nLab
Theta category

Idea

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

For instance ${\Theta }_2$ contains an object that is depicted

$$\begin{array}{cccccccccc}& \nearrow & \Downarrow & \searrow & & & & \nearrow & & \searrow \\ a& & \to & & b& \to & c& & \Downarrow & & b\\ & \searrow & \Downarrow & \nearrow & & & & \searrow & & \nearrow & & \end{array}$$\array{ & \nearrow &\Downarrow& \searrow && && \nearrow && \searrow \\ a && \to && b &\to & c && \Downarrow && b \\ & \searrow &\Downarrow& \nearrow && && \searrow && \nearrow && }

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

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

Accordingly, ${\Theta }_n$ is the category of planar rooted trees of level $\le n$.

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

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

  Dually, this is the planar rooted tree of the form

  $$\begin{array}{ccc}\nwarrow & \uparrow & \cdots \nearrow \\ & *\end{array}$$\array{ \nwarrow &\uparrow & \cdots \nearrow \\ &{*} }

  with $n$-branches.

Definition

Let $T(1)$ denote the free strict ω-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 from 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

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

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

$$\tau (n)\to \tau (n-1)\to \dots \to \tau (0)=1,$$\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 $\tau \prime$ where we adjoin a new bottom and top $x_0$, $x_1$ to every fiber $i^{-1}(x)$ of $\tau$, for every $x\in \tau (k)$:

$$i_{\tau \prime }^{-1}(x)=\{x_0\}\cup i_{\tau }^{-1}(x)\cup \{x_1\}$$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 $i_{\tau \prime }^{-1}(x)$. A $k$-cell of the globular set $[\tau ]$ is a $\tau$-sector $(x,y,z)$ where $x\in \tau (k)$. If $k\ge 1$, 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 $u<x<v$ are consecutive elements in $i_{\tau \prime }^{-1}(i(x))$. 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 $\omega$-Cat determined by $\omega$-categories $T([\tau ])$, as $\tau$ ranges over cells in the underlying globular set of $T(1)$.

Examples

In ${\Theta }_0$ write $O_0$ for the unique object. Then write in ${\Theta }_n$

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

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

Properties

Groupoidal version

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

Theta-spaces

A local model structure on simplicial presheaves on the Theta categories is called Theta spaces and models (n,r)-categories.

References

The $\Theta$-categories were introduced in

• Andre Joyal, Disks, duality and Theta-categories

An discussion with lots off pictures is in chapter 7 of

• Eugenia Cheng, Aaron Lauda, Higher-dimensional categories: an illustrated guidebook (pdf)

A useful discussion is in

• Clemens Berger, Iterated wreath product of the simplex category and iterated loop spaces (arXiv)

and in section 3 of

• Charles Rezk, A cartesian presentation of weak $n$-categories (arXiv)

there leading over to the notion of Theta space.

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)