Abstract: Let $X$ be a compact topological manifold with underlying anima $|X|$, and let $G$ be a group in topological manifolds with underlying group in anima $|G|$. A classical result in algebraic topology gives a canonical identification $H^1(X,G) \simeq H^1(|X|,|G|)$ of the pointed set of isomorphism classes of $G$-torsors on $X$ and the pointed set of $|G|$-torsors on $|X|$. In this talk, I will explain joint work with Peter Haine aiming to prove this result by identifying the localization that makes a $\mathcal{K}$-sheaf of anima on topological manifolds homotopy invariant.