# Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?

Let $C$ be a site and $CAT$ the 2-category of categories. Given a contravariant 2-functor $A:C\rightarrow CAT$, we can of course consider the associated stack. This is done by first considering the associated prestack, denoted by $\hat{A}$, (i.e. sheafifying the hom) and then considering $2$-$lim_\mathfrak{U}Des(\mathfrak{U},\hat{A})$, the descent data of $\hat{A}$ with respect to the covering $\mathfrak{U}$.

EDIT: My question is, could we stackify $A$ purely with the descent data? In other words, define $A'(U):=2$-$lim_\mathfrak{U}(Des(\mathfrak{U},A))$,
$A''(U):=2$-$lim_\mathfrak{U}(Des(\mathfrak{U},A'))$ and
$A'''(U):=2$-$lim_\mathfrak{U}(Des(\mathfrak{U},A''))$.

Would $A'''$ be equivalent to the associated stack of $A$?.

• This is already not working with sheaficiation: if you try to sheafify directl a non separated sheaf this does not neccearly gives a sheaf, but a separated presheaf and you have to go to the process twice to abtain the sheafification. – Simon Henry Oct 11 '14 at 11:47
• Thanks for the answer. But then, what about doing the descant data twice, or probably 3 times, i.e. defined $A''(U):=2$-$lim_{\mathfrak{U}}(Des(\mathfrak{U},A'))$ and then again $A'''$. Would that work? – Modnar Oct 11 '14 at 11:53
• I don't Know, but I'm sure somebody will. Maybe you should consider editing your question if it is what you want to know ? – Simon Henry Oct 11 '14 at 12:05
• This seems to be a folklore result. It is alluded to in [Higher topos theory, §6.5.3]. – Zhen Lin Oct 11 '14 at 12:52
• Hmm.. Thank you very much. So there is at least a 'high chance' that it works in 'good situations'. That is indeed good to know. Anyways, if anyone has an actual proof, it would be much appreciated if you could give a citation. Thanks again. – Modnar Oct 11 '14 at 13:17