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I keep wanting to point to properties of the terminal geometric morphism. While we had this scattered around in various entries (such as at global sections, at (infinity,1)-topos and elsewhere – but not for instance at (infinity,1)-geometric morphism) I am finally giving it its own entry, for ease of hyperlinking.
So far this contains the (elementary) proofs that the geometric morphism to the base $Set$/$Grpd_\infty$ is indeed essentially unique, and that the right adjoint is equivalently given by homs out of the terminal object.
Some ideas for improvement/review:
In idea, first line, remove the $\infty$ prefix of the base $Grpd_\infty$.
In equaiton (1), change Grp to Grpd.
It’d be nice to have a mention of the simpler (1-) topos case. Even if just a link (for instance here).
I came here from Local System, talking about the terminal $(\infty, 1)$-geometric morphism. Is it the same notion as in here? If so there’s a clash of notation ($\infty$-topos vs $(\infty, 1)$-topos, $\infty$-functor vs $(\infty, 1)$-functor, etc.).
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