Derived Stone Embedding

A classical result, the Stone embedding, characterizes profinite sets as totally disconnected, compact Hausdorff spaces. Building on “Pyknotic objects, I. Basic notions”, which introduced a derived Stone embedding of the pro-category of $\pi$-finite spaces into pyknotic spaces, this paper uses the $\mathrm{\infty }$-topoi machinery to partially characterize the essential image of this embedding, extending the classical characterization to the derived setting.