\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*{quasitopos}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{characterization}{Characterization}\dotfill \pageref*{characterization} \linebreak
\noindent\hyperlink{extensivity_and_exactness}{Extensivity and exactness}\dotfill \pageref*{extensivity_and_exactness} \linebreak
\noindent\hyperlink{internal_logic}{Internal logic}\dotfill \pageref*{internal_logic} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concept}{Related concept}\dotfill \pageref*{related_concept} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A quasitopos is a particular kind of [[category]] that has properties similar to that of a [[topos]], but is not quite a [[topos]]. A major difference is that it need not be [[balanced category|balanced]]: a [[morphism]] that is both [[monomorphism|monic]] and [[epimorphism|epic]] is not necessarily invertible. A quasitopos that is balanced is a topos.

Instead of the usual [[subobject classifier]], it has a classifier only for \emph{[[strong monomorphism|strong]]} [[subobject]]s. It satisfies the uniqueness, but not the existence, part of the sheaf axioms ([[Elephant]] A2.6).

Note that some of the literature definitions use the notion of a [[regular monomorphism]]. Since every regular monomorphism is as strong one, this article only uses [[strong monomorphism]].

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

\begin{udefn}
A \textbf{quasitopos} is a [[finitely complete category|finitely complete]], [[finitely cocomplete category|finitely cocomplete]], [[locally cartesian closed category]] $E$ in which there exists an [[object]] $\Omega$ that classifies [[strong monomorphism|strong monomorphisms]].

\end{udefn}
In particular, this means

\begin{itemize}%
\item Every finite [[limit]] and [[colimit]] exists;
\item For each morphism $f: A \to B$, the [[base change|pullback functor]] between [[over category|slice quasitoposes]],\begin{displaymath}
f^*: E/B \to E/A,
\end{displaymath}
admits a [[adjoint functor|right adjoint]];


\item There is a map $t: 1 \to \Omega$ such that every [[strong monomorphism]] $i: A \to X$ occurs as the [[pullback]] of $t$ along some unique morphism $\chi_i: X \to \Omega$:\begin{displaymath}
\itexarray{
A & \to & 1\\
i \downarrow & & \downarrow t\\
X & \overset{\chi_i}{\to} & \Omega
}
\end{displaymath}


\end{itemize}
The object $\Omega$ above is sometimes called a \textbf{strong-subobject classifier}, since it classifies strong subobjects, but also sometimes called a \textbf{weak subobject classifier}, since it satisfies a weaker property than an ordinary [[subobject classifier]].

\begin{uremark}
Equivalently, in addition to finite limits and colimits and local cartesian closure, one may ask only that there exists a classifier $t\colon 1\to\Omega$ as above for \emph{some} class $\mathcal{M}$ of [[monomorphisms]] which contains the [[regular monomorphisms]] and is closed under composition and pullback. From this one can show that every morphism factors as an epimorphism followed by a regular monomorphism (see \hyperlink{Wyler}{Wyler, proposition 12.5}). It then follows that every strong monomorphism is regular, and therefore $\mathcal{M}$ is precisely the class of strong monomorphisms.

\end{uremark}
\begin{uprop}
Let $C$ be a category with two [[Grothendieck topologies]] $J$ and $K$ such that $J\subseteq K$. The [[full subcategory]] $BiSep(C,J,K) \hookrightarrow PSh(C)$ of the [[category of presheaves]] over $C$ consisting of the [[sheaves]] for $J$ that are also [[separated presheaves|separated]] for $K$ is a quasitopos. A category equivalent to such a category is called a \textbf{Grothendieck quasitopos}, by analogy with the notion of [[Grothendieck topos]].

\end{uprop}
In particular, this includes the category of separated presheaves on a given site (if we take $J$ to be the trivial topology), and also includes all Grothendieck toposes (if we take $K=J$).

Equivalently, a Grothendieck quasitopos is a category of the form $Sep_k(\mathbf{E})$, the category of $k$-separated objects for a [[Lawvere-Tierney topology]] $k$ on a Grothendieck topos $\mathbf{E}$.

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

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

\begin{lemma}
\label{PushoutOfStrongMonos}\hypertarget{PushoutOfStrongMonos}{}
\textbf{(pushout of strong monos)}

In a quasitopos the [[pushout]] of a [[strong monomorphism]] is again a strong mono, and the resulting square is also a [[pullback]] square.

\end{lemma}
This appears as \hyperlink{Elephant}{Elephant, Lemma A.2.6.2}, using the synonym \emph{cocover} for \emph{strong monomorphism}. Since a [[topos]] is a quasitopos in which all monomorphisms are strong, this implies that the pushout of a mono in a topos is again a mono and that the resulting square is a pullback. Together with the fact that colimits are universal in a topos, this implies that a topos is an [[adhesive category]].

\begin{corollary}
\label{}\hypertarget{}{}
A quasitopos that is also a [[balanced category]] is a [[topos]].

\end{corollary}
This is \hyperlink{Elephant}{Elephant, corollary 2.6.3}.

\begin{lemma}
\label{}\hypertarget{}{}
A quasitopos has [[disjoint coproduct]]s precisely if the unique morphism $\emptyset \to *$ from the [[initial object]] to the [[terminal object]] is a [[strong monomorphism]].

\end{lemma}
This is \hyperlink{Elephant}{Elephant, corollary 2.6.5}.

\begin{defn}
\label{}\hypertarget{}{}
An [[object]] $C$ in a quasitopos is called \textbf{coarse} if for every [[bimorphism|monic epic]] morphism $f : A \to B$ every morphism $A \to C$ factors uniquely through $f$.

\end{defn}
So the coarse objects are those that see monic epic morphisms as [[isomorphism]]s, hence that do not see the failure of the quasitopos to be a [[balanced category]].

\begin{prop}
\label{}\hypertarget{}{}
In a quasitopos $\mathcal{E}$ the [[full subcategory]] on coarse objects is a [[topos]] and a [[reflective subcategory]]

\begin{displaymath}
Coarse(\mathcal{E}) \stackrel{\leftarrow}{\hookrightarrow} \mathcal{E}
  \,.
\end{displaymath}
\end{prop}
This is \hyperlink{Elephant}{Elephant, prop 2.6.12}.

\begin{lemma}
\label{}\hypertarget{}{}
If $\mathcal{E} \simeq SepPSh(C)$ is a Grothendieck quasitopos of [[separated presheaves]] on a [[site]] $C$, then $Coarse(\mathcal{E}) \simeq Sh(C)$ is the [[sheaf topos]] on $C$.

\end{lemma}
This is in \hyperlink{Elephant}{Elephant, section A4.4}.

\hypertarget{characterization}{}\subsubsection*{{Characterization}}\label{characterization}

There is a [[Giraud theorem]] characterizing Grothendieck quasitoposes:

\begin{utheorem}
Grothendieck quasitoposes are those quasitoposes which are [[locally small category|locally small]], [[cocomplete category|cocomplete]], and have a [[generating set]], or equivalently as the [[locally presentable categories]] which are locally cartesian closed and in which every \emph{strong} [[congruence]] has a [[effective quotient]].

\end{utheorem}
see C2.2.13 of the (\hyperlink{Elephant}{Elephant})

\hypertarget{extensivity_and_exactness}{}\subsubsection*{{Extensivity and exactness}}\label{extensivity_and_exactness}

A topos is always [[extensive category|extensive]] and [[exact category|exact]], but this is not the case for quasitopoi.

A quasitopos is a [[coherent category]], since it has finite colimits which are stable under pullback (since it is locally cartesian closed), and so in particular its initial object is [[strict initial object|strict]], and it has finite coproducts which are pullback-stable, but they need not be disjoint: for objects $A$ and $B$, in the pullback

\begin{displaymath}
\itexarray{P & \overset{}{\to} & B\\
  \downarrow && \downarrow\\
  A & \underset{}{\to} & A+B}
\end{displaymath}
the object $P$ need not be initial. This is easy to see from the fact that any Heyting category is a quasitopos, since then $A+B$ is the join $A\vee B$, and so the pullback is the meet $A\wedge B$, which is not in general the bottom element.

It is true, however, that such a $P$ is always a \emph{quotient} of the initial object, i.e. the unique map $0\to P$ is epic. If the map $0\to 1$ is strong monic, as it is in the ``topological'' examples, then $0$ can have no proper epimorphic images, and so coproducts are disjoint. The converse also holds, since if coproducts are disjoint then $0\to 1$ is an equalizer of the two injections $1\rightrightarrows 1+1$. A quasitopos with this property is sometimes called \textbf{solid}.

More generally, in any quasitopos $E$, we can factor $0\to 1$ into an epic followed by a strong monic, $0\to \bar{0} \to 1$. One can prove that then the [[slice category]] $E/\bar{0}$ is a Heyting algebra (i.e. a posetal quasitopos), while the [[co-slice category]] $\bar{0}/E$ is a solid quasitopos, and moreover $E$ itself is recoverable via [[Artin gluing]] from a particular functor $E/\bar{0} \to \bar{0}/E$. Thus, to a certain extent, the only interest in the theory of quasitoposes, above and beyond the theory of Heyting algebras, is in the solid ones.

By contrast, if a solid quasitopos is additionally [[exact category|exact]], and hence a [[pretopos]], then in particular it is [[balanced category|balanced]], which implies that it is in fact a topos. One can prove, however, that a quasitopos is always \emph{quasi-exact}, meaning that every \emph{strong} [[congruence]] has an [[effective quotient]].

\hypertarget{internal_logic}{}\subsubsection*{{Internal logic}}\label{internal_logic}

Like a topos, a quasitopos has an [[internal logic]], for which the usual choice is to represent propositions by \emph{strong} subobjects. The resulting internal logic fails to satisfy the [[function comprehension principle]], forcing one to distinguish between [[functions]] and [[anafunctions]].

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

\begin{itemize}%
\item Any (elementary) [[topos]] is a quasitopos. The first two properties are known, and in a topos every monomorphism is strong, so the ordinary subobject classifier works.

Conversely, if a quasitopos is also a [[balanced category]], then it is also a topos.


\item Any [[Heyting algebra]] is a quasitopos. This is in notable contrast to the case of topoi, since no nontrivial poset is a topos. The crucial distinction is that every morphism in a poset is both monic and epic, but only the identities are strong monic or strong epic.


\item The category of [[pseudotopological spaces]] is a quasitopos, as is the category of [[subsequential spaces]]. (The latter is Grothendieck, but not the former.) The category of [[topological spaces]] fails only to be locally cartesian closed. In such ``topological'' quasitopoi, the strong monics are the ``subspace inclusions'' (i.e. those monics whose source has the topology induced from the target), and the strong-subobject classifier is the two-point space with the indiscrete topology. (In particular, we cannot demand any sort of [[separation axiom]] and still have a quasitopos.)


\item The category of [[marked simplicial set]]s.


\item A category of [[concrete sheaves]] on a [[concrete site]] is a Grothendieck quasitopos. See [[local topos]].

This includes the following examples:

\begin{itemize}%
\item The category of [[simplicial complex|simplicial complexes]].


\item The category of [[diffeological space|diffeological spaces]].



\end{itemize}

\item The following examples are categories of separated presheaves for the $\neg\neg$-topology on various presheaf toposes:

\begin{itemize}%
\item The category [[M-category\#def|Mono]] of [[monomorphisms]] between sets (as presheaves on the [[interval category]] - the [[Sierpinski topos]]).


\item The category [[relation\#endorel|EndoRel or Bin]] of sets equipped with a [[relation]] (as presheaves on [[Quiv]]

\begin{displaymath}
G_1 = (0 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} 1),
\end{displaymath}
a truncation of the [[globular category]]).


\item The category of sets equipped with a reflexive relation (as presheaves on a truncated reflexive globular category).


\item The category of sets equipped with a symmetric relation (as presheaves on the full subcategory of finite sets and injections consisting of just the objects $1$, $2$).


\item The category of sets equipped with a reflexive symmetric relation (as presheaves on the full subcategory of finite sets consisting of just the objects $1$, $2$). See [[category of simple graphs]].



\end{itemize}

\item The category of [[bornological set|bornological sets]].


\item The category of assemblies of a [[partial combinatory algebra]].


\item The category of Spanier's [[quasi-topological space|quasi-topological spaces]], the category of concrete sheaves on the site consisting of compact Hausdorff spaces with the finite covering topology. See \hyperlink{DE}{Dubuc-Espanol}.



\end{itemize}
\hypertarget{related_concept}{}\subsection*{{Related concept}}\label{related_concept}

\begin{itemize}%
\item \textbf{quasitopos}


\item [[(∞,1)-quasitopos]]



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

Original articles include

\begin{itemize}%
\item [[Jacques Penon]], \emph{Quasi-topos} , C.R. Acad. Sci. Paris \textbf{276} S\'e{}rie A (1973) pp.237--240. (\href{http://gallica.bnf.fr/ark:/12148/bpt6k6217213f/f251.image}{gallica})


\item [[Jacques Penon]], \emph{Sur le quasi-topos} , Cahiers Top. G\'e{}om. Diff. 18 (1977), 181--218.



\end{itemize}
Standard textbook references are

\begin{itemize}%
\item Oswald Wyler, \emph{Lecture Notes on Topoi and Quasitopoi} , World Scientific Singapore 1991.

\end{itemize}
\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}, Oxford UP 2002.  (section A2.6)


\item J. Adamek, H. Herrlich, G. E. Strecker, \emph{Abstract and Concrete Categories} , Dover Mineola 2009. (Available online as \href{http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html}{TAC Reprint no.17} (2006) pp.1-507; section 28)



\end{itemize}
Quasi-toposes of [[concrete sheaves]] are considered in

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{Concrete quasitopoi} , Lecture Notes in Math. 753 (1979), 239--254


\item [[Eduardo Dubuc]], L. Espanol, \emph{Quasitopoi over a base category} (\href{http://arxiv.org/abs/math.CT/0612727}{arXiv:math.CT/0612727})



\end{itemize}
A review is in

\begin{itemize}%
\item [[John Baez]], [[Alex Hoffnung]], \emph{Convenient categories of smooth spaces}, to appear, Trans. AMS, (\href{http://arxiv.org/abs/0807.1704}{arXiv})

\end{itemize}
More generally, quasi-[[sheaf toposes]] are discussed in

\begin{itemize}%
\item [[Richard Garner]], [[Stephen Lack]], \emph{Grothendieck quasitoposes} , arXiv:1106.5331 (2012). (\href{http://arxiv.org/abs/1106.5331}{abstract})

\end{itemize}
[[!redirects quasitopoi]] [[!redirects quasitoposes]] [[!redirects quasi-topos]] [[!redirects quasi-topoi]] [[!redirects quasi-toposes]] [[!redirects quasi-exact category]] [[!redirects strong-subobject classifier]] [[!redirects weak subobject classifier]] [[!redirects Grothendieck quasitopos]] [[!redirects Grothendieck quasitoposes]] [[!redirects Grothendieck quasitopoi]]



\end{document}