# Moduli of spaces with prescribed homotopy groups

## TL;DR

This paper develops a homotopy-theoretic framework for understanding the moduli of spaces with prescribed homotopy groups, simplifying the obstruction theory for realizing algebraic data as topological spaces.

## Contribution

It introduces a new approach using $"infty$-categories of product-preserving presheaves, providing more conceptual and streamlined proofs for realization obstructions.

## Key findings

- Reproduces Blanc-Dwyer-Goerss obstructions within a homotopy-theoretic context.
- Uses $\
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## Abstract

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