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I would like to know whether the restriction functor commutes with the inverse image functor.

(I basically follow the terminology in "Topological and Smooth Stacks".)

Let $X$ be a topological space. The small and big sites of $X$ are denoted $\mathrm{Op}(X)$ and $\mathbf{T}/X$ respectively.

A small sheaf over $X$ is a sheaf over the small site $\mathrm{Op}(X)$ and a big sheaf over $X$ is a sheaf over the big site $\mathbf{T}/X$. The former coincides with the ordinary definition of sheaf over $X$.

Using a natural functor $\mathrm{Op}(X) \to \mathbf{T}/X$, we can define a functor $R$ from $\mathrm{Sh}(\mathbf{T}/X)$ to $\mathrm{Sh}(\mathrm{Op}(X))$. Here for a site $\mathcal{C}$, $\mathrm{Sh}(\mathcal{C})$ is the category of sheaves over the site $\mathcal{C}$.

On the other hand, if we have a continuous map $f:Y \to X$, we have the inverse image functor $f^*:\mathrm{Sh}(\mathrm{Op}(X))\to\mathrm{Sh}(\mathrm{Op}(Y))$ for small sheaves. We can also define the inverse image functor $f^*:\mathrm{Sh}(\mathbf{T}/X)\to\mathrm{Sh}(\mathbf{T}/Y)$ for big sheaves. (For a big sheaf $F$ over $X$, $f^*F$ is defined by just the composition $\mathbf{T}/Y \to \mathbf{T}/X \to \mathrm{(Set)}$.)

Then does the identity $R \circ f^* = f^* \circ R$ hold? If not, how about applying the global section functor $\Gamma$ to the identity?

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Disregard my previous comment, I forgot about sheafification again... – Zhen Lin Mar 17 '12 at 23:02

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