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\section*{n-fold complete Segal 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{fold_segal_objects}{$n$-fold Segal objects}\dotfill \pageref*{fold_segal_objects} \linebreak
\noindent\hyperlink{complete_fold_segal_spaces}{Complete $n$-fold Segal spaces}\dotfill \pageref*{complete_fold_segal_spaces} \linebreak
\noindent\hyperlink{definition_via_the_model_category_of_simplicial_sets}{Definition via the model category of simplicial sets}\dotfill \pageref*{definition_via_the_model_category_of_simplicial_sets} \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}

\emph{$n$-fold complete Segal spaces} are a model for [[(∞,n)-category|(∞,n)-categories]], i.e. the [[homotopy theory|homotopical]] version of [[n-category|n-categories]].

We can view [[strict n-category|strict n-categories]] as [[n-fold category|n-fold categories]] where part of the structure is trivial; for example, strict 2-categories can be described as double categories where the only vertical morphisms are identities. $n$-fold Segal spaces similarly result from viewing $(\infty,n)$-categories as a special class of $n$-fold internal $(\infty,1)$-categories in [[∞Grpd|∞-groupoids]].

If $\mathcal{C}$ is an $(\infty,1)$-category, then $(\infty,1)$-categories [[category object in an (infinity,1)-category|internal]] to $\mathcal{C}$ can be defined as certain simplicial objects in $\mathcal{C}$ (namely those satisfying the ``Segal condition''). Thus $n$-fold internal $(\infty,1)$-categories in $\infty$-groupoids correspond to a class of $n$-simplicial $\infty$-groupoids, and $n$-fold Segal spaces are defined by additionally specifying certain constancy conditions.

To describe the correct homotopy theory of $(\infty,n)$-categories we also want to regard the fully faithful and essentially surjective morphisms between $n$-fold Segal spaces as equivalences. It turns out that, just as in the case of [[Segal spaces]], the localization at these maps can be accomplished by restricting to a full subcategory of \emph{complete} objects.

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

\hypertarget{fold_segal_objects}{}\subsubsection*{{$n$-fold Segal objects}}\label{fold_segal_objects}

If $\mathcal{C}$ is an $(\infty,1)$-category with pullbacks, we say that a simplicial object $X_\bullet : \Delta^{op} \to \mathcal{C}$ satisfies the [[Segal condition]] if the squares

\begin{displaymath}
\itexarray{
     X_{m+n} &\to& X_{n}
     \\
     \downarrow && \downarrow
     \\
     X_{m} &\to& X_{0}
  }
\end{displaymath}
are all pullbacks. Such \emph{Segal objects} give the $(\infty,1)$-categorical version of [[internal category|internal categories]] as \emph{algebraic} structures. (I.e. we have not inverted a class of fully faithful and essentially surjective morphisms.)

If $Seg(\mathcal{C})$ denotes the full subcategory of $Fun(\Delta^{op}, \mathcal{C})$ spanned by the Segal objects, then this is again an $(\infty,1)$-category with pullbacks, so we can iterated the definition to obtain a full subcategory $Seg^{n}(\mathcal{C})$ of $Fun(\Delta^{n,op}, \mathcal{C})$ of \emph{Segal $\Delta^{n}$-objects} in $\mathcal{C}$.

We can now inductively define $n$-fold Segal objects by imposing constancy conditions: An \emph{$n$-fold Segal object} in $\mathcal{C}$ is a Segal $\Delta^{n}$-object $X$ such that

\begin{enumerate}%
\item The $(n-1)$-simplicial object $X_0 : \Delta^{n-1,op} \to \mathcal{C}$ is constant
\item The $(n-1)$-simplicial objects $X_i$ are $(n-1)$-fold Segal objects for all $i$.

\end{enumerate}
When $\mathcal{C}$ is the $(\infty,1)$-category of spaces (or $\infty$-groupoids) we refer to $n$-fold Segal objects as \emph{$n$-fold Segal spaces}.

\hypertarget{complete_fold_segal_spaces}{}\subsubsection*{{Complete $n$-fold Segal spaces}}\label{complete_fold_segal_spaces}

We now define fully faithful and essentially surjective morphisms between $n$-fold Segal inductively in terms of the corresponding notions for [[Segal spaces]]:

\begin{defn}
\label{}\hypertarget{}{}
A morphism $X \to Y$ between $n$-fold Segal spaces is \emph{fully faithful and essentially surjective} if:

\begin{enumerate}%
\item $X_{\bullet,0,\ldots,0} \to Y_{\bullet,0,\ldots,0}$ is a fully faithful and essentially surjective morphism of Segal spaces
\item $X_{1,\bullet,\ldots,\bullet} \to Y_{1,\bullet,\ldots,\bullet}$ is a fully faithful and essentially surjective morphism of $(n-1)$-fold Segal spaces

\end{enumerate}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
An $n$-fold Segal space $X$ is \emph{complete} if:

\begin{enumerate}%
\item The Segal space $X_{\bullet,0,\ldots,0}$ is complete.
\item The $(n-1)$-fold Segal space $X_{1,\bullet,\ldots,\bullet}$ is complete.

\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
There are several equivalent ways to reformulate these inductive definitions. For example, a morphism $f : X \to Y$ is fully faithful and essentially surjective if and only if:

\begin{enumerate}%
\item $X_{\bullet,0,\ldots,0} \to Y_{\bullet,0,\ldots,0}$ is essentially surjective.
\item For all objects $x,x' \in X_{0,\ldots,0}$ the induced map on $(n-1)$-fold Segal spaces of morphisms $X(x,x') \to Y(fx,fx')$ is fully faithful and essentially surjective.

\end{enumerate}
\end{remark}
\begin{theorem}
\label{}\hypertarget{}{}
The complete $n$-fold Segal spaces are precisely the $n$-fold Segal spaces that are local with respect to the fully faithful and essentially surjective morphisms. Thus the localization of the $(\infty,1)$-category of $n$-fold Segal spaces at this class of morphisms is equivalent to the full subcategory of complete $n$-fold Segal spaces.

\end{theorem}
This was first proved in Barwick's \hyperlink{BarwickThesis}{thesis}, generalizing Rezk's proof in the case $n=1$. Later, \hyperlink{Lurie09}{Lurie} extended the notion of complete Segal objects to more general contexts than spaces, which allows an inductive definition of complete $n$-fold Segal spaces as complete Segal objects in complete $(n-1)$-fold Segal spaces. The theorem for $n$-fold Segal spaces then follows by inductively applying the generalization of Rezk's theorem (for the case $n=1$) to this setting.

\hypertarget{definition_via_the_model_category_of_simplicial_sets}{}\subsubsection*{{Definition via the model category of simplicial sets}}\label{definition_via_the_model_category_of_simplicial_sets}

(\ldots{})

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

\begin{itemize}%
\item [[Theta-spaces]], [[Segal n-categories]]


\item [[higher Segal space]]


\item [[semi-Segal space]]



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

The definition originates in the thesis

\begin{itemize}%
\item [[Clark Barwick]], \emph{$(\infty,n)$-$Cat$ as a closed model category} PhD (2005)

\end{itemize}
which however remains unpublished. It appears in print in section 12 of

\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}
The basic idea was being popularized and put to use in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}

\end{itemize}
A detailed discussion in the general context of [[internal categories in an (∞,1)-category]] is in section 1 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{$(\infty,2)$-Categories and the Goodwillie Calculus I} (\href{http://arxiv.org/abs/0905.0462}{arXiv:0905.0462})

\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[Simona Paoli]], \emph{Simplicial Methods for Higher Categories -- Segal-type Models of Weak $n$-Categories}, Springer 2019 (\href{https://doi.org/10.1007/978-3-030-05674-2}{doi:10.1007/978-3-030-05674-2}, \href{https://link.springer.com/content/pdf/bfm%3A978-3-030-05674-2%2F1.pdf}{toc pdf})

\end{itemize}
For related references see at \emph{[[(∞,n)-category]]} .

[[!redirects n-fold Segal space]]

[[!redirects n-fold complete Segal spaces]] [[!redirects n-fold Segal spaces]]



\end{document}