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.


1. 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.


2. 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.



Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition?
You said it yourself in the question! The reason that sheaves of abelian groups are not $\infty$-sheaves in general, when considered as presheaves taking values in the $\infty$-category $\mathsf{Mod}\_{\mathbb Z}$, is that the $\infty$-functor $\mathsf {Ab} \longrightarrow \mathsf{Mod}\_{\mathbb Z}$ does not preserve limits. And the reason that sheaves of sets are $\infty$-sheaves is that the $\infty$-functor $\mathsf{Set} \longrightarrow \mathsf{Spaces}$ does preserve limits: it has a left adjoint, given by $\pi\_0$. The same holds for the inclusion of $k$-truncated spaces into $(k+1)$-truncated spaces, in which case the adjoint is Postnikov truncation, and this answers your question about higher stacks. More generally the full subcategory of $k$-truncated objects in any $\infty$-category is closed under all limits.


You wrote: "Is it some fundamental difference between the abelian and non-abelian settings?". Perhaps not. The inclusion of the 1-category of abelian groups into the $\infty$-category of *nonnegatively graded* chain complexes preserves all limits, and this is arguably the "true" abelian analogue of the inclusion of sets into spaces via Dold-Kan.


