homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
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:
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.
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 $\Delta \hookrightarrow Cat$ that defines the ordinary nerve of categories.
There is also the canonical embedding of categories into bicategories. Combined this gives the inclusion
The bicategorical nerve is the nerve induced from that. So for $C$ a bicategory we have
There are also an oplax version and two non-normalized versions.
The simplicial sets in the image of the geometric nerve are 3-coskeletal.
The geometric nerve identifies precisely bigroupoids with 2-hypergroupoids: those Kan complexes for which the horn fillers in dimension $\geq 3$ are unique . (Theorem 8.6 in (Duskin))
(This shows in particular that bigroupoids model all homotopy 2-types.)
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 $sSet$-category. This can also be applied to 2-groupoids and, thus, results in a classifying space construction for crossed modules.
geometric nerve
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.