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In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+1\right)$ times, and in general, for a presheaf of $\infty$-groupoids, one needs to apply a transfinite iteration. However, not much detail is given about this. Does anyone know where I can read more about this? Thanks.

flag	asked Nov 28 2010 at 22:42 David Carchedi 7,747●10●27

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For n=2 it can be found in Igor Bakovic's thesis. For n=1 this was known to Grothendieck and his crew. I would perhaps add to the question the clarifying remark that this +-construction is not Quillen's, but Grothendieck's. – David Roberts Nov 28 2010 at 23:04

The Quillen +-construction is fairly well-known; what's the Grothendieck one? – Romeo Nov 28 2010 at 23:18

See Mac Lane--Moerdijk, Sheaves in Geometry and Logic, section III.5. – Finn Lawler Nov 29 2010 at 0:03

Thanks for the comments thus far. I should mention I am most interested in finding a reference where this is done in full generality, not just at the cases $n=1$ or $n=2$, as I need to apply this in an $\infty$-setting. (Although, of course, looking at these special cases will be illuminating) – David Carchedi Nov 29 2010 at 0:51

I should also mention that is in fact important for me to break this up into substeps- I'm not interested in just a way of computing the stackification, but I am interested in understanding all the different levels in between being a presheaf and stack. – David Carchedi Nov 29 2010 at 0:53

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Konrad Waldorf · Answer 1 · 2010-11-29 07:08:51Z UTC

Thomas Nikolaus and Christoph Schweigert discuss the +-construction for $n=2$ in their paper Equivariance in Higher Geometry. They split it up into two steps (I think): first producing a pre-2-stack out of a presheaf of 2-groupoids, and then making it a 2-stack.

Applied to the pre-2-stack obtained by delooping the monoidal stack of principal $U(1)$-bundles, one gets exactly the definition of a bundle gerbe.

The plus construction for stacks of n-types

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