homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
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
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:
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
Dually, this is the planar rooted tree of the form
with $n$-branches.
We discuss two equivalent definitions
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
($\Delta$ 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 $\Delta$,
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 $x \in \tau(k)$:
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 $[\tau]$.
$\Theta$ is the full subcategory of $Str \omega Cat$ on the strict ω-categories $T([\tau])$, as $\tau$ ranges over cells in the underlying globular set of $T(1)$.
$\Theta_n$ is the $n$-fold categorical wreath product of the simplex category with itself
So
etc.
For all $n \in \mathbb{N}$ there is a canonical embedding
given by $\sigma : a \mapsto ([1], a)$.
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.
(…)
For any small category $A$ there is a canonical functor
given by
By iteration, this induces a canonical functor
Write $Str n Cat$ for the category of strict n-categories.
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).
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 $a_i$:
The pasting diagram
corresponds to the objects of $\Theta_2 = \Delta \wr \Delta$ given by
where in turn
$a_1 = [2]$
$a_2 = [0]$
$a_3 = [1]$.
Composing with the functor $\delta_n$ from remark 1 we obtain an embedding of $n$-fold simplices into strict $n$-categories
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
for the inclusion of the gaunt strict $n$-categoeries into all strict n-categories.
$\Theta_n$ 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 3, and closed under retracts.
The groupoidal version $\tilde \Theta$ of $\Theta$ is a test category (Ara).
In $\Theta_0$ write $O_0$ for the unique object. Then write in $\Theta_n$
This is the strict n-category free on a single $n$-globe.
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.
The $\Theta$-categories were introduced in
A discussion with lots of pictures is in chapter 7 of
More discussion is in
Discussion of its embedding into strict $n$-categories is in
The characterization in terms of $n$-fold categorical wreath products is in
see also section 3 of
there leading over to the notion of Theta space.
The groupoidal version $\tilde \Theta$ is discussed in
The relation of $\Theta_n$ to configuration spaces of points in the Euclidean space $\mathbb{R}^n$ is discussed in
Related discussion in the context of (infinity,n)-categories is also in