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If you take the nerve of a groupoid, you get a Kan complex.


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.

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?

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Dear Alan, how do you define the nerve of a bicategory. –  Harry Gindi Feb 16 '11 at 22:26
Well my deleted comment was silly. @Harry - he's taking the hom-wise nerve to get a (weakened) simplicial category and the the other nerve to get a bisimplicial set. Personally I would take the Duskin nerve, which is the '2-simplices are 2-commuting triangles, etc.' version. –  David Roberts Feb 16 '11 at 22:34
@David: I thought you could only apply functors homwise in the case where the enrichment is strict (strict 2-categories are categories enriched in the cartesian monoidal category $Cat$.) –  Harry Gindi Feb 16 '11 at 23:17
I just read about Duskin nerve on nLab. I'm not sure I want that because I would like to end up in bisimplicial sets. If there's a nice answer for Duskin nerve though, I would love to hear it! –  Alan Wilder Feb 16 '11 at 23:37
What do you mean by "has adjoints for 1-morphisms"? Every morphism has a left adjoint? a right adjoint? both? either? or that composition with a given morphism induces an adjunction between hom-categories? –  Steve Lack Feb 17 '11 at 4:49

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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.

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There is the paper ugr.es/~bullejos/geometryampl.pdf which works out the comparison between nerves for 2-categories. –  Jonathan Chiche Feb 23 '11 at 10:34

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