\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{two-out-of-six property}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{relation_to_other_concepts}{Relation to other concepts}\dotfill \pageref*{relation_to_other_concepts} \linebreak
\noindent\hyperlink{2outof3}{2-out-of-3}\dotfill \pageref*{2outof3} \linebreak
\noindent\hyperlink{identities_and_isomorphisms}{Identities and isomorphisms}\dotfill \pageref*{identities_and_isomorphisms} \linebreak
\noindent\hyperlink{closure_under_retracts}{Closure under retracts}\dotfill \pageref*{closure_under_retracts} \linebreak
\noindent\hyperlink{saturation}{Saturation}\dotfill \pageref*{saturation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In a [[category]] $C$, a [[class]] $W\subseteq Mor(C)$ of [[morphisms]] is said to satisfy \textbf{2-out-of-6} if for any sequence of three composable morphisms

\begin{displaymath}
X\xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} K
\end{displaymath}
if $w v$ and $v u$ are in $W$, then so are $u$, $v$, $w$, and $w v u$.

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

\begin{itemize}%
\item The class of [[isomorphisms]] in any category satisfies 2-out-of-6. This case is the archetype of most of the cases in which the property is invoked: 2-out-of-6 is characteristic of morphisms that have a notion of [[inverse]].


\item A category equipped with a class of ``[[category with weak equivalences|weak equivalences]]'' containing the [[identity morphisms]] and satisfying 2-out-of-6 is called a [[homotopical category]]. In particular, this includes any [[model category]].



\end{itemize}
\hypertarget{relation_to_other_concepts}{}\subsection*{{Relation to other concepts}}\label{relation_to_other_concepts}

\hypertarget{2outof3}{}\subsubsection*{{2-out-of-3}}\label{2outof3}

The 2-out-of-6 property implies the [[two-out-of-three]] property. For on the one hand, if $f$ and $g$ are in $W$, then applying 2-out-of-6 to $\xrightarrow{f} \xrightarrow{1} \xrightarrow{g}$, we find that $g f\in W$. On the other hand, if $f$ and $g f$ are in $W$, then applying 2-out-of-6 to $\xrightarrow{1} \xrightarrow{f} \xrightarrow{g}$, we find that $g\in W$, and similarly if $g$ and $g f$ are in $W$.

\hypertarget{identities_and_isomorphisms}{}\subsubsection*{{Identities and isomorphisms}}\label{identities_and_isomorphisms}

If $W$ satisfies 2-out-of-6 and contains the identities (i.e. $C$ is a homotopical category), then $W$ contains all isomorphisms. For if $f$ has inverse $g$, then applying 2-out-of-6 to $\xrightarrow{g} \xrightarrow{f} \xrightarrow{g}$ we find that $g$ and $f$ are in $W$.

\hypertarget{closure_under_retracts}{}\subsubsection*{{Closure under retracts}}\label{closure_under_retracts}

The 2-out-of-6 property is closely related to the property that $W$ is closed under [[retracts]], as a [[subcategory]] of the [[arrow category]]. For instance, we have the following theorem due to \hyperlink{BlumbergMandell}{Blumberg-Mandell} (stated there in the context of [[Waldhausen categories]]):

\begin{theorem}
\label{CofibRetracts}\hypertarget{CofibRetracts}{}
Suppose a [[category with weak equivalences]] $\mathcal{C}$ has an additional class of maps called [[cofibrations]] which satisfy the following properties:

\begin{itemize}%
\item All [[pushouts]] of cofibrations exist.


\item The pushout of a cofibration that is also a weak equivalence is again a cofibration and a weak equivalence.


\item Every weak equivalence factors as a weak equivalence that is a cofibration followed by a weak equivalence that is a [[retraction]].



\end{itemize}
Then if the weak equivalences in $\mathcal{C}$ are closed under retracts, they also satisfy 2-out-of-6.

\end{theorem}
\begin{proof}
Suppose the first three assumptions on the cofibrations, and let

\begin{displaymath}
A \xrightarrow{u} B \xrightarrow{v} C \xrightarrow{w} D
\end{displaymath}
be a sequence of composable maps, with $w v$ and $v u$ weak equivalences. Factor $v u\colon A\to C$ as $A \xrightarrow{i} C' \xrightarrow{p} C$ where $i$ is a cofibration weak equivalence and $p$ is a weak equivalence with a [[section]]] $s\colon C\to C'$. Let $B'$ be the pushout

\begin{displaymath}
\itexarray{
    A & \overset{i}{\to} & C'\\
  ^u\downarrow && \downarrow^h\\
  B& \underset{k}{\to} & B'}
\end{displaymath}
Since $p i = v u$, we have a unique map $g\colon B' \to C$ such that $g h = p$ and $g k = v$. Define $f = h s$; then $g f = g h s = p s = 1_C$.

Since $i$ is a cofibration weak equivalence, so is $k$. And since $w g k = w v\colon B\to D$ is a weak equivalence, by two-out-of-three, $w g\colon B' \to D$ is also a weak equivalence. But now we have a commutative diagram

\begin{displaymath}
\itexarray{C & \overset{f}{\to} & B' & \xrightarrow{g} & C \\
  ^w\downarrow && \downarrow^{w g} && \downarrow^w\\
  D& \underset{=}{\to} & D & \underset{=}{\to} & D}
\end{displaymath}
exhibiting $w$ as a retract of $w g$ in the arrow category. Thus, by assumption $w$ is a weak equivalence. By successive applications of two-out-of-three, so are $v$, $u$, and $w v u$.

\end{proof}
Of course, there is a dual theorem for fibrations. Note that the fibrations in a [[category of fibrant objects]] satisfy (the duals of) all the above conditions. They are not implied by the axioms for the cofibrations in a [[Waldhausen category]] (the factorization axiom is what is missing), but many Waldhausen categories do satisfy them.

\hypertarget{saturation}{}\subsubsection*{{Saturation}}\label{saturation}

The 2-out-of-6 property is also closely related to the property that $W$ is \emph{saturated}, in the sense that any morphism which becomes an isomorphism in the [[localization]] $C[W^{-1}]$ is already a weak equivalence. (This is unrelated to the notion of [[saturated class of maps]] used in the theory of [[weak factorization systems]].)

Clearly saturation implies 2-out-of-6, but we also have the following two converses.

\begin{theorem}
\label{FracSat}\hypertarget{FracSat}{}
Suppose $W$ admits a [[calculus of fractions]]. Then $W$ satisfies two-out-of-six if and only if it is saturated.

\end{theorem}
\begin{proof}
This is from 7.1.20 of \emph{[[Categories and Sheaves]]}. Suppose $f\colon X\to Y$ becomes an isomorphism in $\mathcal{C}[W^{-1}]$, and represent its inverse by $Y \xrightarrow{g} X' \overset{s}{\leftarrow} X$ with $s\in W$. Then since $g f$ and $s$ represent the same morphism in $\mathcal{C}[W^{-1}]$, there is a morphism $t\colon X'\to X''$ in $W$ such that $t g f = t s$. Since $t s\in W$, it follows by 2-out-of-3 that $g f\in W$.

Now applying this same argument to $g$, we obtain an $h$ such that $h g \in W$. But then by 2-out-of-6, we have $f\in W$ as desired.

\end{proof}
\begin{theorem}
\label{CofSat}\hypertarget{CofSat}{}
Suppose $C$ has a class of ``cofibrations'' satisfying the properties in Theorem \ref{CofibRetracts}, and moreover the pushout of any weak equivalence along a cofibration is a weak equivalence. Then $W$ satisfies two-out-of-six if and only if it is saturated (and hence, if and only if it is closed under retracts).

\end{theorem}
\begin{proof}
See \hyperlink{BlumbergMandell}{Blumberg-Mandell} for details; an outline follows.

We first observe that $W$ admits a [[homotopy calculus of left fractions]], and in particular that every morphism in $\mathcal{C}[W^{-1}]$ can be represented by a zigzag $A \to C \overset{\sim}{\leftarrow} B$ in which $B\xrightarrow{\sim} C$ is a cofibration and a weak equivalence. See \hyperlink{BlumbergMandell}{Blumberg-Mandell}, section 5 for a detailed proof. The idea is that given any zigzag $A \overset{\sim}{\leftarrow} D \to B$, we factor $D\to A$ as a cofibration weak equivalence followed by a retraction weak equivalence, then push out the cofibration along $D\to B$ and use the section to obtain a map from $A$ into the pushout.

Now suppose $a\colon A\to B$ becomes an isomorphism in $C[W^{-1}]$, and represent its inverse by $B \xrightarrow{b} C \overset{c}{\leftarrow} A$ with $c$ a cofibration weak equivalence. Since the composite $A \xrightarrow{b a} C \overset{c}{\leftarrow} A$ represents $1_A$, we have $b a \in W$. Consider the following diagram where the squares are pushouts:

\begin{displaymath}
\itexarray{ && A & \overset{a}{\to} & B & \xrightarrow{b} & C \\
 &&  ^c\downarrow && \downarrow && \downarrow\\
B & \underset{b}{\to} & C& \underset{}{\to} & B' & \underset{}{\to} & C'}
\end{displaymath}
All the vertical maps are cofibration weak equivalences, by assumption. Moreover, the bottom map $C\to C'$ is a weak equivalence, since it is the pushout of the weak equivalence $b a$ along the cofibration $c$. And since the zigzag

\begin{displaymath}
B \xrightarrow{b} C \to B' \overset{\sim}{\leftarrow} B
\end{displaymath}
represents the same morphism as

\begin{displaymath}
B \xrightarrow{b} C \overset{c}{\leftarrow} A \xrightarrow{a} B
\end{displaymath}
which represents $1_B$, we have that $B\xrightarrow{b} C \to B'$ is a weak equivalence. Thus, by 2-out-of-6, $b$ is a weak equivalence, hence so is $a$ by 2-out-of-3.

\end{proof}
Of course, there is a dual theorem for fibrations.

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

\begin{itemize}%
\item [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], \emph{[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]}


\item [[Andrew Blumberg]] and [[Michael Mandell]], \emph{Algebraic $K$-theory and abstract homotopy theory}



\end{itemize}
\begin{itemize}%
\item [[Masaki Kashiwara]], [[Pierre Schapira]], \emph{[[Categories and Sheaves]]}

\end{itemize}
[[!redirects two-out-of-six]] [[!redirects two-out-of-six-property]] [[!redirects 2-out-of-6 property]] [[!redirects 2-out-of-6-property]] [[!redirects 2-out-of-6]]



\end{document}