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\begin{document}

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\section*{complete Segal space}

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

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

[[!include internal infinity-categories contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-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{CompleteSegalSpaces}{Complete Segal spaces}\dotfill \pageref*{CompleteSegalSpaces} \linebreak
\noindent\hyperlink{CompleteSegalSpaceObjects}{Complete Segal space objects}\dotfill \pageref*{CompleteSegalSpaceObjects} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{CharacterizationOfCompleteness}{Characterization of Completeness}\dotfill \pageref*{CharacterizationOfCompleteness} \linebreak
\noindent\hyperlink{ModelCategoryStructure}{Model category structure}\dotfill \pageref*{ModelCategoryStructure} \linebreak
\noindent\hyperlink{RelationToSimplicialLocalization}{Relation to simplicial localization}\dotfill \pageref*{RelationToSimplicialLocalization} \linebreak
\noindent\hyperlink{model_categories_for_presheaves}{Model categories for presheaves}\dotfill \pageref*{model_categories_for_presheaves} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{OrdinaryCategoriesAsCompleteSegalSpaces}{Ordinary categories as complete Segal spaces}\dotfill \pageref*{OrdinaryCategoriesAsCompleteSegalSpaces} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{the_construction}{The construction}\dotfill \pageref*{the_construction} \linebreak
\noindent\hyperlink{PropertiesOfTheInclusionOfCategories}{Properties of the inclusion}\dotfill \pageref*{PropertiesOfTheInclusionOfCategories} \linebreak
\noindent\hyperlink{RelativeCategoryAsCompleteSegalSpace}{Relative and Model categories as complete Segal spaces}\dotfill \pageref*{RelativeCategoryAsCompleteSegalSpace} \linebreak
\noindent\hyperlink{QuasiCategoriesAsCompleteSegal}{Quasi-categories as complete Segal spaces}\dotfill \pageref*{QuasiCategoriesAsCompleteSegal} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesGroupoidalVersion}{Groupoidal version}\dotfill \pageref*{ReferencesGroupoidalVersion} \linebreak


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

A \emph{complete Segal space} is a model for an [[internal category in an (∞,1)-category]] in [[∞Grpd]], with the latter [[presentable (∞,1)-category|presented]] by [[sSet]]/[[Top]]. So complete Segal spaces present [[(∞,1)-categories]]. They are also called \emph{Rezk categories} after [[Charles Rezk]].

More in detail, a complete Segal space $X$ is

\begin{itemize}%
\item for each $n \in \mathbb{N}$ a [[Kan complex]] $X_n$, thought of as the \emph{space of composable sequences of $n$-morphisms and their composites};


\item forming a [[simplicial object]] $X_\bullet$ in [[sSet]] (a [[bisimplicial set]]);



\end{itemize}
such that

\begin{enumerate}%
\item there is a [[composition]] operation well defined up to [[coherence|coherent]] [[homotopy]]: exibited by the [[Segal maps]]

\begin{displaymath}
X_k \to X_1 \times_{X_0} \cdots \times_{X_0} X_1
\end{displaymath}
(into the iterated [[homotopy pullback]] of the [[∞-groupoid]] of 1-morphisms over the $\infty$-groupoid of objects) being [[homotopy equivalences]]

(so far this defines a \emph{[[Segal space]]});


\item the notion of [[equivalence]] in $X_\bullet$ is compatible with that in the ambient [[∞Grpd]] (``completeness''): the sub-simplicial object $Core(X_\bullet)$ on the invertible morphisms in each degree is homotopy constant: it has all face and degeneracy maps being [[homotopy equivalences]].

(this says that if a morphism is an equivalence under the explicit composition operation then it is already a morphism in $X_0$ ).



\end{enumerate}
\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

We first discuss

\begin{itemize}%
\item \hyperlink{CompleteSegalSpaces}{Complete Segal spaces}

\end{itemize}
as such, and then the more general notion of

\begin{itemize}%
\item \hyperlink{CompleteSegalSpaceObjects}{Complete Segal space objects}

\end{itemize}
internal to a suitable model category/$(\infty,1)$-category $\mathcal{C}$ -- this reduces to the previous notion for $\mathcal{C} = sSet_{Quillen}$.

\hypertarget{CompleteSegalSpaces}{}\subsubsection*{{Complete Segal spaces}}\label{CompleteSegalSpaces}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{[[Segal space]]} is a [[simplicial object]] in [[simplicial sets]]

\begin{displaymath}
X \in [\Delta^{op}, sSet]
\end{displaymath}
such that

\begin{itemize}%
\item it is fibrant in the [[Reedy model structure]] $[\Delta^{op}, sSet_{Quillen}]_{Reedy}$;


\item it is a [[local object]] with respect to the [[spine]] inclusions $\{Sp[n] \hookrightarrow \Delta[n]\}_{n \in \mathbb{N}}$;

equivalently: for all $n \in \mathbb{N}$ the [[Segal map]]

\begin{displaymath}
X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1
\end{displaymath}
is a [[weak homotopy equivalence]] (of [[Kan complexes]], in fact).



\end{itemize}
\end{defn}
(\hyperlink{Rezk}{Rezk, 4.1}).

\begin{defn}
\label{HomotopyCategory}\hypertarget{HomotopyCategory}{}
For $X$ a Segal space, its \textbf{[[homotopy category]]} $Ho(X)$ is the [[Ho(Top)]]-[[enriched category]] whose [[objects]] are the vertices of $X_0$

\begin{displaymath}
Obj(X) = (X_0)_0
\end{displaymath}
and for $x,y \in Obj(X)$ the [[hom object]] is the [[homotopy type]] of the [[homotopy fiber product]]

\begin{displaymath}
Ho(X)(x,y) := \pi_0 \Big(\{x\} \times_{X_0} X_1 \times_{X_0} \{y\}\Big)
  \,.
\end{displaymath}
The [[composition]]

\begin{displaymath}
Ho_X(x,y) \times Ho_X(y,z) \to Ho_X(x,z)
\end{displaymath}
is the (uniquely defined) action of the [[infinity-anafunctor]]

\begin{displaymath}
X_1 \times_{X_0} X_1
  \underoverset{\simeq}{(d_2, d_0)}{\leftarrow}
  X_2
  \stackrel{d_1}{\to}
  X_1
\end{displaymath}
on these connected components.

\end{defn}
(\hyperlink{Rezk}{Rezk, 5.3})

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a Segal space, write

\begin{displaymath}
X_{hoequ} \hookrightarrow X_1
\end{displaymath}
for the inclusion of the connected components of those vertices that become [[isomorphisms]] in the homotopy category, def. \ref{HomotopyCategory}.

\end{defn}
(\hyperlink{Rezk}{Rezk, 5.7})

\begin{defn}
\label{}\hypertarget{}{}
A Segal space $X$ is called a \textbf{complete Segal space} if

\begin{displaymath}
s_0 : X_0 \to X_{hoequ}
\end{displaymath}
is a weak equivalence.

\end{defn}
(\hyperlink{Rezk}{Rezk, 6.})

\begin{remark}
\label{}\hypertarget{}{}
This condition is equivalent to $X$ being a [[local object]] with respect to the morphism $N(\{0 \stackrel{\simeq}{\to} 1\}) \to *$. This is discussed \hyperlink{CharacterizationOfCompleteness}{below}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The completeness condition may also be thought of as \emph{[[univalence]]}. See there for more.

\end{remark}
\hypertarget{CompleteSegalSpaceObjects}{}\subsubsection*{{Complete Segal space objects}}\label{CompleteSegalSpaceObjects}

(\ldots{})

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

\hypertarget{CharacterizationOfCompleteness}{}\subsubsection*{{Characterization of Completeness}}\label{CharacterizationOfCompleteness}

\begin{theorem}
\label{}\hypertarget{}{}
A Segal space $X$ is a complete Segal space precisely if it is a [[local object]] with respect to the morphism $N(0 \stackrel{\simeq}{\to} 1) \to *$, hence precisely if with respect to the canonical [[sSet]]-[[enriched category|enriched]] [[hom objects]] we have that

\begin{displaymath}
X_0 
   \simeq 
  [\Delta^{op}, sSet](*, X) 
    \to 
  [\Delta^{op}, sSet](N(0 \stackrel{\simeq}{\to} 1), X)
\end{displaymath}
is a weak equivalence.

\end{theorem}
(\hyperlink{Rezk}{Rezk, theorem 6.2})

\hypertarget{ModelCategoryStructure}{}\subsubsection*{{Model category structure}}\label{ModelCategoryStructure}

The category $[\Delta^{op}, sSet]$ of [[simplicial presheaves]] on the [[simplex category]] ([[bisimplicial sets]]) supports a [[model category]] structure whose fibrant objects are precisely the complete Segal spaces: the \emph{[[model structure for complete Segal spaces]]}. This presents the [[(∞,1)-category of (∞,1)-categories]].

\hypertarget{RelationToSimplicialLocalization}{}\subsubsection*{{Relation to simplicial localization}}\label{RelationToSimplicialLocalization}

Given $\mathcal{C}$ a [[category with weak equivalences]] $\mathcal{W} \subset Mor(\mathcal{C})$ or more generally a ``[[relative category]]'', then there is canonically a complete Segal space associated with it by the ``relative nerve'' construction, def. \ref{RelativeNerve} followed by fibrant replacement, and hence by the \hyperlink{ModelCategoryStructure}{model structure} this determines an [[(∞,1)-category]] $N_{Rezk}(\mathcal{C},\mathcal{W})$ .

On the other hand classical [[simplicial localization]]-theory provides several ways (e.g. [[hammock localization]]) to turn $(\mathcal{C}, \mathcal{W})$ into an [[(∞,1)-category]] $\mathcal{C}[\mathcal{W}]^{-1}$ which universally turns the elements in $\mathcal{W}$ into [[homotopy equivalences]].

\begin{theorem}
\label{}\hypertarget{}{}
These constructions are compatible in that there is an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
N_{Rezk}(\mathcal{C},\mathcal{W})
  \simeq
  \mathcal{C}[\mathcal{W}^{-1}]
  \,.
\end{displaymath}
\end{theorem}
For [[simplicial model categories]] this is (\hyperlink{{#Rezk}}{Rezk, theorem 8.3}. For general [[model categories]] this is (\hyperlink{Bergner07}{Bergner 07, theorem 6.2}). For the fully general case this follows from results by [[Clark Barwick]], [[Daniel Kan]] and [[Bertrand Toën]] as pointed out by [[Chris Schommer-Pries]] \href{http://mathoverflow.net/a/93139/381}{here on MathOverflow}.

\hypertarget{model_categories_for_presheaves}{}\subsubsection*{{Model categories for presheaves}}\label{model_categories_for_presheaves}

There is a notion of [[right/left fibration of complete Segal spaces]] analogous to [[right/left Kan fibrations]] for [[quasi-categories]].

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

We discuss some examples. For more and more basic examples see also at \emph{\href{Segal+space#Examples}{Segal space -- Examples}}.

\hypertarget{OrdinaryCategoriesAsCompleteSegalSpaces}{}\subsubsection*{{Ordinary categories as complete Segal spaces}}\label{OrdinaryCategoriesAsCompleteSegalSpaces}

We discuss how an ordinary [[small category]] is naturally regarded as a complete Segal space. (\hyperlink{Rezk}{Rezk, 3.5})

\hypertarget{preliminaries}{}\paragraph*{{Preliminaries}}\label{preliminaries}

We need the following basic ingredients.

Write $(-)^{(-)} : Cat^{op} \times Cat \to Cat$ for the [[internal hom]] in [[Cat]], sending two categories $A$, $X$ to the [[functor category]] $X^A = Func(A,X)$.

By the discussion at [[nerve]] we have a canonical functor

\begin{displaymath}
\Delta \hookrightarrow Cat
\end{displaymath}
including the [[simplex category]] into [[Cat]] by regarding the [[simplex]] $\Delta[n]$ as the category generated from $n$ consecutive morphisms.

The [[nerve]] itself is then then functor

\begin{displaymath}
N : Cat \to sSet
\end{displaymath}
to [[sSet]] sending a category $C$ to

\begin{displaymath}
N(C) : k \mapsto C^{\Delta[k]}
  \,.
\end{displaymath}
Its restriction along $Grpd \hookrightarrow Cat$ to [[groupoids]] lands in [[Kan complexes]] $KanCplx \hookrightarrow$ [[sSet]].

The [[core]] operation is the functor

\begin{displaymath}
Core : Cat \to Grpd
\end{displaymath}
[[right adjoint]] to the inclusion of [[Grpd]] into [[Cat]]. It sends a category to the groupoid obtained by discarding all non-invertible morphisms.

\hypertarget{the_construction}{}\paragraph*{{The construction}}\label{the_construction}

Let $C$ be a [[small category]]. Define

\begin{displaymath}
\mathbf{C} \in [\Delta^{op}, sSet]
\end{displaymath}
by

\begin{displaymath}
\mathbf{C}_k := N(Core(C^{\Delta[k]}))  
  \,.
\end{displaymath}
In degree 0 this is the the [[core]] of $C$ itself. In degree 1 it is the groupoid $\mathbf{C}_1$ underlying the [[arrow category]] of $C$.

One sees that the [[source]] and [[target]] functors $s, t : C^{\Delta[1]} \to C$ are [[isofibrations]] and hence their image under [[core]] and [[nerve]] are [[Kan fibrations]]. Therefore it follows that the [[homotopy pullback]] (see there) $\mathbf{C}_1 \times_{\mathbf{C}_0} \cdots \times_{\mathbf{C}_0} \mathbf{C}_1$ is given already be the ordinary [[pullback]] in the [[1-category]] [[Grpd]]. Using this, it is immediate that for all $k$ the functors

\begin{displaymath}
Core(C^{\Delta[k]}) \to Core(C^{\Delta[1]}) \times_{Core(C)} \cdots \times_{Core(C)} Core(C^{\Delta[1]})
\end{displaymath}
are [[isomorphisms]], and so in particular

\begin{displaymath}
\mathbf{C}_k 
   \to 
  \mathbf{C}_1 \times_{\mathbf{C}_0} \cdots \times_{\mathbf{C}_0} \mathbf{C}_1
\end{displaymath}
is an equivalence.

It is clear that the [[composition]] operation in the complete Segal space defined this way ``is'' the composition in $C$. In particular the morphisms that are invertible under this composition are precisely those that are already invertible in $C$. Therefore we have the core simplicial object

\begin{displaymath}
Core(\mathbf{C}) : k \mapsto N(Core(C)^{\Delta[k]})
   = N(Core(C))^{\Delta[k]}
  \,,
\end{displaymath}
where, note, now we \emph{first} take the core of $C$ and then form morphism categories.

This simplicial Kan complex has in each positive degree a [[path space object]] for the [[Kan complex]] $N(Core(C))$.

Notably (since $\Delta[k]$ is [[weak homotopy equivalence|weak homotopy equivalent]] to the point) it follows that indeed all the face and degeneracy maps are weak homotopy equivalences.

So for every category $C$, the simplicial object $\mathbf{C}$ constructed as above is a complete Segal space. This construction extends to a functor $Cat \to completeSegalSpace$ and this is homotopy full and faithful.

\hypertarget{PropertiesOfTheInclusionOfCategories}{}\paragraph*{{Properties of the inclusion}}\label{PropertiesOfTheInclusionOfCategories}

Write

\begin{displaymath}
Sing_J : Cat \to [\Delta^{op}, sSet]
\end{displaymath}
for the functor just defined

\begin{prop}
\label{}\hypertarget{}{}
For $C$ and $D$ two categories, there are [[natural isomorphisms]]

\begin{displaymath}
Sing_J(C \times D)
  \simeq
  Sing_J(C) \times Sing_J(D)
\end{displaymath}
and

\begin{displaymath}
Sing_J(D^C) \simeq (Sing_J D)^{Sing_J C}
  \,.
\end{displaymath}
A functor $f : C \to D$ is an [[equivalence of categories]] precisely if $Sing_J(f)$ is an equivalence in the [[Reedy model structure]] $[\Delta^{op}, sSet]_{Reedy}$ (hence is degreewise a [[weak homotopy equivalence]] of Kan complexes).

\end{prop}
This appears as (\hyperlink{Rezk}{Rezk, theorem 3.7}).

\hypertarget{RelativeCategoryAsCompleteSegalSpace}{}\subsubsection*{{Relative and Model categories as complete Segal spaces}}\label{RelativeCategoryAsCompleteSegalSpace}

Let $C$ be a category with a class $W \subset Mor(C)$ of [[category with weak equivalences|weak equivalences]]. For instance, $C$ could be a [[model category]] or (much) more generally a ``[[relative category]]''. Then the \hyperlink{OrdinaryCategoriesAsCompleteSegalSpaces}{above} construction has the following evident variant.

\begin{prop}
\label{RelativeNerve}\hypertarget{RelativeNerve}{}
Let $N(C,W) \in [\Delta^{op}, sSet]$ be given by

\begin{displaymath}
N(C,W) : n \mapsto N(Core_W(C^{\Delta[n]})) \,,
\end{displaymath}
where now $Core_W(-)$ denotes the [[subcategory]] on those [[natural transformations]] whose components are [[weak equivalences]] in $C$.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
The typical [[model category]] is not a [[small category]] with respect to the base choice of [[universe]]. In this case $N(C,W)$ will be a ``[[large set|large]]'' bisimplicial set. In other words, one needs to employ some [[universe enlargement]] to interpret this definition.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
If $C$ is a model category, then $Core_W(C^{\Delta[n]})$ is the subcategory of weak equivalences in any of the standard [[model structures on functors]] on $C^{\Delta[n]}$. By a \href{/nlab/show/%28infinity%2C1%29-categorical+hom-space#SpacesOfEquivalences}{classical fact} discssed at \emph{[[(∞,1)-categorical hom-space]]}, its [[nerve]] is a model for the [[core]] of the corresponding [[(∞,1)-category of (∞,1)-functors]].

\end{remark}
The bisimplicial set $N(C,W)$ is not, in general, a complete Segal space. It does, however, represent the same [[(∞,1)-category]] as the [[simplicial localization]] of $C$ at $W$; see \href{http://mathoverflow.net/questions/92916/does-the-classification-diagram-localize-a-category-with-weak-equivalences/93139}{this MO question}.

We can, of course, always reflect $N(C,W)$ into a complete Segal space by passing to a [[fibrant replacement]] in the [[model structure for complete Segal spaces]]. But something better is true here: it suffices to make a \emph{[[Reedy model structure|Reedy]] fibrant} replacement (which does not change the [[homotopy type]] of the simplicial sets $N(Core_W(C^{\Delta[n]}))$, but only ``arranges them more nicely'').

\begin{prop}
\label{}\hypertarget{}{}
Any [[Reedy model structure|Reedy fibrant replacement]] of $N(C,W)$ is a complete Segal space.

\end{prop}
This is (\hyperlink{Rezk}{Rezk, theorem 8.3}).

\hypertarget{QuasiCategoriesAsCompleteSegal}{}\subsubsection*{{Quasi-categories as complete Segal spaces}}\label{QuasiCategoriesAsCompleteSegal}

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
\Delta_J : \Delta \to sSet
\end{displaymath}
for the [[cosimplicial object|cosimplicial]] [[simplicial set]] that sends $[n]$ to the [[nerve]] of the [[codiscrete groupoid]] on $n+1$ objects

\begin{displaymath}
\Delta_J[n] = N(0 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n)
  \,.
\end{displaymath}
Write

\begin{displaymath}
Sing_J : sSet \to [\Delta^{op}, sSet]
\end{displaymath}
for the functor given by

\begin{displaymath}
Sing_J(X)_n = Hom_{sSet}(\Delta[n] \times \Delta_J[\bullet], X)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
For $X \in sSet$ a [[quasi-category]]/[[inner Kan complex]], $Sing_J(X)$ is a complete Segal space.

\end{prop}
See at \emph{[[model structure for dendroidal complete Segal spaces]]} the section \emph{\href{model+structure+for+dendroidal+complete+Segal+spaces#QuasiOperadsToDendroidalCompleteSegal}{Quasi-operads to dendroidal complete Segal spaces}}

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

\begin{itemize}%
\item [[semi-Segal space]], [[Segal space]]


\item [[model structure for complete Segal spaces]]


\item [[higher complete Segal space]], [[dendroidal complete Segal space]]


\item [[table - models for (infinity,1)-operads]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Complete Segal spaces were originally defined in

\begin{itemize}%
\item [[Charles Rezk]], \emph{A model for the homotopy theory of homotopy theory} , Trans. Amer. Math. Soc., 353(3), 973-1007 (\href{http://www.math.uiuc.edu/~rezk/rezk-ho-models-final-changes.pdf}{pdf})

\end{itemize}
The relation to [[quasi-categories]] is discussed in

\begin{itemize}%
\item [[André Joyal]], [[Myles Tierney]], \emph{Quasi-categories vs Segal spaces} (\href{http://arxiv.org/abs/math/0607820}{arXiv:0607820})

\end{itemize}
Further discussion of the relation to [[simplicial localization]] is in

\begin{itemize}%
\item [[Julia Bergner]], \emph{Complete Segal spaces arising from simplicial categories} (\href{http://arxiv.org/abs/0704.1624}{arXiv:0704.1624})

\end{itemize}
A survey of the definition and its relation to equivalent definitions is in section 4 of

\begin{itemize}%
\item [[Julia Bergner]], \emph{A survey of $(\infty, 1)$-categories} (\href{http://arxiv.org/abs/math.AT/0610239}{arXiv}).

\end{itemize}
See also pages 29 to 31 of

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

\end{itemize}
For literature on the variants and refinements see at \emph{[[Theta space]]} and \emph{[[n-fold complete Segal space]]}.

Related MathOverflow discussion includes

\begin{itemize}%
\item \href{http://mathoverflow.net/q/92916/381}{Does the classification diagram localize a category with weak equivalences?}

\end{itemize}
\hypertarget{ReferencesGroupoidalVersion}{}\subsubsection*{{Groupoidal version}}\label{ReferencesGroupoidalVersion}

The groupoidal version of complete Segal spaces (that modelling just [[groupoid objects in an (∞,1)-category]] instead of general [[category objects in an (∞,1)-category]]) is discussed in

\begin{itemize}%
\item [[Julia Bergner]], \emph{Adding inverses to diagrams encoding algebraic structures}, Homology, Homotopy Appl. 10(2), 2008, 149-174 (\href{http://arxiv.org/abs/math/0610291}{arXiv:math/0610291})


\item [[Julia Bergner]], \emph{Adding inverses to diagrams II: Invertible homotopy theories are spaces}, Homology, Homotopy and Applications, vol. 10(1) 2008 (\href{http://www.math.ucr.edu/~jbergner/Groupoid.pdf}{pdf})



\end{itemize}
[[!redirects complete Segal spaces]] [[!redirects Rezk category]] [[!redirects Rezk categories]]



\end{document}