In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{Z})$ as $\infty$-categories. This has the advantage of getting easier proofs of known results and even improve them. For instance, I heard one can prove that singular cohomology and sheaf cohomology agree not just for locally contractible spaces, but for all locally weakly contractible spaces. However, I cannot find an exposition of why this is true or examples of working with it, it is only mentioned, like in these wonderful notes by Adeel Khan https://www.preschema.com/lecture-notes/2023-kias/dagkias.pdf.
Is there a reference detailing why singular cohomology of a space is a sheaf of $\infty$-categories? Ideally also working with it, for example proving that it agrees with sheaf cohomology for locally weakly contractible spaces?
