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geometric nerve of a tricategory

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Contents

Idea

The geometric nerve is a natural nerve operation on tricategories, associating to every tricategory 𝒞\mathcal{C} a simplicial set N S(𝒞)\mathrm{N}^{\mathsf{S}}_{\bullet}(\mathcal{C}).

Also called the Street nerve of 𝒞\mathcal{C}, this notion, like the Duskin nerve of a bicategory, is implicit in (Ross Street‘s work on orientals). While this construction was announced in (Duskin, 2002) as to appear in a (then) forthcoming paper, the latter never appeared. Instead, the notion was developed by Cegarra–Heredia in (Cegarra–Heredia, 2012).

Roughly, the Street nerve of 𝒞\mathcal{C} may be thought of as the simplicial set N S(𝒞)\mathrm{N}^{\mathsf{S}}_{\bullet}(\mathcal{C}) whose nn-simplices are lax functors from the locally discrete tricategory associated to [n][n] to 𝒞\mathcal{C} satisfying a variety of unitality conditions. In detail, however, the definition is more involved, as one faces problems with the definition of lax functors between tricategories found in (Gordon–Powers–Street); see (Definition 5.1.1 of Cegarra–Heredia).

There are other nerve constructions for tricategories besides the Street nerve. One of them is the Grothendieck nerve, which to every tricategory 𝒞\mathcal{C} associates a “pseudosimplicial bicategoryN G(𝒞):Δ opBicats\mathrm{N}^\mathsf{G}_\bullet(\mathcal{C})\colon\Delta^\op\longrightarrow\mathsf{Bicats}; meaning a pseudofunctor from the locally discrete tricategory associated to Δ op\Delta^\op to the tricategory Bicats\mathsf{Bicats}. Its bicategory of nn-simplices has as objects nn-tuples of composable morphisms of 𝒞\mathcal{C}.

One also has the Segal nerve N Segal(𝒞):Δ opBicats\mathrm{N}^\mathsf{Segal}_\bullet(\mathcal{C})\colon\Delta^\op\longrightarrow\mathsf{Bicats} of 𝒞\mathcal{C}, which is a simplicial bicategory, and a kind of “rectification” of N G(𝒞)\mathrm{N}^\mathsf{G}_\bullet(\mathcal{C}). When 𝒞\mathcal{C} is the locally discrete tricategory associated to a bicategory \mathcal{B}, the Segal nerve of 𝒞\mathcal{C} agrees with the 22-nerve of \mathcal{B} introduced in (Lack–Paoli, 2006).

All of these nerve constructions are equivalent in the sense that their classifying spaces are homotopy equivalent to each other. For more details, see (Cegarra–Heredia, 2012).

Properties

Picturing the Street nerve

Let (𝒞\mathcal{C},Hom 𝒞(,)\mathsf{Hom}_{\mathcal{C}}(-,-),\otimes,1 𝒞1^\mathcal{C},α\alpha,α \alpha^{\bullet},ϕ\phi,ϕ \phi^{\bullet},λ\lambda,λ \lambda^{\bullet},η\eta,η \eta^{\bullet},ρ\rho,ρ \rho^{\bullet},ϵ\epsilon,ϵ \epsilon^{\bullet},π\mathbf{\pi},λ\mathbf{\lambda},μ\mathbf{\mu},ρ\mathbf{\rho}) be a tricategory, where

  • (α\alpha,α \alpha^{\bullet},ϕ\phi,ϕ \phi^{\bullet}) is the associator adjoint equivalence of 𝒞\mathcal{C},
  • (λ\lambda,λ \lambda^{\bullet},η\eta,η \eta^{\bullet}) is the left unitor adjoint equivalence of 𝒞\mathcal{C},
  • (ρ\rho,ρ \rho^{\bullet},ϵ\epsilon,ϵ \epsilon^{\bullet}) is the right unitor adjoint equivalence of 𝒞\mathcal{C}, and
  • π\mathbf{\pi},λ\mathbf{\lambda},μ\mathbf{\mu},ρ\mathbf{\rho} are the pentagonator, left, middle, and right 22-unitors of 𝒞\mathcal{C}.

(Below we also write these with “𝒞^\mathcal{C}” superscripts, and write π\mathbf{\pi},λ\mathbf{\lambda},μ\mathbf{\mu},ρ\mathbf{\rho} in blackboard bold font.)

The Street nerve of the tricategory 𝒞\mathcal{C} is then the simplicial set N S(𝒞)\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C}) where

  1. The 00-simplices of N S(𝒞)\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C}) are the objects of 𝒞\mathcal{C};

  2. The 11-simplices of N S(𝒞)\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C}) are the 11-morphisms of 𝒞\mathcal{C};

  3. The 22-simplices of N S(𝒞)\mathrm{N}^\mathsf{S}_{\bullet}(\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 S(𝒞)\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C}) are 1515-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,Γ 0123)(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},\Gamma_{0123})

    as in the diagram

  5. The 44-simplices of N S(𝒞)\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C}) are 2828-tuples

    • (A 0A_{0},A 1A_{1},A 2A_{2},A 3A_{3},f 01f_{01},f 02f_{02},f 03f_{03},f 04f_{04},f 12f_{12},f 13f_{13},f 14f_{14},f 23f_{23},f 24f_{24},f 34f_{34},θ 012\theta_{012},θ 013\theta_{013},θ 014\theta_{014},θ 023\theta_{023}, θ 024\theta_{024},θ 123\theta_{123},θ 124\theta_{124},θ 134\theta_{134},θ 234\theta_{234},Γ 0123\Gamma_{0123},Γ 0124\Gamma_{0124},Γ 0134\Gamma_{0134},Γ 0234\Gamma_{0234},Γ 1234\Gamma_{1234})

    with objects, 11-morphisms, and 22-morphisms as in the diagram and 33-morphisms as in the diagrams such that the diagram corresponding to Street‘s fourth oriental, commutes. (See [here] for a zoomable PDF).

  6. The nn-simplices of N S(𝒞)\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C}), similarly to the 44-simplices of N S(𝒞)\mathrm{N}^\mathsf{S}_{\bullet}(\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},
    • 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}, and
    • A collection {Γ ijkl} 0i<j<k<ln\{\Gamma_{ijkl}\}_{0\leq i\lt j\lt k\lt l\leq n} of 33-morphisms of 𝒞\mathcal{C}

    such that, for each i,j,k,lNi,j,k,l\in\N with 0i<j<k<ln0\leq i\lt j\lt k\lt l\leq n, the diagram corresponding to Street‘s fourth oriental above commutes.

  7. The degeneracy map

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

    of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C}) in degree 00 is the map sending a 00-simplex AA of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\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.

  8. The degeneracy maps

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

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

  9. The degeneracy maps in degree 22

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

    of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C}) in degree 22 are the maps described as follows: given a 22-simplex of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C}), we have (page author’s note: please take the following 33-morphisms with a grain of salt; I’m quite unsure about whether they are correct or not. In any case, note that we must use the left, middle, and right 22-unitors of 𝒞\mathcal{C} here—this is where they appear in the Street nerve!) For the details regarding these pastings, see [this PDF].

  10. The face maps

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

    of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\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
  11. The face maps

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

    of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C}) in degree 22 are described as follows: given a 22-simplex

of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C}), we have

  1. The face maps
    d 0 3:N 3 S(𝒞)N 2 S(𝒞), \mathrm{d}^{3}_{0} \colon \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
    d 1 3:N 3 S(𝒞)N 2 S(𝒞), \mathrm{d}^{3}_{1} \colon \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
    d 2 3:N 3 S(𝒞)N 2 S(𝒞), \mathrm{d}^{3}_{2} \colon \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),
    d 3 3:N 3 S(𝒞)N 2 S(𝒞), \mathrm{d}^{3}_{3} \colon \mathrm{N}^{\mathsf{S}}_{3}(\mathcal{C})\longrightarrow \mathrm{N}^{\mathsf{S}}_{2}(\mathcal{C}),

    of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C}) in degree 33 are described as follows: given a 33-simplex σ\sigma of N S(𝒞)\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C}) as in the diagram

we have

References

  • Antonio M. Cegarra and Benjamín A. Heredia, Geometric Realizations of Tricategories. Algebraic & Geometric Topology 14, no. 4 (2014): 1997-2064. [arXiv:1203.3664]

  • Pilar Carrasco, Nerves of Trigroupoids as Duskin-Glenn’s 33-Hypergroupoids, Applied Categorical Structures 23.5 (2015): 673–707.

  • Stephen Lack and Simona Paoli, 2-nerves for bicategories, Journal of KK-Theory 38.2 (2008): 153–175. [arXiv:0607271].

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

  • Robert Gordon?, John Power, Ross Street, Coherence for tricategories, Mem. Amer. Math Soc. 117 (1995) no 558.

  • 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.

Last revised on July 3, 2020 at 00:45:11. See the history of this page for a list of all contributions to it.