# nLab Theta category

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

For $n \in \mathbb{N}$ the category $\Theta_n$Joyal’s disk category or cell category – may be thought of as the full subcategory of the category $Str n Cat$ of strict n-categories on those $n$-categories that are free on pasting diagrams of $n$-globes.

For instance $\Theta_2$ contains an object that is depicted as

$\array{ & \nearrow &\Downarrow& \searrow && && \nearrow && \searrow \\ a && \to && b &\to & c && \Downarrow && b \\ & \searrow &\Downarrow& \nearrow && && \searrow && \nearrow && } \,,$

being the pasting diagram of two 2-globes along a common 1-globe and of the result with a 1-globe and another 2-globe along common 0-globes.

Such pasting diagrams may be alternatively encoded in planar trees, the above one corresponds to the tree:

$\array{ \nwarrow \nearrow & & & \uparrow &&& 2 \\ & \nwarrow & \uparrow & \nearrow &&& 1 \\ && {*} } \,.$

Accordingly, $\Theta_n$ is also the category of planar rooted trees of level $\leq n$.

In low degree we have

• $\Theta_0 = *$ is the point.

• $\Theta_1 = \Delta$ is the simplex category: the $n$-simplex $[n]$ is thought of as a linear quiver and as such the pasting diagram of $n$ 1-morphisms

$0 \to 1 \to \cdots \to n \,.$

Dually, this is the planar rooted tree of the form

$\array{ \nwarrow &\uparrow & \cdots \nearrow \\ &{*} }$

with $n$-branches.

## Definition

We discuss two equivalent definitions

### Via the free strict ω-category

Let $T(1)$ denote the free strict ∞-category generated from the terminal globular set $1$.

Notice that this terminal globular set consists of precisely one $k$-globe for each $k \in \mathbb{N}$: one point, one edge from the point to itself, one disk from the edge to itself, and so on.

So $T(1)$ is freely generated under composition from these cells. As described above, this means that each element of the underlying globular set of $T(1)$ may be depicted by a pasting diagram made out of globes, and such a pasting diagram itself may be considered as a globular set whose $k$-cells are instances of the $k$-globes appearing in the diagram.

We now describe this formally.

The n-cells of $T(1)$ may be identified with planar trees $\tau$ of height $n$, which by definition are functors

$\tau: [n]^{op} \to \Delta$

($\Delta$ is the category of simplices and $[n] \in \Delta$ is a simplex, i.e., ordered set $\{0 \lt 1 \lt \ldots \lt n\}$, regarded as a category) such that $\tau(0) = 1$. Such a $\tau$ is exhibited as a chain of morphisms in $\Delta$,

$\tau(n) \to \tau(n-1) \to \ldots \to \tau(0) = 1,$

and we will denote each of the maps in the chain by $i$. Thus, for each $x \in \tau(k)$, there is a fiber $i^{-1}(x)$ which is a linearly ordered set. (Need to fill in how $\circ_j$ composition of such trees is defined.)

To each planar tree $\tau$ we associate an underlying globular set $[\tau]$, as follows. Given $\tau$, define a new tree $\tau'$ where we adjoin a new bottom and top $x_0$, $x_1$ to every fiber $i^{-1}(x)$ of $\tau$, for every $x \in \tau(k)$:

$i_{\tau'}^{-1}(x) = \{x_0\} \cup i_{\tau}^{-1}(x) \cup \{x_1\}$

Now define a $\tau$-sector to be a triple $(x, y, z)$ where $i(y) = x = i(z)$ and $y, z$ are consecutive edges of $i_{\tau'}^{-1}(x)$. A $k$-cell of the globular set $[\tau]$ is a $\tau$-sector $(x, y, z)$ where $x \in \tau(k)$. If $k \geq 1$, the source of a $k$-cell $(x, y, z)$ is the $(k-1)$-cell $(i(x), u, x)$ and the target is the $(k-1)$-cell $(i(x), x, v)$ where $u \lt x \lt v$ are consecutive elements in $i_{\tau'}^{-1}(i(x))$. It is trivial to check that the globular axioms are satisfied.

Now let $T([\tau])$ denote the free strict $\omega$-category generated by the globular set $[\tau]$.

###### Definition

$\Theta$ is the full subcategory of $Str \omega Cat$ on the strict ∞-categories $T([\tau])$, as $\tau$ ranges over cells in the underlying globular set of $T(1)$.

### Via iterated wreath product

###### Proposition/Definition

$\Theta_n$ is the $n$-fold categorical wreath product of the simplex category with itself

$\Theta_n \simeq \Delta^{\wr n} \,.$
###### Examples

So

$\Theta_1 = \Delta$
$\Theta_2 = \Delta \wr \Delta$

etc.

###### Corollary

For all $n \in \mathbb{N}$ there is a canonical embedding

$\sigma : \Theta_n \hookrightarrow \Theta_{n+1}$

given by $\sigma : a \mapsto (, a)$.

### Via duals of disks

In analogy to how the simplex category is equivalent to the opposite category of finite strict linear intervals, $\Delta \simeq \mathbb{I}^{op}$, so the $\Theta$-category is equivalent to the opposite of the category of Joyal’s combinatorial finite disks.

$\Theta \coloneqq \mathbb{D}^{op} \,.$

(…)

## Properties

### Embedding of grids (products of the simplex category)

###### Definition

For any small category $A$ there is a canonical functor

$\delta_A : \Delta \times A \to \Delta \wr A$

given by

$\delta_A([n], a) = ([n], a^n) \,.$
###### Remark

By iteration, this induces a canonical functor

$\delta_n : \Delta^{\times n} \to \Theta_n \,.$

### Embedding into strict $n$-categories

Write $Str n Cat$ for the category of strict n-categories.

###### Proposition

There is a dense full embedding

$\Theta_n \hookrightarrow Str n Cat \,.$

This was conjectured in (Batanin-Street) and shown in terms of free $n$-categories on $n$-graphs in (Makkai-Zawadowsky, theorem 5.10) and (Berger 02, prop. 2.2). In terms of the wreath product presentation, prop. this is (Berger 05, theorem 3.7).

###### Proposition

Under this embedding an object $([k], (a_1, \cdots, a_k)) \in \Delta \wr \Delta^{\wr (n-1)}$ is identified with the $k$-fold horizontal composition of the pasting composition of the $(n-1)$-morphisms $a_i$:

$([k], (a_1, \cdots, a_k)) = a_1 \cdot a_2 \cdot \cdots \cdot a_k \,.$
###### Example
$\array{ & \nearrow &\Downarrow& \searrow && && \nearrow && \searrow \\ a && \to && b &\to & c && \Downarrow && b \\ & \searrow &\Downarrow& \nearrow && && \searrow && \nearrow && }$

corresponds to the objects of $\Theta_2 = \Delta \wr \Delta$ given by

$(, (a_1, a_2, a_3)) \,,$

where in turn

• $a_1 = $

• $a_2 = $

• $a_3 = $.

###### Example

Composing with the functor $\delta_n$ from remark we obtain an embedding of $n$-fold simplices into strict $n$-categories

$\Delta^{\times n} \stackrel{\delta_n}{\to} \Theta_n \hookrightarrow Str n Cat \,.$

Under this embedding an object $([k_1], [k_2], \cdots, [k_n])$ is sent to the $n$-category which looks like (a globular version of) a $k_1 \times k_2 \times \cdots \times k_n$ grid of $n$-cells.

Write

$Str n Cat_{gaunt} \hookrightarrow Str n Cat$

for the inclusion of the gaunt strict $n$-categories into all strict n-categories.

###### Proposition

$\Theta_n$ is the smallest full subcategory of $Str n Cat_{gaunt}$ containing the grids, the image of $\delta_n : \Delta^{\times n} \to Str n Cat$, example , and closed under retracts.

### Groupoidal version

The groupoidal version $\tilde \Theta$ of $\Theta$ is a test category (Ara).

## Examples

In $\Theta_0$ write $O_0$ for the unique object. Then write in $\Theta_n$

$O_n := (O_{n-1}) \,.$

This is the strict n-category free on a single $n$-globe.

The $\Theta$-categories were introduced in

A discussion with lots of pictures is in chapter 7 of

More discussion is in

• David Oury, On the duality between trees and disks, TAC vol. 24 (pdf)

The following paper proves that $\Theta$ is a test category

Discussion of embedding of $\Theta$ into strict $n$-categories is in

• Michael Makkai, M. Zawadowsky, Duality for simple $\omega$-categories and disks, Theory Appl. Categories 8 (2001), 114–243
• Clemens Berger, A cellular nerve for higher categories, Adv. Math. 169 (2002), 118–175.

The characterization in terms of $n$-fold categorical wreath products is in

The groupoidal version $\tilde \Theta$ is discussed in
• Dimitri Ara, The groupoidal analogue $\tilde \Theta$ to Joyal’s category $\Theta$ is a test category (arXiv:1012.4319)
The relation of $\Theta_n$ to configuration spaces of points in the Euclidean space $\mathbb{R}^n$ is discussed in