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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