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The Stacks project

Remark 101.5.2. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. In Properties of Stacks, Remark 100.3.7 we have seen that the 2-category of morphisms \mathcal{Z} \to \mathcal{X} representable by algebraic spaces with target \mathcal{X} forms a category. In this category the inertia stack of \mathcal{X}/\mathcal{Y} is a group object. Recall that an object of \mathcal{I}_{\mathcal{X}/\mathcal{Y}} is just a pair (x, \alpha ) where x is an object of \mathcal{X} and \alpha is an automorphism of x in the fibre category of \mathcal{X} that x lives in with f(\alpha ) = \text{id}. The composition

c : \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \times _\mathcal {X} \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \longrightarrow \mathcal{I}_{\mathcal{X}/\mathcal{Y}}

is given by the rule on objects

((x, \alpha ), (x', \alpha '), \beta ) \mapsto (x, \alpha \circ \beta ^{-1} \circ \alpha ' \circ \beta )

which makes sense as \beta : x \to x' is an isomorphism in the fibre category by our definition of fibre products. The neutral element e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}} is given by the functor x \mapsto (x, \text{id}_ x). We omit the proof that the axioms of a group object hold.


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