Definition 85.5.1. Let $S$ be a scheme. We say a sheaf $X$ on $(\mathit{Sch}/S)_{fppf}$ is an affine formal algebraic space if there exist

1. a directed set $\Lambda$,

2. a system $(X_\lambda , f_{\lambda \mu })$ over $\Lambda$ in $(\mathit{Sch}/S)_{fppf}$ where

1. each $X_\lambda$ is affine,

2. each $f_{\lambda \mu } : X_\lambda \to X_\mu$ is a thickening,

such that

$X \cong \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } X_\lambda$

as fppf sheaves and $X$ satisfies a set theoretic condition (see Remark 85.7.5). A morphism of affine formal algebraic spaces over $S$ is a map of sheaves.

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