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\section*{Theta-space}

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

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

[[!include higher category theory - contents]]

\hypertarget{internal_categories}{}\paragraph*{{Internal $(\infty,1)$-Categories}}\label{internal_categories}

[[!include internal infinity-categories contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{special_values_of_}{Special values of $(n,k)$}\dotfill \pageref*{special_values_of_} \linebreak
\noindent\hyperlink{cartesian_monoidal_and_enriched_structure}{Cartesian monoidal and enriched structure}\dotfill \pageref*{cartesian_monoidal_and_enriched_structure} \linebreak
\noindent\hyperlink{OfAll}{$(n+1,r+1)$-$\Theta$-space of $(n,r)$-$\Theta$-spaces}\dotfill \pageref*{OfAll} \linebreak
\noindent\hyperlink{homotopy_hypothesis}{Homotopy hypothesis}\dotfill \pageref*{homotopy_hypothesis} \linebreak
\noindent\hyperlink{relation_to_cellular_sets}{Relation to cellular sets}\dotfill \pageref*{relation_to_cellular_sets} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

The notion of \emph{$\Theta_n$-space} is one model for the notion of \emph{[[(∞,n)-category]]}. A $\Theta_n$-space may be thought of as a [[globe category|globular]] $n$-category \emph{up to coherent homotopy}, a globular $n$-category \emph{[[internalization|internal]]} to the [[(∞,1)-category]] [[∞Grpd]].

Concretely, a \emph{$\Theta_n$-space is a [[simplicial presheaf]] on the [[Theta category|Theta}n category]], hence a ``[[cellular object|cellular space]]'' that satisfies

\begin{enumerate}%
\item the globular [[Segal condition]] as a [[weak homotopy equivalence]];


\item and a completeness condition analogous to that of [[complete Segal spaces]].



\end{enumerate}
In fact for $n = 1$ $\Theta_n = \Delta$ is the simplex category and a $\Theta_1$-space is the same as a [[complete Segal space]].

Noticing that a presheaf of \emph{sets} on $\Theta_n$ which satisfies the cellular [[Segal condition]] is equivalently a [[strict n-category]], $Theta_n$-spaces may be thought of as [[n-categories]] [[internal category in an (infinity,1)-category|internal to]] the [[(∞,1)-category]] [[∞Grpd]], defined in the [[cell category|cellular]] way.

\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

There is a [[cartesian closed category|cartesian closed]] [[category with weak equivalences]] $\Theta_n Sp_k^{fib}$ of \textbf{$(n+k,n)$-$\Theta$-spaces} for all

\begin{itemize}%
\item $0 \leq n \leq \infty$;


\item $-2 \leq k \leq \infty$



\end{itemize}
as the [[category of fibrant objects]] in a [[model category]] $\Theta_n Sp_k$,\newline being a [[Bousfield localization of model categories|left Bousfield localization]] of the injective [[model structure on simplicial presheaves]] on the $n$th [[Theta category]].

The [[weak equivalences]] in $\Theta_n Sp_k^{fib}$ are then (by the standard result discussed at [[Bousfield localization of model categories]]) just the objectwise weak equivalences in the standard [[model structure on simplicial sets]] $sSet_{Quillen}$.

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

For $J$ a [[category]], write $\Theta J$ for the [[categorical wreath product]] over the [[simplex category]] $\Delta$ \href{http://arxiv.org/abs/math/0512575}{Ber05}.

Then with $\Theta_0 := {*}$ we have inductively

\begin{displaymath}
\Theta_n = \Theta \Theta_{n-1}
  \,.
\end{displaymath}
For $D = SPSh(C)^{inj}_{S}$ a [[model structure on simplicial presheaves]] on a category $C$ obtained by left [[Bousfield localization]] at a set of morphisms $S \subset Mor(SPSh(C)^{inj})$ from the global injective model structure, write

\begin{displaymath}
D-\Theta Sp := SPSh(\Theta C)^{inj}_{S_\Theta}
  \,,
\end{displaymath}
where $S_\Theta$ is the set of morphisms given by \ldots{}. .

Set

\begin{displaymath}
\Theta_0 Sp_k := SSet_k
  \,,
\end{displaymath}
the left [[Bousfield localization]] of the standard [[model structure on simplicial sets]] such that fibrant objects are the [[Kan complex]]es that are [[homotopy n-type|homotopy k-type]]s. Then finally define inductively

\begin{displaymath}
\Theta_{n+1} Sp_k := (\Theta_n Sp_k)-\Theta Sp
  \,.
\end{displaymath}
Unwinding this definition we see that

\begin{displaymath}
\Theta_{n} Sp_k = SPSh(\Theta_n)^{inj}_{S_{n}}
  \,,
\end{displaymath}
for some set $S_n \subset Mor(SPSh(C))$ of morphisms.

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

\hypertarget{special_values_of_}{}\subsubsection*{{Special values of $(n,k)$}}\label{special_values_of_}

I would have started $k$ at $-1$. What does Rezk's notion do with $k = -2$? ---Toby

$-1$-groupoids are spaces which are either empty or contractible. $-2$-groupoids are spaces which are contractible. So $k=-2$ is the completely trivial case; it's included for completeness. -- Charles

I do know what a $(-2,0)$-category is, a triviality as you say. But for $n \gt 0$, an $(n-2,n)$-category is the same as an $(n-2,n-1)$-category as far as I can see. (Note: I say this \emph{without} having worked through your version, but just thinking about what $(n,r)$-categories should be, as at [[(n,r)-category]].) ---Toby

I would say: $(n-2,n)$-category is a trivial concept, for every $n$, though $(n-2,n-1)$ isn't. An $(n+1+k,n+1)$-category should amount to a category enriched over $(n+k,n)$-categories. An $(-2,0)$-category is trivial (a point); an $(-1,1)$-category is a category enriched over the point, and so equivalent to the terminal category; an $(0,2)$-category is a category enriched over categories which are equivalent to the terminal category (and so equivalent to the terminal $2$-category, etc.) -- Charles R.



(Sorry for not noticing before that you \emph{are} Charles Rezk; for some reason I though of [[Charles Wells]].) ---Toby

[[David Roberts]]: I'm a little confused. The way I think about it, and I may have the indexing wrong, is that in an $(n,n+2)$-category $C$, for \emph{all} pairs of $n$-arrows $x,y$, there is a unique $n+1$-arrow between them. This implies that $x$ and $y$ are parallel, in particular, that $C$ has a single $(n-1)$-arrow.

\emph{Toby}: Wait, I don't buy Charles's argument after all. Yes, a $(-1,1)$-category is a category enriched over the point, but that doesn't make it necessarily the terminal category; it makes it a truth value. If it has an object, then it's trivial, but it might be empty instead. The difference between a $(-1,1)$-category and a $(-1,0)$-category is that every $0$-morphism in the latter must be invertible, which is no difference at all; that's why we have this repetition. (And thereafter it propagates indefinitely.)

Similarly, with David's argument, what if $C$ has no $n$-arrows at all?

[[David Roberts]]: Yes - that should then be 'Assuming $C$ has an object, then it has a single $(n-1)$-arrow'. Assuming I got the indexing right, I must stress. I think I grasp $(n,n+1)$-categories, but I'm not solid on these new beasties.

Toby, I guess you are right. I don't know what I was thinking. -- Charles R.

Thanks for joining, in, Charles. Toby is, by the way, our esteemed expert for -- if not the inventor of -- [[negative thinking]]. :-) - [[Urs Schreiber|USc]]

\emph{Toby}: All right, so we allow $k = -2$, since $n$ might be $0$; but for an $(n-2,n)$-$\Theta$-space is the same as an $(n-1,n)$-$\Theta$-space for $n \gt 0$. OK, I'm happy with that; now to understand the definition! ({\tt \symbol{94}}\_{\tt \symbol{94}})

\hypertarget{cartesian_monoidal_and_enriched_structure}{}\subsubsection*{{Cartesian monoidal and enriched structure}}\label{cartesian_monoidal_and_enriched_structure}

The [[model category]] $\Theta_n Sp_k$ is a [[cartesian monoidal category|cartesian]] [[monoidal model category]].

The idea is that $\Theta_n Sp_k$ is naturally an [[enriched model category]] over itself.

\hypertarget{OfAll}{}\subsubsection*{{$(n+1,r+1)$-$\Theta$-space of $(n,r)$-$\Theta$-spaces}}\label{OfAll}

Here is the idea on how to implement the notion $(n+1,r+1)$-category of all $(n,r)$-categories in the context of Theta-spaces. At the time of this writing, this hasn't been spelled out in total.

As mentioned above regard $\Theta_k Sp_n$ as a category enriched over itself. Then define a presheaf $\mathbf{X}$ on $\Theta_{n+1}$ by setting

\begin{itemize}%
\item $\mathbf{X}[0] =$ collection of objects of $\Theta_n Sp_k$


\item $\mathbf{X}([m](\theta_1, \cdots, \theta_m))
 = \coprod_{a_0, \cdots, a_m}
   C(a_0,a_1)(\theta_1) \times \cdots \times
   C(a_{m-1},a_m)(\theta_m)$



\end{itemize}
This object satisfies the [[Segal conditions]] (its descent conditions) in all degrees except degree 0. A suitable localization operation ca-n fix this. The resulting object should be the ``$(n+1,k+1)$-$\Theta$-space of $(n,k)$-$\Theta$-spaces''.

\hypertarget{homotopy_hypothesis}{}\subsubsection*{{Homotopy hypothesis}}\label{homotopy_hypothesis}

The definition of weak $(n,r)$-categories modeled by $\Theta$-spaces does satisfy the [[homotopy hypothesis]]: there is an evident notion of groupoid objects in $\Theta_n Sp_k$ and the full subcategory on these models [[homotopy n-types]].

(\hyperlink{Rezk}{Rez09, 11.25}).

\hypertarget{relation_to_cellular_sets}{}\subsubsection*{{Relation to cellular sets}}\label{relation_to_cellular_sets}

There is a [[model structure on cellular sets]] (see there), hence on set-valued presheaves on $\Theta_n$ (instead of simplicial presheaves) which is [[Quillen equivalence|Quillen equivalent]] to the Rezk model structure on $\Theta_n$-spaces.

In fact the Theta-space model structure is the \href{Cisinski+model+structure#SimplicialCompletion}{simplicial completion} of the [[Cisinski model structure]] on presheaves on $\Theta_n$ (\hyperlink{Ara}{Ara})

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

For low values of $n,k$ this reproduces the following cases:

\begin{itemize}%
\item for $n=0$ we have $\Theta_0 Sp_\infty = sSet_{Quillen}$ with its [[model structure on simplicial sets|standard model structure]] and hence $\Theta_0 Sp_\infty^{fib} =$ [[∞Grpd]].


\item for $n=1$ objects in $\Theta_1 Sp_\infty^{fib}$ are [[complete Segal spaces]], hence [[(∞,1)-category|(∞,1)-categories]].



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

\begin{itemize}%
\item [[cellular set]]


\item [[omega-category]]


\item [[globular operad]], [[globular theory]],



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

The notion of $\Theta$-spaces was introduced in

\begin{itemize}%
\item [[Charles Rezk]], \emph{A cartesian presentation of weak $n$-categories} Geom. Topol. 14 (2010), no. 1, 521--571 (\href{http://arxiv.org/abs/0901.3602}{arXiv:0901.3602})

\emph{Correction to ``A cartesian presentation of weak $n$-categories''} Geom. Topol. 14 (2010), no. 4, 2301--2304. MR 2740648 (\href{http://www.math.uiuc.edu/~rezk/cs-objects-correction.pdf}{pdf})



\end{itemize}
\begin{itemize}%
\item [[Charles Rezk]], \emph{Cartesian presentations of weak n-categories An introduction to $\Theta_n$-spaces} (2009) (\href{http://www.math.uiuc.edu/~rezk/northwestern-2009-n-cat-handout.pdf}{pdf})

\end{itemize}
The definition of the categories $\Theta_n$ goes back to [[Andre Joyal]] who also intended to define [[n-category|n-categories]] using it. This has been achieved at about the same time by Simpson:

\begin{itemize}%
\item [[Simpson, Carlos]], \emph{A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert-Van Kampen} (\href{https://arxiv.org/abs/alg-geom/9704006}{arXiv:alg-geom/9704006})


\item [[Simpson, Carlos]], \emph{On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak n-categories} (\href{https://arxiv.org/abs/math/9810058}{arXiv:math/9810058})



\end{itemize}
Discussion comparing $\Theta_{n+1}$-spaces to [[enriched (infinity,1)-categories]] in $\Theta_n$-spaces is in

\begin{itemize}%
\item [[Julie Bergner]], [[Charles Rezk]], \emph{Comparison of models for $(\infty,n)$-categories} (\href{http://arxiv.org/abs/1204.2013}{arXiv:1204.2013})


\item [[Julie Bergner]], [[Charles Rezk]], \emph{Comparison of models for $(\infty,n)$-categories II} (\href{http://arxiv.org/abs/1406.4182}{arXiv:1406.4182})



\end{itemize}
The note on the $(n+1,k+1)$-$\Theta$-space of all $(n,k)$-$\Theta$-spaces comes from communication with [[Charles Rezk]] \href{http://mathoverflow.net/questions/5867/n1-r1-theta-space-of-n-r-theta-spaces}{here}.

Relation to simplicial completion of the [[Cisinski model structure]] [[model structure on cellular sets|on cellular sets]] is in

\begin{itemize}%
\item [[Dimitri Ara]], \emph{Higher quasi-categories vs higher Rezk spaces} (\href{http://arxiv.org/abs/1206.4354}{arXiv:1206.4354})

\end{itemize}
[[!redirects theta space]] [[!redirects theta-space]] [[!redirects theta spaces]] [[!redirects theta-spaces]]

[[!redirects Theta space]] [[!redirects Theta-Space]] [[!redirects ? Space]] [[!redirects ∞-Space]] [[!redirects ? space]] [[!redirects ∞-space]]

[[!redirects Theta spaces]] [[!redirects Theta-Spaces]] [[!redirects Theta-spaces]] [[!redirects ? Spaces]] [[!redirects ∞-Spaces]] [[!redirects ? spaces]] [[!redirects ∞-spaces]]

[[!redirects Theta\_n space]] [[!redirects Theta\_n spaces]]



\end{document}