# Spahn Clemens Berger, A Cellular Nerve for Higher Categories (Rev #5, changes)

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## 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.

###### 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 $i_T^{-1}$.

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 $s,t:T_*\to G\overline T_*$. 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).

This $*$-construction is due to Batanin.

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 $n$ is $n_G$, for $n\in \mathbb{N}$ let $\mathbf{n}$ denotes the linear $n$-level tree.

Then we have $\mathbf{n}_*\simeq G(-,n_G)$ is the standard $n$-globe.

Revision on November 18, 2012 at 19:47:08 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.