nLab closed monoidal structure on sheaves

Definition

Let $X$ be a site with underlying category $S_X$. Write $PSh(X) = [S^{op}, Set]$ for the presheaf category of $X$ and $Sh(X)$ for the corresponding subcategory of sheaves.

The closed monoidal structure on presheaves restricts under the inclusion $Sh(X) \hookrightarrow PSh(X)$ to a closed monoidal structure on sheaves.

Properties

• For $f : X \to Y$ a left axact morphism of sites, there is for $F \in Sh(X)$, $F \in Sh(Y)$ a natural isomorphism

$$f_* hom(f^{-1}G, F) \simeq hom(G, f_* F) \,.$$

Here $f_*$ is the direct image and $f^{-1}$ the inverse image operation.