Definition 89.19.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda $. Let $x' \to x$ be a morphism in $\mathcal{F}$ lying over $A' \to A$. The kernel

is the *group of infinitesimal automorphisms of $x'$ over $x$*.

Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. Given a morphism $x' \to x$ in $\mathcal{F}$ lying over $A' \to A$, there is an induced homomorphism

\[ \text{Aut}_{A'}(x') \to \text{Aut}_ A(x). \]

Lemma 89.16.7 says that the cokernel of this homomorphism determines whether condition (RS) on $\mathcal{F}$ passes to $\overline{\mathcal{F}}$. In this section we study the kernel of this homomorphism. We will see that it also gives a measure of how far $\mathcal{F}$ is from $\overline{\mathcal{F}}$.

Definition 89.19.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda $. Let $x' \to x$ be a morphism in $\mathcal{F}$ lying over $A' \to A$. The kernel

\[ \text{Inf}(x'/x) = \mathop{\mathrm{Ker}}(\text{Aut}_{A'}(x') \to \text{Aut}_ A(x)) \]

is the *group of infinitesimal automorphisms of $x'$ over $x$*.

Definition 89.19.2. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda $. Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Assume a choice of pushforward $x_0 \to x_0'$ of $x_0$ along the map $k \to k[\epsilon ], a \mapsto a$ has been made. Then there is a unique map $x'_0 \to x_0$ such that $x_0 \to x_0' \to x_0$ is the identity on $x_0$. Then

\[ \text{Inf}_{x_0}(\mathcal F) = \text{Inf}(x'_0/x_0) \]

is the *group of infinitesimal automorphisms of $x_0$*

Remark 89.19.3. Up to canonical isomorphism $\text{Inf}_{x_0}(\mathcal{F})$ does not depend on the choice of pushforward $x_0 \to x_0'$ because any two pushforwards are canonically isomorphic. Moreover, if $y_0 \in \mathcal{F}(k)$ and $x_0 \cong y_0$ in $\mathcal{F}(k)$, then $\text{Inf}_{x_0}(\mathcal{F}) \cong \text{Inf}_{y_0}(\mathcal{F})$ where the isomorphism depends (only) on the choice of an isomorphism $x_0 \to y_0$. In particular, $\text{Aut}_ k(x_0)$ acts on $\text{Inf}_{x_0}(\mathcal{F})$.

Remark 89.19.4. Assume $\mathcal{F}$ is a predeformation category. Then

for $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$ the automorphism group $\text{Aut}_ k(x_0)$ is trivial and hence $\text{Inf}_{x_0}(\mathcal{F}) = \text{Aut}_{k[\epsilon ]}(x'_0)$, and

for $x_0, y_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$ there is a unique isomorphism $x_0 \to y_0$ and hence a canonical identification $\text{Inf}_{x_0}(\mathcal{F}) = \text{Inf}_{y_0}(\mathcal{F})$.

Since $\mathcal{F}(k)$ is nonempty, choosing $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$ and setting

\[ \text{Inf}(\mathcal{F}) = \text{Inf}_{x_0}(\mathcal{F}) \]

we get a well defined *group of infinitesimal automorphisms of $\mathcal{F}$*. With this notation we have $\text{Inf}(\mathcal{F}_{x_0}) = \text{Inf}_{x_0}(\mathcal{F})$. Please compare with the equality $T\mathcal{F}_{x_0} = T_{x_0}\mathcal{F}$ in Remark 89.12.5.

We will see that $\text{Inf}_{x_0}(\mathcal{F})$ has a natural $k$-vector space structure when $\mathcal{F}$ satisfies (RS). At the same time, we will see that if $\mathcal{F}$ satisfies (RS), then the infinitesimal automorphisms $\text{Inf}(x'/x)$ of a morphism $x' \to x$ lying over a small extension are governed by $\text{Inf}_{x_0}(\mathcal{F})$, where $x_0$ is a pushforward of $x$ to $\mathcal{F}(k)$. In order to do this, we introduce the automorphism functor for any object $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F})$ as follows.

Definition 89.19.5. Let $p : \mathcal{F} \to \mathcal{C}$ be a category cofibered in groupoids over an arbitrary base category $\mathcal{C}$. Assume a choice of pushforwards has been made. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F})$ and let $U = p(x)$. Let $U/\mathcal{C}$ denote the category of objects under $U$. The *automorphism functor of $x$* is the functor $\mathit{Aut}(x) : U/\mathcal{C} \to \textit{Sets}$ sending an object $f : U \to V$ to $\text{Aut}_ V(f_*x)$ and sending a morphism

\[ \xymatrix{ V' \ar[rr] & & V\\ & U \ar[ul]^{f'} \ar[ur]_ f & } \]

to the homomorphism $\text{Aut}_{V'}(f'_*x) \to \text{Aut}_ V(f_*x)$ coming from the unique morphism $f'_*x \to f_*x$ lying over $V' \to V$ and compatible with $x \to f'_*x$ and $x \to f_*x$.

We will be concerned with the automorphism functors of objects in a category cofibered in groupoids $\mathcal{F}$ over $\mathcal{C}_\Lambda $. If $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda )$, then the category $A/\mathcal{C}_\Lambda $ is nothing but the category $\mathcal{C}_ A$, i.e. the category defined in Section 89.3 where we take $\Lambda = A$ and $k = A/\mathfrak m_ A$. Hence the automorphism functor of an object $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$ is a functor $\mathit{Aut}(x) : \mathcal{C}_ A \to \textit{Sets}$.

The following lemma could be deduced from Lemma 89.16.12 by thinking about the “inertia” of a category cofibred in groupoids, see for example Stacks, Section 8.7 and Categories, Section 4.34. However, it is easier to see it directly.

Lemma 89.19.6. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. Then $\mathit{Aut}(x): \mathcal{C}_ A \to \textit{Sets}$ satisfies (RS).

**Proof.**
It follows that $\mathit{Aut}(x)$ satisfies (RS) from the fully faithfulness of the functor $\mathcal{F}(A_1 \times _ A A_2) \to \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$ in Lemma 89.16.4.
$\square$

Lemma 89.19.7. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. Let $x_0$ be a pushforward of $x$ to $\mathcal{F}(k)$.

$T_{\text{id}_{x_0}} \mathit{Aut}(x)$ has a natural $k$-vector space structure such that addition agrees with composition in $T_{\text{id}_{x_0}} \mathit{Aut}(x)$. In particular, composition in $T_{\text{id}_{x_0}} \mathit{Aut}(x)$ is commutative.

There is a canonical isomorphism $T_{\text{id}_{x_0}} \mathit{Aut}(x) \to T_{\text{id}_{x_0}} \mathit{Aut}(x_0)$ of $k$-vector spaces.

**Proof.**
We apply Remark 89.6.4 to the functor $\mathit{Aut}(x) : \mathcal{C}_ A \to \textit{Sets}$ and the element $\text{id}_{x_0} \in \mathit{Aut}(x)(k)$ to get a predeformation functor $F = \mathit{Aut}(x)_{\text{id}_{x_0}}$. By Lemmas 89.19.6 and 89.16.11 $F$ is a deformation functor. By definition $T_{\text{id}_{x_0}} \mathit{Aut}(x) = TF = F(k[\epsilon ])$ which has a natural $k$-vector space structure specified by Lemma 89.11.8.

Addition is defined as the composition

\[ F(k[\epsilon ]) \times F(k[\epsilon ]) \longrightarrow F(k[\epsilon ] \times _ k k[\epsilon ]) \longrightarrow F(k[\epsilon ]) \]

where the first map is the inverse of the bijection guaranteed by (RS) and the second is induced by the $k$-algebra map $k[\epsilon ] \times _ k k[\epsilon ] \to k[\epsilon ]$ which maps $(\epsilon , 0)$ and $(0, \epsilon )$ to $\epsilon $. If $A \to B$ is a ring map in $\mathcal{C}_\Lambda $, then $F(A) \to F(B)$ is a homomorphism where $F(A) = \mathit{Aut}(x)_{\text{id}_{x_0}}(A)$ and $F(B) = \mathit{Aut}(x)_{\text{id}_{x_0}}(B)$ are groups under composition. We conclude that $+ : F(k[\epsilon ]) \times F(k[\epsilon ])\to F(k[\epsilon ])$ is a homomorphism where $F(k[\epsilon ])$ is regarded as a group under composition. With $\text{id} \in F(k[\epsilon ])$ the unit element we see that $+(v, \text{id}) = +(\text{id}, v) = v$ for any $v \in F(k[\epsilon ])$ because $(\text{id}, v)$ is the pushforward of $v$ along the ring map $k[\epsilon ] \to k[\epsilon ] \times _ k k[\epsilon ]$ with $\epsilon \mapsto (\epsilon , 0)$. In general, given a group $G$ with multiplication $\circ $ and $+ : G \times G \to G$ is a homomorphism such that $+(g, 1) = +(1, g) = g$, where $1$ is the identity of $G$, then $+ = \circ $. This shows addition in the $k$-vector space structure on $F(k[\epsilon ])$ agrees with composition.

Finally, (2) is a matter of unwinding the definitions. Namely $T_{\text{id}_{x_0}} \mathit{Aut}(x)$ is the set of automorphisms $\alpha $ of the pushforward of $x$ along $A \to k \to k[\epsilon ]$ which are trivial modulo $\epsilon $. On the other hand $T_{\text{id}_{x_0}} \mathit{Aut}(x_0)$ is the set of automorphisms of the pushforward of $x_0$ along $k \to k[\epsilon ]$ which are trivial modulo $\epsilon $. Since $x_0$ is the pushforward of $x$ along $A \to k$ the result is clear. $\square$

Remark 89.19.8. We point out some basic relationships between infinitesimal automorphism groups, liftings, and tangent spaces to automorphism functors. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. Let $x' \to x$ be a morphism lying over a ring map $A' \to A$. Then from the definitions we have an equality

\[ \text{Inf}(x'/x) = \text{Lift}(\text{id}_ x, A') \]

where the liftings are of $\text{id}_ x$ as an object of $\mathit{Aut}(x')$. If $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$ and $x'_0$ is the pushforward to $\mathcal{F}(k[\epsilon ])$, then applying this to $x'_0 \to x_0$ we get

\[ \text{Inf}_{x_0}(\mathcal{F}) = \text{Lift}(\text{id}_{x_0}, k[\epsilon ]) = T_{\text{id}_{x_0}} \mathit{Aut}(x_0), \]

the last equality following directly from the definitions.

Lemma 89.19.9. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Then $\text{Inf}_{x_0}(\mathcal{F})$ is equal as a set to $T_{\text{id}_{x_0}} \mathit{Aut}(x_0)$, and so has a natural $k$-vector space structure such that addition agrees with composition of automorphisms.

**Proof.**
The equality of sets is as in the end of Remark 89.19.8 and the statement about the vector space structure follows from Lemma 89.19.7.
$\square$

Lemma 89.19.10. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibred in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Then $\varphi $ induces a $k$-linear map $\text{Inf}_{x_0}(\mathcal{F}) \to \text{Inf}_{\varphi (x_0)}(\mathcal{G})$.

**Proof.**
It is clear that $\varphi $ induces a morphism from $\mathit{Aut}(x_0) \to \mathit{Aut}(\varphi (x_0))$ which maps the identity to the identity. Hence this follows from the result for tangent spaces, see Lemma 89.12.4.
$\square$

Lemma 89.19.11. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Let $x' \to x$ be a morphism lying over a surjective ring map $A' \to A$ with kernel $I$ annihilated by $\mathfrak m_{A'}$. Let $x_0$ be a pushforward of $x$ to $\mathcal{F}(k)$. Then $\text{Inf}(x'/x)$ has a free and transitive action by $T_{\text{id}_{x_0}} \mathit{Aut}(x') \otimes _ k I = \text{Inf}_{x_0}(\mathcal{F}) \otimes _ k I$.

**Proof.**
This is just the analogue of Lemma 89.17.5 in the setting of automorphism sheaves. To be precise, we apply Remark 89.6.4 to the functor $\mathit{Aut}(x') : \mathcal{C}_{A'} \to \textit{Sets}$ and the element $\text{id}_{x_0} \in \mathit{Aut}(x)(k)$ to get a predeformation functor $F = \mathit{Aut}(x')_{\text{id}_{x_0}}$. By Lemmas 89.19.6 and 89.16.11 $F$ is a deformation functor. Hence Lemma 89.17.5 gives a free and transitive action of $TF \otimes _ k I$ on $\text{Lift}(\text{id}_ x, A')$, because as $\text{Lift}(\text{id}_ x, A')$ is a group it is always nonempty. Note that we have equalities of vector spaces

\[ TF = T_{\text{id}_{x_0}} \mathit{Aut}(x') \otimes _ k I = \text{Inf}_{x_0}(\mathcal{F}) \otimes _ k I \]

by Lemma 89.19.7. The equality $\text{Inf}(x'/x) = \text{Lift}(\text{id}_ x, A')$ of Remark 89.19.8 finishes the proof. $\square$

Lemma 89.19.12. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Let $x' \to x$ be a morphism in $\mathcal{F}$ lying over a surjective ring map. Let $x_0$ be a pushforward of $x$ to $\mathcal{F}(k)$. If $\text{Inf}_{x_0}(\mathcal{F}) = 0$ then $\text{Inf}(x'/x) = 0$.

Lemma 89.19.13. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $ satisfying (RS). Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Then $\text{Inf}_{x_0}(\mathcal{F}) = 0$ if and only if the natural morphism $\mathcal{F}_{x_0} \to \overline{\mathcal{F}_{x_0}}$ of categories cofibered in groupoids is an equivalence.

**Proof.**
The morphism $\mathcal{F}_{x_0} \to \overline{\mathcal{F}_{x_0}}$ is an equivalence if and only if $\mathcal{F}_{x_0}$ is fibered in setoids, cf. Categories, Section 4.39 (a setoid is by definition a groupoid in which the only automorphism of any object is the identity). We prove that $\text{Inf}_{x_0}(\mathcal{F}) = 0$ if and only if this condition holds for $\mathcal{F}_{x_0}$. Obviously if $\mathcal{F}_{x_0}$ is fibered in setoids then $\text{Inf}_{x_0}(\mathcal{F}) = 0$. Conversely assume $\text{Inf}_{x_0}(\mathcal{F}) = 0$. Let $A$ be an object of $\mathcal{C}_\Lambda $. Then by Lemma 89.19.12, $\text{Inf}(x/x_0) = 0$ for any object $x \to x_0$ of $\mathcal{F}_{x_0}(A)$. Since by definition $\text{Inf}(x/x_0)$ equals the group of automorphisms of $x \to x_0$ in $\mathcal{F}_{x_0}(A)$, this proves $\mathcal{F}_{x_0}(A)$ is a setoid.
$\square$

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