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.
Theory and Applications of Categories, Vol. 9, 2002, No. 10, pp 198-308 http://www.tac.mta.ca/tac/volumes/9/n10/n10.dvi http://www.tac.mta.ca/tac/volumes/9/n10/n10.ps http://www.tac.mta.ca/tac/volumes/9/n10/n10.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/9/n10/n10.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/9/n10/n10.psTAC Home