The main theorem of this chapter is Theorem 90.26.4 above. It describes completely those categories cofibred in groupoids over $\mathcal{C}_\Lambda $ which have a presentation by a smooth prorepresentable groupoid in functors. In this section we briefly discuss how the minimality discussed in Sections 90.14 and 90.15 can be used to obtain a “minimal” smooth prorepresentable presentation.
Definition 90.27.1. Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $.
We say $(U, R, s, t, c)$ is normalized if the groupoid $(U(k[\epsilon ]), R(k[\epsilon ]), s, t, c)$ is totally disconnected, i.e., there are no morphisms between distinct objects.
We say $(U, R, s, t, c)$ is minimal if the $U \to [U/R]$ is given by a minimal versal formal object of $[U/R]$.
The difference between the two notions is related to the difference between conditions (90.15.0.1) and (90.15.0.2) and disappears when $k' \subset k$ is separable. Also a normalized smooth prorepresentable groupoid in functors is minimal as the following lemma shows. Here is a precise statement.
Lemma 90.27.2. Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $.
$(U, R, s, t, c)$ is normalized if and only if the morphism $U \to [U/R]$ induces an isomorphism on tangent spaces, and
$(U, R, s, t, c)$ is minimal if and only if the kernel of $TU \to T[U/R]$ is contained in the image of $\text{Der}_\Lambda (k, k) \to TU$.
Proof. Part (1) follows immediately from the definitions. To see part (2) set $\mathcal{F} = [U/R]$. Since $\mathcal{F}$ has a presentation it is a deformation category, see Theorem 90.26.4. In particular it satisfies (RS), (S1), and (S2), see Lemma 90.16.6. Recall that minimal versal formal objects are unique up to isomorphism, see Lemma 90.14.5. By Theorem 90.15.5 a minimal versal object induces a map $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ satisfying (90.15.0.2). Since $U \cong \underline{R}|_{\mathcal{C}_\Lambda }$ over $\mathcal{F}$ we see that $TU \to T\mathcal{F} = T[U/R]$ satisfies the property as stated in the lemma. $\square$
The quotient of a minimal prorepresentable groupoid in functors on $\mathcal C_\Lambda $ does not admit autoequivalences which are not automorphisms. To prove this, we first note the following lemma.
Lemma 90.27.3. Let $U: \mathcal{C}_\Lambda \to \textit{Sets}$ be a prorepresentable functor. Let $\varphi : U \to U$ be a morphism such that $d\varphi : TU \to TU$ is an isomorphism. Then $\varphi $ is an isomorphism.
Proof. If $U \cong \underline{R}|_{\mathcal{C}_\Lambda }$ for some $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$, then completing $\varphi $ gives a morphism $\underline{R} \to \underline{R}$. If $f: R \to R$ is the corresponding morphism in $\widehat{\mathcal{C}}_\Lambda $, then $f$ induces an isomorphism $\text{Der}_\Lambda (R, k) \to \text{Der}_\Lambda (R, k)$, see Example 90.11.14. In particular $f$ is a surjection by Lemma 90.4.6. As a surjective endomorphism of a Noetherian ring is an isomorphism (see Algebra, Lemma 10.31.10) we conclude $f$, hence $\underline{R} \to \underline{R}$, hence $\varphi : U \to U$ is an isomorphism. $\square$
Lemma 90.27.4. 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 90.21.1. Denote $\phi : U \to U$ and $\psi : R \to R$ the corresponding morphisms. Because the diagram
is commutative, since $d\varphi $ is bijective, and since we have the characterization of minimality in Lemma 90.27.2 we conclude that $d\phi $ is injective (hence bijective by dimension reasons). Thus $\phi : U \to U$ is an isomorphism by Lemma 90.27.3. We can use a similar argument, using the exact sequence
of Lemma 90.26.2 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. $\square$
Lemma 90.27.5. Let $(U, R, s, t, c)$ and $(U', R', s', t', c')$ be minimal smooth prorepresentable groupoids 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. Let $\psi : [U'/R'] \to [U/R]$ be a quasi-inverse to $\varphi $. Then $\psi \circ \varphi $ and $\varphi \circ \psi $ are isomorphisms by Lemma 90.27.4, hence $\varphi $ and $\psi $ are isomorphisms. $\square$
The following lemma summarizes some of the things we have seen earlier in this chapter.
Lemma 90.27.6. Let $\mathcal{F}$ be a deformation category such that $\dim _ k T\mathcal{F} <\infty $ and $\dim _ k \text{Inf}(\mathcal{F}) < \infty $. Then there exists a minimal versal formal object $\xi $ of $\mathcal{F}$. Say $\xi $ lies over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. Let $U = \underline{R}|_{\mathcal{C}_\Lambda }$. Let $f = \underline{\xi } : U \to \mathcal{F}$ be the associated morphism. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}_\Lambda $ constructed from $f : U \to \mathcal{F}$ in Lemma 90.25.2. Then $(U, R, s, t, c)$ is a minimal smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $ and there is an equivalence $[U/R] \to \mathcal{F}$.
Proof. As $\mathcal{F}$ is a deformation category it satisfies (S1) and (S2), see Lemma 90.16.6. By Lemma 90.13.4 there exists a versal formal object. By Lemma 90.14.5 there exists a minimal versal formal object $\xi /R$ as in the statement of the lemma. 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 90.25.2 from the map $\underline{\xi }$. By Lemma 90.26.1 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 90.26.3 we see that $(U, R, s, t, c)$ is prorepresentable. Finally, $(U, R, s, t, c)$ is minimal because $U \to [U/R] = \mathcal{F}$ corresponds to the minimal versal formal object $\xi $. $\square$
Presentations by minimal prorepresentable groupoids in functors satisfy the following uniqueness property.
Lemma 90.27.7. Let $\mathcal{F}$ be category cofibered in groupoids over $\mathcal{C}_\Lambda $. Assume there exist presentations of $\mathcal{F}$ by minimal smooth prorepresentable groupoids in functors $(U, R, s, t, c)$ and $(U', R', s', t', c')$. Then $(U, R, s, t, c)$ and $(U', R', s', t', c')$ are isomorphic.
Proof. Follows from Lemma 90.27.5 and the observation that a morphism $[U/R] \to [U'/R']$ is the same thing as a morphism of groupoids in functors (by our explicit construction of $[U/R]$ in Definition 90.21.9). $\square$
In summary we have proved the following theorem.
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
$\mathcal{F}$ is a deformation category.
$\dim _ k T\mathcal{F}$ is finite.
$\dim _ k \text{Inf}(\mathcal{F})$ 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. $\square$
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