Definition 87.11.1. Let $S$ be a scheme. We say a sheaf $X$ on $(\mathit{Sch}/S)_{fppf}$ is a *formal algebraic space* if there exist a family of maps $\{ X_ i \to X\} _{i \in I}$ of sheaves such that

$X_ i$ is an affine formal algebraic space,

$X_ i \to X$ is representable by algebraic spaces and étale,

$\coprod X_ i \to X$ is surjective as a map of sheaves

and $X$ satisfies a set theoretic condition (see Remark 87.11.5). A *morphism of formal algebraic spaces* over $S$ is a map of sheaves.

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