Mixing
Cosheaf homology Global Sections
$begingroup$
Let X be a topological space and U={Ui} be some open covering of X.
Let F be a cosheaf of abelian groups on X.
Is the 0th cech homology group the same as the global sections on X?
sheaf-theory sheaf-cohomology
asked Dec 31 '15 at 18:30
user84563user84563
39429
$endgroup$
add a comment |
$begingroup$
Let X be a topological space and U={Ui} be some open covering of X.
Let F be a cosheaf of abelian groups on X.
Is the 0th cech homology group the same as the global sections on X?
sheaf-theory sheaf-cohomology
asked Dec 31 '15 at 18:30
user84563user84563
39429
$endgroup$
add a comment |
$begingroup$
Let X be a topological space and U={Ui} be some open covering of X.
Let F be a cosheaf of abelian groups on X.
Is the 0th cech homology group the same as the global sections on X?
sheaf-theory sheaf-cohomology
asked Dec 31 '15 at 18:30
user84563user84563
39429
$endgroup$
Let X be a topological space and U={Ui} be some open covering of X.
Let F be a cosheaf of abelian groups on X.
Is the 0th cech homology group the same as the global sections on X?
sheaf-theory sheaf-cohomology
sheaf-theory sheaf-cohomology
asked Dec 31 '15 at 18:30
user84563user84563
39429
asked Dec 31 '15 at 18:30
user84563user84563
39429
asked Dec 31 '15 at 18:30
user84563user84563
39429
asked Dec 31 '15 at 18:30
user84563user84563
39429
asked Dec 31 '15 at 18:30
user84563user84563
39429
39429
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
I am not very familiar with this area, but wouldn't the following cosheaf give something distinct from a 0th Čech homology group? Let $X$ be a disconnected space with two components $A$ and $B$. Let the cosheaf $F$ be defined by $F(U)=mathbb Z$ if $Uneqemptyset$ and $Usubset A$. Let $F(U)=mathbb Z_2$ if $U$ is nonempty and $Usubset B$. If $U$ meets both $A$ and $B$ then $F(U)=mathbb Zoplusmathbb Z_2$ and if $U=emptyset$ then $F(U)={0}$. This seems to satisfy the cosheaf axioms.
The global sections are isomorphic to $mathbb Zoplusmathbb Z_2$, which is a hybrid between the $0$th homology with coefficients in two different abelian groups.
answered Dec 31 '15 at 18:53
Cheerful ParsnipCheerful Parsnip
21.2k23598
$endgroup$
$begingroup$
Hmm yes that seems to work... I was just wondering because the 0th cech cohomology group yields global sections for sheaves. There should be some dual notion for cosheaves
$endgroup$
– user84563
Dec 31 '15 at 19:13
add a comment |
$begingroup$
The $0$th Čech homology group $ check{mathrm{H}}_0(X,F) $ of a topological space $ X $ with coefficients in an abelian cosheaf $ F $ is isomorphic to the global (co)sections of $ F $, i.e. $ F(X) $, in a natural way. This is actually an immediate consequence of Proposition 4.2. in Section V from the book "Bredon, Sheaf Theory".
A sketch for this statement is as the following:
$ check{mathrm{H}}_0(X,F) $ is the same as cokernel of $ partial_0 $ (the 0th boundary operator). On the other hand, by cosheaf condition, the cokernel of $ partial_0 $ is $ F(X) $. Thus, the result is obtained.
answered Nov 5 '18 at 14:16
ARAARA
144
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
active
oldest
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active
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active
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votes
$begingroup$
I am not very familiar with this area, but wouldn't the following cosheaf give something distinct from a 0th Čech homology group? Let $X$ be a disconnected space with two components $A$ and $B$. Let the cosheaf $F$ be defined by $F(U)=mathbb Z$ if $Uneqemptyset$ and $Usubset A$. Let $F(U)=mathbb Z_2$ if $U$ is nonempty and $Usubset B$. If $U$ meets both $A$ and $B$ then $F(U)=mathbb Zoplusmathbb Z_2$ and if $U=emptyset$ then $F(U)={0}$. This seems to satisfy the cosheaf axioms.
The global sections are isomorphic to $mathbb Zoplusmathbb Z_2$, which is a hybrid between the $0$th homology with coefficients in two different abelian groups.
answered Dec 31 '15 at 18:53
Cheerful ParsnipCheerful Parsnip
21.2k23598
$endgroup$
$begingroup$
Hmm yes that seems to work... I was just wondering because the 0th cech cohomology group yields global sections for sheaves. There should be some dual notion for cosheaves
$endgroup$
– user84563
Dec 31 '15 at 19:13
add a comment |
$begingroup$
I am not very familiar with this area, but wouldn't the following cosheaf give something distinct from a 0th Čech homology group? Let $X$ be a disconnected space with two components $A$ and $B$. Let the cosheaf $F$ be defined by $F(U)=mathbb Z$ if $Uneqemptyset$ and $Usubset A$. Let $F(U)=mathbb Z_2$ if $U$ is nonempty and $Usubset B$. If $U$ meets both $A$ and $B$ then $F(U)=mathbb Zoplusmathbb Z_2$ and if $U=emptyset$ then $F(U)={0}$. This seems to satisfy the cosheaf axioms.
The global sections are isomorphic to $mathbb Zoplusmathbb Z_2$, which is a hybrid between the $0$th homology with coefficients in two different abelian groups.
answered Dec 31 '15 at 18:53
Cheerful ParsnipCheerful Parsnip
21.2k23598
$endgroup$
$begingroup$
Hmm yes that seems to work... I was just wondering because the 0th cech cohomology group yields global sections for sheaves. There should be some dual notion for cosheaves
$endgroup$
– user84563
Dec 31 '15 at 19:13
add a comment |
$begingroup$
I am not very familiar with this area, but wouldn't the following cosheaf give something distinct from a 0th Čech homology group? Let $X$ be a disconnected space with two components $A$ and $B$. Let the cosheaf $F$ be defined by $F(U)=mathbb Z$ if $Uneqemptyset$ and $Usubset A$. Let $F(U)=mathbb Z_2$ if $U$ is nonempty and $Usubset B$. If $U$ meets both $A$ and $B$ then $F(U)=mathbb Zoplusmathbb Z_2$ and if $U=emptyset$ then $F(U)={0}$. This seems to satisfy the cosheaf axioms.
The global sections are isomorphic to $mathbb Zoplusmathbb Z_2$, which is a hybrid between the $0$th homology with coefficients in two different abelian groups.
answered Dec 31 '15 at 18:53
Cheerful ParsnipCheerful Parsnip
21.2k23598
$endgroup$
I am not very familiar with this area, but wouldn't the following cosheaf give something distinct from a 0th Čech homology group? Let $X$ be a disconnected space with two components $A$ and $B$. Let the cosheaf $F$ be defined by $F(U)=mathbb Z$ if $Uneqemptyset$ and $Usubset A$. Let $F(U)=mathbb Z_2$ if $U$ is nonempty and $Usubset B$. If $U$ meets both $A$ and $B$ then $F(U)=mathbb Zoplusmathbb Z_2$ and if $U=emptyset$ then $F(U)={0}$. This seems to satisfy the cosheaf axioms.
The global sections are isomorphic to $mathbb Zoplusmathbb Z_2$, which is a hybrid between the $0$th homology with coefficients in two different abelian groups.
answered Dec 31 '15 at 18:53
Cheerful ParsnipCheerful Parsnip
21.2k23598
answered Dec 31 '15 at 18:53
Cheerful ParsnipCheerful Parsnip
21.2k23598
answered Dec 31 '15 at 18:53
Cheerful ParsnipCheerful Parsnip
21.2k23598
answered Dec 31 '15 at 18:53
Cheerful ParsnipCheerful Parsnip
21.2k23598
21.2k23598
$begingroup$
Hmm yes that seems to work... I was just wondering because the 0th cech cohomology group yields global sections for sheaves. There should be some dual notion for cosheaves
$endgroup$
– user84563
Dec 31 '15 at 19:13
add a comment |
$begingroup$
Hmm yes that seems to work... I was just wondering because the 0th cech cohomology group yields global sections for sheaves. There should be some dual notion for cosheaves
$endgroup$
– user84563
Dec 31 '15 at 19:13
$begingroup$
Hmm yes that seems to work... I was just wondering because the 0th cech cohomology group yields global sections for sheaves. There should be some dual notion for cosheaves
$endgroup$
– user84563
Dec 31 '15 at 19:13
$begingroup$
Hmm yes that seems to work... I was just wondering because the 0th cech cohomology group yields global sections for sheaves. There should be some dual notion for cosheaves
$endgroup$
– user84563
Dec 31 '15 at 19:13
add a comment |
$begingroup$
The $0$th Čech homology group $ check{mathrm{H}}_0(X,F) $ of a topological space $ X $ with coefficients in an abelian cosheaf $ F $ is isomorphic to the global (co)sections of $ F $, i.e. $ F(X) $, in a natural way. This is actually an immediate consequence of Proposition 4.2. in Section V from the book "Bredon, Sheaf Theory".
A sketch for this statement is as the following:
$ check{mathrm{H}}_0(X,F) $ is the same as cokernel of $ partial_0 $ (the 0th boundary operator). On the other hand, by cosheaf condition, the cokernel of $ partial_0 $ is $ F(X) $. Thus, the result is obtained.
answered Nov 5 '18 at 14:16
ARAARA
144
$endgroup$
add a comment |
$begingroup$
The $0$th Čech homology group $ check{mathrm{H}}_0(X,F) $ of a topological space $ X $ with coefficients in an abelian cosheaf $ F $ is isomorphic to the global (co)sections of $ F $, i.e. $ F(X) $, in a natural way. This is actually an immediate consequence of Proposition 4.2. in Section V from the book "Bredon, Sheaf Theory".
A sketch for this statement is as the following:
$ check{mathrm{H}}_0(X,F) $ is the same as cokernel of $ partial_0 $ (the 0th boundary operator). On the other hand, by cosheaf condition, the cokernel of $ partial_0 $ is $ F(X) $. Thus, the result is obtained.
answered Nov 5 '18 at 14:16
ARAARA
144
$endgroup$
add a comment |
$begingroup$
The $0$th Čech homology group $ check{mathrm{H}}_0(X,F) $ of a topological space $ X $ with coefficients in an abelian cosheaf $ F $ is isomorphic to the global (co)sections of $ F $, i.e. $ F(X) $, in a natural way. This is actually an immediate consequence of Proposition 4.2. in Section V from the book "Bredon, Sheaf Theory".
A sketch for this statement is as the following:
$ check{mathrm{H}}_0(X,F) $ is the same as cokernel of $ partial_0 $ (the 0th boundary operator). On the other hand, by cosheaf condition, the cokernel of $ partial_0 $ is $ F(X) $. Thus, the result is obtained.
answered Nov 5 '18 at 14:16
ARAARA
144
$endgroup$
The $0$th Čech homology group $ check{mathrm{H}}_0(X,F) $ of a topological space $ X $ with coefficients in an abelian cosheaf $ F $ is isomorphic to the global (co)sections of $ F $, i.e. $ F(X) $, in a natural way. This is actually an immediate consequence of Proposition 4.2. in Section V from the book "Bredon, Sheaf Theory".
A sketch for this statement is as the following:
$ check{mathrm{H}}_0(X,F) $ is the same as cokernel of $ partial_0 $ (the 0th boundary operator). On the other hand, by cosheaf condition, the cokernel of $ partial_0 $ is $ F(X) $. Thus, the result is obtained.
answered Nov 5 '18 at 14:16
ARAARA
144
edited Feb 7 at 7:46
answered Nov 5 '18 at 14:16
ARAARA
144
answered Nov 5 '18 at 14:16
ARAARA
144
answered Nov 5 '18 at 14:16
ARAARA
144
144
add a comment |
add a comment |
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