Taking the nerve of a groupoid gives a simplicial set. This is functorial $N:{\mathbf{Grpd}}\to {\mathbf{sSet}}$. NLab tells me that, in general, nerve has a left adjoint, which is geometric realization. Does the nerve $N$ have a right adjoint?

Even better, does $N:{\mathbf{Cat}}\to{\mathbf{sSet}}$ have a right adjoint?

I realize I may be asking for too much, as here we are working internally to the topos ${\mathbf{Set}}$. Maybe it is better ask this question internal to an $(\infty,1)$-topos and ask only that the right adjoint exists up to homotopy. I would be interested in the respective answers for the $(\infty,1)$-topos ${\mathbf{sSet}}$ and ${\mathbf{Top}}$ instead of the topos ${\bf{Set}}$, which are basically the same up to homotopy.

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    $\begingroup$ In the groupoid case, the "geometric realization" of a simplicial set is its homotopy groupoid? $\endgroup$ – Ma Ming Nov 30 '11 at 8:04
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    $\begingroup$ @Colin: It does not have a right adjoint, because as you will see with toy examples, it does not preserve colimits. $\endgroup$ – Martin Brandenburg Nov 30 '11 at 8:36
  • $\begingroup$ To complement Martin's comment, see Zhen Lin's answer here. $\endgroup$ – user62675 Aug 11 '14 at 17:03

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