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

Question:

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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  • $\begingroup$ Dear Alan, how do you define the nerve of a bicategory. $\endgroup$
    – Harry Gindi
    Commented Feb 16, 2011 at 22:26
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    $\begingroup$ 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. $\endgroup$
    – David Roberts ♦
    Commented Feb 16, 2011 at 22:34
  • $\begingroup$ @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 $Cat$.) $\endgroup$
    – Harry Gindi
    Commented Feb 16, 2011 at 23:17
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    $\begingroup$ 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! $\endgroup$
    – Alan Wilder
    Commented Feb 16, 2011 at 23:37
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    $\begingroup$ 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? $\endgroup$
    – Steve Lack
    Commented Feb 17, 2011 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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