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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 $A$?.

Thank you in advance.

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  • $\begingroup$ 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. $\endgroup$
    – Simon Henry
    Commented Oct 11, 2014 at 11:47
  • $\begingroup$ 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? $\endgroup$
    – Modnar
    Commented Oct 11, 2014 at 11:53
  • $\begingroup$ 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 ? $\endgroup$
    – Simon Henry
    Commented Oct 11, 2014 at 12:05
  • $\begingroup$ This seems to be a folklore result. It is alluded to in [Higher topos theory, §6.5.3]. $\endgroup$
    – Zhen Lin
    Commented Oct 11, 2014 at 12:52
  • $\begingroup$ 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. $\endgroup$
    – Modnar
    Commented Oct 11, 2014 at 13:17

1 Answer 1

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The three-step process is given as Theorem 3.8 in Ross Street, Two dimensional sheaf theory, J. Pure Appl. Algebra 23 (1982) 251–270 under some smallness assumptions. Without some smallness hypotheses, you may run into set-theoretic problems. For example, Waterhouse produced a presheaf with no flat sheafification for Theorem 5.5 in Basically bounded functors and flat sheaves Pac. J. Math. 57 no. 2 (1975) 597-610.

There is a one-step process using hypercovers instead of covers. As Zhen Lin mentioned in a comment, this is discussed in the beginning of section 6.5.3 in Lurie's Higher Topos Theory.

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  • $\begingroup$ Thank you very much (can't yet upvote, but will when i can :) ). I will have a look at that paper, but I think this should answer my question. $\endgroup$
    – Modnar
    Commented Oct 11, 2014 at 15:19

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