Simplicial Matrices and the Nerves of Weak n-Categories I: Nerves of Bicategories

By Jean Benabou, J. Duskin. Theory and Applications of Categories, 2001

129 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/951b97883df9a955ff1ac66e2d65b4f7c9c425aa

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Citations in the corpus

n-Groupoids and Stacky Groupoids (2008, background): “Set-theoretically, these extra data are worked out when n = 1, 2 in [15].”
Homotopy Theory of Higher Categories: THE MODEL STRUCTURE (2010, background): “See also Bergner [34], Dwyer, Kan and Smith [92], Berger [30], Duskin [86], and others.”
Higher-Dimensional Algebra V: 2-Groups (2003, unknown): “Let |K0| be the geometric realization of the nerve of K0 as defined by Duskin [23].”
Lax Colimits and Free Fibrations in $\infty$-Categories (2015, background): “denote the composite Catstr 2 −−→N∗ Cat ∆ −→N Set ∆. This functor has a left adjoint C2, which is the composite Set∆ −→C Cat ∆ −−→C∗ Catstr 2 . Remark A.13. It is obvious from the deﬁnitions given in [Dus02] that the functor N2 as we have deﬁned it is simply the restriction of Duskin’s nerve for bic”
Lie theory for nilpotent L∞-algebras (2004, background): “n example of a lax ℓ-groupoid; for ℓ= 1, it is the nerve of the fundamental groupoid of X. A lax 0-groupoid is a discrete set, and a lax 1-groupoids is the nerve of a groupoid, by Proposition 2.2. In [10], Duskin identiﬁes lax 2-groupoids with the nerves of bigroupoids. A bigroupoid Gis a bicategory”
Presenting higher stacks as simplicial schemes (2013, methodology): “ct X0 ×h X X0 as an (n −1)-hypergroupoid, since a model for it is given by the (n−1)-hypergroupoid Dec+(X) ×X X0, as Dec+(X) →X is a ﬁbration. This object is usually called the path-homotopy complex ([Dus2] §2.6). Lemma 2.26. If f : X →Y is a relative n-hypergroupoid, then Dec+(X) →Dec+(Y )×Y0 X0 is”
The Classifying Space of a Topological 2-Group (2008, background): “roup G. In their approach, G is regarded as a topological 2-groupoid with one object. There is a well-known nerve construction that turns any 2-groupoid (or even any 2-category) into a simplicial set [15]. Internalizing this construction, these authors turn the topological 2-groupoid G into a simpli”
The relationship between the diagonal and the bar constructions on a bisimplicial set (2005, methodology): “1], the classifying space BC of a 2-category C can also be realized by another simplicial set ∆C, the so-called geometric nerve of the 2-category [24,9].”
2-nerves for bicategories (2006, background): “The image of this nerve functor was characterized explicitly in [4].”