\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*{internally projective object}

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

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

[[!include 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{equivalent_characterizations}{Equivalent characterizations}\dotfill \pageref*{equivalent_characterizations} \linebreak
\noindent\hyperlink{projectivity_versus_internal_projectivity}{Projectivity versus internal projectivity}\dotfill \pageref*{projectivity_versus_internal_projectivity} \linebreak
\noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

An [[object]] of a [[category]] (usually a [[topos]] or [[pretopos]]) is \emph{internally projective} if it satisfies the [[internalization]] of the condition on a [[projective object]].

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

An object $E$ of a [[topos]] $\mathcal{T}$ is called \textbf{internally projective} if the [[internal hom]]/[[exponential object]] [[functor]]

\begin{displaymath}
(-)^E \colon \mathcal{T} \to \mathcal{T}
\end{displaymath}
preserves [[epimorphisms]].

\hypertarget{equivalent_characterizations}{}\subsection*{{Equivalent characterizations}}\label{equivalent_characterizations}

\begin{theorem}
\label{DepProd}\hypertarget{DepProd}{}
$E$ is internally projective if and only if the [[dependent product]] functor (the [[right adjoint]] to [[pullback]])

\begin{displaymath}
\Pi_E \colon \mathcal{T}/E \to \mathcal{T}
\end{displaymath}
preserves epimorphisms.

\end{theorem}
\begin{proof}
The functor $(-)^E$ is the composite $\Pi_E \circ E^*$, and pullback $E^*\colon \mathcal{T} \to \mathcal{T}/E$ preserves epis, so if $\Pi_E$ preserves epis then so does $(-)^E$.

Conversely, for any $f:A\to B$ in $\mathcal{T}/E$, we have a pair of pullback squares:

\begin{displaymath}
\itexarray{
  \Pi_E A & \to & A^E \\
  \downarrow & & \downarrow\\
  \Pi_E B & \to & B^E \\
  \downarrow & & \downarrow\\
  1 & \xrightarrow{id_A} & A^A
}
\end{displaymath}
The lower square and outer rectangle are easily seen to be pullbacks, hence so is the upper square. Since epimorphisms are closed under pullback, the other implication follows.

\end{proof}
\begin{lemma}
\label{Stable}\hypertarget{Stable}{}
If $E$ is internally projective in $\mathcal{T}$, then $I^* E$ is internally projective in $\mathcal{T}/I$.

\end{lemma}
\begin{proof}
This proof was given by [[Alex Simpson]] \href{http://mathoverflow.net/a/140262/49}{here}. Let $q:B\to A$ be an epimorphism from $v:B\to I$ to $u:A\to I$. Then the exponential $u^{I^*E}$ in $\mathcal{T}/I$ is the equalizer of

\begin{displaymath}
I\times A^E \xrightarrow{pr_2} A^E \xrightarrow{u} I^E
\end{displaymath}
and

\begin{displaymath}
I\times A^E \xrightarrow{pr_1} I \xrightarrow{const} I^E
\end{displaymath}
equipped with the obvious projection to $I$. Now we have a pullback square

\begin{displaymath}
\itexarray{
  v^{I^*E} & \to & I\times B^E \\
  \downarrow & & \downarrow \\
  u^{I^*E} & \to & I\times A^E
}
\end{displaymath}
in which the right-hand map is epi since $E$ is internally projective; hence so is the left-hand map.

\end{proof}
\begin{theorem}
\label{StackSem}\hypertarget{StackSem}{}
$E$ is internally projective if and only if the statement ``$E$ is projective'' is true in the [[stack semantics]] of $\mathcal{T}$.

\end{theorem}
\begin{proof}
By definition, truth of ``$E$ is projective'' in the stack semantics means that for any $I$ and any epimorphism $A\to I\times E$, there exists an epimorphism $p:J\to I$ and a [[section]] of $(p\times id)^*A \to J\times E$. (We have used the characterization that an object $X$ is projective just when every epimorphism with codomain $X$ is split.)

If $E$ is internally projective, then given an epimorphism $A\to I\times E$ as above, let $J = \Pi_{pr_1}(A)$, where $pr_1:I\times E \to I$ is the projection. Since $I\times E$ is internally projective in $\mathcal{T}/I$ by Lemma \ref{Stable}, $p:J\to I$ is an epimorphism. And $(p\times id)^*A \to J\times E$ is split by the [[counit]] of the adjunction $(pr_1)^* \dashv \Pi_{pr_1}$.

Conversely, suppose the above condition holds, and let $e:B\to A$ be an epimorphism in $\mathcal{T}/E$. Let $I = \Pi_E(A)$, and let $C$ be the pullback of $e$ along the counit $\Pi_E(A) \times E\to A$. Then $\Pi_E(B)$ is equivalently $\Pi_{pr_1}(C)$, where $pr_1:I \times E \to I$ is the projection.

Morevoer, $C \to I\times E$ is epi, so by assumption there is an epi $p:J\to I$ such that $(p\times id)^*C \to J\times E$ is split. Since pullback along epis reflects epis, it suffices to show that $p^* \Pi_{pr_1}(C)$ is split. However, we have a pullback square

\begin{displaymath}
\itexarray{
  J\times E & \to & I\times E \\
  ^{pr_1}\downarrow & & \downarrow^{pr_1} \\
  J & \to & I
}
\end{displaymath}
so by the [[Beck-Chevalley condition]], $p^* \Pi_{pr_1}(C)$ is equivalently $\Pi_{pr_1}(p\times id)^* C$. But $(p\times id)^*C$ is split, and all functors preserve split epis.

\end{proof}
Note that Lemma 4.5.3(iii) of [[Sketches of an Elephant]] is the special case of the above stack-semantics version of internal projectivity when $I=1$. This is insufficient for the implication (iii)$\Rightarrow$(ii) of that lemma to hold, since if so, then every projective object would be internally projective, which as we show below is not the case.

\hypertarget{projectivity_versus_internal_projectivity}{}\subsection*{{Projectivity versus internal projectivity}}\label{projectivity_versus_internal_projectivity}

If the [[terminal object]] in $\mathcal{T}$ is [[projective object|projective]], then every internally projective object is projective. In the converse direction,

\begin{prop}
\label{enough}\hypertarget{enough}{}
If $\mathcal{T}$ has [[projective object|enough projectives]] and projectives are closed under binary [[products]], then every projective object is internally projective. (In particular, if all objects of $\mathcal{T}$ are projective then all objects are internally projective.)

\end{prop}
\begin{proof}
Let $P$ be a projective object. To show that $e^P \colon E^P \to B^P$ is epic whenever $e \colon E \to B$ is epic, choose an epi $\phi \colon P' \to B^P$ where $P'$ is projective (using the assumption of enough projectives). Since $P \times P'$ is projective, there exists a lift through $e$ of the horizontal composite as shown:

\begin{displaymath}
\itexarray{
 & & & & E \\
 & & & & \downarrow ^\mathrlap{e} \\ 
P' \times P & \underset{\phi \times 1}{\to} & B^P \times P & \underset{eval}{\to} & B; 
}
\end{displaymath}
this, by [[currying]], provides a lift of $\phi \colon P' \to B^P$ through $e^P$. Since $\phi$ is epic, this immediately implies $e^P$ is epic, as desired.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Proposition \ref{enough} may fail without the assumption that projective objects are closed under binary products. An example is given \href{/nlab/show/presentation+axiom#counter}{here}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The \emph{internal axiom of choice} (that is, the [[axiom of choice]] interpreted in the [[internal logic]] of the topos) is equivalent to the statement that every object is internally projective. This is strictly weaker than the ``external'' axiom of choice that every [[epimorphism]] in the topos is [[split epimorphism|split]].

\end{remark}
\begin{cor}
\label{presheaf}\hypertarget{presheaf}{}
In a presheaf topos $Set^{C^{op}}$, if $C$ has binary products, then every projective presheaf is internally projective.

\end{cor}
\begin{proof}
Representables, and arbitrary coproducts of representables, are projective, and every presheaf is covered by some coproduct of representables. This implies that projective presheaves are precisely retracts of coproducts of representables. Under the assumption that $C$ has binary products, coproducts of representables, and also their retracts, are also closed under binary products. Thus projective presheaves are closed under binary products. Now apply Proposition \ref{enough}.

\end{proof}
\hypertarget{discussion}{}\subsection*{{Discussion}}\label{discussion}

\begin{itemize}%
\item Forum discussions \href{http://nforum.ncatlab.org/discussion/3008/internally-projective-objects/}{one}, \href{http://nforum.ncatlab.org/discussion/4342/internally-projective-objects/}{two}.

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

\begin{itemize}%
\item [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

\begin{itemize}%
\item [[projective module]]


\item [[internally projective object]]



\end{itemize}

\item [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

\begin{itemize}%
\item [[injective module]]

\end{itemize}

\item [[free object]], [[free resolution]]

\begin{itemize}%
\item [[free module]]

\end{itemize}

\item flat object, [[flat resolution]]

\begin{itemize}%
\item [[flat module]]

\end{itemize}


\end{itemize}
[[!redirects internally projective objects]]



\end{document}