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For

𝒳:𝒞 opGrpd \mathcal{X} \;\colon\; \mathcal{C}^{op} \xrightarrow{\;} Grpd_\infty

an \infty-presheaf and

c𝒞𝒳(c)Grpd \overset{ c \in \mathcal{C} }{\int} \mathcal{X}(c) \;\; \in \;\; Grpd_\infty

its \infty-Grothendieck construction, it ought to be the case that its projection onto the \infty-Grothendieck construction of its nn-truncation is the \infty-Grothendieck construction on the system

(c,x𝒳(c))cn n(𝒳(c),x) \big( c, \, x \in \mathcal{X}(c) \big) \;\mapsto\; cn_n \big( \mathcal{X}(c), x \big)

of nn-connected covers:

Is this discussed anywhere?


Here some more elaboration:

For n{}n \in \mathbb{N} \sqcup \{\infty\}, we write

for the universal fibration of nn-groupoids.

Via the naturality squares

we may canonically regard any pointed \infty-groupoid (𝒳,x)Grpd ^(\mathcal{X}, x) \,\in\, \widehat{\mathrm{Grpd}_\infty} as an element in the homotopy pullback of the nnth universal fibration along τ n\tau_n.

Lemma

The resulting further homotopy pullback along its classifying map is the nn-connected cover cn n(𝒳,s)\mathrm{cn}_n(\mathcal{X},s) of (𝒳,x)(\mathcal{X},x):

Proof

Since

  1. left fibrations are preserved by pullback

  2. a left Fibrations over the \infty-groupoid *\ast are Kan fibrations

the pullback in question is guaranteed to be an \infty-groupoid. Therefore, since passage to the core is a right adjoint, we may equivalently compute the \infty-pullback of cores.

Finally, since Grpd Grpd_\infty is extensive, we may compute these pullbacks on the respective connected components of the cores.

In conclusion, this means that our \infty-pullback is equivalently the following:

That the top right square is a homotopy pullback follows, for instance by the pasting law and the characterization of \infty -actions?.

That the top left square is homotopy pullback as shown is seen by observing that it is the homotopy colimit of an evident homotopy pullback square of simplicial \infty-groupoids, whose right morphism is, evidently, a homotopy Kan fibration. This implies the claim by this Prop.. (The same kind of argument also applies to the bottom right square.)

Now consider the following tower of \infty-Grothendieck constructions on the nn-truncations of an \infty-presheaf 𝒳()\mathcal{X}(-), here shown in the first few stages:

By the pasting law, and the previous diagram, the top left square ought to be as shown, exhibiting the homotopy fibers of P 1P_1 as the 1-connected covers as claimed. The same argument would apply to any P nP_n.