# The pullback of the “space of sections” is the same as the “space of sections” of the pullback

I want to verify the sentence:

...can check pullback of the “space of sections” is the “space of sections” of the pullback

in Vakil's Exercise 2.7.E, the latest version.

Let $f:X\to Y$ be a continuous map and $\mathcal G$ be a (pre)sheaf over $Y$ and $G=\bigsqcup_q\mathcal G_q$ be the space of sections (espace etale) as defined in his 2.2.11.

I think the "space of sections of the pullback" is $F=\bigsqcup_p(f^{-1}\mathcal G)_p$, where $f^{-1}\mathcal G$ is the inverse image sheaf.

But I don't know what is the "pullback of the space of sections". I guess I must also prove that they are isomorphic as topological spaces, not just as sets.

If my interpretation is totally wrong, please ignore it and show me the correct interpretation. I only know the definition of the space of sections (espace etale).

• The "pull-back of the space of sections" is the cartesian product $G\times_Y X$. – Roland Dec 19 '17 at 22:27
• @Roland Thank you! It is a little hard for me to see why this fibre product is isomorphic to space of sections of the pullback... if $W\to X \to Y=W \to G \to Y$, how do we construct a map $W\to F$? – No One Dec 19 '17 at 22:45
• I would suggest trying to show that the stalks $f^{-1}\mathcal{G}_p$ are the same as the stalks $\mathcal{G}_{f(p)}$. After this, the rest is a simple topological argument. – leibnewtz Dec 20 '17 at 10:03