This question arose from Peter Scholze's notes on six functor formalisms, specifically lecture VII in the proof of proposition 7.1.

We fix a LCH space $X$ and consider the functor $D(\mathrm{Ab}(X)) \to \mathrm{Shv}^{\mathrm{hyp}}(X,D(\mathbb Z))$ from the derived $\infty$-category of abelian sheaves to the infinite category of $D(\mathbb Z)$-valued hypersheaves on $X$, which (at the level of complexes) sends a complex $C^*$ of sheaves to the hypersheafification of the presheaf $U \mapsto C^*(U)$.

Scholze claims

When restricted to K-injective complexes, it turns out that the hypersheafification step is unnecessary. In other words, if $U \subset X$ has a hypercover $U_\bullet \to U$, and $C \in Ch(Ab(X))$ is $K$-injective, then the map $C(U) \to \mathrm{lim}\,C(U_\bullet)$ in $D(\mathbb Z)$ is an isomorphism. This follows by noting that the hypercover induces a resolution of $j_! \mathbb Z$ (where $j : U ⊂ X$ is the open immersion), and taking the associated $R\mathrm{Hom}$ into $C$ (which vanishes by assumption that $C$ is $K$-injective)

I had a couple of questions about this argument:

1. How is this resolution of $j_! \mathbb Z$ actually constructed from the hypercover of $U$? I assume it's something along the lines of using the hypercover to create a sequence of constant sheaves on each of the $U_\bullet$, and then using the exceptional pushforward along the inclusions $U_\bullet \hookrightarrow X$ to create a resolution of $j_! \mathbb Z$. But I don't see why this should be a `good' (e.g. flasque) resolution.

2. How do we finally conclude? I don't see what $R\mathrm{Hom}$ming into $C$ and getting $0$ tells us about our original map being an isomorphism.

Sorry for the long question! I thought that this step in the notes used some interesting techniques (resolutions from hypercovers, and using $K$-injective complexes) and I'd like to understand them better.

This question arose from Peter Scholze's notes on six functor formalisms, specifically lecture VII in the proof of proposition 7.1.

We fix a LCH space $X$ and consider the functor $D(\mathrm{Ab}(X)) \to \mathrm{Shv}^{\mathrm{hyp}}(X,D(\mathbb Z))$ from the derived $\infty$-category of abelian sheaves to the infinite category of $D(\mathbb Z)$-valued hypersheaves on $X$, which (at the level of complexes) sends a complex $C^*$ of sheaves to the hypersheafification of the presheaf $U \mapsto C^*(U)$.

Scholze claims

When restricted to K-injective complexes, it turns out that the hypersheafification step is unnecessary. In other words, if $U \subset X$ has a hypercover $U_\bullet \to U$, and $C \in \operatorname{Ch}(\operatorname{Ab}(X))$ is $K$-injective, then the map $C(U) \to \lim C(U_\bullet)$ in $D(\mathbb Z)$ is an isomorphism. This follows by noting that the hypercover induces a resolution of $j_! \mathbb Z$ (where $j : U \subseteq X$ is the open immersion), and taking the associated $R{\operatorname{Hom}}$ into $C$ (which vanishes by assumption that $C$ is $K$-injective)

I had a couple of questions about this argument:

1. How is this resolution of $j_! \mathbb Z$ actually constructed from the hypercover of $U$? I assume it's something along the lines of using the hypercover to create a sequence of constant sheaves on each of the $U_\bullet$, and then using the exceptional pushforward along the inclusions $U_\bullet \hookrightarrow X$ to create a resolution of $j_! \mathbb Z$. But I don't see why this should be a ‘good’ (e.g. flasque) resolution.

2. How do we finally conclude? I don't see what $R{\operatorname{Hom}}$ming into $C$ and getting $0$ tells us about our original map being an isomorphism.

Sorry for the long question! I thought that this step in the notes used some interesting techniques (resolutions from hypercovers, and using $K$-injective complexes) and I'd like to understand them better.