Lemma 8.13.1. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then $j_ U : \mathcal{C}/U \to \mathcal{C}$ is a stack over $\mathcal{C}$ if and only if $h_ U$ is a sheaf.

## 8.13 Stacks and localization

Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. We want to understand stacks over $\mathcal{C}/U$ as stacks over $\mathcal{C}$ together with a morphism towards $U$. The following lemma is the reason why this is easier to do when the presheaf $h_ U$ is a sheaf.

**Proof.**
Combine Lemma 8.6.3 with Categories, Example 4.38.7.
$\square$

Assume that $\mathcal{C}$ is a site, and $U$ is an object of $\mathcal{C}$ whose associated representable presheaf is a sheaf. We denote $j : \mathcal{C}/U \to \mathcal{C}$ the localization functor.

**Construction A.** Let $p : \mathcal{S} \to \mathcal{C}/U$ be a stack over the site $\mathcal{C}/U$. We define a stack $j_!p : j_!\mathcal{S} \to \mathcal{C}$ as follows:

As a category $j_!\mathcal{S} = \mathcal{S}$, and

the functor $j_!p : j_!\mathcal{S} \to \mathcal{C}$ is just the composition $j \circ p$.

We omit the verification that this is a stack (hint: Use that $h_ U$ is a sheaf to glue morphisms to $U$). There is a canonical functor

namely the functor $p$ which is a $1$-morphism of stacks over $\mathcal{C}$.

**Construction B.** Let $q : \mathcal{T} \to \mathcal{C}$ be a stack over $\mathcal{C}$ which is endowed with a morphism of stacks $p : \mathcal{T} \to \mathcal{C}/U$ over $\mathcal{C}$. In this case it is automatically the case that $p : \mathcal{T} \to \mathcal{C}/U$ is a stack over $\mathcal{C}/U$.

Lemma 8.13.2. Assume that $\mathcal{C}$ is a site, and $U$ is an object of $\mathcal{C}$ whose associated representable presheaf is a sheaf. Constructions A and B above define mutually inverse (!) functors of $2$-categories

**Proof.**
This is clear.
$\square$

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