Remark 87.9.8. The classical affine formal algebraic spaces correspond to the affine formal schemes considered in EGA ([EGA]). To explain this we assume our base scheme is \mathop{\mathrm{Spec}}(\mathbf{Z}). Let \mathfrak X = \text{Spf}(A) be an affine formal scheme. Let h_\mathfrak X be its functor of points as in Lemma 87.2.1. Then h_\mathfrak X = \mathop{\mathrm{colim}}\nolimits h_{\mathop{\mathrm{Spec}}(A/I)} where the colimit is over the collection of ideals of definition of the admissible topological ring A. This follows from (87.2.0.1) when evaluating on affine schemes and it suffices to check on affine schemes as both sides are fppf sheaves, see Lemma 87.2.2. Thus h_\mathfrak X is an affine formal algebraic space. In fact, it is a classical affine formal algebraic space by Definition 87.9.7. Thus Lemma 87.2.1 tells us the category of affine formal schemes is equivalent to the category of classical affine formal algebraic spaces.
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