',...$
Is it possible to reverse this process ("integrate" $F$) or stated reversely: Are there nice conditions on a functor $F$ to be a derived functor?
category-theory homological-algebra derived-functors
asked Jul 16 at 14:00
Takirion
661211
Yoneda observed that $L_p(F)(A) = textNat(textExt^p(texthom_A), F)$, this might be helpful.
– Ivan Di Liberti
Jul 16 at 15:34
Could you explain what $L_p$ is? And do you mean $textExt^p(texthom_A)$ to be $Ext^p(underline ,A)$?
– Takirion
Jul 17 at 8:34
@Takirion, I guess in this context $L_p$ is the left derivation of functors (see Wikipedia). So $L_p(F)$ is the derived functor, which you can apply to an object A.
– Babelfish
Jul 23 at 16:40
Ah, yes, this seems plausible.
– Takirion
Jul 23 at 17:17
add a comment |Â
up vote
9
down vote
favorite
up vote
9
down vote
favorite
Let $F:mathcalCrightarrow mathcalC'$ be a functor between "nice" (e.g. abelian with enough injectives) categories. If F is not exact we can form the derived functors $F',F'',...$
Is it possible to reverse this process ("integrate" $F$) or stated reversely: Are there nice conditions on a functor $F$ to be a derived functor?
category-theory homological-algebra derived-functors
asked Jul 16 at 14:00
Takirion
661211
Let $F:mathcalCrightarrow mathcalC'$ be a functor between "nice" (e.g. abelian with enough injectives) categories. If F is not exact we can form the derived functors $F',F'',...$
Is it possible to reverse this process ("integrate" $F$) or stated reversely: Are there nice conditions on a functor $F$ to be a derived functor?
category-theory homological-algebra derived-functors
asked Jul 16 at 14:00
Takirion
661211
asked Jul 16 at 14:00
Takirion
661211
asked Jul 16 at 14:00
Takirion
661211
asked Jul 16 at 14:00
Takirion
661211
661211
Yoneda observed that $L_p(F)(A) = textNat(textExt^p(texthom_A), F)$, this might be helpful.
– Ivan Di Liberti
Jul 16 at 15:34
Could you explain what $L_p$ is? And do you mean $textExt^p(texthom_A)$ to be $Ext^p(underline ,A)$?
– Takirion
Jul 17 at 8:34
@Takirion, I guess in this context $L_p$ is the left derivation of functors (see Wikipedia). So $L_p(F)$ is the derived functor, which you can apply to an object A.
– Babelfish
Jul 23 at 16:40
Ah, yes, this seems plausible.
– Takirion
Jul 23 at 17:17
add a comment |Â
Yoneda observed that $L_p(F)(A) = textNat(textExt^p(texthom_A), F)$, this might be helpful.
– Ivan Di Liberti
Jul 16 at 15:34
Could you explain what $L_p$ is? And do you mean $textExt^p(texthom_A)$ to be $Ext^p(underline ,A)$?
– Takirion
Jul 17 at 8:34
@Takirion, I guess in this context $L_p$ is the left derivation of functors (see Wikipedia). So $L_p(F)$ is the derived functor, which you can apply to an object A.
– Babelfish
Jul 23 at 16:40
Ah, yes, this seems plausible.
– Takirion
Jul 23 at 17:17
Yoneda observed that $L_p(F)(A) = textNat(textExt^p(texthom_A), F)$, this might be helpful.
– Ivan Di Liberti
Jul 16 at 15:34
Yoneda observed that $L_p(F)(A) = textNat(textExt^p(texthom_A), F)$, this might be helpful.
– Ivan Di Liberti
Jul 16 at 15:34
Could you explain what $L_p$ is? And do you mean $textExt^p(texthom_A)$ to be $Ext^p(underline ,A)$?
– Takirion
Jul 17 at 8:34
Could you explain what $L_p$ is? And do you mean $textExt^p(texthom_A)$ to be $Ext^p(underline ,A)$?
– Takirion
Jul 17 at 8:34
@Takirion, I guess in this context $L_p$ is the left derivation of functors (see Wikipedia). So $L_p(F)$ is the derived functor, which you can apply to an object A.
– Babelfish
Jul 23 at 16:40
@Takirion, I guess in this context $L_p$ is the left derivation of functors (see Wikipedia). So $L_p(F)$ is the derived functor, which you can apply to an object A.
– Babelfish
Jul 23 at 16:40
Ah, yes, this seems plausible.
– Takirion
Jul 23 at 17:17
Ah, yes, this seems plausible.
– Takirion
Jul 23 at 17:17
add a comment |Â
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Yoneda observed that $L_p(F)(A) = textNat(textExt^p(texthom_A), F)$, this might be helpful.
– Ivan Di Liberti
Jul 16 at 15:34
Could you explain what $L_p$ is? And do you mean $textExt^p(texthom_A)$ to be $Ext^p(underline ,A)$?
– Takirion
Jul 17 at 8:34
@Takirion, I guess in this context $L_p$ is the left derivation of functors (see Wikipedia). So $L_p(F)$ is the derived functor, which you can apply to an object A.
– Babelfish
Jul 23 at 16:40
Ah, yes, this seems plausible.
– Takirion
Jul 23 at 17:17