Hypercovers of sheaves in classical and quasi-categories

Question

I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck site.

The definition for hypercovers in an $\infty$-topos that I am using is from Higher Topos Theory: In an $\infty$-topos $\mathfrak{X}$ a hypercovering of an element $X$ is the structure map $|U_\bullet|\to X$ of the geometric realization of a simplicial object $U_\bullet\in s\mathfrak{X}_{/X}$ such that the map $$U_n\to (cosk_{n-1}U_\bullet)_n$$ is an effective epimorphism in $\mathfrak{X}_{/X}$ (its Cech nerve is a simplicial resolution of the target) for all $n\geq 1.$

My question now is if this definition gets any easier if we restrict to an $\infty$-topos that is given by the $\infty$-sheaves on a (small) quasi-category with Grothendieck topology. Ideally, I want to relate this to the classical definition that a hypercovering on the presheaves $\mathcal{P(C)}$ of a Grothendieck site is an augmented simplicial object $U_\bullet\in s_+\mathcal{P(C)}$ (with $X\cong U_{-1}$) such that the maps $$U_n\to (cosk_{n-1}U_\bullet)_n$$ are local epimorphisms.

So it all boils down to the question if there is some way of viewing the effective epimorphisms of the $\infty$-sheaf topos on an $\infty$-site as local epimorphisms. Is there any source that does this example? Lastly, I just want to note that we get $\infty$-sheaves from $\infty$-presheaves by localizing along those local epimorphisms the target of which is representable. But I don't see how this would imply my question.

Answer 1

Local epimorphisms are precisely those morphisms in $\mathcal{P}\left(\mathcal{C}\right)$ which become effective epimorphisms after applying the sheafification functor. In particular, if $f$ is an effective epimorphism in $\mathfrak{X}=Sh_\infty\left(\mathcal{C}\right),$ then when regarded as a morphism in $\mathcal{P}\left(C\right)$ via the full and faithful embedding $$Sh_\infty\left(\mathcal{C}\right) \hookrightarrow \mathcal{P}\left(\mathcal{C}\right)$$ it is a local epimorphism. That is to say, you can rephrase the second definition you give (the "classical one") as an augmented simplicial object $$U:\Delta_+^{op} \to \mathcal{P}\left(\mathcal{C}\right)$$ such that, $a \circ U$ is a hypercover in the first sense (where $a$ denotes sheafification). More precisely, $$\left(\Delta^{op}\right)^{\triangleright}\cong \Delta_{+}^{op}$$ so the augmented simplicial object $a \circ U$ of $Sh_\infty\left(\mathcal{C}\right)$ corresponds to a simplicial object in $Sh_\infty\left(\mathcal{C}\right)/a\left(X\right)$ which is a hypercover (this correspondence between an augmented simplicial object and a simplicial object in the slice category uses the definition of the Joyal's join construction as a left adjoint). Note we are also using that sheafification is a left exact functor, and the functors $cosk_n$ etc. are all computed using finite limits.

In response to commented question: Why are local epimorphisms precisely those morphisms which become effective epimorphisms after sheafification?

One way of phrasing what a local epimorphism is as follows:

Given $f:F \to G,$ one can form its Cech nerve $C(f):\Delta^{op} \to \mathcal{P(C)}.$ $f$ is said to be an effective epimorphism if the canonical map $\operatorname{hocolim} C(f) \simeq G.$ Being a local epimorphism is equivalent to the canonical map $\operatorname{hocolim} C(f) \to G$ (which is a sieve always) to be a covering sieve. But, being a covering sieve is equivalent to being a subobject whose sheafification becomes an equivalence, i.e. $a\operatorname{hocolim} C(f) \to aG$ is an equivalence. But since $a$ is left exact and a left adjoint, one has $$a\operatorname{hocolim} C(f) \simeq \operatorname{hocolim} C(af),$$ so one sees directly that $f$ is a local epimorphism if and only if its sheafification is an effective epi.