The symmetric power and étale realisation functors commute

We show that under mild hypotheses on a proper algebraic space $X$, the functors of taking its symmetric powers and its étale realisation commute up to weak equivalence. We conclude an effective version of the Dold-Thom theorem for the étale site and discuss the stabilisation results for the natural morphisms of étale homotopy groups $π_k \mathrm{Sym}^n X \to π_k \mathrm{Sym}^{n+1} X$ in the context of the Weil conjectures.

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