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.

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