nLab Theta category

Context

Higher category theory

higher category theory

Contents

Idea

For $n\in ℕ$ 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

$\begin{array}{cccccccccc}& ↗& ⇓& ↘& & & & ↗& & ↘\\ a& & \to & & b& \to & c& & ⇓& & b\\ & ↘& ⇓& ↗& & & & ↘& & ↗& & \end{array}\phantom{\rule{thinmathspace}{0ex}},$\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:

$\begin{array}{ccccccc}↖↗& & & ↑& & & 2\\ & ↖& ↑& ↗& & & 1\\ & & *\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left[n\right]$ is thought of as a linear quiver and as such the pasting diagram of $n$ 1-morphisms

$0\to 1\to \cdots \to n\phantom{\rule{thinmathspace}{0ex}}.$0 \to 1 \to \cdots \to n \,.

Dually, this is the planar rooted tree of the form

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

with $n$-branches.

Definition

We discuss two equivalent definitions

Via the free strict ω-category

Let $T\left(1\right)$ 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 $k\in ℕ$: one point, one edge from the point to itself, one disk from the edge to itself, and so on.

So $T\left(1\right)$ is freely generated under composition from these cells. As described above, this means that each element of the underlying globular set of $T\left(1\right)$ 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 $T\left(1\right)$ may be identified with planar trees $\tau$ of height $n$, which by definition are functors

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

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

$\tau \left(n\right)\to \tau \left(n-1\right)\to \dots \to \tau \left(0\right)=1,$\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 $x\in \tau \left(k\right)$, there is a fiber ${i}^{-1}\left(x\right)$ 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 $\left[\tau \right]$, 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 ${i}^{-1}\left(x\right)$ of $\tau$, for every $x\in \tau \left(k\right)$:

${i}_{\tau \prime }^{-1}\left(x\right)=\left\{{x}_{0}\right\}\cup {i}_{\tau }^{-1}\left(x\right)\cup \left\{{x}_{1}\right\}$i_{\tau'}^{-1}(x) = \{x_0\} \cup i_{\tau}^{-1}(x) \cup \{x_1\}

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

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

Definition

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

Via iterated wreath product

Proposition/Definition

${\Theta }_{n}$ is the $n$-fold categorical wreath product of the simplex category with itself

${\Theta }_{n}\simeq {\Delta }^{\wr n}\phantom{\rule{thinmathspace}{0ex}}.$\Theta_n \simeq \Delta^{\wr n} \,.
Examples

So

${\Theta }_{1}=\Delta$\Theta_1 = \Delta
${\Theta }_{2}=\Delta \wr \Delta$\Theta_2 = \Delta \wr \Delta

etc.

Corollary

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

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

given by $\sigma :a↦\left(\left[1\right],a\right)$.

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.

$\Theta ≔{𝔻}^{\mathrm{op}}\phantom{\rule{thinmathspace}{0ex}}.$\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

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

given by

${\delta }_{A}\left(\left[n\right],a\right)=\left(\left[n\right],\left(A,A,\cdots ,A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\delta_A([n], a) = ([n], (A, A, \cdots, A)) \,.
Remark

By iteration, this induces a canonical functor

${\delta }_{n}:{\Delta }^{×n}\to {\Theta }_{n}\phantom{\rule{thinmathspace}{0ex}}.$\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

${\Theta }_{n}↪\mathrm{Str}n\mathrm{Cat}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(\left[k\right],\left({a}_{1},\cdots ,{a}_{k}\right)\right)\in \Delta \wr {\Delta }^{\wr \left(n-1\right)}$ is identified with the $k$-fold horizontal composition of the pasting composition of the $\left(n-1\right)$-morphisms ${a}_{i}$:

$\left(\left[k\right],\left({a}_{1},\cdots ,{a}_{k}\right)\right)={a}_{1}\cdot {a}_{2}\cdot \cdots \cdot {a}_{k}\phantom{\rule{thinmathspace}{0ex}}.$([k], (a_1, \cdots, a_k)) = a_1 \cdot a_2 \cdot \cdots \cdot a_k \,.
Example
$\begin{array}{cccccccccc}& ↗& ⇓& ↘& & & & ↗& & ↘\\ a& & \to & & b& \to & c& & ⇓& & b\\ & ↘& ⇓& ↗& & & & ↘& & ↗& & \end{array}$\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

$\left(\left[3\right],\left({a}_{1},{a}_{2},{a}_{3}\right)\right)\phantom{\rule{thinmathspace}{0ex}},$([3], (a_1, a_2, a_3)) \,,

where in turn

• ${a}_{1}=\left[2\right]$

• ${a}_{2}=\left[0\right]$

• ${a}_{3}=\left[1\right]$.

Example

Composing with the functor ${\delta }_{n}$ from remark 1 we obtain an embedding of $n$-fold simplices into strict $n$-categories

${\Delta }^{×n}\stackrel{{\delta }_{n}}{\to }{\Theta }_{n}↪\mathrm{Str}n\mathrm{Cat}\phantom{\rule{thinmathspace}{0ex}}.$\Delta^{\times n} \stackrel{\delta_n}{\to} \Theta_n \hookrightarrow Str n Cat \,.

Under this embedding an object $\left(\left[{k}_{1}\right],\left[{k}_{2}\right],\cdots ,\left[{k}_{n}\right]\right)$ is sent to the $n$-category which looks like (a globular version of) a ${k}_{1}×{k}_{2}×\cdots ×{k}_{n}$ grid of $n$-cells.

Write

$\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}↪\mathrm{Str}n\mathrm{Cat}$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 ${\delta }_{n}:{\Delta }^{×n}\to \mathrm{Str}n\mathrm{Cat}$, example 3, and closed under retracts.

Groupoidal version

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

Examples

In ${\Theta }_{0}$ write ${O}_{0}$ for the unique object. Then write in ${\Theta }_{n}$

${O}_{n}:=\left[1\right]\left({O}_{n-1}\right)\phantom{\rule{thinmathspace}{0ex}}.$O_n := [1](O_{n-1}) \,.

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

References

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)

Discussion of its embedding 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 $\stackrel{˜}{\Theta }$ is discussed in
• Dimitri Ara, The groupoidal analogue $\stackrel{˜}{\Theta }$ 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