## 89.26 Presentations of deformation categories

According to the next lemma, a smooth morphism from a predeformation functor to a predeformation category $\mathcal{F}$ gives rise to a presentation of $\mathcal{F}$ by a smooth groupoid in functors.

Lemma 89.26.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a smooth morphism of categories cofibered in groupoids. Then:

1. If $(U, R, s, t, c)$ is the groupoid in functors on $\mathcal{C}_\Lambda$ constructed from $f : U \to \mathcal{F}$ in Lemma 89.25.2, then $(U, R, s, t, c)$ is smooth.

2. If $f : U(k) \to \mathcal{F}(k)$ is essentially surjective, then the morphism $[f] : [U/R] \to \mathcal{F}$ of Lemma 89.25.3 is an equivalence.

Proof. From the construction of Lemma 89.25.2 we have a commutative diagram

$\xymatrix{ R = U \times _{f, \mathcal{F}, f} U \ar[r]_-s \ar[d]_ t & U \ar[d]^ f \\ U \ar[r]^ f & \mathcal{F} }$

where $t, s$ are the first and second projections. So $t, s$ are smooth by Lemma 89.8.7. Hence (1) holds.

If the assumption of (2) holds, then by Lemma 89.8.8 the morphism $f : U \to \mathcal{F}$ is essentially surjective. Hence by Lemma 89.25.3 the morphism $[f] : [U/R] \to \mathcal{F}$ is an equivalence. $\square$

Lemma 89.26.2. Let $\mathcal{F}$ be a deformation category. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a deformation functor. Let $f: U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids. Then $U \times _{f, \mathcal{F}, f} U$ is a deformation functor with tangent space fitting into an exact sequence of $k$-vector spaces

$0 \to \text{Inf}(\mathcal{F}) \to T(U \times _{f, \mathcal{F}, f} U) \to TU \oplus TU$

Proof. Follows from Lemma 89.20.1 and the fact that $\text{Inf}(U) = (0)$. $\square$

Lemma 89.26.3. Let $\mathcal{F}$ be a deformation category. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a prorepresentable functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}_\Lambda$ constructed from $f : U \to \mathcal{F}$ in Lemma 89.25.2. If $\dim _ k \text{Inf}(\mathcal{F}) < \infty$, then $(U, R, s, t, c)$ is prorepresentable.

Proof. Note that $U$ is a deformation functor by Example 89.16.10. By Lemma 89.26.2 we see that $R = U \times _{f, \mathcal{F}, f} U$ is a deformation functor whose tangent space $TR = T(U \times _{f, \mathcal{F}, f} U)$ sits in an exact sequence $0 \to \text{Inf}(\mathcal{F}) \to TR \to TU \oplus TU$. Since we have assumed the first space has finite dimension and since $TU$ has finite dimension by Example 89.11.11 we see that $\dim TR < \infty$. The map $\gamma : \text{Der}_\Lambda (k, k) \to TR$ see (89.12.6.1) is injective because its composition with $TR \to TU$ is injective by Theorem 89.18.2 for the prorepresentable functor $U$. Thus $R$ is prorepresentable by Theorem 89.18.2. It follows from Lemma 89.22.4 that $(U, R, s, t, c)$ is prorepresentable. $\square$

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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