## 98.14 Axioms

Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Here are the axioms we will consider on $\mathcal{X}$.

a set theoretic condition

^{1}to be ignored by readers who are not interested in set theoretical issues,$\mathcal{X}$ is a stack in groupoids for the étale topology,

$\mathcal{X}$ is limit preserving,

$\mathcal{X}$ satisfies the Rim-Schlessinger condition (RS),

the spaces $T\mathcal{F}_{\mathcal{X}, k, x_0}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ are finite dimensional for every $k$ and $x_0$, see (98.8.0.1) and (98.8.0.2),

the functor (98.9.3.1) is an equivalence,

$\mathcal{X}$ and $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ satisfy openness of versality.

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