descent implies hyperdescent

Question

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 $\mathcal{C}$.

Answer 1

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.