\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*{Toposes, Triples, and Theories}

Toposes, Triples, and Theories by [[Michael Barr]], and [[Charles Wells]], was published in as Grundlehren der math. Wissenschaften 278. Springer-Verlag, 1983, and has been \href{http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html}{republished} in the [[TAC reprints series]].

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

\hypertarget{preface}{}\subsection*{{Preface}}\label{preface}

\hypertarget{1_categories}{}\subsubsection*{{1. Categories}}\label{1_categories}

\hypertarget{1_definition_of_category__________1}{}\paragraph*{{1 Definition of category \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 1}}\label{1_definition_of_category__________1}

\hypertarget{2_functors__________10}{}\paragraph*{{2 Functors \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{}\ldots{}10}}\label{2_functors__________10}

\hypertarget{3_natural_transformations_________14}{}\paragraph*{{3 Natural transformations \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 14}}\label{3_natural_transformations_________14}

\hypertarget{4_elements_and_subob_jects__________17}{}\paragraph*{{4 Elements and Subob jects \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 17}}\label{4_elements_and_subob_jects__________17}

\hypertarget{5_the_yoneda_lemma___________22}{}\paragraph*{{5 The Yoneda Lemma \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 22}}\label{5_the_yoneda_lemma___________22}

\hypertarget{6_pullbacks_____________25}{}\paragraph*{{6 Pullbacks \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 25}}\label{6_pullbacks_____________25}

\hypertarget{7_limits_____________30}{}\paragraph*{{7 Limits \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 30}}\label{7_limits_____________30}

\hypertarget{8_colimits____________40}{}\paragraph*{{8 Colimits \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 40}}\label{8_colimits____________40}

\hypertarget{9_adjoint_functors___________46}{}\paragraph*{{9 Adjoint functors \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 46}}\label{9_adjoint_functors___________46}

\hypertarget{10_filtered_colimits___________57}{}\paragraph*{{10 Filtered colimits \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 57}}\label{10_filtered_colimits___________57}

\hypertarget{11_notes_to_chapter_i__________60}{}\paragraph*{{11 Notes to Chapter I \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 60}}\label{11_notes_to_chapter_i__________60}

\hypertarget{2_toposes_62}{}\subsubsection*{{2. Toposes 62}}\label{2_toposes_62}

\hypertarget{1_basic_ideas_about_toposes_________62}{}\paragraph*{{1 Basic Ideas about Toposes \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 62}}\label{1_basic_ideas_about_toposes_________62}

\hypertarget{2_sheaves_on_a_space__________65}{}\paragraph*{{2 Sheaves on a Space \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 65}}\label{2_sheaves_on_a_space__________65}

\hypertarget{3_properties_of_toposes__________72}{}\paragraph*{{3 Properties of Toposes \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 72}}\label{3_properties_of_toposes__________72}

\hypertarget{4_the_beck_conditions__________77}{}\paragraph*{{4 The Beck Conditions \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 77}}\label{4_the_beck_conditions__________77}

\hypertarget{5_notes_to_chapter_2___________80}{}\paragraph*{{5 Notes to Chapter 2 \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 80}}\label{5_notes_to_chapter_2___________80}

\hypertarget{3_triples_82}{}\subsubsection*{{3. Triples 82}}\label{3_triples_82}

\hypertarget{1_definition_and_examples__________82}{}\paragraph*{{1 Definition and Examples \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 82}}\label{1_definition_and_examples__________82}

\hypertarget{2_the_kleisli_and_eilenbergmoore_categories_____87}{}\paragraph*{{2 The Kleisli and Eilenberg-Moore Categories \ldots{} \ldots{} \ldots{} \ldots{} 87}}\label{2_the_kleisli_and_eilenbergmoore_categories_____87}

\hypertarget{3_tripleability____________92}{}\paragraph*{{3 Tripleability \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 92}}\label{3_tripleability____________92}

\hypertarget{4_properties_of_tripleable_functors_______103}{}\paragraph*{{4 Properties of Tripleable Functors \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 103}}\label{4_properties_of_tripleable_functors_______103}

\hypertarget{5_sufficient_conditions_for_tripleability_______108}{}\paragraph*{{5 Sufficient Conditions for Tripleability \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 108}}\label{5_sufficient_conditions_for_tripleability_______108}

\hypertarget{6_morphisms_of_triples__________110}{}\paragraph*{{6 Morphisms of Triples \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 110}}\label{6_morphisms_of_triples__________110}

\hypertarget{7_adjoint_triples____________114}{}\paragraph*{{7 Adjoint Triples \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 114}}\label{7_adjoint_triples____________114}

\hypertarget{8_historical_notes_on_triples_________120}{}\paragraph*{{8 Historical Notes on Triples \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 120}}\label{8_historical_notes_on_triples_________120}

\hypertarget{4_theories_122}{}\subsubsection*{{4. Theories 122}}\label{4_theories_122}

\hypertarget{1_sketches____________123}{}\paragraph*{{1 Sketches \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 123}}\label{1_sketches____________123}

\hypertarget{2_the_ehresmannkennison_theorem________127}{}\paragraph*{{2 The Ehresmann-Kennison Theorem \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 127}}\label{2_the_ehresmannkennison_theorem________127}

\hypertarget{3_finiteproduct_theories_________129}{}\paragraph*{{3 Finite-Product Theories \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 129}}\label{3_finiteproduct_theories_________129}

\hypertarget{4_left_exact_theories___________135}{}\paragraph*{{4 Left Exact Theories \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 135}}\label{4_left_exact_theories___________135}

\hypertarget{5_notes_on_theories__________144}{}\paragraph*{{5 Notes on Theories \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 144}}\label{5_notes_on_theories__________144}

\hypertarget{5_properties_of_toposes_147}{}\subsubsection*{{5. Properties of Toposes 147}}\label{5_properties_of_toposes_147}

\hypertarget{1_tripleability_of_p___________147}{}\paragraph*{{1 Tripleability of P \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 147}}\label{1_tripleability_of_p___________147}

\hypertarget{2_slices_of_toposes___________149}{}\paragraph*{{2 Slices of Toposes \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 149}}\label{2_slices_of_toposes___________149}

\hypertarget{3_logical_functors___________151}{}\paragraph*{{3 Logical Functors \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 151}}\label{3_logical_functors___________151}

\hypertarget{4_toposes_are_cartesian_closed_________156}{}\paragraph*{{4 Toposes are Cartesian Closed \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 156}}\label{4_toposes_are_cartesian_closed_________156}

\hypertarget{5_exactness_properties_of_toposes________158}{}\paragraph*{{5 Exactness Properties of Toposes \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 158}}\label{5_exactness_properties_of_toposes________158}

\hypertarget{6_the_heyting_algebra_structure_on________165}{}\paragraph*{{6 The Heyting Algebra Structure on $\Omega$ \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 165}}\label{6_the_heyting_algebra_structure_on________165}

\hypertarget{6_permanence_properties_of_toposes_169}{}\subsubsection*{{6. Permanence Properties of Toposes 169}}\label{6_permanence_properties_of_toposes_169}

\hypertarget{1_topologies_____________169}{}\paragraph*{{1 Topologies \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 169}}\label{1_topologies_____________169}

\hypertarget{2_sheaves_for_a_topology_________174}{}\paragraph*{{2 Sheaves for a Topology \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 174}}\label{2_sheaves_for_a_topology_________174}

\hypertarget{3_sheaves_form_a_topos__________179}{}\paragraph*{{3 Sheaves form a topos \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 179}}\label{3_sheaves_form_a_topos__________179}

\hypertarget{4_left_exact_cotriples__________181}{}\paragraph*{{4 Left exact cotriples \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 181}}\label{4_left_exact_cotriples__________181}

\hypertarget{5_left_exact_triples___________184}{}\paragraph*{{5 Left exact triples \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 184}}\label{5_left_exact_triples___________184}

\hypertarget{6_categories_in_a_topos__________188}{}\paragraph*{{6 Categories in a Topos \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 188}}\label{6_categories_in_a_topos__________188}

\hypertarget{7_grothendieck_topologies__________194}{}\paragraph*{{7 Grothendieck Topologies \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 194}}\label{7_grothendieck_topologies__________194}

\hypertarget{8_girauds_theorem__________198}{}\paragraph*{{8 Giraud's Theorem \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 198}}\label{8_girauds_theorem__________198}

\hypertarget{7_representation_theorems_206}{}\subsubsection*{{7. Representation Theorems 206}}\label{7_representation_theorems_206}

\hypertarget{1_freyds_representation_theorems_______206}{}\paragraph*{{1 Freyd's Representation Theorems \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 206}}\label{1_freyds_representation_theorems_______206}

\hypertarget{2_the_axiom_of_choice__________210}{}\paragraph*{{2 The Axiom of Choice \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 210}}\label{2_the_axiom_of_choice__________210}

\hypertarget{3_morphisms_of_sites__________214}{}\paragraph*{{3 Morphisms of Sites \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 214}}\label{3_morphisms_of_sites__________214}

\hypertarget{4_delignes_theorem__________220}{}\paragraph*{{4 Deligne's Theorem \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 220}}\label{4_delignes_theorem__________220}

\hypertarget{5_natural_number_objects__________221}{}\paragraph*{{5 Natural Number Objects \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 221}}\label{5_natural_number_objects__________221}

\hypertarget{6_countable_toposes_and_separable_toposes______229}{}\paragraph*{{6 Countable Toposes and Separable Toposes \ldots{} \ldots{} \ldots{} \ldots{} . 229}}\label{6_countable_toposes_and_separable_toposes______229}

\hypertarget{7_barrs_theorem____________234}{}\paragraph*{{7 Barr's Theorem \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 234}}\label{7_barrs_theorem____________234}

\hypertarget{8_notes_to_chapter_7___________236}{}\paragraph*{{8 Notes to Chapter 7 \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 236}}\label{8_notes_to_chapter_7___________236}

\hypertarget{8_cocone_theories_238}{}\subsubsection*{{8. Cocone Theories 238}}\label{8_cocone_theories_238}

\hypertarget{1_regular_theories___________238}{}\paragraph*{{1 Regular Theories \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 238}}\label{1_regular_theories___________238}

\hypertarget{2_finite_sum_theories___________241}{}\paragraph*{{2 Finite Sum Theories \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 241}}\label{2_finite_sum_theories___________241}

\hypertarget{3_geometric_theories___________242}{}\paragraph*{{3 Geometric Theories \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . . 242}}\label{3_geometric_theories___________242}

\hypertarget{4_properties_of_model_categories________244}{}\paragraph*{{4 Properties of Model Categories \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 244}}\label{4_properties_of_model_categories________244}

\hypertarget{9_more_on_triples_250}{}\subsubsection*{{9. More on Triples 250}}\label{9_more_on_triples_250}

\hypertarget{1_duskins_tripleability_theorem________250}{}\paragraph*{{1 Duskin's Tripleability Theorem \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 250}}\label{1_duskins_tripleability_theorem________250}

\hypertarget{2_distributive_laws__________257}{}\paragraph*{{2 Distributive Laws \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} 257}}\label{2_distributive_laws__________257}

\hypertarget{3_colimits_of_triple_algebras_________262}{}\paragraph*{{3 Colimits of Triple Algebras \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 262}}\label{3_colimits_of_triple_algebras_________262}

\hypertarget{4_free_triples____________267}{}\paragraph*{{4 Free Triples \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} \ldots{} . 267}}\label{4_free_triples____________267}

\hypertarget{bibliography_273}{}\subsubsection*{{Bibliography 273}}\label{bibliography_273}

\hypertarget{index_of_exercises_278}{}\subsubsection*{{Index of exercises 278}}\label{index_of_exercises_278}

\hypertarget{index}{}\subsubsection*{{Index}}\label{index}



\end{document}