The Stacks project

Lemma 100.5.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $Z$ be an algebraic space and let $x_ i : Z \to \mathcal{X}$, $i = 1, 2$ be morphisms. Then

  1. $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$ is a group algebraic space over $Z$,

  2. there is an exact sequence of groups

    \[ 0 \to \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2) \to \mathit{Isom}_\mathcal {X}(x_2, x_2) \to \mathit{Isom}_\mathcal {Y}(f \circ x_2, f \circ x_2) \]
  3. there is a map of algebraic spaces $ \mathit{Isom}_\mathcal {X}(x_1, x_2) \to \mathit{Isom}_\mathcal {Y}(f \circ x_1, f \circ x_2) $ such that for any $2$-morphism $\alpha : f \circ x_1 \to f \circ x_2$ we obtain a cartesian diagram

    \[ \xymatrix{ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2) \ar[d] \ar[r] & Z \ar[d]^\alpha \\ \mathit{Isom}_\mathcal {X}(x_1, x_2) \ar[r] & \mathit{Isom}_\mathcal {Y}(f \circ x_1, f \circ x_2) } \]
  4. for any $2$-morphism $\alpha : f \circ x_1 \to f \circ x_2$ the algebraic space $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2)$ is a pseudo torsor for $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$ over $Z$.

Proof. Part (1) follows from Definition 100.5.3. Part (2) comes from the exact sequence (100.5.2.1) ├ętale locally on $Z$. Part (3) can be seen by unwinding the definitions. Locally on $Z$ in the ├ętale topology part (4) reduces to part (2) of Lemma 100.3.2. $\square$


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