Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:44:00Z http://mathoverflow.net/feeds/question/55669 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55669/nerve-groupoids-kan-complexes-nerve-bicategories-w-adjoints Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ? Alan Wilder 2011-02-16T21:57:17Z 2011-02-23T12:44:01Z <p>If you take the nerve of a groupoid, you get a Kan complex.</p> <p>Question:</p> <p>Take a bicategory that has adjoints for 1-morphisms, which is one notion of 'weak' groupoid (if all 2-morphisms are isomorphisms, then such a bicategory is a 2-groupoid), and take its nerve.</p> <p>Is there a name for a bisimplicial set arising in this way? Does it have some nice properties? For example, is there a model structure on $\mathbf{ssSet}$ such that these are fibrant?</p> http://mathoverflow.net/questions/55669/nerve-groupoids-kan-complexes-nerve-bicategories-w-adjoints/56365#56365 Answer by Tim Porter for Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ? Tim Porter 2011-02-23T07:13:00Z 2011-02-23T12:44:01Z <p>I would recommend checking through the various papers by Cegarra and Remedios (look on the archive) They have done a lot of work in this area, but I am not sure if they have an answer for your question. The Duskin nerve as suggested by David Roberts is related to the bisimplicial approach and the relation is explored in various other papers from Granada, e.g. one by Manolo Bullejos and coauthors.</p>