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Let $F$ be a category cofibered in groupoids over category $C$. Given a morphism $x'\to x$ in $F$ lying over a morphism $A′\to A$ in $C$, there is an induced homomorphism $\operatorname{Aut} A'(x')\to \operatorname{Aut} A(x)$.

The kernel $\operatorname{Inf}(x'/x):=\operatorname{ker}(\operatorname{Aut} A'(x') \to \operatorname{Aut} A(x))$

is also called the group of infinitesimal automorphisms. That's the defition from stacks project

Question: What is the origin and motivation of the name 'infinitesimal' here? How is it connected to the naive geometrical/analytic intuition of infinitesimality (of course if there exist a connection at all)?

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    $\begingroup$ Note that in your reference, the base category $C_\Lambda$ is not just some abstract category, but the category of Artinian local $\Lambda$-algebras with residue field $k$ (Tag 06GC). A surjective $A'\to A$ corresponds to an infinitesimal thickening ${\rm Spec}(A)\to {\rm Spec}(A')$. If $F(A)$ is the category of flat schemes over $A$ (or some other type of geometric object), then ${\rm Inf}(X'/X)$ (where now $X$ is a flat scheme over $A$ etc.) is the group of automorphisms of $X'/A'$ restricting to the identity on $X = X'_A$. $\endgroup$
    – Piotr Achinger
    Aug 9, 2020 at 15:41
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    $\begingroup$ For example, for $A=k$, $A'=k[\varepsilon]/(\varepsilon^2)$, $X/A$, $X' = X\otimes_A A' = X[\varepsilon]$, the group ${\rm Inf}(X'/X)$ is the group of $k$-derivations $\mathcal{O}_X\to \mathcal{O}_X$, and hence the familiar "infinitesimal automorphisms of first order = vector fields". $\endgroup$
    – Piotr Achinger
    Aug 9, 2020 at 15:42

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