Definition 86.11.1. 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

1. $X_ i$ is an affine formal algebraic space,

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

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

and $X$ satisfies a set theoretic condition (see Remark 86.11.5). A morphism of formal algebraic spaces over $S$ is a map of sheaves.

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