
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. $\square$

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