Definition 86.9.1. Let $S$ be a scheme. We say a sheaf $X$ on $(\mathit{Sch}/S)_{fppf}$ is an *affine formal algebraic space* if there exist

a directed set $\Lambda $,

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

each $X_\lambda $ is affine,

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

such that

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

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