#
Simplicial matrices and the nerves of weak n-categories I:
nerves of bicategories

##
John W. Duskin

To a bicategory **B**
(in the sense of Bénabou) we
assign a simplicial set **Ner(B)**, the *(geometric) nerve* of
**B**, which completely encodes the structure of **B**
as a bicategory. As a simplicial set **Ner(B)** is a subcomplex of its
2-Coskeleton and itself isomorphic to its 3-Coskeleton, what we
call a 2-dimensional *Postnikov complex*. We then give,
somewhat more delicately, a complete characterization of those
simplicial sets which are the nerves of bicategories as certain
2-dimensional Postnikov complexes which satisfy certain restricted
`exact horn-lifting' conditions whose satisfaction is controlled by
(and here defines) subsets of (abstractly) *invertible* 2 and
1-simplices. Those complexes which have, at minimum, their
degenerate 2-simplices always invertible and have an invertible
2-simplex $\chi_2^1(x_{12}, x_{01})$ present for each `composable
pair' $(x_{12}, \_ , x_{01}) \in \mhorn_2^1$ are exactly the
nerves of bicategories. At the other extreme, where *all* 2
and 1-simplices are invertible, are those Kan complexes in which
the Kan conditions are satisfied exactly in all dimensions >2.
These are exactly the nerves of *bigroupoids* - all 2-cells are
isomorphisms and all 1-cells are equivalences.

Keywords: bicategory, simplicial set, nerve of a bicategory.

2000 MSC: Primary 18D05 18G30; Secondary 55U10 55P15.

*Theory and Applications of Categories*, Vol. 9, 2001, No. 10, pp 198-308.

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