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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}.


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




Trivial consequence of the axioms.

shtuka, Rapoport-Zink space

category: meta

Revised on August 25, 2014 04:47:50 by David Corfield (