Definition 101.5.3. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. Let Z be an algebraic space.
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 101.5.2. We will sometimes refer to \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x, x) as the relative sheaf of automorphisms of x.
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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