I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning.

- Let $M$ be a manifold, and consider the presheaf $C^*(-,\mathbb{Z})$ on $M$ sending $U$ to $C^*(U;\mathbb{Z})$, the complex of (singular) cochains on $U$. Then this association is a homotopy sheaf in the sense that for any $U$ and a covering $\{U_i\}_i$ of $U$, the map $$C^*(U;\mathbb{Z}) \to \mathrm{holim} \, C^*(N(\{U_i\});\mathbb{Z})$$ is an equivalence (e.g., in Lurie's formalism by considering the quasicategory $\operatorname{Mod}_{\mathbb{Z}}$ and ordinary limits in that sense, or homotopy limits in the model category of chain complexes). For a covering $U = U_1 \cup U_2$, the fact that it's a homotopy sheaf is essentially equivalent to Mayer-Vietoris (and more generally, it corresponds to the Mayer-Vietoris spectral sequence for a general covering).

Here's the crux: when viewed as taking values in chain complexes, the constant sheaf $\underline{\mathbb{Z}}$ is NOT a homotopy sheaf (even though it is an ordinary sheaf), and the map $$ \underline{\mathbb{Z}} \to C^*(-,\mathbb{Z}) $$ is a homotopy sheafification.

What this tells us is: even something that's a sheaf in the ordinary $1$-categorical sense is NOT necessarily a homotopy sheaf (e.g. assuming $M$ has nontrivial cohomology).

Since the sheaf condition is a condition on limits, this tells us that the inclusion of quasicategories from $\operatorname{Ab}$ into $\operatorname{Mod}_{\mathbb{Z}}$ does not preserve limits.

Something similar happens with etale cohomology on the etale site.

- It's well-known that schemes are algebraic stacks. In particular, if I have a representable functor on the category of schemes, then if we view it as a functor from schemes to groupoids (via the inclusion of sets into groupoids), then it's a sheaf (aka stack) for the fppf topology. I assume this means that if I have ANY set-valued sheaf for the fppf topology, then if I consider it as a presheaf valued in groupoids, it remains a sheaf (in the appropriate $2$-categorical sense).

More generally, a scheme is an algebraic $n$-stack, and an $n$-stack is an $m$-stack for $m \ge n$ (c.f. https://arxiv.org/pdf/alg-geom/9609014.pdf, p.3). There's no discussion of needing to sheafify when you do this.

So what's the difference? Am I wrong about one of these two situations? (and if so, which one and how?) Is it some fundamental difference between the abelian and non-abelian settings? Or something else? I talked to my advisor, who has some expertise in this area, but they were confused by this as well.