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Am I misunderstanding something, or isn’t $* \amalg *$ the discrete object classifier in the (2,1)-category of groupoids? More generally, in ${\infty}Gpd$, the characterization of fully faithful functors as being the insertion into a coproduct show that $* \amalg *$ is the discrete classifier there.
I assume that the author meant to write “faithful” instead of “fully faithful”: The text does correctly start out speaking about 0-truncated morphisms; and between groupooids these are the faithful functors (I have added a pointer to the discussion there).
I have briefly edited the entry accordingly, but I suppose this deserves to be explained in more detail.
We have ancient discussion in this direction at pointed sets – As the universal set bundle and scattered remarks elsewhere. One day all this ought to be polished up. Maybe this entry here could be the seed.
Thanks for confirming! And thanks for all your edits, much appreciated.
I’m currently blocked from editing the nLab, but David Corfield says at this blog post here that the role of the terminal object $*$ in the definition of a (-1)-truncated object classifier is played by $Set_*$ instead of the terminal object $*$ in the definition of a 0-truncated object classifier, and in a (1,1)-category it just so happens that the object $\Omega_*$ of pointed truth values just happens to be equivalent to the terminal object $*$.
He then goes on to talk about (1,1)-truncated object classifiers, which implies a notion of (n - 1, r)-truncated object classifier in an (n + 1, r + 1)-category.
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