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.










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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.










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      up vote
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      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.










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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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      asked Aug 30 at 4:02









      Yelon

      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
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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






              share|cite|improve this answer
























                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






                share|cite|improve this answer






















                  up vote
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                  up vote
                  0
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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






                  share|cite|improve this answer












                  $Etimes_X E$ 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







                  share|cite|improve this answer












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                  share|cite|improve this answer










                  answered Aug 30 at 4:08









                  Lord Shark the Unknown

                  88.8k955115




                  88.8k955115




















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                      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 $x$.






                      share|cite|improve this answer
























                        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 $x$.






                        share|cite|improve this answer






















                          up vote
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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 $x$.






                          share|cite|improve this answer












                          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 $x$.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Aug 30 at 4:09









                          Tsemo Aristide

                          52.3k11244




                          52.3k11244



























                               

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