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 $\left(n+2\right)$ 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.
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1$\begingroup$ 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. $\endgroup$– David Roberts ♦Commented Nov 28, 2010 at 23:04
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$\begingroup$ The Quillen +-construction is fairly well-known; what's the Grothendieck one? $\endgroup$– RomeoCommented Nov 28, 2010 at 23:18
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$\begingroup$ See Mac Lane--Moerdijk, Sheaves in Geometry and Logic, section III.5. $\endgroup$– Finn LawlerCommented Nov 29, 2010 at 0:03
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$\begingroup$ 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) $\endgroup$– David CarchediCommented Nov 29, 2010 at 0:51
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1$\begingroup$ I discussed the +-construction in the context of higher topos theory a little bit, in some notes I wrote: math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf . See especially sections 3 and 11. $\endgroup$– Charles RezkCommented Nov 29, 2010 at 15:35
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2 Answers
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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.
answered Nov 29, 2010 at 7:08
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$\begingroup$ I don't see any discussion in this paper of the first step, producing a 2-prestack from a 2-presheaf. They describe the +-construction but the only theorem I see about it is that it takes 2-prestacks to 2-stacks. $\endgroup$ Commented May 5, 2021 at 22:42
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$\begingroup$ @Mike: you're right. But I think that this first step is rather easy: the Hom-categories are presheaves of categories and can be stackified in the usual (1-categorical) way. $\endgroup$ Commented May 6, 2021 at 8:11
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$\begingroup$ But that's not quite the same as applying the +-construction $n+2$ times to an $n$-presheaf. $\endgroup$ Commented May 6, 2021 at 17:02
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This is discussed in section 3.4.3 of https://arxiv.org/abs/2004.00731 by Mathieu Anel and Chaitanya Leena Subramaniam.
answered May 5, 2021 at 23:29
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$\begingroup$ Inspired by this paper, I wrote up a very similar argument on the nLab. $\endgroup$ Commented May 28, 2021 at 18:44