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\begin{document}

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\section*{Theta category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{ViaFreeStrictOmegaCategory}{Via the free strict $\omega$-category}\dotfill \pageref*{ViaFreeStrictOmegaCategory} \linebreak
\noindent\hyperlink{ViaIteratedWreathProduct}{Via iterated wreath product}\dotfill \pageref*{ViaIteratedWreathProduct} \linebreak
\noindent\hyperlink{ViaDualsOfDisks}{Via duals of disks}\dotfill \pageref*{ViaDualsOfDisks} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{EmbeddingOfGrids}{Embedding of grids (products of the simplex category)}\dotfill \pageref*{EmbeddingOfGrids} \linebreak
\noindent\hyperlink{embedding_into_strict_categories}{Embedding into strict $n$-categories}\dotfill \pageref*{embedding_into_strict_categories} \linebreak
\noindent\hyperlink{groupoidal_version}{Groupoidal version}\dotfill \pageref*{groupoidal_version} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $n \in \mathbb{N}$ the [[category]] $\Theta_n$ -- \textbf{Joyal's disk category} or \textbf{cell category} -- may be thought of as the [[full subcategory]] of the category $Str n Cat$ of [[strict ∞-category|strict n-categories]] on those $n$-categories that are [[free construction|free]] on [[pasting diagrams]] of $n$-[[globe]]s.

For instance $\Theta_2$ contains an [[object]] that is depicted as

\begin{displaymath}
\itexarray{
    & \nearrow &\Downarrow& \searrow &&
    && \nearrow && \searrow 
    \\
    a && \to && b &\to & c && \Downarrow && b
    \\
    & \searrow &\Downarrow& \nearrow &&
    && \searrow && \nearrow &&
  }
  \,,
\end{displaymath}
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:

\begin{displaymath}
\itexarray{
    \nwarrow \nearrow &  & & \uparrow &&& 2
    \\
    & \nwarrow & \uparrow & \nearrow &&& 1
    \\
    && {*} 
  }
  \,.
\end{displaymath}
Accordingly, $\Theta_n$ is also the category of planar rooted trees of level $\leq n$.

In low degree we have

\begin{itemize}%
\item $\Theta_0 = *$ is the [[point]].


\item $\Theta_1 = \Delta$ 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

\begin{displaymath}
0 \to 1 \to \cdots \to n
  \,.
\end{displaymath}
Dually, this is the planar rooted tree of the form

\begin{displaymath}
\itexarray{
    \nwarrow &\uparrow & \cdots \nearrow
    \\
    &{*}
  }
\end{displaymath}
with $n$-branches.



\end{itemize}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We discuss two equivalent definitions

\begin{itemize}%
\item \hyperlink{ViaFreeStrictOmegaCategory}{Via the free strict $\omega$-category}


\item \hyperlink{ViaIteratedWreathProduct}{Via iterated wreath product}


\item \hyperlink{ViaDualsOfDisks}{Via duals of disks}



\end{itemize}
\hypertarget{ViaFreeStrictOmegaCategory}{}\subsubsection*{{Via the free strict $\omega$-category}}\label{ViaFreeStrictOmegaCategory}

Let $T(1)$ denote the free [[strict ∞-category]] generated from the [[terminal object|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 \emph{instances} of the $k$-globes appearing in the diagram.

We now describe this formally.

The [[k-morphisms|n-cells]] of $T(1)$ may be identified with \textbf{planar [[trees]]} $\tau$ of height $n$, which by definition are functors

\begin{displaymath}
\tau: [n]^{op} \to \Delta
\end{displaymath}
($\Delta$ is the category of [[simplex|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 $\Delta$,

\begin{displaymath}
\tau(n) \to \tau(n-1) \to \ldots \to \tau(0) = 1,
\end{displaymath}
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 $x \in \tau(k)$:

\begin{displaymath}
i_{\tau'}^{-1}(x) = \{x_0\} \cup i_{\tau}^{-1}(x) \cup \{x_1\}
\end{displaymath}
Now define a $\tau$-\textbf{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 $k$-\textbf{cell} of the globular set $[\tau]$ is a $\tau$-sector $(x, y, z)$ where $x \in \tau(k)$. If $k \geq 1$, the \textbf{source} of a $k$-cell $(x, y, z)$ is the $(k-1)$-cell $(i(x), u, x)$ and the \textbf{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 $[\tau]$.

\begin{defn}
\label{}\hypertarget{}{}
$\Theta$ 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 $T(1)$.

\end{defn}
\hypertarget{ViaIteratedWreathProduct}{}\subsubsection*{{Via iterated wreath product}}\label{ViaIteratedWreathProduct}

\begin{prop}
\label{ByWreathProduct}\hypertarget{ByWreathProduct}{}
$\Theta_n$ is the $n$-fold [[categorical wreath product]] of the [[simplex category]] with itself

\begin{displaymath}
\Theta_n \simeq \Delta^{\wr n}
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Berger}{Berger, section 3})

\begin{example}
\label{}\hypertarget{}{}
So

\begin{displaymath}
\Theta_1 = \Delta
\end{displaymath}
\begin{displaymath}
\Theta_2 = \Delta \wr \Delta
\end{displaymath}
etc.

\end{example}
\begin{cor}
\label{}\hypertarget{}{}
For all $n \in \mathbb{N}$ there is a canonical embedding

\begin{displaymath}
\sigma : \Theta_n \hookrightarrow \Theta_{n+1}
\end{displaymath}
given by $\sigma : a \mapsto ([1], a)$.

\end{cor}
\hypertarget{ViaDualsOfDisks}{}\subsubsection*{{Via duals of disks}}\label{ViaDualsOfDisks}

In analogy to how the [[simplex category]] is [[equivalence of categories|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]].

\begin{displaymath}
\Theta \coloneqq \mathbb{D}^{op}
  \,.
\end{displaymath}
(\ldots{})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{EmbeddingOfGrids}{}\subsubsection*{{Embedding of grids (products of the simplex category)}}\label{EmbeddingOfGrids}

\begin{defn}
\label{}\hypertarget{}{}
For any [[small category]] $A$ there is a canonical functor

\begin{displaymath}
\delta_A : \Delta \times A \to \Delta \wr A
\end{displaymath}
given by

\begin{displaymath}
\delta_A([n], a) = ([n], a^n)
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Berger}{Berger, def. 3.8})

\begin{remark}
\label{ProductOfDeltaInTheta}\hypertarget{ProductOfDeltaInTheta}{}
By iteration, this induces a canonical functor

\begin{displaymath}
\delta_n : \Delta^{\times n} \to \Theta_n
  \,.
\end{displaymath}
\end{remark}
\hypertarget{embedding_into_strict_categories}{}\subsubsection*{{Embedding into strict $n$-categories}}\label{embedding_into_strict_categories}

Write $Str n Cat$ for the category of [[strict n-categories]].

\begin{prop}
\label{}\hypertarget{}{}
There is a [[dense subcategory|dense]] [[full and faithful functor|full embedding]]

\begin{displaymath}
\Theta_n \hookrightarrow Str n Cat
  \,.
\end{displaymath}
\end{prop}
This was conjectured in (\hyperlink{BataninStreet}{Batanin-Street}) and shown in terms of free $n$-categories on $n$-graphs in (\hyperlink{MakkaiZawadowsky}{Makkai-Zawadowsky, theorem 5.10}) and (\hyperlink{BergerCellular}{Berger 02, prop. 2.2}). In terms of the wreath product presentation, prop. \ref{ByWreathProduct} this is (\hyperlink{Berger}{Berger 05, theorem 3.7}).

\begin{prop}
\label{}\hypertarget{}{}
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 $a_i$:

\begin{displaymath}
([k], (a_1, \cdots, a_k))
  = 
  a_1 \cdot a_2 \cdot \cdots \cdot a_k
  \,.
\end{displaymath}
\end{prop}
\begin{example}
\label{}\hypertarget{}{}
The [[pasting diagram]]

\begin{displaymath}
\itexarray{
    & \nearrow &\Downarrow& \searrow &&
    && \nearrow && \searrow 
    \\
    a && \to && b &\to & c && \Downarrow && b
    \\
    & \searrow &\Downarrow& \nearrow &&
    && \searrow && \nearrow &&
  }
\end{displaymath}
corresponds to the objects of $\Theta_2 = \Delta \wr \Delta$ given by

\begin{displaymath}
([3], (a_1, a_2, a_3))
  \,,
\end{displaymath}
where in turn

\begin{itemize}%
\item $a_1 = [2]$


\item $a_2 = [0]$


\item $a_3 = [1]$.



\end{itemize}
\end{example}
\begin{example}
\label{Grids}\hypertarget{Grids}{}
Composing with the functor $\delta_n$ from remark \ref{ProductOfDeltaInTheta} we obtain an embedding of $n$-fold simplices into strict $n$-categories

\begin{displaymath}
\Delta^{\times n} \stackrel{\delta_n}{\to}
  \Theta_n \hookrightarrow Str n Cat
  \,.
\end{displaymath}
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 $k_1 \times k_2 \times \cdots \times k_n$ \textbf{grid} of $n$-cells.

\end{example}
Write

\begin{displaymath}
Str n Cat_{gaunt} \hookrightarrow Str n Cat
\end{displaymath}
for the inclusion of the \href{http://ncatlab.org/nlab/show/%28infinity%2Cn%29-category#GauntStrictNCategories}{gaunt} strict $n$-categories into all [[strict n-categories]].

\begin{prop}
\label{}\hypertarget{}{}
$\Theta_n$ 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 \ref{Grids}, and closed under [[retracts]].

\end{prop}
(\hyperlink{BarwickSchommerPries}{B-SP, prop. 10.5})

\hypertarget{groupoidal_version}{}\subsubsection*{{Groupoidal version}}\label{groupoidal_version}

The groupoidal version $\tilde \Theta$ of $\Theta$ is a [[test category]] (\hyperlink{Ara}{Ara}).

\hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2}

In $\Theta_0$ write $O_0$ for the unique object. Then write in $\Theta_n$

\begin{displaymath}
O_n := [1](O_{n-1})
  \,.
\end{displaymath}
This is the strict [[n-category]] free on a single $n$-[[globe]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[globular theory]]


\item [[globular set]]


\item [[globular operad]]


\item [[simplex category]], [[globe category]], [[cube category]]


\item A local [[model structure on simplicial presheaves]] on the Theta categories is called [[Theta space]]s and models [[(n,r)-category|(n,r)-categories]].


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



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The $\Theta$-categories were introduced in

\begin{itemize}%
\item [[Andre Joyal]], \emph{Disks, duality and Theta-categories} ([[JoyalThetaCategories.pdf:file]])

\end{itemize}
A discussion with lots of pictures is in \href{http://cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf#page=131}{chapter 7} of

\begin{itemize}%
\item [[Eugenia Cheng]], [[Aaron Lauda]], \emph{Higher-dimensional categories: an illustrated guidebook} (\href{http://cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf}{pdf})

\end{itemize}
More discussion is in

\begin{itemize}%
\item David Oury, \emph{On the duality between trees and disks}, TAC vol. 24 (\href{http://www.tac.mta.ca/tac/volumes/24/16/24-16.pdf}{pdf})

\end{itemize}
The following paper proves that $\Theta$ is a test category

\begin{itemize}%
\item [[Denis-Charles Cisinski]], [[Georges Maltsiniotis]], \emph{La cat\'e{}gorie $\Theta$ de Joyal est une cat\'e{}gorie test} , JPAA \textbf{215} no.5 (2011) pp.962-982. (\href{webusers.imj-prg.fr/~georges.maltsiniotis/ps/dec_web.pdf}{draft}) 

\end{itemize}
Discussion of embedding of $\Theta$ into strict $n$-categories is in

\begin{itemize}%
\item [[Michael Batanin]], [[Ross Street]], \emph{The universal property of the multitude of trees}, J. Pure Appl. Alg. 154 (2000), 3--13.

\end{itemize}
\begin{itemize}%
\item [[Michael Makkai]], M. Zawadowsky, \emph{Duality for simple $\omega$-categories and disks}, Theory Appl. Categories 8 (2001), 114--243

\end{itemize}
\begin{itemize}%
\item [[Clemens Berger]], \emph{A cellular nerve for higher categories}, Adv. Math. 169 (2002), 118--175.

\end{itemize}
The characterization in terms of $n$-fold [[categorical wreath products]] is in

\begin{itemize}%
\item [[Clemens Berger]], \emph{Iterated wreath product of the simplex category and iterated loop spaces} (\href{http://arxiv.org/abs/math/0512575}{arXiv:math/0512575}),

\end{itemize}
see also section 3 of

\begin{itemize}%
\item [[Charles Rezk]], \emph{A cartesian presentation of weak $n$-categories} (\href{http://arxiv.org/abs/0901.3602}{arXiv:0901.3602})

\end{itemize}
there leading over to the notion of [[Theta space]].

The groupoidal version $\tilde \Theta$ is discussed in

\begin{itemize}%
\item [[Dimitri Ara]], \emph{The groupoidal analogue $\tilde \Theta$ to Joyal's category $\Theta$ is a test category} (\href{http://arxiv.org/abs/1012.4319}{arXiv:1012.4319})

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

\begin{itemize}%
\item [[David Ayala]], [[Richard Hepworth]], \emph{Configurations spaces and $\Theta_n$} (\href{http://de.arxiv.org/abs/1202.2806}{arXiv:1202.2806})

\end{itemize}
Related discussion in the context of [[(infinity,n)-categories]] is also in

\begin{itemize}%
\item [[Clark Barwick]], [[Chris Schommer-Pries]], \emph{On the Unicity of the Homotopy Theory of Higher Categories} (\href{http://arxiv.org/abs/1112.0040}{arXiv:1112.0040}, \href{http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/}{slides})

\end{itemize}
[[!redirects ? category]] [[!redirects Theta-category]] [[!redirects ? categories]] [[!redirects Theta-categories]] [[!redirects disk category]] [[!redirects disk categories]]

[[!redirects cell category]]



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