Which simplicial objects are Čech nerves?

Question

In 1-categories, a regular epimorphism is a coequalizer of some parallel pair. An effective epimorphism is one which coequalizes its kernel pair. In the presence of kernel pairs, regular and effective epis coincide: if a coequalizer has a kernel pair, it is the coequalizer of its kernel pair.

The coequalizer of a kernel pair is just its colimit.

For $(\infty,1)$-categories, it is no longer true that whenever the Čech nerve of an arrow exists, the arrow is its $(\infty,1)$-colimit. If this is the case, we say the arrow is an $(\infty,1)$-effective epimorphism.

A reasonable definition for an $(\infty,1)$-regular epimorphism as a colimit of some simplicial diagram. This seems reasonable, but leads me to wonder:

• Which simplicial objects are Čech nerves, and how to identify a Čech nerve?
• If the Čech nerve of an $(\infty,1)$-regular epi exists, is it its $(\infty,1)$-colimit?

(I actually just noticed the nlab asks the same questions.)

Answer 1

The analogous 1-categorical version of your first question would be "which parallel pairs are kernel pairs?" As far as I know this does not have a non-tautological answer in an arbitrary category, but in a Barr-exact (= effective regular) category the answer is "the internal equivalence relations" (although this is more or less part of the definition of "Barr-exact", so it may not be very satisfying). The $\infty$-version of an internal equivalence relation is an internal groupoid object, and the corresponding equivalence can be found for $\infty$-toposes and similar categories in Higher Topos Theory, though I don't know whether an exact $(\infty,1)$-categorical analogue of "Barr-exact" has been defined yet.

The answer to your second question is also yes in good $(\infty,1)$-categories such as $\infty$-toposes (and presumably also "Barr-exact" ones, whatever those are), because in that case the effective epis are the left class in a factorization system whose right class are the monomorphisms, and it's easy to see that any $(\infty,1)$-regular epi is left orthogonal to monomorphisms, hence is effective. Here, of course, the analogous 1-categorical question has the answer "yes" in all categories, not just the Barr-exact ones, and I don't know whether that is still true in the $\infty$-case.

Answer 2

Here is an answer to a question you didn't ask, but which I thought you were going to ask from the setup:

Q: Is every regular epimorphism effective?

A: Yes, in a semitopos.

Proof: Let $\mathcal C$ be a semitopos in Lurie's sense (Definition 6.2.3.1 in HTT). That is, $\mathcal C$ is a presentable, locally caretesian closed $\infty$-category where the Cech nerve of any morphism is effective. Note that bey Corollary 6.2.3.12 in HTT, if $fg$ is an effective epimorphism, then so is $f$.

Let $X_\bullet$ be a simplicial object in $\mathcal C$, and $|X_\bullet|$ its colimit. Let $X_\bullet'$ be the Cech nerve of $X_0 \to |X_\bullet|$, and $|X_\bullet'|$ its geometric realization. Then $X_0 \twoheadrightarrow |X_\bullet'|$ is an effective epimorphism. There is a map of simplicial objects $X_\bullet \to X_\bullet'$ which is the identity map on the zeroth level. The induced map $|X_\bullet| \twoheadrightarrow |X_\bullet'|$ is an effective epimorphism because it factors the effective epimorphism $X_0 \twoheadrightarrow |X_\bullet'|$. There is another map of simplicial objects from $X'_\bullet$ to the constant diagram at $|X_\bullet|$. There is an induced map $|X_\bullet'| \to |X_\bullet|$ and the composite $|X_\bullet | \twoheadrightarrow |X_\bullet'| \to |X_\bullet|$ is the identity. So $|X_\bullet| \twoheadrightarrow |X_\bullet'|$ is both an effective epimorphism and a split mono and hence it is an isomorphism (by Proposition 6.2.3.10 in HTT).