Remark 89.12.5. 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 89.6.4, we get a predeformation category $\mathcal{F}_ x$. We define

$T_ x\mathcal{F} = T\mathcal{F}_ x$

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 89.10.6 and 89.12.4).

There are also:

• 2 comment(s) on Section 89.12: Tangent spaces of predeformation categories

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).