The Stacks project

Remark 89.17.3. Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda $. Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Let $V$ be a finite dimensional vector space. Then $\text{Lift}(x_0, k[V])$ is the set of isomorphism classes of $\mathcal{F}_{x_0}(k[V])$ where $\mathcal{F}_{x_0}$ is the predeformation category of objects in $\mathcal{F}$ lying over $x_0$, see Remark 89.6.4. Hence if $\mathcal{F}$ satisfies (S2), then so does $\mathcal{F}_{x_0}$ (see Lemma 89.10.6) and by Lemma 89.12.2 we see that

\[ \text{Lift}(x_0, k[V]) = T\mathcal{F}_{x_0} \otimes _ k V \]

as $k$-vector spaces.

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