notation in vector bundles
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In the definition of the family of vector spaces, or in vector bundles, pullback there is something that confused me.
We have a map $p:Erightarrow X$ together with operations $+ : Etimes_X Erightarrow E$ and with the multiplication. What is $times_X$? I mean the subscripted $X$? It seems to be the subset of $Etimes E$ but don't know exactly.
differential-geometry notation vector-bundles
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up vote
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In the definition of the family of vector spaces, or in vector bundles, pullback there is something that confused me.
We have a map $p:Erightarrow X$ together with operations $+ : Etimes_X Erightarrow E$ and with the multiplication. What is $times_X$? I mean the subscripted $X$? It seems to be the subset of $Etimes E$ but don't know exactly.
differential-geometry notation vector-bundles
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In the definition of the family of vector spaces, or in vector bundles, pullback there is something that confused me.
We have a map $p:Erightarrow X$ together with operations $+ : Etimes_X Erightarrow E$ and with the multiplication. What is $times_X$? I mean the subscripted $X$? It seems to be the subset of $Etimes E$ but don't know exactly.
differential-geometry notation vector-bundles
In the definition of the family of vector spaces, or in vector bundles, pullback there is something that confused me.
We have a map $p:Erightarrow X$ together with operations $+ : Etimes_X Erightarrow E$ and with the multiplication. What is $times_X$? I mean the subscripted $X$? It seems to be the subset of $Etimes E$ but don't know exactly.
differential-geometry notation vector-bundles
differential-geometry notation vector-bundles
asked Aug 30 at 4:02
Yelon
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394212
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2 Answers
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$Etimes_X E$ is the fibre product of $E$ with itself over $X$. It's the
set of all $(e_1,e_2)in Etimes E$ with $p(e_1)=p(e_2)$. Equivalently it's
the pullback in the category of topological spaces of the map $p:Xto E$
with itself.
Here is the pullback diagram:
$requireAMScd$
beginCD
Etimes_XE @>>> E\
@VVV @VV p V\
E @>>p> X
endCD
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It is the fiber product: $Etimes_XE=(x,y)in Etimes E:p(x)=p(y)$. It is a vector bundle and its fiber at $x$ is $E_xtimes E_x$ where $E_x$ is the fiber of $x$.
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
$Etimes_X E$ is the fibre product of $E$ with itself over $X$. It's the
set of all $(e_1,e_2)in Etimes E$ with $p(e_1)=p(e_2)$. Equivalently it's
the pullback in the category of topological spaces of the map $p:Xto E$
with itself.
Here is the pullback diagram:
$requireAMScd$
beginCD
Etimes_XE @>>> E\
@VVV @VV p V\
E @>>p> X
endCD
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up vote
0
down vote
$Etimes_X E$ is the fibre product of $E$ with itself over $X$. It's the
set of all $(e_1,e_2)in Etimes E$ with $p(e_1)=p(e_2)$. Equivalently it's
the pullback in the category of topological spaces of the map $p:Xto E$
with itself.
Here is the pullback diagram:
$requireAMScd$
beginCD
Etimes_XE @>>> E\
@VVV @VV p V\
E @>>p> X
endCD
add a comment |Â
up vote
0
down vote
up vote
0
down vote
$Etimes_X E$ is the fibre product of $E$ with itself over $X$. It's the
set of all $(e_1,e_2)in Etimes E$ with $p(e_1)=p(e_2)$. Equivalently it's
the pullback in the category of topological spaces of the map $p:Xto E$
with itself.
Here is the pullback diagram:
$requireAMScd$
beginCD
Etimes_XE @>>> E\
@VVV @VV p V\
E @>>p> X
endCD
$Etimes_X E$ is the fibre product of $E$ with itself over $X$. It's the
set of all $(e_1,e_2)in Etimes E$ with $p(e_1)=p(e_2)$. Equivalently it's
the pullback in the category of topological spaces of the map $p:Xto E$
with itself.
Here is the pullback diagram:
$requireAMScd$
beginCD
Etimes_XE @>>> E\
@VVV @VV p V\
E @>>p> X
endCD
answered Aug 30 at 4:08
Lord Shark the Unknown
88.8k955115
88.8k955115
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up vote
0
down vote
It is the fiber product: $Etimes_XE=(x,y)in Etimes E:p(x)=p(y)$. It is a vector bundle and its fiber at $x$ is $E_xtimes E_x$ where $E_x$ is the fiber of $x$.
add a comment |Â
up vote
0
down vote
It is the fiber product: $Etimes_XE=(x,y)in Etimes E:p(x)=p(y)$. It is a vector bundle and its fiber at $x$ is $E_xtimes E_x$ where $E_x$ is the fiber of $x$.
add a comment |Â
up vote
0
down vote
up vote
0
down vote
It is the fiber product: $Etimes_XE=(x,y)in Etimes E:p(x)=p(y)$. It is a vector bundle and its fiber at $x$ is $E_xtimes E_x$ where $E_x$ is the fiber of $x$.
It is the fiber product: $Etimes_XE=(x,y)in Etimes E:p(x)=p(y)$. It is a vector bundle and its fiber at $x$ is $E_xtimes E_x$ where $E_x$ is the fiber of $x$.
answered Aug 30 at 4:09
Tsemo Aristide
52.3k11244
52.3k11244
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