The Stacks project

Theorem 89.26.4. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. Then $\mathcal{F}$ admits a presentation by a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $ if and only if the following conditions hold:

  1. $\mathcal{F}$ is a deformation category.

  2. $\dim _ k T\mathcal{F}$ is finite.

  3. $\dim _ k \text{Inf}(\mathcal{F})$ is finite.

Proof. Recall that a prorepresentable functor is a deformation functor, see Example 89.16.10. Thus if $\mathcal{F}$ is equivalent to a smooth prorepresentable groupoid in functors, then conditions (1), (2), and (3) follow from Lemma 89.24.2 (1), (2), and (3).

Conversely, assume conditions (1), (2), and (3) hold. Condition (1) implies that (S1) and (S2) are satisfied, see Lemma 89.16.6. By Lemma 89.13.4 there exists a versal formal object $\xi $. Setting $U = \underline{R}|_{\mathcal{C}_\Lambda }$ the associated map $\underline{\xi } : U \to \mathcal{F}$ is smooth (this is the definition of a versal formal object). Let $(U, R, s, t, c)$ be the groupoid in functors constructed in Lemma 89.25.2 from the map $\underline{\xi }$. By Lemma 89.26.1 we see that $(U, R, s, t, c)$ is a smooth groupoid in functors and that $[U/R] \to \mathcal{F}$ is an equivalence. By Lemma 89.26.3 we see that $(U, R, s, t, c)$ is prorepresentable. Hence $[U/R] \to \mathcal{F}$ is the desired presentation of $\mathcal{F}$. $\square$


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