Theorem 90.27.8 (06TE)—The Stacks project

Theorem 90.27.8. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. Consider the following conditions

$\mathcal{F}$ admits a presentation by a normalized smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $,
$\mathcal{F}$ admits a presentation by a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $,
$\mathcal{F}$ admits a presentation by a minimal smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $, and
$\mathcal{F}$ satisfies the following conditions
1. $\mathcal{F}$ is a deformation category.
2. $\dim _ k T\mathcal{F}$ is finite.
3. $\dim _ k \text{Inf}(\mathcal{F})$ is finite.

Then (2), (3), (4) are equivalent and are implied by (1). If $k' \subset k$ is separable, then (1), (2), (3), (4) are all equivalent. Furthermore, the minimal smooth prorepresentable groupoids in functors which provide a presentation of $\mathcal{F}$ are unique up to isomorphism.

Proof. We see that (1) implies (3) and is equivalent to (3) if $k' \subset k$ is separable from Lemma 90.27.2. It is clear that (3) implies (2). We see that (2) implies (4) by Theorem 90.26.4. We see that (4) implies (3) by Lemma 90.27.6. This proves all the implications. The final uniqueness statement follows from Lemma 90.27.7. $\square$

The Stacks project

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