The local weak equivalence–hypercover conjecture

Let $(\mathcal{C},\tau)$ be a site, let $\mathbb{A}$ be an augmentation category, and let $f\colon\mathcal{F}\to\mathcal{F}'$ be a map in $\mathbb{A}\textbf{-Pr}(\mathcal{C})$. A local weak equivalence is a map inducing an isomorphism on the sheaves $\pi_0^\tau$ and satisfying the stated pullback condition on augmented homotopy groups. Local weak equivalence–hypercover conjecture. A map $f\colon\mathcal{F}\to\mathcal{F}'$ is a local weak equivalence if and only if it is a hypercover. This would identify the proposed local weak equivalences with hypercovers, providing the expected description of the local model structure; the source does not establish the equivalence.