
## 8.9 Stackification of categories fibred in groupoids

Here is the result.

Lemma 8.9.1. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids over $\mathcal{C}$. There exists a stack in groupoids $p' : \mathcal{S}' \to \mathcal{C}$ and a $1$-morphism $G : \mathcal{S} \to \mathcal{S}'$ of categories fibred in groupoids over $\mathcal{C}$ (see Categories, Definition 4.34.6) such that

1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and any $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ the map

$\mathit{Mor}(x, y) \longrightarrow \mathit{Mor}(G(x), G(y))$

induced by $G$ identifies the right hand side with the sheafification of the left hand side, and

2. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and any $x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}'_ U)$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ such that for every $i \in I$ the object $x'|_{U_ i}$ is in the essential image of the functor $G : \mathcal{S}_{U_ i} \to \mathcal{S}'_{U_ i}$.

Moreover the stack in groupoids $\mathcal{S}'$ is determined up to unique $2$-isomorphism by these conditions.

Proof. Apply Lemma 8.8.1. The result will be a stack in groupoids by applying Lemma 8.5.2. $\square$

Lemma 8.9.2. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids over $\mathcal{C}$. Let $p' : \mathcal{S}' \to \mathcal{C}$ and $G : \mathcal{S} \to \mathcal{S}'$ the stack in groupoids and $1$-morphism constructed in Lemma 8.9.1. This construction has the following universal property: Given a stack in groupoids $q : \mathcal{X} \to \mathcal{C}$ and a $1$-morphism $F : \mathcal{S} \to \mathcal{X}$ of categories over $\mathcal{C}$ there exists a $1$-morphism $H : \mathcal{S}' \to \mathcal{X}$ such that the diagram

$\xymatrix{ \mathcal{S} \ar[rr]_ F \ar[rd]_ G & & \mathcal{X} \\ & \mathcal{S}' \ar[ru]_ H }$

is $2$-commutative.

Proof. This is a special case of Lemma 8.8.2. $\square$

Lemma 8.9.3. Let $\mathcal{C}$ be a site. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Z} \to \mathcal{Y}$ be morphisms of categories fibred in groupoids over $\mathcal{C}$. In this case the stackification of the $2$-fibre product is the $2$-fibre product of the stackifications.

Proof. This is a special case of Lemma 8.8.4. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02ZO. Beware of the difference between the letter 'O' and the digit '0'.