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- W144654815 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 χ2(x12, x01) present for each “composable pair” (x12, , x01) ∈ ∧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." @default.
- W144654815 created "2016-06-24" @default.
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- W144654815 date "2002-01-01" @default.
- W144654815 modified "2023-10-17" @default.
- W144654815 title "SIMPLICIAL MATRICES AND THE NERVES OF WEAK n-CATEGORIES I : NERVES OF BICATEGORIES" @default.
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