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

John W. Duskin

Theory and applications of categories, CT2000, Tome 9 (2000), pp. 198-308.

Résumé

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' $(x_{12},\mathrm{\_},x_{01})\in {\text{mhorn}}_2^1$ 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.

Détail

Classification : Primary 18D05 18G30, Secondary 55U10 55P15.
Mots-clés : bicategory, simplicial set, nerve of a bicategory.