The Stacks project

Lemma 89.27.4. Let $(U, R, s, t, c)$ be a minimal smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $. If $\varphi : [U/R] \to [U/R]$ is an equivalence of categories cofibered in groupoids, then $\varphi $ is an isomorphism.

Proof. A morphism $\varphi : [U/R] \to [U/R]$ is the same thing as a morphism $\varphi : (U, R, s, t, c) \to (U, R, s, t, c)$ of groupoids in functors over $\mathcal{C}_\Lambda $ as defined in Definition 89.21.1. Denote $\phi : U \to U$ and $\psi : R \to R$ the corresponding morphisms. Because the diagram

\[ \xymatrix{ & \text{Der}_\Lambda (k, k) \ar[dr]_\gamma \ar[dl]^\gamma \\ TU \ar[rr]_{d\phi } \ar[d] & & TU \ar[d] \\ T[U/R] \ar[rr]^{d\varphi } & & T[U/R] } \]

is commutative, since $d\varphi $ is bijective, and since we have the characterization of minimality in Lemma 89.27.2 we conclude that $d\phi $ is injective (hence bijective by dimension reasons). Thus $\phi : U \to U$ is an isomorphism by Lemma 89.27.3. We can use a similar argument, using the exact sequence

\[ 0 \to \text{Inf}([U/R]) \to TR \to TU \oplus TU \]

of Lemma 89.26.2 to prove that $\psi : R \to R$ is an isomorphism. But is also a consequence of the fact that $R = U \times _{[U/R]} U$ and that $\varphi $ and $\phi $ are isomorphisms. $\square$

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