\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*{super algebra}

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

\hypertarget{superalgebra_and_supergeometry}{}\paragraph*{{Super-Algebra and Super-Geometry}}\label{superalgebra_and_supergeometry}

[[!include supergeometry - contents]]

\begin{quote}%
from \href{Ausdehnungslehre#Grassmann44}{Grassmann 1844, p. 61 and 84}


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

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{basic_idea}{Basic idea}\dotfill \pageref*{basic_idea} \linebreak
\noindent\hyperlink{AbstractIdea}{Abstract idea}\dotfill \pageref*{AbstractIdea} \linebreak
\noindent\hyperlink{AssociativeSuperalgebras}{Associative superalgebras}\dotfill \pageref*{AssociativeSuperalgebras} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{superalgebras}{Superalgebras}\dotfill \pageref*{superalgebras} \linebreak
\noindent\hyperlink{RelatedNotions}{Related notions}\dotfill \pageref*{RelatedNotions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{endomorphisms_algebras_matrix_algebras}{Endomorphisms algebras, matrix algebras}\dotfill \pageref*{endomorphisms_algebras_matrix_algebras} \linebreak
\noindent\hyperlink{grassmann_algebra}{Grassmann algebra}\dotfill \pageref*{grassmann_algebra} \linebreak
\noindent\hyperlink{clifford_algebra}{Clifford algebra}\dotfill \pageref*{clifford_algebra} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{RelationToOrdinaryCommutativeAlgebra}{Relation to ordinary commutative algebras}\dotfill \pageref*{RelationToOrdinaryCommutativeAlgebra} \linebreak
\noindent\hyperlink{relation_to_matrix_algebras}{Relation to matrix algebras}\dotfill \pageref*{relation_to_matrix_algebras} \linebreak
\noindent\hyperlink{Picard2Groupoid}{Picard 3-group, Brauer group}\dotfill \pageref*{Picard2Groupoid} \linebreak
\noindent\hyperlink{AlgebraOverSuperpoints}{Algebra in the topos over superpoints}\dotfill \pageref*{AlgebraOverSuperpoints} \linebreak
\noindent\hyperlink{the_topos}{The topos}\dotfill \pageref*{the_topos} \linebreak
\noindent\hyperlink{AlgebraInToposOverSuperPoints-TheLineObject}{The line object $\mathbb{R}$}\dotfill \pageref*{AlgebraInToposOverSuperPoints-TheLineObject} \linebreak
\noindent\hyperlink{SuperModulesAsbbKModules}{$\mathbb{R}$-Modules}\dotfill \pageref*{SuperModulesAsbbKModules} \linebreak
\noindent\hyperlink{associative_and_lie_superalgebras}{Associative and Lie Superalgebras}\dotfill \pageref*{associative_and_lie_superalgebras} \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{definition_15}{Definition}\dotfill \pageref*{definition_15} \linebreak
\noindent\hyperlink{brauer_groups_and_picard_2groupoid}{Brauer groups and Picard 2-groupoid}\dotfill \pageref*{brauer_groups_and_picard_2groupoid} \linebreak


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

\hypertarget{basic_idea}{}\subsubsection*{{Basic idea}}\label{basic_idea}

In the general sense, \emph{superalgebra} is the study of ([[higher algebra|higher]]) [[algebra]]

\begin{itemize}%
\item [[internalization|internal]] to the [[symmetric monoidal category]] of $\mathbb{Z}_2$-[[graded vector spaces]] ([[super vector spaces]]);

\end{itemize}
equivalently

\begin{itemize}%
\item over the [[base topos]] on [[superpoints]].

\end{itemize}
More specifically, a \emph{[[supercommutative superalgebra]]} is an [[commutative algebra]] in the context of superalgebra.

See at \emph{[[geometry of physics -- superalgebra]]} for more on this.

In the following we first discuss

\begin{itemize}%
\item \emph{\hyperlink{AssociativeSuperalgebras}{Associative superalgebras}}

\end{itemize}
as [[monoids]] in the [[symmetric monoidal category]] of [[super vector spaces]]. Then we pass to the perspective of

\begin{itemize}%
\item \emph{\hyperlink{AlgebraOverSuperpoints}{Algebra in the topos over superpoints}}

\end{itemize}
and consider systematically [[algebra]] in the [[sheaf topos]] over the [[site]] of [[superpoints]] and show how this reproduces and generalizes the previous notions.

See (\hyperlink{Sachse}{Sachse}) and the references at \emph{[[super ∞-groupoid]]} for some history of the topos-theoretic perspective on superalgebra.

\hypertarget{AbstractIdea}{}\subsubsection*{{Abstract idea}}\label{AbstractIdea}

We discuss the general abstract \emph{raison d' \^e{}tre} of super algebra. Readers looking for just the plain definition should probably \hyperlink{AssociativeSuperalgebras}{skip to below} on first reading.

One way to understand the relevance of [[supercommutative superalgebra]] is [[Deligne's theorem on tensor categories]], which states that well-behaved [[tensor categories]] over the [[complex numbers]] are [[equivalence of categories|equivalent]] to [[categories of representations]] of [[supergroups]]. From this perspective the crucial sign rule is related to the [[symmetric monoidal category|symmetric]] [[braided monoidal category|braiding]] in [[tensor categories]]. This in turn may itself be understood from a more general perspective as follows.

Superalgebra is [[universal property|universal]] in the following sense. The crucial super-grading rule (the ``Koszul sign rule'', \hyperlink{Grassmann1844}{Grassmann 1844, \S{}37, \S{}55})

\begin{displaymath}
a \otimes b = (-1)^{deg(a) deg(b)} b \otimes a
\end{displaymath}
in the [[symmetric monoidal category]] of $\mathbb{Z}$-[[graded vector spaces]] is induced from the [[subcategory]] which is the [[abelian 2-group]] of metric graded [[lines]]. This in turn is the [[free construction|free]] [[abelian 2-group]] (groupal [[symmetric monoidal category]]) on a single generator. (This point of view is amplified in the first part of (\hyperlink{Kapranov13}{Kapranov 13}), whose second part is about [[super 2-algebra]], more details in \hyperlink{Kapranov15}{Kapranov 15}). Generally then super-grading and hence super-algebra arises from the [[truncated object|2-truncation]] (3-[[coskeleton]]) of the free [[abelian ∞-group]] on a single generator, which is the [[sphere spectrum]] $\mathbb{S}$. So the $\mathbb{Z}_2$-grading of superalgebra comes from the [[stable homotopy groups of spheres]] $\pi_n(\mathbb{S})$ in degree 1 and 2:

\newline | [[free construction|free object]] on single [[generators and relations|generator]]: | [[abelian group]] | [[abelian 2-group]] | [[abelian 3-group]] | [[abelian 4-group]] | | | [[abelian 7-group]] | [[abelian ∞-group|abelian 8-group]] | [[abelian ∞-group]] |

This suggests (as indicated in (\hyperlink{Kapranov13}{Kapranov 13}, \hyperlink{Kapranov15}{Kapranov 15})) that in full generality [[higher geometry|higher]] [[supergeometry]] is to be thought of as $\mathbb{S}$-[[graded object|graded]] geometry, hence [[Isbell duality|dually]] as [[higher algebra]] with [[∞-group of units]] [[augmented ∞-group|augmented]] over the [[sphere spectrum]].

But notice that this is canonically so for every [[E-∞ ring]], see at \emph{\href{infinity-group+of+units#AugmentedDefinition}{∞-group of units -- Augmented definition}}. This would mean: In [[higher geometry]]/[[higher algebra]] supergeometry/superalgebra is intrinsic, canonically given.

Using this together with [[Hisham Sati|Sati]]`s \emph{[[Geometric and topological structures related to M-branes]]} and the [[image of the J-homomorphism]]

[[!include image of J -- table]]

we can derive the terminology in the above table as indicated now.

\begin{quote}%
The following uses the notions of \emph{[[motivic quantization]]} as indicated there, to be expanded.


\end{quote}
\begin{itemize}%
\item \textbf{$d = 1$ [[sigma-model]]}

The [[local coefficients]] for [[quantization|quantizing]] the ([[spinning particle|spinning]]) [[particle]] on the [[boundary]] of the [[string]] ending on a [[D-brane]] (by K-theoretic [[geometric quantization by push-forward]]/[[D-brane charge]]) are

\begin{displaymath}
\itexarray{   
    B BU(1)
    &\simeq&
    B K(\mathbb{Z},2)
    &\to& 
    B gl_1(KU) 
    &\to&
    KU \text{-}Mod
  }
\end{displaymath}
for $KU$ the [[complex K-theory spectrum]] [[E-∞ ring]], and hence the characteristic twists are in degree 2 of the group of units, hence of the \href{infinity-group%20of%20units#AugmentedDefinition}{graded ∞-group of units}

\begin{displaymath}
gl^{gr}_1(KU) \to \mathbb{S}
\end{displaymath}
hence are graded by the second [[homotopy group]]

\begin{displaymath}
\pi_2(\mathbb{S}) \simeq \mathbb{Z}_2
\end{displaymath}
of the sphere spectrum.


\item \textbf{$d = 2$ [[sigma-model]]}

The [[local coefficients]] for [[quantization|quantizing]] the [[string]] (on the [[boundary]] of the [[M2-brane]] ending on an [[M9-brane]]) are

\begin{displaymath}
\itexarray{   
    B B^2 U(1)
    &\simeq&
    B K(\mathbb{Z},3)
    &\to& 
    B gl_1(tmf) 
    &\to&
    tmf \text{-}Mod
  }
\end{displaymath}
for the [[tmf]] [[E-∞ ring]], and hence the characteristic twists are in degree 3 of the group of units, hence of the graded group of units

\begin{displaymath}
gl^{gr}_1(tmf) \to \mathbb{S}
\end{displaymath}
hence are graded by the third [[homotopy group]]

\begin{displaymath}
\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}
\end{displaymath}
of the sphere spectrum.


\item \textbf{$d = 5$ [[sigma-model]]}

The [[local coefficients]] for [[quantization|quantizing]] the [[Yang monopole]] (on the [[boundary]] of the [[M5-brane]] ending on an [[M9-brane]]) are

\begin{displaymath}
\itexarray{   
    B B^5 U(1)
    &\simeq&
    B K(\mathbb{Z},6)
    &\to& 
    B gl_1(K(5)) 
    &\to&
    K(5) \text{-}Mod
  }
  \,,
\end{displaymath}
and hence the characteristic twists are in degree 6 of the group of units, hence of the graded group of units

\begin{displaymath}
gl^{gr}_1(K(5)) \to \mathbb{S}
\end{displaymath}
hence are graded by the sixth [[homotopy group]]

\begin{displaymath}
\pi_6(\mathbb{S}) \simeq \mathbb{Z}_{2}
\end{displaymath}
of the sphere spectrum.



\end{itemize}
\hypertarget{AssociativeSuperalgebras}{}\subsection*{{Associative superalgebras}}\label{AssociativeSuperalgebras}

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

\hypertarget{superalgebras}{}\paragraph*{{Superalgebras}}\label{superalgebras}

An ordinary [[associative algebra]] (a [[vector space]] with a linear and associative and unital product operation) is a [[monoid]] in the [[monoidal category]] [[Vect]] of [[vector spaces]].

Throughout, fix a [[field]] $k$ of [[characteristic]] 0.

\begin{defn}
\label{SuperVectorSpaces}\hypertarget{SuperVectorSpaces}{}
Write [[SVect]] for the [[symmetric monoidal category]] of [[super vector space]]s over $k$. This is the [[category]] of $\mathbb{Z}_2$-[[graded vector space]]s equipped with the unique non-trivial symmetric [[braided monoidal category|braided monoidal structure]].

Objects are vector spaces with a [[direct sum]] decomposition

\begin{displaymath}
V = V_{even} \oplus V_{odd}
\end{displaymath}
and the [[tensor product]] is given in terms of that on vector spaces by

\begin{displaymath}
V \otimes W = 
  (V_{even} \otimes W_{even} \oplus V_{odd}\otimes W_{odd})
  \oplus
  (V_{even} \otimes W_{odd} \oplus V_{odd} \otimes W_{even})
\end{displaymath}
but equipped with the non-trivial braiding morphism

\begin{displaymath}
b_{V, W} : V \otimes W \to W \otimes V
\end{displaymath}
that is the usual braiding isomorphism of [[Vect]] on $V_{even} \otimes W_{even}$ and on $V_{even} \otimes W_{odd} \oplus V_{odd} \otimes W_{even}$ but is $(-1)$ times this on $V_{odd}\otimes W_{odd}$.

\end{defn}
\begin{defn}
\label{SuperAlgebras}\hypertarget{SuperAlgebras}{}
A \textbf{super (associative) algebra} over $K$ is a [[monoid]] in the [[symmetric monoidal category]] [[SVect]] of [[super vector space]]s.

A \textbf{[[graded commutative algebra|(graded)-commutative (associative) algebra]]} over $K$ is a [[monoid]] in the [[symmetric monoidal category]] [[SVect]] of [[super vector space]]s.

\end{defn}
This means that a commutative superalgebra is a [[super vector space]]

\begin{displaymath}
A = A_{even} \oplus A_{odd}
\end{displaymath}
equipped with a morphism of super vector spaces

\begin{displaymath}
(-)\cdot (-) : A \otimes A \to A
\end{displaymath}
that is associative and commutative in the usual sense. Specifically for the commutativity this means that with $a,b  \in A_{odd}$ we have

\begin{displaymath}
a \cdot b = - b \cdot a
  \,.
\end{displaymath}
Whereas if either of $a$ or $b$ is in $A_{even}$ we have

\begin{displaymath}
a \cdot b = b \cdot a
  \,.
\end{displaymath}
\hypertarget{RelatedNotions}{}\paragraph*{{Related notions}}\label{RelatedNotions}

\begin{defn}
\label{Center}\hypertarget{Center}{}
The \textbf{[[center]]} of a superalgebra $A$ is the sub-superalgebra $Z(A) \hookrightarrow A$ spanned by all those elements $z \in A$ of homogeneous degree which graded-commute with all other homogeneois elements $a$.

\end{defn}
\begin{defn}
\label{Opposite}\hypertarget{Opposite}{}
For $A$ a superalgebra, its \textbf{opposite} $A^{op}$ is the superalgebra with the same underlying [[super vector space]] as $A$, and with multiplication defined on homogeneous elements by

\begin{displaymath}
a_1 \cdot_{A^{op}} a_2 \coloneqq (-1)^{{\vert a_1\vert}{\vert a_2\vert}} a_2 \cdot_{A} a_1
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{CentralSimple}\hypertarget{CentralSimple}{}
A superalgebra $A$ is called \textbf{central simple} if

\begin{enumerate}%
\item its [[center]], def. \ref{Center} is the ground field;


\item its only 2-sided graded [[ideals]] are $0$ and $A$ itself.



\end{enumerate}
\end{defn}
\begin{defn}
\label{AlgWithBimodules}\hypertarget{AlgWithBimodules}{}
Write $2sVect \simeq sAlg$ for the [[2-category]] equivalent to the one whose [[objects]] are superalgebra, [[1-morphisms]] are [[bimodules]] and [[2-morphisms]] are intertwiners. This is naturally a [[monoidal 2-category]].

\end{defn}
\begin{defn}
\label{2sVect}\hypertarget{2sVect}{}
By the discussion at \emph{[[2-vector space]]} this is equivalently the 2-category of \textbf{super 2-vector spaces}. [[equivalence|Equivalence]] in $2sVect \simeq sAlg$ is also called \emph{[[Morita equivalence]]} of super-algebras.

\end{defn}
\begin{defn}
\label{Azumaya}\hypertarget{Azumaya}{}
A superalgebra is an [[Azumaya algebra]] if it is an invertible object in the [[monoidal 2-category]] $s2Vect \simeq sAlg$, def. \ref{AlgWithBimodules}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The group of [[equivalence classes]] of Azumaya super algebras is called the super \emph{[[Brauer group]]}, see there for more details.

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

\hypertarget{endomorphisms_algebras_matrix_algebras}{}\paragraph*{{Endomorphisms algebras, matrix algebras}}\label{endomorphisms_algebras_matrix_algebras}

\begin{defn}
\label{EndomorphismSuperalgebra}\hypertarget{EndomorphismSuperalgebra}{}
For $V \in SVect$ a [[super vector space]], its [[endomorphism ring]] is canonically a super-algebra. Superalgebras isomorphic to ones of this form, are also called \textbf{matrix super algebras}.

\end{defn}
\begin{prop}
\label{EndomorphismSuperalgebrasAreCentralSimple}\hypertarget{EndomorphismSuperalgebrasAreCentralSimple}{}
A matrix superalgebra, def. \ref{EndomorphismSuperalgebra} is central simple, def. \ref{CentralSimple}.

\end{prop}
\hypertarget{grassmann_algebra}{}\paragraph*{{Grassmann algebra}}\label{grassmann_algebra}

\begin{itemize}%
\item For $V$ a [[vector space]] or [[graded vector space]] the [[Grassmann algebra]] $\wedge^\bullet V$ is a super algebra. This are the \emph{free} superalgebras.

\end{itemize}
\hypertarget{clifford_algebra}{}\paragraph*{{Clifford algebra}}\label{clifford_algebra}

An class of examples of non-(graded)-commutative superalgebra are [[Clifford algebra]].

In fact, let $V$ be a [[vector space]] equipped with symmetric [[inner product]] $\langle -,- \rangle$. Write $\wedge^\bullet V$ be the [[Grassmann algebra]] on $V$. The inner product makes this a super [[Poisson algebra]]. The [[Clifford algebra]] $Cl(V, \langle -,- \rangle)$ is the [[deformation quantization]] of this.

\begin{example}
\label{ComplexCl1}\hypertarget{ComplexCl1}{}
There is a superalgebra over the [[complex numbers]] of the form

\begin{displaymath}
A = \mathbb{C} \oplus \mathbb{C}\langle u\rangle
  \,,
\end{displaymath}
where the single odd generator satisfies $u \cdot u = 1$.

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

\hypertarget{RelationToOrdinaryCommutativeAlgebra}{}\paragraph*{{Relation to ordinary commutative algebras}}\label{RelationToOrdinaryCommutativeAlgebra}

\begin{defn}
\label{InclusionOfCommutativeSuperalgebras}\hypertarget{InclusionOfCommutativeSuperalgebras}{}
Given some [[ground field]] $k$, write

\begin{displaymath}
\iota \colon CAlg_k \hookrightarrow SCAlg_k
\end{displaymath}
for the [[full subcategory]] of ordinary [[commutative algebras]] over $k$ into [[supercommutative superalgebras]] (as those having trivial odd part).

\end{defn}
\begin{prop}
\label{AdjointsToInclusionOfPlainAlgebra}\hypertarget{AdjointsToInclusionOfPlainAlgebra}{}
The inclusion $\iota$ of def. \ref{InclusionOfCommutativeSuperalgebras} has

\begin{enumerate}%
\item a [[right adjoint]] $(-)_{even}$ given by restricting a superalgebra to its even part;


\item a [[left adjoint]] $(-)/((-)_{odd})$ given by forming the ``[[body]]'', the [[quotient]] by the ideal generated by the odd part (by the ``[[soul]]'').



\end{enumerate}
\end{prop}
This is immediate, but conceptually important, it is made explicit for instance in (\hyperlink{CarchediRoytenberg12}{Carchedi-Roytenberg 12, example 3.18}).

\begin{remark}
\label{AdjointTripleBetweenPlainAndSuperalgebra}\hypertarget{AdjointTripleBetweenPlainAndSuperalgebra}{}
Prop. \ref{AdjointsToInclusionOfPlainAlgebra} gives an [[adjoint triple]] of the form

\begin{displaymath}
CAlg_k \stackrel{\overset{(-)/((-)_{odd})}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{(-)_{even}}{\longleftarrow}}}
  SCAlg_k
\end{displaymath}
and hence an [[adjoint cylinder]], which induces a pair of [[adjoint modalities]] ([[fermionic modality]] $\dashv$ [[bosonic modality]]). See at \emph{[[super smooth infinity-groupoid]]} for more on this.

\end{remark}
\hypertarget{relation_to_matrix_algebras}{}\paragraph*{{Relation to matrix algebras}}\label{relation_to_matrix_algebras}

\begin{prop}
\label{}\hypertarget{}{}
A superalgebra is isomorphic to a matrix algebra, def. \ref{EndomorphismSuperalgebra}, precisely if it is [[equivalence|equivalent]] in $2 sVect \simeq Alg$, def. \ref{AlgWithBimodules}, ([[Morita equivalence|Morita equivalent]]) to the ground field super algebra.

\end{prop}
\hypertarget{Picard2Groupoid}{}\paragraph*{{Picard 3-group, Brauer group}}\label{Picard2Groupoid}

We discuss the [[Picard 3-group]] of $2sVect \simeq sAlg$, def. \ref{AlgWithBimodules}, hence the corresponding [[Brauer group]]. See also at \emph{[[super line 2-bundle]]}.

\begin{theorem}
\label{AzumayaAreCentralSimple}\hypertarget{AzumayaAreCentralSimple}{}
A superalgebra is invertible/Azumaya, def. \ref{Azumaya}, precisely if it is finite dimensional and central simple, def. \ref{CentralSimple}.

\end{theorem}
This is due to (\hyperlink{Wall}{Wall}).

\begin{theorem}
\label{}\hypertarget{}{}
The [[Brauer group]] of superalgebras over the [[complex numbers]] is the [[cyclic group of order 2]]. That over the [[real numbers]] is cyclic of order 8:

\begin{displaymath}
sBr(\mathbb{C}) \simeq \mathbb{Z}_2
\end{displaymath}
\begin{displaymath}
sBr(\mathbb{R}) \simeq \mathbb{Z}_8
  \,.
\end{displaymath}
The non-trivial element in $sBr(\mathbb{R})$ is that presented by the superalgebra $\mathbb{C} \oplus \mathbb{C} u$ of example \ref{ComplexCl1}, with $u \cdot u = 1$.

\end{theorem}
This is due to (\hyperlink{Wall}{Wall}).

The following generalizes this to the higher [[homotopy groups]].

\begin{prop}
\label{}\hypertarget{}{}
The [[homotopy groups]] of the [[braided 3-group]] $sAlg^\times$ of Azumaya superalgebra are

\begin{tabular}{l|l|l}
&$sAlg^\times_{\mathbb{C}}$&$sAlg^\times_{\mathbb{R}}$\\
\hline 
$\pi_2$&$\mathbb{C}^\times$&$\mathbb{R}^\times$\\
$\pi_1$&$\mathbb{Z}_2$&$\mathbb{Z}_2$\\
$\pi_0$&$\mathbb{Z}_2$&$\mathbb{Z}_8$\\
\end{tabular}

where the [[groups of units]] $\mathbb{C}^\times$ and $\mathbb{R}^\times$ are regarded as [[discrete groups]].

\end{prop}
This appears in (\hyperlink{Freed}{Freed, (1.38)}).

\hypertarget{AlgebraOverSuperpoints}{}\subsection*{{Algebra in the topos over superpoints}}\label{AlgebraOverSuperpoints}

We now consider [[higher algebra]] in the [[(∞,1)-topos]] over [[super points]]: the [[cohesive (∞,1)-topos]] $\mathbf{H} =$ [[Super∞Grpd]].

\hypertarget{the_topos}{}\subsubsection*{{The topos}}\label{the_topos}

\begin{defn}
\label{}\hypertarget{}{}
Write $SuperPoint$ for the [[site]] of [[superpoint]]s. Write

\begin{displaymath}
SuperSet := Sh(SuperPoint)
\end{displaymath}
for the [[sheaf topos]] (a [[presheaf topos]]) over this site. Write

\begin{displaymath}
Super \infty Grpd := Sh_{(\infty,1)}(SuperPoint)
\end{displaymath}
for the [[(∞,1)-sheaf (∞,1)-topos]] over this site: the [[(∞,1)-topos]] of [[super ∞-groupoid]]s.

\end{defn}
\hypertarget{AlgebraInToposOverSuperPoints-TheLineObject}{}\subsubsection*{{The line object $\mathbb{R}$}}\label{AlgebraInToposOverSuperPoints-TheLineObject}

\begin{defn}
\label{EmbeddingOfSupermanifoldsIntoSheavesOnSuperpoints}\hypertarget{EmbeddingOfSupermanifoldsIntoSheavesOnSuperpoints}{}
Write

\begin{displaymath}
j \colon SuperSmthMfd \hookrightarrow Sh(SuperPoint)
\end{displaymath}
for the restricted [[Yoneda embedding]] of [[supermanifolds]] given by the canonical inclusion $SuperPoint \hookrightarrow SuperSmoothManifold$.

\end{defn}
\begin{defn}
\label{TheRealLine}\hypertarget{TheRealLine}{}
Write

\begin{displaymath}
\mathbb{R} := j(\mathbb{R}) \in Sh(SuperPoint)
\end{displaymath}
for the presheaf represented by the [[real line]], regarded as a [[supermanifold]]. Equipped with its canonical [[internalization|internal]] [[ring]] structure this is

\begin{displaymath}
\mathbb{R} \in Ring(Sh(SuperPoint))
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
By the discussion at [[supermanifold]] (in the section \href{http://ncatlab.org/nlab/show/supermanifold#AsLocallyRingedSpacesProperties}{As locally ringed spaces - Properties}) $\mathbb{R}$ sends the formal dual of a [[Grassmann algebra]] to its even subalgebra

\begin{displaymath}
\mathbb{R}  : Spec \wedge^\bullet V \mapsto (\wedge^\bullet V)_{even}
  \,.
\end{displaymath}
This is canonically equipped with the structure of a (unital) [[commutative ring]] in $SuperSet$.

\end{remark}
In (\hyperlink{Sachse}{Sachse}) this appears around (21).

\hypertarget{SuperModulesAsbbKModules}{}\subsubsection*{{$\mathbb{R}$-Modules}}\label{SuperModulesAsbbKModules}

\begin{defn}
\label{}\hypertarget{}{}
Write $Mod_{\mathbb{R}}(SuperSet)$ for the [[Mod|category of modules]] over $\mathbb{R}$ of def. \ref{TheRealLine} in $SuperSet$.

\end{defn}
\begin{prop}
\label{EmbeddingOfSVectIntoKMod}\hypertarget{EmbeddingOfSVectIntoKMod}{}
The restriction of the embedding of def. \ref{EmbeddingOfSupermanifoldsIntoSheavesOnSuperpoints} to supermanifolds which are [[super vector spaces]] is a [[functor]]

\begin{displaymath}
j : SVect_{\mathbb{R}} = Mod_{\mathbb{R}}(Set) \hookrightarrow Mod_{\mathbb{K}}(SuperSet)
\end{displaymath}
from real [[super vector spaces]] to internal modules over $\mathbb{R}$ that sends $V \in SVect_{\mathbb{R}}$ to

\begin{displaymath}
j(V) : Spec \Lambda \mapsto (\Lambda \otimes_\mathbb{R} V)_{even}
  = 
  (\Lambda_{even} \otimes_\mathbb{R} V_{even})
    \oplus
  (\Lambda_{odd} \otimes_\mathbb{R} V_{odd})
  \,.
\end{displaymath}
This is a [[full and faithful functor]].

\end{prop}
This appears as (\hyperlink{Sachse}{Sachse, corollary 3.2}).

\begin{proof}
The proof is a variation on the [[Yoneda lemma]].

\end{proof}
This means that ordinary [[super vector spaces]] are embedded as a [[full subcategory]] in $\mathbb{K}$-modules in the topos over [[super points]].

\hypertarget{associative_and_lie_superalgebras}{}\subsubsection*{{Associative and Lie Superalgebras}}\label{associative_and_lie_superalgebras}

\begin{prop}
\label{}\hypertarget{}{}
The functor $j$ from prop \ref{EmbeddingOfSVectIntoKMod} induces a [[full and faithful functor]]

\begin{displaymath}
SAlg_{\mathbb{R}}(Set) \hookrightarrow Alg_{\mathbb{R}}(SuperSet)
\end{displaymath}
of superalgebras over $\mathbb{R}$ as in def. \ref{SuperAlgebras} and internal [[associative algebra]]s over $\mathbb{R}$ in $SuperSet$.

Similarly we have a faithful embedding

\begin{displaymath}
SLieAlg_{\mathbb{R}}(Set) \hookrightarrow LieAlg_{\mathbb{R}}(SuperSet)
\end{displaymath}
of ordinary [[super Lie algebras]] over $\mathbb{R}$ into the internal [[Lie algebras]] over $\mathbb{R}$.

\end{prop}
This appears as (\hyperlink{Sachse}{Sachse, corollary 3.3}).

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

(\ldots{})

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

\begin{itemize}%
\item [[supercommutative algebra]], [[differential graded-commutative superalgebra]]

[[model structure on differential graded-commutative superalgebras]]


\item [[smooth superalgebra]]


\item [[supermanifold]]


\item [[super 2-algebra]]



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

\hypertarget{definition_15}{}\subsubsection*{{Definition}}\label{definition_15}

The concept of [[Grassmann algebra]] and the super-sign rule is due to

\begin{itemize}%
\item [[Hermann Grassmann]], \emph{[[Ausdehnungslehre]]}, 1844

\end{itemize}
Review of basic superalgebra includes

\begin{itemize}%
\item [[Yuri Manin]], chapter 3 of \emph{[[Gauge Field Theory and Complex Geometry]]}, Grundlehren der Mathematischen Wissenschaften 289, Springer 1988


\item [[Dennis Westra]], \emph{Superrings and supergroups}, 2009 (\href{http://www.mat.univie.ac.at/~michor/westra_diss.pdf}{pdf})



\end{itemize}
Discussion of superalgebra as algebra in certain [[symmetric monoidal category|symmetric monoidal]] [[tensor categories]] is in

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Cat\'e{}gorie Tensorielle} (\href{https://www.math.ias.edu/files/deligne/Tensorielles.pdf}{pdf})

\end{itemize}
(see also at [[Deligne's theorem on tensor categories]]).

Lecture notes include

\begin{itemize}%
\item [[Daniel Freed]], \emph{[[Five lectures on supersymmetry]]}

\end{itemize}
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 \emph{Supersymmetry and Quantum Field Theory} ([[Dmitry Volkov]] memorial volume) Springer-Verlag, 1998, Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(\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}
For more along these lines see also the references at \emph{[[supermanifold]]} and at \emph{[[super infinity-groupoid]]}.

Discussion in terms of [[smooth algebras]] ([[synthetic differential supergeometry]]) is in

\begin{itemize}%
\item [[David Carchedi]], [[Dmitry Roytenberg]], \emph{On theories of superalgebras of differentiable functions}, Theory and Applications of Categories, Vol. 28, 2013, No. 30, pp 1022-1098. (\href{http://arxiv.org/abs/1211.6134}{arxiv:1211.6134}, \href{http://www.tac.mta.ca/tac/volumes/28/30/28-30abs.html}{TAC})

\end{itemize}
\hypertarget{brauer_groups_and_picard_2groupoid}{}\subsubsection*{{Brauer groups and Picard 2-groupoid}}\label{brauer_groups_and_picard_2groupoid}

[[Brauer groups]] of superalgebras are discussed in

\begin{itemize}%
\item [[Daniel Freed]], Lectures on twisted K-theory and orientifolds (\href{http://www.ma.utexas.edu/users/dafr/ESI.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[C. T. C. Wall]], \emph{Graded Brauer groups}, J. Reine Angew. Math. 213 (1963/1964), 187-199.

\end{itemize}
\begin{itemize}%
\item [[Pierre Deligne]], \emph{Notes on spinors} in \emph{[[Quantum Fields and Strings]]}


\item [[Peter Donovan]], [[Max Karoubi]], \emph{Graded Brauer groups and K-theory with local coefficients}, Publications Math. IHES 38 (1970), 5-25 (\href{http://www.math.jussieu.fr/~karoubi/Donavan.K.pdf}{pdf})



\end{itemize}
See also at \emph{[[super line 2-bundle]]} for more on this.

Discussion of superalgebra as induced from free groupal symmetric monoidal categories ([[abelian 2-groups]]) and hence ultimately from the [[sphere spectrum]] is in

\begin{itemize}%
\item [[Mikhail Kapranov]], \emph{Categorification of supersymmetry and stable homotopy groups of spheres}, talk at \emph{\href{http://mathserver.neu.edu/~bwebster/ACRT/}{Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)}} April 2013 (\href{http://mathserver.neu.edu/~bwebster/ACRT/calendar-with-abstracts.pdf}{abstract pdf}, \href{https://youtu.be/StbRti1fV7A}{video})


\item [[Mikhail Kapranov]], \emph{Supergeometry in mathematics and physics}, in [[Gabriel Catren]], [[Mathieu Anel]], (eds.) \emph{[[New Spaces for Mathematics and Physics]]} (\href{http://arxiv.org/abs/1512.07042}{arXiv:1512.07042})


\item [[Mikhail Kapranov]], \emph{Super-geometry}, talk at \emph{\href{http://ercpqg-espace.sciencesconf.org/program}{New Spaces for Mathematics \& Physics}}, IHP Paris, Oct-Sept 2015 (\href{https://www.youtube.com/watch?v=bjsNwKYT8JE}{video recording})



\end{itemize}
[[!redirects super algebras]]

[[!redirects superalgebra]] [[!redirects super-algebra]]

[[!redirects superalgebras]]

[[!redirects even rules]]

[[!redirects commutative superalgebra]] [[!redirects commutative superalgebras]] [[!redirects commutative super-algebra]] [[!redirects commutative super-algebras]] [[!redirects super-commutative superalgebra]] [[!redirects super-commutative superalgebras]] [[!redirects super-commutative super-algebra]] [[!redirects super-commutative super-algebras]]



\end{document}