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\begin{document}

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\section*{higher 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{warning}{Warning}\dotfill \pageref*{warning} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{higher_associativity_parameterized_by_polyhedra}{Higher associativity parameterized by polyhedra}\dotfill \pageref*{higher_associativity_parameterized_by_polyhedra} \linebreak
\noindent\hyperlink{categories_with_multivalued_composition}{Categories with multivalued composition}\dotfill \pageref*{categories_with_multivalued_composition} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{DyckerhoffKapranov}{Dyckerhoff-Kapranov}\dotfill \pageref*{DyckerhoffKapranov} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{warning}{}\subsection*{{Warning}}\label{warning}

There are several unrelated generalizations of the concept of a [[Segal space]] which might be thought of as ``higher Segal spaces''. For example, one might discuss

\begin{itemize}%
\item [[$n$-fold Segal spaces]], a model for $(\infty,n)$-categories.


\item [[$n$-uple Segal spaces]], a model for cubical $(\infty,n)$-categories.


\item [[$d$-Segal spaces]] in the sense of Dyckerhoff and Kapranov, a model for something like an $(\infty,1)$-category, but without uniqueness of composites (for $d \geq 2$) and with higher associativity only in dimension $d$ and above.



\end{itemize}
This article discusses $d$-Segal spaces in the sense of Dyckerhoff and Kapranov.

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

There are several ways to think about $d$-Segal spaces:

\hypertarget{higher_associativity_parameterized_by_polyhedra}{}\subsubsection*{{Higher associativity parameterized by polyhedra}}\label{higher_associativity_parameterized_by_polyhedra}

A $1$-Segal space $C$ is a [[Segal space]], i.e. a simplicial space satisfying the [[Segal condition]]. We think of the Segal condition in the following way. For every subdivision of an interval $I$ into subintervals $I_1,\dots,I_n$, and for any choice of labelings of the endpoints of these intervals by objects $c_0,\dots,c_n$, and any choice of labelings $\gamma_1 \in C(c_0,c_1),\dots,\gamma_n \in C(c_{n-1},c_n)$ of the intervals $I_1, \dots, I_n$, the Segal condition provides us with a ``composite'' labeling $\gamma_n \circ \dots \circ \gamma_1$ of the whole interval $I$, in a coherent way. ``Coherence'' here means that the composition is continuous in the $\gamma_i$`s, but moreover that it is associative: if we compose our labelings in two steps, for example, we get the same result as if we compose our labelings in one step: $\gamma_3 \circ (\gamma_2 \circ \gamma_1) = \gamma_3 \circ \gamma_2 \circ \gamma_1$.

A $2$-Segal space is, like a $1$-Segal space, a simplicial space, but it satisfies only a weakened version of the Segal condition. Instead of stipulating that labelings of triangualtions of 1-dimensional intervals may be coherently composed, we stipulate that labelings of triangulations of 2-dimensional polygons may be coherently composed.

Similarly $d$-Segal spaces are simplicial spaces with higher associativity data parameterized by triangulations of $d$-dimensional polyhedra.

\hypertarget{categories_with_multivalued_composition}{}\subsubsection*{{Categories with multivalued composition}}\label{categories_with_multivalued_composition}

A 2-Segal space is a ``category with multivalued composition'', or a category enriched in [[Span]]. A composite of two morphisms $a \to b \to c$ need not exist, and if it does it may not be unique. But whatever composites there are satisfy all ``higher associativity conditions'' one could want.

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

\hypertarget{DyckerhoffKapranov}{}\subsubsection*{{Dyckerhoff-Kapranov}}\label{DyckerhoffKapranov}

In (\hyperlink{DyckerhoffKapranov12}{DyckerhoffKapranov 12}) a 2-Segal space is defined to be a simplicial space with a higher analog of the weak composition operation known from [[Segal spaces]].

Let $X$ be a [[simplicial topological space]] or [[bisimplicial set]] or generally a [[simplicial object]] in a suitable [[simplicial model category]].

For $n \in \mathbb{N}$ let $P_n$ be the $n$-[[polygon]]. For any [[triangulation]] $T$ of $P_n$ let $\Delta^T$ be the corresponding [[simplicial set]]. Regarding $\Delta^n$ as the cellular [[boundary]] of that polygon provides a morphism of simplicial sets $\Delta^T \to \Delta^n$.

Say that $X$ is a \textbf{2-Segal object} if for all $n$ and all $T$ as above, the induced [[morphisms]]

\begin{displaymath}
X_n := [\Delta^n, X]
  \to 
  X_T := [\Delta^T,X]
\end{displaymath}
are [[weak equivalences]].

\textbf{Warning.} A Dyckerhoff-Kapranov ``2-Segal spaces'' is not itself a model for an [[(∞,2)-category]]. Instead, it is a model for an [[(∞,1)-operad]] (\hyperlink{DyckerhoffKapranov12}{Dyckerhoff-Kapranov 12, section 3.6}).

Under some conditions DW 2-Segal spaces $X_\bullet$ induce [[Hall algebra]] structures on $X_1$ (\hyperlink{DyckerhoffKapranov12}{Dyckerhoff-Kapranov 12, section 8}).

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

A central motivating example comes from $K$-theory. If $C$ is a [[Quillen-exact category]], then $S_\bullet C$ is a 2-Segal space. Here $S_\bullet$ is the [[Waldhausen S-construction]]. There is one object of $S_\bullet C$, denoted $0$. There is a morphism $0 \to 0$ for each object of $C$. A composite of in $S_\bullet C$ of two objects $c,c' \in C$ is an object $c'' \in C$ equipped with a short exact sequence $0 \to c \to c'' \to c' \to 0$. Thus the composite is generally not unique, but it does satisfy all the higher associativity conditions required of a 2-Segal space.

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

For more references along these lines see at \emph{[[n-fold complete Segal space]]}

The Dyckerhoff-Kapranov ``higher Segal spaces'' \hyperlink{DyckerhoffKapranov}{above} are discussed in

\begin{itemize}%
\item Tobias Dyckerhoff, [[Mikhail Kapranov]], \emph{Higher Segal spaces I}, (\href{http://arxiv.org/abs/1212.3563}{arxiv:1212.3563})

\end{itemize}
\begin{itemize}%
\item Tobias Dyckerhoff, \emph{Higher Segal spaces}, talk at Steklov Mathematical Institute (2011) (\href{http://www.mathnet.ru/php/presentation.phtml?presentid=3718&option_lang=eng}{video})

\end{itemize}
\begin{itemize}%
\item [[Mikhail Kapranov]], \emph{Higher Segal spaces}, talk at IHES (2012) (\href{http://www.dailymotion.com/video/xpm3at_mikhail-kapranov-a-meeting-on-the-occasion-of-the-75th-birthday-of-yuri-ivanovich-manin_tech}{video})

\end{itemize}
[[!redirects higher Segal space]] [[!redirects higher Segal spaces]]

[[!redirects 2-Segal space]] [[!redirects 2-Segal spaces]]

[[!redirects higher complete Segal space]]



\end{document}