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.)



Which simplicial objects are Čech nerves?
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.


