Does the nerve of a category have a right adjoint? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:37:42Z http://mathoverflow.net/feeds/question/82251 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82251/does-the-nerve-of-a-category-have-a-right-adjoint Does the nerve of a category have a right adjoint? Colin Tan 2011-11-30T05:55:16Z 2011-11-30T05:55:16Z <p>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, <a href="http://ncatlab.org/nlab/show/nerve+and+realization" rel="nofollow">nerve has a left adjoint</a>, which is geometric realization. Does the nerve $N$ have a right adjoint?</p> <p>Even better, does $N:{\mathbf{Cat}}\to{\mathbf{sSet}}$ have a right adjoint?</p> <p>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. </p>