nLab geometric nerve of a tricategory

Idea

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

Also called the Street nerve of $\mathcal{C}$, 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 $\mathcal{C}$ may be thought of as the simplicial set $\mathrm{N}^{\mathsf{S}}_{\bullet}(\mathcal{C})$ whose $n$-simplices are lax functors from the locally discrete tricategory associated to $[n]$ to $\mathcal{C}$ 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 $\mathcal{C}$ associates a “pseudosimplicial bicategory” $\mathrm{N}^\mathsf{G}_\bullet(\mathcal{C})\colon\Delta^\op\longrightarrow\mathsf{Bicats}$; meaning a pseudofunctor from the locally discrete tricategory associated to $\Delta^\op$ to the tricategory $\mathsf{Bicats}$. Its bicategory of $n$-simplices has as objects $n$-tuples of composable morphisms of $\mathcal{C}$.

One also has the Segal nerve $\mathrm{N}^\mathsf{Segal}_\bullet(\mathcal{C})\colon\Delta^\op\longrightarrow\mathsf{Bicats}$ of $\mathcal{C}$, which is a simplicial bicategory, and a kind of “rectification” of $\mathrm{N}^\mathsf{G}_\bullet(\mathcal{C})$. When $\mathcal{C}$ is the locally discrete tricategory associated to a bicategory $\mathcal{B}$, the Segal nerve of $\mathcal{C}$ agrees with the $2$-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

• The Street nerve identifies precisely nerves of trigroupoid?s with $3$-hypergroupoids: those Kan complexes for which the horn fillers in dimension $\geq 4$ are unique; see (Carrasco, 2014).

Picturing the Street nerve

Let ($\mathcal{C}$,$\mathsf{Hom}_{\mathcal{C}}(-,-)$,$\otimes$,$1^\mathcal{C}$,$\alpha$,$\alpha^{\bullet}$,$\phi$,$\phi^{\bullet}$,$\lambda$,$\lambda^{\bullet}$,$\eta$,$\eta^{\bullet}$,$\rho$,$\rho^{\bullet}$,$\epsilon$,$\epsilon^{\bullet}$,$\mathbf{\pi}$,$\mathbf{\lambda}$,$\mathbf{\mu}$,$\mathbf{\rho}$) be a tricategory, where

• ($\alpha$,$\alpha^{\bullet}$,$\phi$,$\phi^{\bullet}$) is the associator adjoint equivalence of $\mathcal{C}$,
• ($\lambda$,$\lambda^{\bullet}$,$\eta$,$\eta^{\bullet}$) is the left unitor adjoint equivalence of $\mathcal{C}$,
• ($\rho$,$\rho^{\bullet}$,$\epsilon$,$\epsilon^{\bullet}$) is the right unitor adjoint equivalence of $\mathcal{C}$, and
• $\mathbf{\pi}$,$\mathbf{\lambda}$,$\mathbf{\mu}$,$\mathbf{\rho}$ are the pentagonator, left, middle, and right $2$-unitors of $\mathcal{C}$.

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

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

1. The $0$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are the objects of $\mathcal{C}$;

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

3. The $2$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are quadruples $(i,j,k,\theta)$ as in the diagram

   

   where $A,B,C\in\mathrm{Obj}(\mathcal{C})$, $i,j,k\in\mathrm{Mor}_1(\mathcal{C})$ and $\theta\colon j\circ i\Rightarrow k$ is a $2$-morphism of $\mathcal{C}$;

4. The $3$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are $15$-tuples

   $$(A_{0},A_{1},A_{2},A_{3},f_{01},f_{02},f_{03},f_{12},f_{13},f_{23},\theta_{012},\theta_{013},\theta_{023},\theta_{123},\Gamma_{0123})$$

   as in the diagram

5. The $4$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$ are $28$-tuples

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

   with objects, $1$-morphisms, and $2$-morphisms as in the diagram and $3$-morphisms as in the diagrams such that the diagram corresponding to Street‘s fourth oriental, commutes. (See [here] for a zoomable PDF).

6. The $n$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$, similarly to the $4$-simplices of $\mathrm{N}^\mathsf{S}_{\bullet}(\mathcal{C})$, consist of

   • A collection $\{A_{i}\}_{0\leq i\leq n}$ of objects of $\mathcal{C}$,
   • A collection $\{f_{ij}\colon A_{i}\longrightarrow A_{j}\}_{0\leq i\lt j\leq n}$ of $1$-morphisms of $\mathcal{C}$,
   • A collection $\{\theta_{ijk}\colon f_{jk}\circ f_{ij}\Rightarrow f_{ik}\}_{0\leq i\lt j\lt k\leq n}$ of $2$-morphisms of $\mathcal{C}$, and
   • A collection $\{\Gamma_{ijkl}\}_{0\leq i\lt j\lt k\lt l\leq n}$ of $3$-morphisms of $\mathcal{C}$

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

7. The degeneracy map

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

   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $0$ is the map sending a $0$-simplex $A$ of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ (i.e. an object $A$ of $\mathcal{C}$) to the $1$-simplex $\mathrm{id}_{A}\colon A\to A$.

8. The degeneracy maps

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

   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $1$ are the maps described as follows: given a $1$-simplex $\sigma=(A\xrightarrow{f}B)$ of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$, we have

9. The degeneracy maps in degree $2$

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

   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $2$ are the maps described as follows: given a $2$-simplex of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$, we have (page author’s note: please take the following $3$-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 $2$-unitors of $\mathcal{C}$ here—this is where they appear in the Street nerve!) For the details regarding these pastings, see [this PDF].

10. The face maps

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

    of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $1$ are given by

    $$\mathrm{d}^{1}_{0}(A\xrightarrow{f}B)=B$$
    $$\mathrm{d}^{1}_{1}(A\xrightarrow{f}B)=A$$

11. The face maps

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

    of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $2$ are described as follows: given a $2$-simplex

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

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

   of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ in degree $3$ are described as follows: given a $3$-simplex $\sigma$ of $\mathrm{N}^\mathsf{S}_\bullet(\mathcal{C})$ as in the diagram

we have

Related concepts

• nerve

• geometric nerve of a bicategory

• geometric nerve of a tricategory

• ∞-nerve

• homotopy coherent nerve

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 $3$-Hypergroupoids, Applied Categorical Structures 23.5 (2015): 673–707.

• Stephen Lack and Simona Paoli, 2-nerves for bicategories, Journal of $K$-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. pdf

• 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.