A Remark in Weibel's “Introduction to Homological Algebra”
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In the section on the derived functors of the inverse limit(with $...3rightarrow 2 rightarrow 1 rightarrow 0$ as index category), Weibel constructs the inverse limit using the map $Delta$ in the photograph.
Later, he remarks that everything goes through for any $AB4*$ category. But I can't even construct the map in the general case. My question is, how does one do this for any $AB4*$ category?
homological-algebra abelian-categories
asked Jan 6 at 10:25
Jehu314Jehu314
1549
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add a comment |
$begingroup$
In the section on the derived functors of the inverse limit(with $...3rightarrow 2 rightarrow 1 rightarrow 0$ as index category), Weibel constructs the inverse limit using the map $Delta$ in the photograph.
Later, he remarks that everything goes through for any $AB4*$ category. But I can't even construct the map in the general case. My question is, how does one do this for any $AB4*$ category?
homological-algebra abelian-categories
asked Jan 6 at 10:25
Jehu314Jehu314
1549
$endgroup$
2
$begingroup$
The question is just about how to define $Delta$ without elements ? This is easy then, $Delta$ is the difference between the identity $prod_{iinmathbb{N}}A_itoprod_{iinmathbb{N}}A_i$ and the map such that on the $A_j$ component, it is $prod_{iinmathbb{N}}A_ito A_{j+1}to A_j$.
$endgroup$
– Roland
Jan 6 at 12:32
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Thanks, it was so silly of me to miss that!
$endgroup$
– Jehu314
Jan 6 at 14:04
add a comment |
$begingroup$
In the section on the derived functors of the inverse limit(with $...3rightarrow 2 rightarrow 1 rightarrow 0$ as index category), Weibel constructs the inverse limit using the map $Delta$ in the photograph.
Later, he remarks that everything goes through for any $AB4*$ category. But I can't even construct the map in the general case. My question is, how does one do this for any $AB4*$ category?
homological-algebra abelian-categories
asked Jan 6 at 10:25
Jehu314Jehu314
1549
$endgroup$
In the section on the derived functors of the inverse limit(with $...3rightarrow 2 rightarrow 1 rightarrow 0$ as index category), Weibel constructs the inverse limit using the map $Delta$ in the photograph.
Later, he remarks that everything goes through for any $AB4*$ category. But I can't even construct the map in the general case. My question is, how does one do this for any $AB4*$ category?
homological-algebra abelian-categories
homological-algebra abelian-categories
asked Jan 6 at 10:25
Jehu314Jehu314
1549
asked Jan 6 at 10:25
Jehu314Jehu314
1549
asked Jan 6 at 10:25
Jehu314Jehu314
1549
asked Jan 6 at 10:25
Jehu314Jehu314
1549
asked Jan 6 at 10:25
Jehu314Jehu314
1549
1549
2
$begingroup$
The question is just about how to define $Delta$ without elements ? This is easy then, $Delta$ is the difference between the identity $prod_{iinmathbb{N}}A_itoprod_{iinmathbb{N}}A_i$ and the map such that on the $A_j$ component, it is $prod_{iinmathbb{N}}A_ito A_{j+1}to A_j$.
$endgroup$
– Roland
Jan 6 at 12:32
$begingroup$
Thanks, it was so silly of me to miss that!
$endgroup$
– Jehu314
Jan 6 at 14:04
add a comment |
2
$begingroup$
The question is just about how to define $Delta$ without elements ? This is easy then, $Delta$ is the difference between the identity $prod_{iinmathbb{N}}A_itoprod_{iinmathbb{N}}A_i$ and the map such that on the $A_j$ component, it is $prod_{iinmathbb{N}}A_ito A_{j+1}to A_j$.
$endgroup$
– Roland
Jan 6 at 12:32
$begingroup$
Thanks, it was so silly of me to miss that!
$endgroup$
– Jehu314
Jan 6 at 14:04
2
2
$begingroup$
The question is just about how to define $Delta$ without elements ? This is easy then, $Delta$ is the difference between the identity $prod_{iinmathbb{N}}A_itoprod_{iinmathbb{N}}A_i$ and the map such that on the $A_j$ component, it is $prod_{iinmathbb{N}}A_ito A_{j+1}to A_j$.
$endgroup$
– Roland
Jan 6 at 12:32
$begingroup$
The question is just about how to define $Delta$ without elements ? This is easy then, $Delta$ is the difference between the identity $prod_{iinmathbb{N}}A_itoprod_{iinmathbb{N}}A_i$ and the map such that on the $A_j$ component, it is $prod_{iinmathbb{N}}A_ito A_{j+1}to A_j$.
$endgroup$
– Roland
Jan 6 at 12:32
$begingroup$
Thanks, it was so silly of me to miss that!
$endgroup$
– Jehu314
Jan 6 at 14:04
$begingroup$
Thanks, it was so silly of me to miss that!
$endgroup$
– Jehu314
Jan 6 at 14:04
add a comment |
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$begingroup$
The question is just about how to define $Delta$ without elements ? This is easy then, $Delta$ is the difference between the identity $prod_{iinmathbb{N}}A_itoprod_{iinmathbb{N}}A_i$ and the map such that on the $A_j$ component, it is $prod_{iinmathbb{N}}A_ito A_{j+1}to A_j$.
$endgroup$
– Roland
Jan 6 at 12:32
$begingroup$
Thanks, it was so silly of me to miss that!
$endgroup$
– Jehu314
Jan 6 at 14:04