The Stacks project

Remark 90.12.5. We can globalize the notions of tangent space and differential to arbitrary categories cofibered in groupoids as follows. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $, and let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. As in Remark 90.6.4, we get a predeformation category $\mathcal{F}_ x$. We define

to be the tangent space of $\mathcal{F}$ at $x$. If $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda $ and $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$, then there is an induced morphism $\varphi _ x: \mathcal{F}_ x \to \mathcal{G}_{\varphi (x)}$. We define the differential $d_ x \varphi : T_ x \mathcal{F} \to T_{\varphi (x)} \mathcal{G}$ of $\varphi $ at $x$ to be the map $d \varphi _ x: T \mathcal{F}_ x \to T \mathcal{G}_{\varphi (x)}$. If both $\mathcal{F}$ and $\mathcal{G}$ satisfy (S2) then all of these tangent spaces have a natural $k$-vector space structure and all the differentials $d_ x \varphi : T_ x \mathcal{F} \to T_{\varphi (x)} \mathcal{G}$ are $k$-linear (use Lemmas 90.10.6 and 90.12.4).