# nLab Duskin nerve

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

The Duskin nerve is the nerve operation on bicategories. It is a functor from BiCat to sSet.

$N:\mathrm{BiCat}\to \mathrm{sSet}\phantom{\rule{thinmathspace}{0ex}}.$N : BiCat \to sSet \,.

## Definition

We may think of the simplex category $\Delta$ as the full subcategory of Cat on the categories free on non-empty finite linear graphs. This gives the canonical inclusion $\Delta ↪\mathrm{Cat}$ that defines the ordinary nerve of categories.

There is also the canonical embedding of categories into bicategories. Combined this gives the inclusion

$\Delta ↪\mathrm{Cat}↪\mathrm{BiCat}\phantom{\rule{thinmathspace}{0ex}}.$\Delta \hookrightarrow Cat \hookrightarrow BiCat \,.

The bicategorical nerve is the nerve induced from that. So far $C$ a bicategory we have

$N\left(C\right):\left[k\right]↦{\mathrm{Hom}}_{\mathrm{BiCat}}\left(\Delta \left[k\right],C\right)\phantom{\rule{thinmathspace}{0ex}}.$N(C) : [k] \mapsto Hom_{BiCat}(\Delta[k], C) \,.

## Example

Any strict 2-category determines both a ‘bicategory’ in the above sense (since a ‘strict’ thing is also a ‘weak’ one) and a simplicially enriched category. The latter is found by taking the nerve of each ‘hom-category’. The Duskin nerve of a 2-category is the same as the homotopy coherent nerve of the corresponding $\mathrm{sSet}$-category. This can also be applied to 2-groupoids and results in a classifying space construction for crossed modules.

## References

• John Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories (tac)
• V. Blanco, M. Bullejos, E. Faro, A Full and faithful Nerve for 2-categories, Applied Categorical Structures, Vol 13-3, 223-233, 2005. (see also arxiv

Revised on September 14, 2011 19:17:52 by Spelling Corrector? (128.59.193.192)