notation in vector bundles

Question

In the definition of the family of vector spaces, or in vector bundles, pullback there is something that confused me. We have a map $p:E\rightarrow X$ together with operations $+ : E\times_X E\rightarrow E$ and with the multiplication. What is $\times_X$? I mean the subscripted $X$? It seems to be the subset of $E\times E$ but don't know exactly.

Answer 1

$E\times_X E$ is the fibre product of $E$ with itself over $X$. It's the set of all $(e_1,e_2)\in E\times E$ with $p(e_1)=p(e_2)$. Equivalently it's the pullback in the category of topological spaces of the map $p:X\to E$ with itself.

Here is the pullback diagram: $\require{AMScd}$ \begin{CD} E\times_XE @>>> E\\ @VVV @VV p V\\ E @>>p> X \end{CD}

Answer 2

It is the fiber product: $E\times_XE=\{(x,y)\in E\times E:p(x)=p(y)\}$. It is a vector bundle and its fiber at $x$ is $E_x\times E_x$ where $E_x$ is the fiber of $x$.