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Lemma

Given XHX \in \mathbf{H} such that X\flat X is 0-truncated, then H / relX\mathbf{H}_{/\flat_{rel}X} is strongly \infty-connected over (H inf) / relX(\mathbf{H}_{inf})_{/\flat_{rel}X}.

Proof

We have an essential geometric morphism given by a composite of adjoint triples

H / relX Disc relΠ rel (H inf) / relXη X *DiscΠΓGrpd /X, \array{ \mathbf{H}_{/\flat_{rel}X} & \stackrel{\overset{\Pi_{rel}}{\longrightarrow}}{\stackrel{\overset{Disc_{rel}}{\longleftarrow}}{\underset{}{\longrightarrow}}} & (\mathbf{H}_{inf})_{/\flat_{rel} X} \stackrel{\overset{\Pi \simeq \Gamma}{\longrightarrow}}{\stackrel{\overset{\eta_X^\ast \circ Disc}{\longleftarrow}}{\underset{}{\longrightarrow}}} } \infty Grpd_{/\flat X} \,,

where the top pairs come from the formula (here) for localization of adjunctions to slices, and the bottom one exists in each case by the adjoint (∞,1)-functor theorem, since the middle one preserves (∞,1)-colimits (since colimits in slices are computed on the dependent sums, since DiscDisc preserves colimits, and since pullbacks preserve colimits in an \infty-topos). The fact that two top composite preserves the terminal object follows now by the idempotency of the various adjunctions Π rel relX relX\Pi_{rel} \flat_{rel} X \simeq \flat_{rel}X and then by infinitesimal cohesion Π relX relXX\Pi \flat_{rel} X \simeq \flat \flat_{rel} X \simeq \flat X. Finally using that X\flat X is 0-connected, hence a set it follows from Π**\Pi \ast \simeq \ast that the composite right adjoint is fully faithful over over xXx\in \flat X, hence is fully faithful on all of Grpd\infty Grpd.

Other Stuff

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category: meta

Revised on August 30, 2014 23:52:10 by Adeel Khan (77.182.85.162)