The Stacks project

Lemma 87.34.5. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Assume $X$ is McQuillan, i.e., equal to $\text{Spf}(A)$ for some weakly admissible topological $S$-algebra $A$. Then $(X_{affine, {\acute{e}tale}})^{opp}$ is equivalent to the category whose

  1. objects are $A$-algebras of the form $B^\wedge = \mathop{\mathrm{lim}}\nolimits B/JB$ where $A \to B$ is an étale ring map and $J$ runs over the weak ideals of definition of $A$, and

  2. morphisms are continuous $A$-algebra homomorphisms.

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