# nLab geometric nerve of a tricategory

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## 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

## 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.

• 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.