## Simplicial Matrices And The Nerves Of Weak n-Categories I: Nerves Of Bicategories (2002)

Citations: | 26 - 1 self |

### BibTeX

@MISC{Duskin02simplicialmatrices,

author = {John W. Duskin},

title = {Simplicial Matrices And The Nerves Of Weak n-Categories I: Nerves Of Bicategories},

year = {2002}

}

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### Abstract

To a bicategory B (in the sense of Benabou) 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 # 1 2 (x 12 , x 01 ) present for each "composable pair" (x 12 , , x 01 ) # # 1 2 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.