\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*{supermanifold}

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

\hypertarget{supergeometry}{}\paragraph*{{Supergeometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{as_locally_representable_sheaves_on_super_cartesian_spaces}{As locally representable sheaves on super Cartesian spaces}\dotfill \pageref*{as_locally_representable_sheaves_on_super_cartesian_spaces} \linebreak
\noindent\hyperlink{AsLocallyRingedSpace}{As locally ringed spaces}\dotfill \pageref*{AsLocallyRingedSpace} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{AsLocallyRingedSpacesProperties}{Properties}\dotfill \pageref*{AsLocallyRingedSpacesProperties} \linebreak
\noindent\hyperlink{ModeledOnGrassmannAlgebra}{As manifolds modeled on Grassman algebras}\dotfill \pageref*{ModeledOnGrassmannAlgebra} \linebreak
\noindent\hyperlink{OverSuperpoints}{As manifolds over the base topos on superpoints}\dotfill \pageref*{OverSuperpoints} \linebreak
\noindent\hyperlink{definition_3}{Definition}\dotfill \pageref*{definition_3} \linebreak
\noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \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{as_locally_ringed_spaces_2}{As locally ringed spaces}\dotfill \pageref*{as_locally_ringed_spaces_2} \linebreak
\noindent\hyperlink{ReferencesOverSuperpoints}{As manifolds over superpoints}\dotfill \pageref*{ReferencesOverSuperpoints} \linebreak
\noindent\hyperlink{as_manifolds_modelled_on_grassmann_algebras}{As manifolds modelled on Grassmann algebras}\dotfill \pageref*{as_manifolds_modelled_on_grassmann_algebras} \linebreak
\noindent\hyperlink{other}{Other}\dotfill \pageref*{other} \linebreak


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

A [[supermanifold]] is a [[space]] locally modeled on [[Cartesian space]]s and [[superpoints]].

There are different approaches to the definition and theory of supermanifolds in the literature. The definition

\begin{itemize}%
\item \hyperlink{AsLocallyRingedSpace}{As locally ringed spaces}

\end{itemize}
is popular. The definition

\begin{itemize}%
\item \hyperlink{OverSuperpoints}{As manifolds over superpoints}

\end{itemize}
has been argued to have advantages, see also the references at [[super ∞-groupoid]].

\hypertarget{as_locally_representable_sheaves_on_super_cartesian_spaces}{}\subsection*{{As locally representable sheaves on super Cartesian spaces}}\label{as_locally_representable_sheaves_on_super_cartesian_spaces}

See at \emph{[[geometry of physics -- supergeometry]]} the section \emph{\href{geometry+of+physics+--+supergeometry#Supermanifolds}{Supermanifolds}}.

\hypertarget{AsLocallyRingedSpace}{}\subsection*{{As locally ringed spaces}}\label{AsLocallyRingedSpace}

We discuss a description of supermanifolds that goes back to (\href{BerezinLeites}{BerezinLeites}).

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

\begin{defn}
\label{SupermanifoldLocallyRingedSpace}\hypertarget{SupermanifoldLocallyRingedSpace}{}
A \textbf{supermanifold} $X$ of dimension $p|q$ is a [[ringed space]] $(|X|, O_X)$ where

\begin{itemize}%
\item the [[topological space]] $|X|$ is [[second countable space]], [[Hausdorff space]],


\item $O_X$ is a [[sheaf]] of commutative [[super algebras]] that is locally on small enough open subsets $U \subset |X|$ isomorphic to one of the form $C^\infty(\mathbb{R}^p) \otimes \wedge^\bullet \mathbb{R}^q$.



\end{itemize}
A [[morphism]] of supermanifolds is a [[homomorphism]] of [[ringed spaces]] (\ldots{}).

\end{defn}
Forgetting the graded part by projecting out the [[nilpotent ideal]] in $O_X$ (i.e. applying the [[bosonic modality]]) yields the underlying ordinary [[smooth manifold]] $X_{red}$.

One just writes $C^\infty(X)$ for the [[super algebra]] $O_X(X)$ of global sections.

With the obvious morphisms of [[ringed space]] this forms the [[category]] [[SDiff]] of supermanifolds.

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

\begin{itemize}%
\item [[super Cartesian space]]

\begin{itemize}%
\item [[superpoint]]

\begin{itemize}%
\item [[odd line]]

\end{itemize}


\end{itemize}


\end{itemize}
\begin{example}
\label{}\hypertarget{}{}
For $E \to X$ a smooth finite-rank [[vector bundle]] the [[manifold]] $X$ equipped with the [[Grassmann algebra]] over $C^\infty(X)$ of the sections of the dual bundle

\begin{displaymath}
O_X(U) := \Gamma (\wedge^\bullet(E^*))
\end{displaymath}
is a supermanifold. This is usually denoted by $\Pi E$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
In particular, let $\mathbb{R}^{p+q} \to \mathbb{R}^p$ be the trivial rank $q$ [[vector bundle]] on $\mathbb{R}^p$ then one writes

\begin{displaymath}
\mathbb{R}^{p|q} := \Pi (\mathbb{R}^{p+q} \to \mathbb{R}^p)
\end{displaymath}
for the corresponding supermanifold.

\end{example}
\hypertarget{AsLocallyRingedSpacesProperties}{}\subsubsection*{{Properties}}\label{AsLocallyRingedSpacesProperties}

\begin{theorem}
\label{BatchelorTheorem}\hypertarget{BatchelorTheorem}{}
\textbf{(Batchelor's theorem)}

Every supermanifold is [[isomorphism|isomorphic]] to one of the form $\Pi E$ where $E$ is an ordinary smooth [[vector bundle]].

\end{theorem}
\begin{remark}
\label{}\hypertarget{}{}
Nevertherless, the category of supermanifolds is far from being [[equivalence of categories|equivalent]] to that of [[vector bundles]]: a morphism of vector bundles translates to a morphism of supermanifolds that is strictly homogeneous in degrees, while a general morphism of supermanifolds need not be of this form.

\end{remark}
But we have the following useful characterization of morphisms of supermanifolds:

\begin{theorem}
\label{MorphismsOfSupermanifoldsByAlgebras}\hypertarget{MorphismsOfSupermanifoldsByAlgebras}{}
\begin{itemize}%
\item There is a [[natural bijection]]

\begin{displaymath}
SDiff(X,Y) \simeq SAlgebras(C^\infty(Y), C^\infty(X)),
\end{displaymath}
so the contravariant embedding of supermanifolds into superalgebra is a [[full and faithful functor]].


\item Composition with the standard coordinate functions on $\mathbb{R}^{p|q}$ yields an [[isomorphism]]

\begin{displaymath}
SDiff(X, \mathbb{R}^{p|q})
  \simeq
  \underbrace{
  (C^\infty(X)^{ev} \times \cdots 
  \times C^\infty(X)^{ev})}_{p\; times}
  \times
  \underbrace{
    (C^\infty(X)^{odd} \times \cdots 
    \times C^\infty(X)^{odd})}_{q\; times}
\end{displaymath}


\end{itemize}
\end{theorem}
\begin{proof}
The first statement is a direct extension of the classical fact that [[smooth manifolds embed into formal duals of R-algebras]].

\end{proof}
\hypertarget{ModeledOnGrassmannAlgebra}{}\subsection*{{As manifolds modeled on Grassman algebras}}\label{ModeledOnGrassmannAlgebra}

We discuss a desription of supermanifolds that goes back to (\hyperlink{DeWitt92}{DeWitt 92}) and (\hyperlink{Rogers}{Rogers}).

(\ldots{})

\hypertarget{OverSuperpoints}{}\subsection*{{As manifolds over the base topos on superpoints}}\label{OverSuperpoints}

Let $SuperPoint$ be the [[category]] of [[superpoint]]s. And $Sh(SuperPoint) = PSh(SuperPoint)$ its [[presheaf topos]].

We discuss a definition of supermanifolds that characterizes them, roughly, as manifolds over this [[base topos]]. See (\hyperlink{Sachse}{Sachse}) and the references at [[super ∞-groupoid]].

See also \href{http://theoreticalatlas.wordpress.com/2013/07/26/john-huerta-supermanifolds/}{this post} at Theoretical Atlas.

\hypertarget{definition_3}{}\subsubsection*{{Definition}}\label{definition_3}

\begin{defn}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
SuperSet := Sh(SuperPoint)
\end{displaymath}
be the [[sheaf topos]] over [[superpoints]]. Let

\begin{displaymath}
\mathbb{R} \in Ring(SuperSet)
\end{displaymath}
be the canonical [[continuum]] [[real line]] under the restricted [[Yoneda embedding]] of supermanifolds and equipped with its canonical internal algebra structure, hence by prop. \ref{MorphismsOfSupermanifoldsByAlgebras} the presheaf of algebras which sends a Grassmann algebra to its even subalgebra, as discussed at [[superalgebra]].

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{superdomain} is an open subfunctor (\ldots{}) of a [[locally convex vector space|locally convex]] $\mathbb{K}$-module.

\end{defn}
This appears as (\hyperlink{Sachse}{Sachse, def. 4.6}).

We now want to describe supermanifolds as [[manifold]]s in $SuperSet$ modeled on superdomains.

Write [[SmoothMfd]] for the [[category]] of ordinary [[smooth manifold]]s.

\begin{defn}
\label{SupermanifoldOverSuperpoint}\hypertarget{SupermanifoldOverSuperpoint}{}
A \emph{supermanifold} is a functor $X : SuperPoint^{op} \to SmoothMfd$ equipped with an equivalence class of \hyperlink{}{supersmooth atlases}.

A [[morphism]] of supermanifolds is a [[natural transformation]] $f : X \to X'$, such that for each pair of [[chart]]s $u : U \to X$ and $u' : U' \to X'$ the [[pullback]]

\begin{displaymath}
\itexarray{
    U \times_{X'} U' &&\stackrel{p'}{\to}&& U'
    \\
    {}^{\mathllap{p_1}}\downarrow & & && \downarrow^{\mathrlap{u'}}
    \\
    U &\stackrel{u}{\to}& X &\stackrel{f}{\to}& X'
  }
\end{displaymath}
can be equipped with the structture of a \hyperlink{}{Banach superdomain} such that $p_1$ and $p_2$ are supersmooth (\ldots{})

\end{defn}
This appears as (\hyperlink{Sachse}{Sachse, def. 4.13, 4.14}).

\hypertarget{properties_2}{}\subsubsection*{{Properties}}\label{properties_2}

\begin{prop}
\label{}\hypertarget{}{}
The categories of supermanifolds defined as locally ringed spaces, def. \ref{SupermanifoldLocallyRingedSpace} and as manifolds over superpoints, def. \ref{SupermanifoldOverSuperpoint} are [[equivalence of categories|equivalent]].

\end{prop}
This appears as (\hyperlink{Sachse}{Sachse, theorem 5.1}). See section 5.2 there for a discussion of the relation to the \hyperlink{ModeledOnGrassmannAlgebra}{DeWitt-definition}.

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

\begin{itemize}%
\item [[super-scheme]], [[spectral super-scheme]]


\item [[integration over supermanifolds]]


\item [[super vector bundle]]


\item [[complex supermanifold]], [[super Riemann surface]]


\item [[superconnection]]


\item [[superdifferential form]]


\item [[superalgebra]], [[smooth superalgebra]]


\item [[NQ-supermanifold]]



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

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

A brief survey is in

\begin{itemize}%
\item [[Henning Hohnhold]], [[Stephan Stolz]], [[Peter Teichner]], \emph{Super manifolds: an incomplete survey}, Bulletin of the Manifold Atlas (2011) 1--6 (\href{http://people.mpim-bonn.mpg.de/teichner/Papers/Survey-Journal.pdf}{pdf})

\end{itemize}
Discussion with an eye towards [[supergravity]] is in

\begin{itemize}%
\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section II.2.4 of \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991)

\end{itemize}
Discussion with an eye on [[integration over supermanifolds]] is in

\begin{itemize}%
\item [[Edward Witten]], \emph{Notes On Supermanifolds and Integration} (\href{http://arxiv.org/abs/1209.2199}{arXiv:1209.2199})

\end{itemize}
Global properties are discussed in

\begin{itemize}%
\item [[Louis Crane]], Jeffrey M. Rabin, \emph{Global properties of supermanifolds}, Comm. Math. Phys. Volume 100, Number 1 (1985), 141-160. (\href{http://projecteuclid.org/euclid.cmp/1103943340}{Euclid})

\end{itemize}
\hypertarget{as_locally_ringed_spaces_2}{}\subsubsection*{{As locally ringed spaces}}\label{as_locally_ringed_spaces_2}

\begin{itemize}%
\item [[Felix Berezin]], D. A. Letes, \emph{Supermanifolds}, (Russian) Dokl. Akad. Nauk SSSR \textbf{224} (1975), no. 3, 505--508; English transl.: Soviet Math. Dokl. \textbf{16} (1975), no. 5, 1218--1222 (1976).


\item I. L. Buchbinder, S. M. Kuzenko, \emph{Ideas and methods of supersymmetry and supergravity; or A walk through superspace}



\end{itemize}
A more general variant of this in the spirit of [[locally algebra-ed topos]]es is in

\begin{itemize}%
\item Alexander Alldridge, \emph{A convenient category of supermanifolds} (\href{http://arxiv.org/abs/1109.3161}{arXiv:1109.3161})

\end{itemize}
\hypertarget{ReferencesOverSuperpoints}{}\subsubsection*{{As manifolds over superpoints}}\label{ReferencesOverSuperpoints}

The observation that the study of super-structures in mathematics is usefully regarded as taking place over the [[base topos]] on the [[site]] of [[super points]] has been made around 1984 in

\begin{itemize}%
\item [[Albert Schwarz]], \emph{On the definition of superspace}, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37--42, (\href{http://www.mathnet.ru/links/b12306f831b8c37d32d5ba8511d60c93/tmf5111.pdf}{russian original pdf})


\item [[Alexander Voronov]], \emph{Maps of supermanifolds} , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43--48



\end{itemize}
and in

\begin{itemize}%
\item V. Molotkov., \emph{Infinite-dimensional $\mathbb{Z}_2^k$-supermanifolds} , ICTP preprints, IC/84/183, 1984.

\end{itemize}
A summary/review is in the appendix of

\begin{itemize}%
\item Anatoly Konechny and [[Albert Schwarz]],

\emph{On $(k \oplus l|q)$-dimensional supermanifolds}, in: [[Julius Wess]], V. Akulov (eds.) \emph{Supersymmetry and Quantum Field Theory} (D. Volkov memorial volume) Springer-Verlag, 1998 , Lecture Notes in Physics, 509 (\href{http://arxiv.org/abs/hep-th/9706003}{arXiv:hep-th/9706003})

\emph{Theory of $(k \oplus l|q)$-dimensional supermanifolds} Sel. math., New ser. 6 (2000) 471 - 486


\item [[Albert Schwarz]], I- Shapiro, \emph{Supergeometry and Arithmetic Geometry} (\href{http://arxiv.org/abs/hep-th/0605119}{arXiv:hep-th/0605119})



\end{itemize}
A review with more emphasis on the relevant [[category theory]]/[[topos theory]] is in

\begin{itemize}%
\item [[Christoph Sachse]], \emph{A Categorical Formulation of Superalgebra and Supergeometry} (\href{http://arxiv.org/abs/0802.4067}{arXiv:0802.4067})

\end{itemize}
The site of formal duals not just to [[Grassmann algebras]] but to all super-[[infinitesimally thickened points]] is discussed in (\hyperlink{KonechnySchwarz}{Konechny-Schwarz}) above and also in

\begin{itemize}%
\item L. Balduzzi, C. Carmeli, R. Fioresi, \emph{The local functors of points of Supermanifolds} (\href{http://arxiv.org/abs/0908.1872}{arXiv:0908.1872})

\end{itemize}
\hypertarget{as_manifolds_modelled_on_grassmann_algebras}{}\subsubsection*{{As manifolds modelled on Grassmann algebras}}\label{as_manifolds_modelled_on_grassmann_algebras}

\begin{itemize}%
\item [[Bryce DeWitt]], \emph{Supermanifolds}, Cambridge Monographs on Mathematical Physics, 1984


\item [[Alice Rogers]], Supermanifolds: Theory and Applications, World Scientific, (2007)



\end{itemize}
Alice Rogers claims, in Chapter 1, that the smooth-manifold-of-(infinite-dimensional)-Grassmann-algebras approach (the ``concrete approach'') is identical to the sheaf-of-ringed-spaces approach (the ``algebro-geometric'' approach) and that this equivalence is shown in Chapter 8. DeWitt seems unsure of this, but is writing more than 20 years earlier, before the ringed-space approach has been fully developed.

\hypertarget{other}{}\subsubsection*{{Other}}\label{other}

\begin{itemize}%
\item [[Yuri Manin]], \emph{Topics in noncommutative geometry}, Princeton Univ. Press 1991.


\item [[Pierre Deligne]], P. Etingof, [[Daniel Freed]], L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and [[Edward Witten]] (eds.) \emph{[[Quantum Fields and Strings]], A course for mathematicians}, 2 vols. Amer. Math. Soc. Providence 1999. (\href{http://www.math.ias.edu/qft}{web version})


\item V. S. Varadarajan, \emph{Supersymmetry for mathematicians: an introduction}, AMS and Courant Institute, 2004.


\item [[Alberto S. Cattaneo]], Florian Schaetz, \emph{Introduction to supergeometry}, \href{http://arxiv.org/abs/1011.3401}{arxiv/1011.3401}



\end{itemize}
There are many books in [[physics]] on [[supersymmetry]] (most well known is by Wess and Barger from 1992), but they emphasise more on the [[supersymmetry algebra]]s rather than on (the superspace and) \emph{supermanifolds}. They should therefore rather be listed under entry [[supersymmetry]].

See also \href{http://www.math.uni-hamburg.de/home/sachse/handoutbatchelor.pdf}{pdf}

[[!redirects super manifold]]

[[!redirects supermanifolds]]



\end{document}