nLab
geometric nerve of a bicategory

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The (unitary) geometric nerve is a natural nerve operation on bicategories. It is a functor from BiCat to sSet. This is also sometimes called the Duskin nerve. The notion is implicit in work by R. Street (1987). The direct approach was used by Duskin in work at about the same time, as explained in both articles. (Duskin’s article directly on the idea was published in 2002.)

The construction, thus, yields a functor:

N:BiCat NLaxsSet. N : BiCat_{NLax} \to sSet \,.

extending the ordinary nerve construction on the category of small categories, where morphisms of BiCat are normal lax 2-functors: these are the lax 2-functors which strictly preserve identities.

Special cases of the construction relate to earlier constructions relating to the homotopy coherent nerve, see below for more detail.

Definition

We may think of the simplex category Δ\Delta as the full subcategory of Cat on the categories free on non-empty finite linear graphs. This gives the canonical inclusion ΔCat\Delta \hookrightarrow Cat that defines the ordinary nerve of categories.

There is also the canonical embedding of categories into bicategories. Combined this gives the inclusion

ΔCatBiCat. \Delta \hookrightarrow Cat \hookrightarrow BiCat \,.

The bicategorical nerve is the nerve induced from that. So for CC a bicategory we have

N(C):[k]BiCat NLax(Δ[k],C). N(C) : [k] \mapsto BiCat_{NLax}(\Delta[k], C) \,.

There are also an oplax version and two non-normalized versions.

Properties

(This shows in particular that bigroupoids model all homotopy 2-types.)

  • The nerve is a full and faithful functor BiCat NLaxsSetBiCat_{NLax}\to sSet.

Example

Any strict 2-category determines both a ‘bicategory’ in the above sense (since a ‘strict’ thing is also a ‘weak’ one) and a simplicially enriched category. The latter is found by taking the nerve of each ‘hom-category’. The Duskin nerve of a 2-category is the same as the homotopy coherent nerve of the corresponding sSetsSet-category. This can also be applied to 2-groupoids and, thus, results in a classifying space construction for crossed modules.

Picturing the Duskin nerve

Following (Johnson–Yau, Section 5.4), one may picture the Duskin nerve N(𝒞)N(\mathcal{C}) of a bicategory (𝒞,1 𝒞, 𝒞,α 𝒞,λ 𝒞,ρ 𝒞)(\mathcal{C},1^{\mathcal{C}},\circ_{\mathcal{C}},\alpha^{\mathcal{C}},\lambda^{\mathcal{C}},\rho^{\mathcal{C}}) as follows:

  1. The 00-simplices of N(𝒞)N(\mathcal{C}) are the objects of 𝒞\mathcal{C};

  2. The 11-simplices of N(𝒞)N(\mathcal{C}) are the 11-morphisms of 𝒞\mathcal{C};

  3. The 22-simplices of N(𝒞)N(\mathcal{C}) are quadruples (i,j,k,θ)(i,j,k,\theta) as in the diagram

    2-simplex of the Duskin nerve of a bicategory

    where A,B,CObj(𝒞)A,B,C\in\mathrm{Obj}(\mathcal{C}), i,j,kMor 1(𝒞)i,j,k\in\mathrm{Mor}_1(\mathcal{C}) and θ:jik\theta\colon j\circ i\Rightarrow k is a 22-morphism of 𝒞\mathcal{C};

  4. The 33-simplices of N(𝒞)N(\mathcal{C}) are 1414-tuples

    (A 0,A 1,A 2,A 3,f 01,f 02,f 03,f 12,f 13,f 23,θ 012,θ 013,θ 023,θ 123)(A_{0},A_{1},A_{2},A_{3},f_{01},f_{02},f_{03},f_{12},f_{13},f_{23},\theta_{012},\theta_{013},\theta_{023},\theta_{123})

    as in the diagram A 3-simplex of the Duskin nerve of a bicategory such that we have an equality The pasting diagram equality for the 3-simplices of the Duskin nerve of a bicategory of pasting diagrams in 𝒞\mathcal{C};

  5. The nn-simplices of N(𝒞)N(\mathcal{C}) consist of

    • A collection {A i} 0in\{A_{i}\}_{0\leq i\leq n} of objects of 𝒞\mathcal{C},
    • A collection {f ij:A iA j} 0i<jn\{f_{ij}\colon A_{i}\longrightarrow A_{j}\}_{0\leq i\lt j\leq n} of 11-morphisms of 𝒞\mathcal{C}, and
    • A collection {θ ijk:f jkf ijf ik} 0i<j<kn\{\theta_{ijk}\colon f_{jk}\circ f_{ij}\Rightarrow f_{ik}\}_{0\leq i\lt j\lt k\leq n} of 22-morphisms of 𝒞\mathcal{C}

    such that, for each i,j,ki,j,k\in\mathbb{N} with 0i<j<kn0\leq i\lt j\lt k\leq n, we have an equality of pasting diagrams in 𝒞\mathcal{C};

  6. The degeneracy map

    s 0 0:N 0(𝒞)N 1(𝒞)\mathrm{s}^{0}_{0}\colon N_{0}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C})

    of N(𝒞)N(\mathcal{C}) in degree 00 is the map sending a 00-simplex AA of N(𝒞)N(\mathcal{C}) (i.e. an object AA of 𝒞\mathcal{C}) to the 11-simplex id A:AA\mathrm{id}_{A}\colon A\to A.

  7. The degeneracy maps

    s 0 1:N 1(𝒞)N 2(𝒞), \mathrm{s}^{1}_{0} \colon N_{1}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
    s 1 1:N 1(𝒞)N 2(𝒞), \mathrm{s}^{1}_{1} \colon N_{1}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),

    of N(𝒞)N(\mathcal{C}) in degree 11 are the maps described as follows: given a 11-simplex σ=(AfB)\sigma=(A\xrightarrow{f}B) of N(𝒞)N(\mathcal{C}), we have

  8. The degeneracy maps in degree 22

    s 0 2:N 2(𝒞)N 3(𝒞), \mathrm{s}^{2}_{0} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
    s 1 2:N 2(𝒞)N 3(𝒞), \mathrm{s}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
    s 2 2:N 2(𝒞)N 3(𝒞), \mathrm{s}^{2}_{2} \colon N_{2}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),

    of N(𝒞)N(\mathcal{C}) in degree 22 are the maps described as follows: given a 22-simplex of N(𝒞)N(\mathcal{C}), we have

  9. The face maps

    d 0 1:N 1(𝒞)N 0(𝒞), \mathrm{d}^{1}_{0} \colon N_{1}(\mathcal{C})\longrightarrow N_{0}(\mathcal{C}),
    d 1 1:N 1(𝒞)N 0(𝒞), \mathrm{d}^{1}_{1} \colon N_{1}(\mathcal{C})\longrightarrow N_{0}(\mathcal{C}),

    of N(𝒞)N(\mathcal{C}) in degree 11 are given by

    d 0 1(AfB)=B\mathrm{d}^{1}_{0}(A\xrightarrow{f}B)=B
    d 1 1(AfB)=A\mathrm{d}^{1}_{1}(A\xrightarrow{f}B)=A
  10. The face maps

    d 0 2:N 2(𝒞)N 1(𝒞), \mathrm{d}^{2}_{0} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),
    d 1 2:N 2(𝒞)N 1(𝒞), \mathrm{d}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),
    d 1 2:N 2(𝒞)N 1(𝒞), \mathrm{d}^{2}_{1} \colon N_{2}(\mathcal{C})\longrightarrow N_{1}(\mathcal{C}),

    of N(𝒞)N(\mathcal{C}) in degree 22 are described as follows: given a 22-simplex

of N(𝒞)N(\mathcal{C}), we have 11. The face maps

d 0 3:N 3(𝒞)N 2(𝒞), \mathrm{d}^{3}_{0} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
d 1 3:N 3(𝒞)N 2(𝒞), \mathrm{d}^{3}_{1} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
d 2 3:N 3(𝒞)N 2(𝒞), \mathrm{d}^{3}_{2} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),
d 3 3:N 3(𝒞)N 2(𝒞), \mathrm{d}^{3}_{3} \colon N_{3}(\mathcal{C})\longrightarrow N_{2}(\mathcal{C}),

of N(𝒞)N(\mathcal{C}) in degree 33 are described as follows: given a 33-simplex of N(𝒞)N(\mathcal{C}), we have 12. The face maps

d 0 4:N 4(𝒞)N 3(𝒞), \mathrm{d}^{4}_{0} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
d 1 4:N 4(𝒞)N 3(𝒞), \mathrm{d}^{4}_{1} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
d 2 4:N 4(𝒞)N 3(𝒞), \mathrm{d}^{4}_{2} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
d 3 4:N 4(𝒞)N 3(𝒞), \mathrm{d}^{4}_{3} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),
d 4 4:N 4(𝒞)N 3(𝒞), \mathrm{d}^{4}_{4} \colon N_{4}(\mathcal{C})\longrightarrow N_{3}(\mathcal{C}),

of N(𝒞)N(\mathcal{C}) in degree 44 are described as follows: given a 44-simplex σ\sigma of N(𝒞)N(\mathcal{C}) as in the diagram we have

References

  • Ross Street, The algebra of oriented simplexes, Journal of Pure and Applied Algebra, Volume 49, Issue 3, December 1987, Pages 283–335

  • John Duskin, Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories (tac), Theory and Applications of Categories, Vol. 9, No. 10, 2002, pp. 198–308.

  • V. Blanco, M. Bullejos, E. Faro, A Full and faithful Nerve for 2-categories, Applied Categorical Structures, Vol 13-3, 223-233, 2005. (See also arxiv).

  • Niles Johnson, Donald Yau, 2-Dimensional Categories (arXiv:2002.06055).

Last revised on July 3, 2020 at 00:20:54. See the history of this page for a list of all contributions to it.