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

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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 $N\mathcal{C}$. However, $\mathcal{C}$ is a 2-category, thus for any two objects $x,y\in\mathcal{C}$, $Hom_{\mathcal{C}}(x,y)$ is a category, applying the nerve functor gives us a simplicial set $N(Hom(x,y))$.

My question is, can these two simplicial set structure compatible in some way, gives us a bisimplicial set $N_{p,q}(\mathcal{C})$, say? Or is there another way to give a bisimplicial structure on a 2-category?

-

## 1 Answer

Yes. This is called the double nerve of a 2-category.

See in particular the first reference cited at that link.

-

Thanks a lot! But I can't open the .pdf file of the first reference. Am I the only one suffering from this problem? – Fei Nov 17 2009 at 13:40
I have added the arXiv link now: arxiv.org/abs/math.CT/0406615 – Urs Schreiber Nov 17 2009 at 13:51
Yeah, it works, thank you very much – Fei Nov 17 2009 at 16:56
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 8, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8798958659172058, "perplexity_flag": "middle"}