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
p : \mathcal{S} \to \mathcal{C} is fibred in groupoids over \mathcal{C} (see Categories, Definition 4.35.1),
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
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
\mathcal{S} is a stack in groupoids over \mathcal{C},
\mathcal{S} is a stack over \mathcal{C} and all fibre categories are groupoids, and
\mathcal{S} is fibred in groupoids over \mathcal{C} and is a stack over \mathcal{C}.
Proof.
Omitted, but see Categories, Lemma 4.35.2.
\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 4.35.3. 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:
Its objects will be stacks in groupoids p : \mathcal{S} \to \mathcal{C}.
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.)
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 4.32.3.
Proof.
This is clear from Categories, Lemma 4.35.7 and Lemmas 8.5.2 and 8.4.6.
\square
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