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 $\mathrm{Str}n\mathrm{Cat}$ 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

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

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 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 also 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

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 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 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

$$\tau :[n{]}^{\mathrm{op}}\to \Delta$$\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 $\mathrm{Str}\omega \mathrm{Cat}$ on the strict ω-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

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

(Berger, section 3)

Examples

So

$${\Theta }_1=\Delta$$\Theta_1 = \Delta

$${\Theta }_2=\Delta \wr \Delta$$\Theta_2 = \Delta \wr \Delta

etc.

Corollary

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

$$\sigma :{\Theta }_n\hookrightarrow {\Theta }_{n+1}$$\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, $\Delta \simeq {𝕀}^{\mathrm{op}}$, so the $\Theta$-category is equivalent to the opposite of the category of Joyal’s combinatorial finite disks.

$$\Theta ≔{𝔻}^{\mathrm{op}}.$$\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

$${\delta }_A:\Delta \times A\to \Delta \wr A$$\delta_A : \Delta \times A \to \Delta \wr A

given by

$${\delta }_A([n],a)=([n],(A,A,\cdots ,A)).$$\delta_A([n], a) = ([n], (A, A, \cdots, A)) \,.

(Berger, def. 3.8)

Remark

By iteration, this induces a canonical functor

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

Embedding into strict $n$-categories

Write $\mathrm{Str}n\mathrm{Cat}$ for the category of strict n-categories.

Proposition

There is a dense full embedding

$${\Theta }_n\hookrightarrow \mathrm{Str}n\mathrm{Cat}.$$\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. 1 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 $a_i$:

$$([k],(a_1,\cdots ,a_k))=a_1\cdot a_2\cdot \cdots \cdot a_k.$$([k], (a_1, \cdots, a_k)) = a_1 \cdot a_2 \cdot \cdots \cdot a_k \,.

Example

The pasting diagram

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

corresponds to the objects of ${\Theta }_2=\Delta \wr \Delta$ given by

$$([3],(a_1,a_2,a_3)),$$([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 1 we obtain an embedding of $n$-fold simplices into strict $n$-categories

$${\Delta }^{\times n}\stackrel{{\delta }_n}{\to }{\Theta }_n\hookrightarrow \mathrm{Str}n\mathrm{Cat}.$$\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

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

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

Proposition

${\Theta }_n$ is the smallest full subcategory of $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}$ containing the grids, the image of ${\delta }_n:{\Delta }^{\times n}\to \mathrm{Str}n\mathrm{Cat}$, example 3, 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 ${\Theta }_n$

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

Discussion of its embedding 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 ${ℝ}^n$ 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)