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\section*{super infinity-groupoid}

\begin{quote}%
This entry is about [[∞-groupoids]] parameterized over [[superpoints]]. Hence [[discrete ∞-groupoids]] equipped with super-structure. For \emph{smoothly} supergeometric [[∞-groupoids]], parameterized over [[supermanifolds]], see \emph{[[super smooth ∞-groupoid]]} and \emph{[[super formal smooth ∞-groupoid]]}.


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

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - contents]]

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

[[!include supergeometry - 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{InfinitesimalCohesion}{Infinitesimal cohesion}\dotfill \pageref*{InfinitesimalCohesion} \linebreak
\noindent\hyperlink{relation_to_smooth_super_groupoids}{Relation to smooth super $\infty$-groupoids}\dotfill \pageref*{relation_to_smooth_super_groupoids} \linebreak
\noindent\hyperlink{superalgebra}{Superalgebra}\dotfill \pageref*{superalgebra} \linebreak
\noindent\hyperlink{supergeometry_2}{Supergeometry}\dotfill \pageref*{supergeometry_2} \linebreak
\noindent\hyperlink{structures_in_}{Structures in $Super \infty Grpd$}\dotfill \pageref*{structures_in_} \linebreak
\noindent\hyperlink{ExponentiatedLieAlgebras}{Exponentiated $\infty$-Lie algebras}\dotfill \pageref*{ExponentiatedLieAlgebras} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

A \emph{super $\infty$-groupoid} is an [[∞-groupoid]] modeled on [[super points]].

The notion subsumes and generalizes that of \emph{bare} [[super groups]], but not that of \emph{[[super Lie groups]]}, the latter are instead examples of [[smooth super ∞-groupoids]] sitting over the base of super $\infty$-groupoids.

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

\begin{defn}
\label{SuperInfinityGroupoids}\hypertarget{SuperInfinityGroupoids}{}
Let $SuperPoints$ be the [[category]] of [[super points]], regarded as a [[site]] with trivial [[coverage]].

The [[(∞,1)-sheaf (∞,1)-topos]] over $SuperPoint$

\begin{displaymath}
Super\infty Grpd := Sh_{(\infty,1)}(SuperPoint)
\end{displaymath}
we call the [[(∞,1)-topos]] of \textbf{super $\infty$-groupoids}.

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

\hypertarget{InfinitesimalCohesion}{}\subsubsection*{{Infinitesimal cohesion}}\label{InfinitesimalCohesion}

\begin{prop}
\label{}\hypertarget{}{}
The [[(∞,1)-topos]] $Super\infty Grpd$, def. \ref{SuperInfinityGroupoids}, is an [[infinitesimal cohesive (∞,1)-topos]] over [[∞Grpd]].

\end{prop}
\begin{proof}
Being an [[(∞,1)-category of (∞,1)-presheaves]] the constant $\infty$-presheaf [[(∞,1)-functor]]

\begin{displaymath}
Disc \;\colon\; \infty Grpd \longrightarrow Super \infty Grpd
\end{displaymath}
has a [[left adjoint]] $\Pi$ given by forming [[(∞,1)-colimits]] and a [[right adjoint]] $\Gamma$ given by [[(∞,1)-limits]]. Since the [[category]] $SuperPoints$ has a [[terminal object]] (the point $\mathbb{R}^{0|0}$) its [[opposite category]] has an [[initial object]] and so $\Gamma$ is just given by evaluation at that object. $\Gamma X \simeq X(\ast)$.

It follows that $\Gamma\circ Disc \simeq Id$ and hence (by the discussion at [[adjoint (∞,1)-functor]]) that $Disc$ is a [[full and faithful (∞,1)-functor]].

Moreover, evaluation preserves [[(∞,1)-limits]] (since for [[(∞,1)-presheaves]] there are computed objectwise for each object of the [[site]]) and so by the [[adjoint (∞,1)-functor theorem]] there does exist a further [[right adjoint]] $coDisc \colon \infty Grpd \hookrightarrow Super \infty Grpd$. By the [[(∞,1)-Yoneda lemma]] and by [[adjoint (∞,1)-functor|adjointness]] this sends an [[∞-groupoid]] $X$ to the [[(∞,1)-presheaf]] given by

\begin{displaymath}
coDisc(X) 
   \;\colon\;
  \mathbb{R}^{0|q}
  \;\mapsto\;
  Super\infty Grpd(\mathbb{R}^{0|q}, coDisc(X))
  \simeq
  \infty Grpd(\Gamma(\mathbb{R}^{0|q}), X)
  \,.
\end{displaymath}
Now the crucial aspect is that $\Gamma(\mathbb{R}^{0|q}) \simeq \ast$ for all $q \in \mathbb{N}$ since every [[superpoint]] has a unique [[global point]], this being the archetypical property of [[infinitesimally thickened points]]. So it follows that

\begin{displaymath}
coDisc \simeq Disc
\end{displaymath}
and hence that $\Pi \simeq \Gamma$. Therefore $\Pi$ preserves in particular [[finite products]] so that $Super \infty Grpd$ is [[cohesion|cohesive]], but of course this shows now that it is in fact [[infinitesimal cohesion|infinitesimally cohesive]].

\end{proof}
\hypertarget{relation_to_smooth_super_groupoids}{}\subsubsection*{{Relation to smooth super $\infty$-groupoids}}\label{relation_to_smooth_super_groupoids}

Let

\begin{displaymath}
\begin{aligned}
    SmoothSuper\infty Grpd & := Sh_{(\infty,1)}(CartSp, Super\infty Grpd)
    \simeq
     Sh_{(\infty,1)}(CartSp\times SuperPoint, \infty Grpd)
    \\
    & =:
     Sh_{(\infty,1)}(CartSp\times SuperPoint)
  \end{aligned}
\end{displaymath}
be the [[(∞,1)-sheaf (∞,1)-topos]] of [[smooth super ∞-groupoids]]. This is cohesive over the [[base topos]] $Super \infty Grpd$.

For more on this see at \emph{[[smooth super ∞-groupoid]]}.

\hypertarget{superalgebra}{}\subsubsection*{{Superalgebra}}\label{superalgebra}

$Super \infty Grpd$ is naturally a [[ringed topos]], with [[commutative ring]]-object

\begin{displaymath}
\mathbb{K} \in Super \infty Grpd
\end{displaymath}
which as a presheaf $\mathbb{K} : SuperPoint^{op} \simeq GrAlg \to Set \hookrightarrow sSet$ is given by

\begin{displaymath}
\mathbb{K} : Spec \Lambda \mapsto \Lambda_{even}
\end{displaymath}
with ring structure induced over each [[super point]] $\mathbb{R}^{0|q} = Spec \Lambda = Spec \wedge^\bullet \mathbb{R}^q$ from the ring structure of the even part $\Lambda_{even}$ of the [[Grassmann algebra]] $\lambda$.

The [[higher algebra]] over this ring object is what is called [[superalgebra]]. See there for details on this.

For $k$ the ground field and $j(k)$ its embedding as a super vector space into the topos by the map discussed at \href{super+algebra#SuperModulesAsbbKModules}{superalgebra -- In the topos over superpoints -- K-modules} we have

\begin{displaymath}
\mathbb{K} \simeq j(k)
  \,.
\end{displaymath}
\hypertarget{supergeometry_2}{}\subsubsection*{{Supergeometry}}\label{supergeometry_2}

(\ldots{}) [[supergeometry]] (\ldots{})

\hypertarget{structures_in_}{}\subsection*{{Structures in $Super \infty Grpd$}}\label{structures_in_}

We discuss the general abstract  realized in $Super \infty Grpd$.

\hypertarget{ExponentiatedLieAlgebras}{}\subsubsection*{{Exponentiated $\infty$-Lie algebras}}\label{ExponentiatedLieAlgebras}

We discuss  in $Super \infty Grpd$.

\begin{defn}
\label{SuperLInfinityAlgebra}\hypertarget{SuperLInfinityAlgebra}{}
A \textbf{[[super L-∞ algebra]]} is an [[L-∞ algebra]] [[internalization|internal to]] [[super vector space]]s.

The [[category]] of super $L_\infty$-algebras is

\begin{displaymath}
S L_\infty Alg := (sdgcAlg^+_{sf})^{op}
\end{displaymath}
the [[opposite category]] of [[semi-free dga|semi-free]] [[dg-algebra]]s in [[super vector space]]s: [[commutative monoid]]s in the category of [[cochain complex]]es of [[super vector space]]s whose underlying commutative [[graded algebra]] is free on generators in positive degree.

For $\mathfrak{g}$ a super $L_\infty$-algebra we write $CE(\mathfrak{g})$ for the corresponding [[dg-algebra]]: its [[Chevalley-Eilenberg algebra]].

\end{defn}
\begin{defn}
\label{Exp}\hypertarget{Exp}{}
For $\mathfrak{g}$ a super $L_\infty$-algebra, its [[Lie integration]] is the super $\infty$-groupoid presented by the [[simplicial presheaf]]

\begin{displaymath}
\exp(\mathfrak{g}) \in [SuperPoint^{op}, sSet]
\end{displaymath}
on superpoints given by the assignment

\begin{displaymath}
\exp(\mathfrak{g}) 
  \;\colon\; 
  (\mathbb{R}^{0|q}, [k]) 
    \mapsto
   Hom_{sdgcAlg_k}\big(
     CE(\mathfrak{g}), 
    \Omega^\bullet_{vert,si}(\mathbb{R}^{0|q} \times \Delta^k)
   \big)
  \,.
\end{displaymath}
Here on the right we have [[vertical differential form]]s with respect to the projection of [[supermanifold]]s $\mathbb{R}^{0|q} \times \Delta^k \to \mathbb{R}^{0|q}$ and with [[sitting instants]] (see [[Lie integration]], \href{Lie+integration#SmoothDifferentialFormsOnSimplicesWithSittingInstants}{this Def.}).

\end{defn}
\begin{lemma}
\label{SuperLieIntegrationOverSuperPointIsLieIntegrationOfEvenPartAfterTensoring}\hypertarget{SuperLieIntegrationOverSuperPointIsLieIntegrationOfEvenPartAfterTensoring}{}
For $q \in \mathbb{N}$ write $\Lambda_q \coloneqq C^\infty(\mathbb{R}^{0|q})$ for the [[Grassmann algebra]] on $q$-generators, being the global functions on the [[super point]] $\mathbb{R}^{0|q}$.

Over $\mathbb{R}^{0|q}$ the super Lie integration from def \ref{Exp} is the ordinary [[Lie integration]] of the ordinary [[L-∞ algebra]] $(\mathfrak{g} \otimes_k \Lambda_q)_{even}$

\begin{displaymath}
\exp\left(
    \mathfrak{g}
  \right)\left(\mathbb{R}^{0|q}\right)
    = 
  \exp\left( 
    (\mathfrak{g}\otimes_k \Lambda_q)_{even} 
  \right)
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
This is the standard [[even rules]] mechanism. Using that the category $sVect$ of finite-dimensional [[super vector spaces]] is a [[compact closed category]], we compute

\begin{displaymath}
\begin{aligned}
    Hom_{sdgcAlg}
    \left(
      CE(\mathfrak{g}), 
      \Omega^\bullet_{vert,si}(\mathbb{R}^{0|q} \times \Delta^n)
    \right)
    & \simeq
    Hom_{sdgcAlg}\left(
      CE(\mathfrak{g}), 
      C^\infty(\mathbb{R}^{0|q}) \otimes \Omega^\bullet( \Delta^n)
    \right)
    \\
    & \simeq 
    Hom_{sdgcAlg}\left(
      CE(\mathfrak{g}), 
      \Lambda_q \otimes \Omega^\bullet( \Delta^n)
    \right)    
    \\
    & \subset
    Hom_{Ch^\bullet(sVect)}\left(
      \mathfrak{g}^*[1], 
      \Lambda_q \otimes \Omega^\bullet( \Delta^n)
    \right)
    \\
    & \simeq
    Hom_{Ch^\bullet(sVect)}\left(
      \mathfrak{g}^*[1]\otimes (\Lambda_q)^*,  
      \Omega^\bullet( \Delta^n)
    \right)
    \\
    & \simeq
    Hom_{Ch^\bullet(sVect)}\left(
      (\mathfrak{g} \otimes \Lambda_q)^*[1],  
      \Omega^\bullet( \Delta^n)
    \right)
    \\
    & \simeq
    Hom_{Ch^\bullet(sVect)}\left(
      (\mathfrak{g} \otimes \Lambda_q)^*[1]_{even},  
      \Omega^\bullet( \Delta^n)
    \right)
    \\
    & \supset
    Hom_{sdgcAlg}\left( 
      CE((\mathfrak{g}\otimes_k \Lambda_q)_{even}),  
      \Omega^\bullet( \Delta^n)
    \right)
  \end{aligned}
  \,.
\end{displaymath}
Here in the third step we used that the underlying dg-algebra of $CE(\mathfrak{g})$ is free to find the space of morphisms of dg-algebras inside that of super-vector spaces (of generators) as indicated. Since the differential on both sides is $\Lambda_q$-linear, the claim follows.

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

\begin{itemize}%
\item [[cohesive (∞,1)-topos]]

\begin{itemize}%
\item [[discrete ∞-groupoid]]


\item [[Euclidean-topological ∞-groupoid]]


\item [[smooth ∞-groupoid]]


\item [[synthetic differential ∞-groupoid]]


\item \textbf{super ∞-groupoid}


\item [[smooth super ∞-groupoid]]


\item [[synthetic differential super ∞-groupoid]]



\end{itemize}


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

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}
An fairly comprehensive and introductory review 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 point]]s is discussed 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}
[[!redirects super ∞-groupoid]]

[[!redirects super infinity-groupoids]] [[!redirects super ∞-groupoids]]

[[!redirects Super∞Grpd]]

[[!redirects super ∞-group]] [[!redirects super ∞-groups]] [[!redirects super infinity-group]] [[!redirects super infinity-groups]]



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