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)

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 $\omegaGraph$.

Let $C$ be $2$-category with underlying reflexive $2$-graph `{(C_i)_{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*.

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.

Contents:

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

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*.

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.

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.)

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 $n$.

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 $S$. (…)

(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.

The forgetful functor

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

is an equivalence of categories.

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.

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: $X_n\mapsto hom_{\omega Graph}(n_*,X)$