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For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.

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Is there a natural way to give a bisimplicial structure on a 2-category?

I mean by the nerve functor. Given a 2-category $\mathcal{C}$, if we forget the 2-category structure (just view $\mathcal{C}$ as a category), the nerve functor will give us a simplicial set $N\mathc …