8.7 The inertia stack
Let p : \mathcal{S} \to \mathcal{C} and p' : \mathcal{S}' \to \mathcal{C} be fibred categories over the category \mathcal{C}. Let F : \mathcal{S} \to \mathcal{S}' be a 1-morphism of fibred categories over \mathcal{C}. Recall that we have defined in Categories, Definition 4.34.2 a relative inertia fibred category \mathcal{I}_{\mathcal{S}/\mathcal{S}'} \to \mathcal{C} as the category whose objects are pairs (x , \alpha ) where x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}) and \alpha : x \to x with F(\alpha ) = \text{id}_{F(x)}. There is also an absolute version, namely the inertia \mathcal{I}_\mathcal {S} of \mathcal{S}. These inertia categories are actually stacks over \mathcal{C} provided that \mathcal{S} and \mathcal{S}' are stacks.
Lemma 8.7.1. Let \mathcal{C} be a site. Let p : \mathcal{S} \to \mathcal{C} and p' : \mathcal{S}' \to \mathcal{C} be stacks over the site \mathcal{C}. Let F : \mathcal{S} \to \mathcal{S}' be a 1-morphism of stacks over \mathcal{C}.
The inertia \mathcal{I}_{\mathcal{S}/\mathcal{S}'} and \mathcal{I}_\mathcal {S} are stacks over \mathcal{C}.
If \mathcal{S}, \mathcal{S}' are stacks in groupoids over \mathcal{C}, then so are \mathcal{I}_{\mathcal{S}/\mathcal{S}'} and \mathcal{I}_\mathcal {S}.
If \mathcal{S}, \mathcal{S}' are stacks in setoids over \mathcal{C}, then so are \mathcal{I}_{\mathcal{S}/\mathcal{S}'} and \mathcal{I}_\mathcal {S}.
Proof.
The first three assertions follow from Lemmas 8.4.6, 8.5.6, and 8.6.6 and the equivalence in Categories, Lemma 4.34.1 part (1).
\square
Lemma 8.7.2. Let \mathcal{C} be a site. If \mathcal{S} is a stack in groupoids, then the canonical 1-morphism \mathcal{I}_\mathcal {S} \to \mathcal{S} is an equivalence if and only if \mathcal{S} is a stack in setoids.
Proof.
Follows directly from Categories, Lemma 4.39.7.
\square
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