# descent implies hyperdescent

My question concerns the Propsition 5.12 of the paper of Bhatt-Mathew on arc-topology where they claim that the functor $$X\mapsto D^b_{\text{cons}}(X,\Lambda)^{[-n,n]}$$ which assigns to a qcqs scheme $$X$$ the subcategory of the full derived category $$D(X_{ét},\Lambda)$$ of étale sheaves of $$\Lambda$$-modules of amplitude in $$[-n,n]$$ spanned by those objects which are bounded with constructible cohomology. Here they claim that this functor satisfies $$v$$-descent and this automatically implies $$v$$-hyperdescent. In the proof they use the fact this functor is bounded(in the derived category) and hence takes values in $$\text{Cat}_{2n+2}$$ the ($$\infty$$)-category of $$2n+2$$-categories. It seems to me that we can generalise the situation as follows.

Let functor $$\mathcal{F}\colon\text{Sch}_R\to\mathcal{C}$$(+ some properties) be a functor where $$\mathcal{C}$$ is an infinity category(+ some properties) and $$\tau$$ be a Grothendieck topology on $$\text{Sch}_R$$. Then I want to study the implication $$\tau$$-descent $$\implies$$ $$\tau$$-hyperdescent''.

• Here I assume that $$\mathcal{F}$$ is locally of finite presentation(finitary'' according to loc. cit.).
• In what generality of $$\mathcal{C}$$ is it is true ? I think I can prove the implication in small cases like when $$\mathcal{C}=\text{Cat}_1$$.

### Definitions

We say that $$\mathcal{F}$$ satisfies $$\tau$$-descent if for each $$\tau$$-covering $$X'\to X$$ the natural map $$\mathcal{F}(X) \to\lim(\mathcal{F}(X')\rightrightarrows\mathcal{F}(X'\times_X X')\rightrightarrows \cdots)$$ is an equivalence in $$\mathcal{C}$$. The functor is said to satisfy $$\tau$$-hyperdescent if for every hypercover $$X_{.}=(\cdots \rightrightarrows X_1 \rightrightarrows X_0 \to X_{-1}=X)$$ of $$X$$ in the $$\tau$$-topology we have that $$\mathcal{F}(X)\to \lim(\mathcal{F}(X_0)\rightrightarrows \mathcal{F}(X_1) \rightrightarrows\cdots)$$ is an equivalence in $$\mathcal{C}$$.

It is certainly true that descent implies hyperdescent whenever $$\mathcal C$$ is a $$n$$-category for some $$n<\infty$$ (it wasn't clear from your question whether you knew this or not). This is because, for any $$\infty$$-site $$\mathcal A$$:
1. A presheaf $$F:\mathcal A^{op}\to\mathcal C$$ is a sheaf or hypersheaf if and only if $$\mathrm{Map}(c, F(-)):\mathcal A^{op}\to\mathcal S$$ is for all $$c\in\mathcal C$$.
2. Every sheaf $$F:\mathcal A^{op} \to \mathcal S_{\leq n}$$ is a hypersheaf, because truncated objects in an $$\infty$$-topos are hypercomplete.
Here is a more general condition. Suppose that $$\mathcal C$$ is generated under colimits by cotruncated objects, i.e., truncated objects in $$\mathcal C^{op}$$. Any $$n$$-category satisfies this, but also the $$\infty$$-category of coconnective spectra in any presentable $$\infty$$-category (and this is usually not an $$n$$-category for any finite $$n$$). For example, if $$R$$ is a ring then $$D(R)_{\leq 0}$$ is compactly generated by cotruncated objects; many sheaves in the paper of Bhatt and Mathew take values in this $$\infty$$-category.
Under this condition on $$\mathcal C$$, every sheaf $$F:\mathcal A^{op}\to\mathcal C$$ is a hypersheaf. Indeed, suppose $$X_\bullet\to X$$ is a hypercover. We want to show that the map $$F(X) \to \mathrm{lim}_{n\in\Delta^{op}} F(X_n)$$ is an equivalence in $$\mathcal C$$. Since $$\mathcal C$$ is generated under colimits by cotruncated objects, it suffices to check after applying $$\mathrm{Map}(c,-)$$ for $$c$$ cotruncated. This means that $$\mathrm{Map}(c,-)$$ takes values in $$n$$-truncated spaces for some $$n$$ (depending on $$c$$). In particular $$\mathrm{Map}(c,F(-))$$ is a hypersheaf, whence the result. For related observations see Definition 3.1.4 in https://arxiv.org/pdf/2002.11647.pdf and the following results.