# Hypercovers of sheaves in classical and quasi-categories

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.

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.

• As a matter of history, the original form of that join construction was in some notes of Jack Duskin and Don van Osdol from the ?1980s. Certainly my student, Phil Ehlers, used and developed their ideas in his thesis in 1993. Duskin and van Osdol gave Bill Lawvere as the source of the construction. (This is not to detract from the development and use that André Joyal made of those ideas, merely to check that the `record' is straightened a bit.) – Tim Porter Jul 29 '14 at 11:07
• @TimPorter: Thanks! I didn't know that, nor did I intentionally misrepresent the origin; I learned it from Joyal's paper. It's harder "being young" to have an informed historical perspective. – David Carchedi Jul 29 '14 at 16:24
• Thank you! I still don't see why local epimorphisms are exactly the effective epimorphisms after sheafification. Is there an easy argument or a source where this is done explicitly? – COhrt Jul 29 '14 at 18:55
• @David: don't worry. Various people find and refind results and ideas constantly. It is good to keep the record straightish but is nearly always impossible to do it 100%. There is an old story of professor A who proved something, but on mentioning it to a colleague was told that there was a paper on that 15 years earlier. Professor A went and checked on Math Revs and eventually found the paper.... It, of course, was by himself! – Tim Porter Jul 30 '14 at 10:45
• @COhrt (and Tim): I edited my answer to include the proof, see above. – David Carchedi Jul 30 '14 at 12:42