# nLab Sandbox

Every wiki needs a sandbox! Just test between the horizontal rules below (*** in the source) and otherwise don't worry about messing things up.

## Stuff

###### Lemma

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

###### Proof

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 $\infty Grpd$.

# Other Stuff

###### Theorem

$2=1$.

###### Proof

Trivial consequence of the axioms.

category: meta

Revised on August 25, 2014 04:47:50 by David Corfield (146.200.41.76)