Definition 100.5.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $Z$ be an algebraic space.

1. Let $x : Z \to \mathcal{X}$ be a morphism. We set

$\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x, x) = Z \times _{x, \mathcal{X}} \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$

We endow it with the structure of a group algebraic space over $Z$ by pulling back the composition law discussed in Remark 100.5.2. We will sometimes refer to $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x, x)$ as the relative sheaf of automorphisms of $x$.

2. Let $x_1, x_2 : Z \to \mathcal{X}$ be morphisms. Set $y_ i = f \circ x_ i$. Let $\alpha : y_1 \to y_2$ be a $2$-morphism. Then $\alpha$ determines a morphism $\Delta ^\alpha : Z \to Z \times _{y_1, \mathcal{Y}, y_2} Z$ and we set

$\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2) = (Z \times _{x_1, \mathcal{X}, x_2} Z) \times _{Z \times _{y_1, \mathcal{Y}, y_2} Z, \Delta ^\alpha } Z.$

We will sometimes refer to $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2)$ as the relative sheaf of isomorphisms from $x_1$ to $x_2$.

If $\mathcal{Y} = \mathop{\mathrm{Spec}}(\mathbf{Z})$ or more generally when $\mathcal{Y}$ is an algebraic space, then we use the notation $\mathit{Isom}_\mathcal {X}(x, x)$ and $\mathit{Isom}_\mathcal {X}(x_1, x_2)$ and we use the terminology sheaf of automorphisms of $x$ and sheaf of isomorphisms from $x_1$ to $x_2$.

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