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 pasing 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

…

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.

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.