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

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

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

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 $Disc$ 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 $\Pi_{rel} \flat_{rel} X \simeq \flat_{rel}X$ and then by infinitesimal cohesion $\Pi \flat_{rel} X \simeq \flat \flat_{rel} X \simeq \flat X$. Finally using that $\flat X$ is 0-connected, hence a set it follows from $\Pi \ast \simeq \ast$ that the composite right adjoint is fully faithful over over $x\in \flat X$, hence is fully faithful on all of $\infty Grpd$.

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shtuka, Rapoport-Zink space presheaf

category: meta

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