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geometric nerve of a bicategory

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Higher category theory

higher category theory

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Idea

The (unitary) geometric nerve is a natural nerve operation on bicategories. It is a functor from BiCat to sSet. This is also sometimes called the Duskin nerve. The notion is implicit in work by R. Street (1987). The direct approach was used by Duskin in work at about the same time, as explained in both articles. (Duskin’s article directly on the idea was published in 2002.)

The construction, thus, yields a functor:

N:BiCat NLaxsSet. N : BiCat_{NLax} \to sSet \,.

extending the ordinary nerve construction on the category of small categories, where morphisms of BiCat are normal lax 2-functors: these are the lax 2-functors which strictly preserve identities.

Special cases of the construction relate to earlier constructions relating to the homotopy coherent nerve, see below for more detail.

Definition

We may think of the simplex category Δ\Delta as the full subcategory of Cat on the categories free on non-empty finite linear graphs. This gives the canonical inclusion ΔCat\Delta \hookrightarrow Cat that defines the ordinary nerve of categories.

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

ΔCatBiCat. \Delta \hookrightarrow Cat \hookrightarrow BiCat \,.

The bicategorical nerve is the nerve induced from that. So for CC a bicategory we have

N(C):[k]BiCat NLax(Δ[k],C). N(C) : [k] \mapsto BiCat_{NLax}(\Delta[k], C) \,.

There are also an oplax version and two non-normalized versions.

Properties

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

  • The nerve is a full and faithful functor BiCat NLaxsSetBiCat_{NLax}\to sSet.

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 sSetsSet-category. This can also be applied to 2-groupoids and, thus, results in a classifying space construction for crossed modules.

References

  • Ross Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra, Volume 49, Issue 3, December 1987, Pages 283–335

  • John Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories (tac), Theory and Applications of Categories, Vol. 9, No. 10, 2002, pp. 198–308.

  • 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 April 2, 2014 00:11:36 by Tim Porter (2.27.158.67)