Clemens Berger, A Cellular Nerve for Higher Categories (Rev #11, changes)

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This entry draws from

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

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

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

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.)

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

Let $\mathrm{fT}:{S}_{*}\to {T}_{*}$~~ f:S_*\to~~ T~~ T_*~~ be a~~ monomorphism.~~ tree.

~~$f$~~A ~~ is called to be ~~*$T$-sector of height $k$* is defined to be a cospan*cartesian*~~ if~~

$$\begin{array}{ccc}({y}^{{\textstyle \prime}}{S}_{*}{)}_{n}& \stackrel{{f}_{n}}{\to}& ({y}^{{\textstyle \prime}{\textstyle \prime}}{T}_{*}{)}_{n}\\ {\downarrow}^{s}\searrow & & {\downarrow}^{t}\swarrow \\ ({S}_{*}{)}_{n-1}& \stackrel{{f}_{n-1}}{\to}y& ({T}_{*}{)}_{n-1}\end{array}$$ \array{~~ (S_*)_n&\stackrel{f_n}{\to}&(T_*)_n~~ y^\prime&&y^{\prime\prime} \\~~ \downarrow^s&&\downarrow^t~~ \searrow&&\swarrow \\~~ (S_*)_{n-1}&\stackrel{f_{n-1}}{\to}&(T_*)_{n-1}~~ &y }

~~ is~~ denoted~~ a~~ by~~ pullback~~~~ for~~~~ all~~$n(y;{y}^{{\textstyle \prime}},{y}^{{\textstyle \prime}{\textstyle \prime}})$~~ n~~ (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 $\mathrm{Sf},:T{S}_{*}\to {T}_{*}$~~ S,T~~ f:S_*\to T_* be~~ level~~ a~~ trees.~~ monomorphism.

~~(1) Any map ~~$f$~~$S_*\to T_*$~~ is called to be ~~ is injective.~~*cartesian* if

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

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

~~ (3)~~ is~~ The~~ a~~ inclusions~~ pullback for all${S}_{*}n\hookrightarrow {T}_{*}$~~ S_*\hookrightarrow~~ n~~ T_*~~~~ ~~ .~~ correspond~~~~ bijectively~~~~ to~~~~ plain~~~~ subtrees~~~~ of~~~~$T$~~~~ with a specific choice of ~~~~$T$~~~~-sector for each input vertex of ~~~~$S$~~~~. (…)~~

~~ (1)~~ Let~~ The~~~~ category~~${\Theta}_{0}S,T$~~ \Theta_0~~ S,T ~~ defined~~ be~~ by~~~~ having~~~~ as~~~~ objects~~~~ the~~ level~~ trees~~ 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)~~ (1)~~ A~~ Any map$S_*\to T_*$ is*globular theory*~~ defined~~ injective.~~ to~~~~ be~~~~ a~~~~ category~~~~$\Theta_A$~~~~ such that~~

$\Theta_0\hookrightarrow \Theta_A$

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

~~ is~~ (3)~~ an~~ The~~ inclusion~~ inclusions~~ of~~~~ a~~~~ wide~~~~ subcategory~~~~ such~~~~ that~~~~ representable~~~~ presheaves~~~~ on~~${\mathrm{\Theta S}}_{A}\hookrightarrow {T}_{*}$~~ \Theta_A~~ S_*\hookrightarrow T_* ~~ restrict~~ correspond bijectively to~~ sheaves~~ plain~~ on~~ subtrees of${\Theta}_{0}T$~~ \Theta_0~~ T~~ .~~ with a specific choice of$T$-sector for each input vertex of $S$. (…)

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

(1) The~~ forgetful~~ category~~ functor~~$\Theta_0$ 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*.

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

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

is an equivalence of categories.

$\Theta_0\hookrightarrow \Theta_A$

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.

~~ Let~~ The forgetful functor~~$X\in Psh(\Theta_0)$~~~~ 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.~~

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

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.