The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

8.5 Stacks in groupoids

Among stacks those which are fibred in groupoids are somewhat easier to comprehend. We redefine them as follows.

Definition 8.5.1. A stack in groupoids over a site $\mathcal{C}$ is a category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$ such that

  1. $p : \mathcal{S} \to \mathcal{C}$ is fibred in groupoids over $\mathcal{C}$ (see Categories, Definition 4.34.1),

  2. for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, for all $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ the presheaf $\mathit{Isom}(x, y)$ is a sheaf on the site $\mathcal{C}/U$, and

  3. for all coverings $\mathcal{U} = \{ U_ i \to U\} $ in $\mathcal{C}$, all descent data $(x_ i, \phi _{ij})$ for $\mathcal{U}$ are effective.

Usually the hardest part to check is the third condition. Here is the lemma comparing this with the notion of a stack.

Lemma 8.5.2. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. The following are equivalent

  1. $\mathcal{S}$ is a stack in groupoids over $\mathcal{C}$,

  2. $\mathcal{S}$ is a stack over $\mathcal{C}$ and all fibre categories are groupoids, and

  3. $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$ and is a stack over $\mathcal{C}$.

Proof. Omitted, but see Categories, Lemma 4.34.2. $\square$

Lemma 8.5.3. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a stack. Let $p' : \mathcal{S}' \to \mathcal{C}$ be the category fibred in groupoids associated to $\mathcal{S}$ constructed in Categories, Lemma 4.34.3. Then $p' : \mathcal{S}' \to \mathcal{C}$ is a stack in groupoids.

Proof. Recall that the morphisms in $\mathcal{S}'$ are exactly the strongly cartesian morphisms of $\mathcal{S}$, and that any isomorphism of $\mathcal{S}$ is such a morphism. Hence descent data in $\mathcal{S}'$ are exactly the same thing as descent data in $\mathcal{S}$. Now apply Lemma 8.4.2. Some details omitted. $\square$

Lemma 8.5.4. Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be categories over $\mathcal{C}$. Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$. Then $\mathcal{S}_1$ is a stack in groupoids over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is a stack in groupoids over $\mathcal{C}$.

Proof. Follows by combining Lemmas 8.5.2 and 8.4.4. $\square$

The $2$-category of stacks in groupoids over $\mathcal{C}$ is defined as follows.

Definition 8.5.5. Let $\mathcal{C}$ be a site. The $2$-category of stacks in groupoids over $\mathcal{C}$ is the sub $2$-category of the $2$-category of stacks over $\mathcal{C}$ (see Definition 8.4.5) defined as follows:

  1. Its objects will be stacks in groupoids $p : \mathcal{S} \to \mathcal{C}$.

  2. Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$. (Since every morphism is strongly cartesian every functor preserves them.)

  3. Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_ x) = \text{id}_{p(x)}$ for all $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$.

Note that any $2$-morphism is automatically an isomorphism, so that in fact the $2$-category of stacks in groupoids over $\mathcal{C}$ is a (strict) $(2, 1)$-category.

Lemma 8.5.6. Let $\mathcal{C}$ be a category. The $2$-category of stacks in groupoids over $\mathcal{C}$ has 2-fibre products, and they are described as in Categories, Lemma 4.31.3.

Proof. This is clear from Categories, Lemma 4.34.7 and Lemmas 8.5.2 and 8.4.6. $\square$


Comments (0)


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 02ZH. Beware of the difference between the letter 'O' and the digit '0'.