nLab
Duskin nerve

Idea

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

$$N:\mathrm{BiCat}\to \mathrm{sSet}.$$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 $\Delta \hookrightarrow \mathrm{Cat}$ that defines the ordinary nerve of categories.

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

$$\Delta \hookrightarrow \mathrm{Cat}\hookrightarrow \mathrm{BiCat}.$$\Delta \hookrightarrow Cat \hookrightarrow BiCat \,.

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

$$N(C):[k]\mapsto {\mathrm{Hom}}_{\mathrm{BiCat}}(\Delta [k],C).$$N(C) : [k] \mapsto Hom_{BiCat}(\Delta[k], C) \,.

Properties

• The simplicial sets in the image of the Duskin nerve are 3-coskeletal.

• The Duskin nerve identifies precisely bigroupoids with 2-hypergroupoids: those Kan complexes for which the horn fillers in dimension $\ge 3$ are unique .

  (Theorem 8.6 in (Duskin))

  (This shows in particular that bigroupoids model all homotopy 2-types.)

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.

Related concepts

• nerve

• Duskin nerve

• ω-nerve

• homotopy coherent nerve

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