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We describe a homotopy-theoretic approach to the theory of moduli ofrealizations of Blanc-Dwyer-Goerss, reproducing their obstructions to realizinga given $\Pi$-algebra as homotopy groups of a pointed space. Our techniques arebased on the $\infty$-category $\mathcal{P}_{\Sigma}(\mathcal{S}ph)$ ofproduct-preserving presheaves on finite wedges of positive-dimensional spheres,leading to more conceptual and streamlined arguments.

Moduli of spaces with prescribed homotopy groups