nLab Theta category

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

• Via iterated wreath product

• Via duals of disks

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

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

(Berger, section 3)

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 $\sigma : a \mapsto ([1], a)$.

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

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

(Berger, def. 3.8)

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

The pasting diagram

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

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

$$([3], (a_1, a_2, a_3)) \,,$$

where in turn

• $a_1 = [2]$

• $a_2 = [0]$

• $a_3 = [1]$.

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.

(B-SP, prop. 10.5)

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 $\Theta_n$

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

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

Related concepts

• globular theory

• globular set

• globular operad

• simplex category, globe category, cube category

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

• A Cisinski model structure on bare presheaves on $\Theta_n$, modelling (∞,n)-categories is the model structure on cellular sets.

References

The $\Theta$-categories were introduced in

• Andre Joyal, Disks, duality and Theta-categories (pdf)

A discussion with lots of pictures is in chapter 7 of

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

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

• Denis-Charles Cisinski, Georges Maltsiniotis, La catégorie $\Theta$ de Joyal est une catégorie test , JPAA 215 no.5 (2011) pp.962-982. (pdf, in French)

  ‎

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

• Michael Batanin, Ross Street, The universal property of the multitude of trees, J. Pure Appl. Alg. 154 (2000), 3–13.

• 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

• Clemens Berger, Iterated wreath product of the simplex category and iterated loop spaces (arXiv:math/0512575),

see also section 3 of

• Charles Rezk, A cartesian presentation of weak $n$-categories (arXiv:0901.3602)

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)

The relation of $\Theta_n$ to configuration spaces of points in the Euclidean space $\mathbb{R}^n$ is discussed in

• David Ayala, Richard Hepworth, Configurations spaces and $\Theta_n$ (arXiv:1202.2806)

Related discussion in the context of (infinity,n)-categories is also in

• Clark Barwick, Chris Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories (arXiv:1112.0040, slides)