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Alan Wilder
Alan Wilder
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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 the onlyall 2-morphisms are identitiesisomorphisms, then such a bicategory is a groupoid2-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?

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 the only 2-morphisms are identities, then such a bicategory is a 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?

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?

Source Link
Alan Wilder
Alan Wilder
  • 569
  • 2
  • 9

Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ?

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 the only 2-morphisms are identities, then such a bicategory is a 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?