Spahn Clemens Berger, A Cellular Nerve for Higher Categories

This entry draws from

• Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)

0. Notation and terminology

0.1

0.2 Higher graphs and higher categories

Definition

The globe category $G$ is defined to be the category with one object in each degree and the globular operators $s,t$ are defined by the identities

$$s\circ s=s\circ t$$

$$t\circ t= t\circ s$$

A presheaf on $G$ is called a globular set or omega graph or $\omega$-graph.

$\omega$-graphs with natural transformations as morphisms form a category denoted by $\omegaGraph$.

Definition (Godement´s interchange rules)

Let $C$ be $2$-category with underlying reflexive $2$-graph $C_i$ for $i=0,1,2$ with globular operators given by source, target, and identity.

Then $(C_i)_i$ comes with three composition laws

$$\circ_i^j: C_j\times_i C_j\to C_j$$

for $0\le i\lt j\le 2$. Spelled out this means:

$i=0, j=1$: composition of $1$-morphism along $0$-morphisms (i.e.objects)

$i=0,j=2$: composition of $2$-morphisms along $0$-morphisms (i.e.objects), also called horizontal composition.

$i=1,j=2$: composition of $2$-morphisms along $1$-morphisms, also called vertical composition.

Then Godement´s interchange rule or Godement´s interchange law or just interchange law is the assertion that the immediate diagrams commute.

Note that there is on more type of composition of a $1$-morphism with a $2$-morphism called whiskering.

Definition ($\omega$-category)

An $\omega$-category is defined to be a reflexive graph $(C_i)_i$ such that for every triple $i\lt j\lt k$, the family $(C_i,C_j,C_k;\circ_i^j,\circ_i^k, \circ_j^)$ has the structure of a $2$-category.

1. Globular theories and cellular nerves

Contents:

Batanin’s $\omega$-operads are described by their operator categories which are called globular theories.

Definition (finite planar level tree)

A finite planar level tree ( or for short just a tree) is a graded set $(T(n))_{n\in \mathbb{N}_0}$ endowed with a map $i_T: T_{\gt 0}$ decreasing the degree by one and such that all fibers $i_T^{-1}(x)$ are linearly ordered.

The collection of trees with maps of graded sets commuting with $i$ defines a category $\mathcal{T}$, called the category of trees.

Example

The finite ordinal $[n]\in \Delta$ we can regard as the 1-level tree with $n$ input edges. Hence the simplex category embeds in the tree category $\Delta\hookrightarrow\mathcal{T}$.

The following ${}_*$-construction is due to Batanin.

Lemma and Definition ($\omega$-graph of sectors of a tree)

Let $T$ be a tree.

A $T$-sector of height $k$ is defined to be a cospan

$$\array{ y^\prime&&y^{\prime\prime} \\ \searrow&&\swarrow \\ &y }$$

denoted by $(y;y^\prime,y^{\prime\prime})$ where $y\in T(k)$ and $y\lt y^{\prime\prime}$ are consecutive vertices in the linear order $i_T^{-1}(y)$.

The set $GT$ of $T$-sector is graded by the height of sectors.

The source of a sector $(y;y^\prime,y^{\prime\prime})$ is defined to be $(i(y);x,y)$ where $x,y$ are consecutive vertices.

The target of a sector $(y;y^\prime,y^{\prime\prime})$ is defined to be $(i(y);y,z)$ where $y,z$ are consecutive vertices.

$$\array{ &y^\prime&&y^{\prime\prime} \\ & \searrow&&\swarrow \\ x&&y&&z \\ \searrow&&\downarrow^i&&\swarrow \\ &&i(y) }$$

To have a source and a target for every sector of $T$ we adjoin in every but the highest degree a lest- and a greatest vertex serving as new minimum and maximum for the linear orders $i^{-1}(x)$. We denote this new tree by $\overline T$ and the set of its sectors by $T_*:=G\overline T(k)$ and obtain source- and target operators $s,t:T_*\to T_*$. This operators satisfy

$$s\circ s=s\circ t$$

$$t\circ t =t\circ s$$

as one sees in the following diagram depicting an “augmented” tree of height $3$

$$\array{ T(3)&&&y^\prime&&y^{\prime\prime} \\ &&& \searrow&&\swarrow \\ T(2)&&x&&y&&z \\ &&\searrow&&\downarrow^i&&\swarrow \\ T(1)&&u&&v&&w \\ &&\searrow&&\downarrow^i&&\swarrow \\ T(0)&&&&r }$$

which means that $T_*$ is an $\omega$-graph (also called globular set).

Now let $G$ denote the globe category whose unique object in degree $n\in \mathbb{N}$ is $n_G$, and let $\mathbf{n}$ denotes the linear $n$-level tree.

Then we have $\mathbf{n}_*\simeq G(-,n_G)$ is the standard $n$-globe. (Note that the previous diagram corresponds to the standard $3$ globe.)

Definition

Let $f:S_*\to T_*$ be a monomorphism.

$f$ is called to be cartesian if

$$\array{ (S_*)_n&\stackrel{f_n}{\to}&(T_*)_n \\ \downarrow^s&&\downarrow^t \\ (S_*)_{n-1}&\stackrel{f_{n-1}}{\to}&(T_*)_{n-1} }$$

is a pullback for all $n$.

Lemma

Let $S,T$ be level trees.

(1) Any map $S_*\to T_*$ is injective.

(2) The inclusions $S_*\hookrightarrow T_*$ correspond bijectively to cartesian subobjects of $T_*$.

(3) The inclusions $S_*\hookrightarrow T_*$ correspond bijectively to plain subtrees of $T$ with a specific choice of $T$-sector for each input vertex of $S$. (…)

Definition

(1) The category $\Theta_0$ defined by having as objects the level trees and as morphisms the maps between the associated $\omega$-graphs. These morphisms are called immersions. This category shall be equipped with the structure of a site by defining the covering sieves by epimorphic families (of immersions). This site is called the globular site.

(2) A globular theory is defined to be a category $\Theta_A$ such that

$$\Theta_0\hookrightarrow \Theta_A$$

is an inclusion of a wide subcategory such that representable presheaves on $\Theta_A$ restrict to sheaves on $\Theta_0$.

(3) Presheaves on $\Theta_A$ which restrict to sheaves on $\Theta_0$ are called $\Theta_A$-models.

Lemma

The forgetful functor

$$Sh (\Theta_0)\to \omega Graph:=Psh (G)$$

is an equivalence of categories.

Proof

Let $X\in Psh(\Theta_0)$ and show that $X\in Sh(\Theta_0)$ iff $X(T)\simeq hom_{Psh(G)}(T_*,X)$ by writing $X$ as a colimit of representables.

Definition

There is a monad $(w,\eta,\mu)$ on $\omega Graph$ defined by

$$w(X)_n:=\coprod_{T:ht(T)\le n}hom_{\omega Graph}(T_*,X)$$

$\eta:id_{Psh(G)}\to w$ is induced by Yoneda: $X_n\mapsto hom_{\omega Graph}(n_*,X)$

2. Cellular sets and their geometric realization

3. A closed model structure for cellular sets

4. Homotopy structure for weak $\omega$-categories