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\begin{document}

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\section*{super vector space}

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

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

[[!include supergeometry - contents]]

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{StructuresInternalToSuperVectorSpaces}{Structures internal to super-vector spaces}\dotfill \pageref*{StructuresInternalToSuperVectorSpaces} \linebreak
\noindent\hyperlink{delignes_theorem_on_tensor_categories}{Deligne's theorem on tensor categories}\dotfill \pageref*{delignes_theorem_on_tensor_categories} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{super vector space} is an [[object]] in the non-trivial [[symmetric monoidal category]] structure on the [[monoidal category]] of $\mathbb{Z}/2$-[[graded vector spaces]]: as an object it is just a $\mathbb{Z}/2$-[[graded vector space]], but the [[braiding]] of the underlying [[tensor product of vector spaces]] is taken to be the non-trivial linear map which on elements of homogeneous degree is given by

\begin{displaymath}
\tau^{super} 
   \;\colon\; 
  v \otimes w \;\mapsto\; (-1)^{deg(v) deg(w)} \, w \otimes v 
  \,.
\end{displaymath}
We make this precise as definition \ref{CategoryOfSuperVectorSpaces} below.

Super vector spaces form the basis of [[superalgebra]] (over [[ground rings]] which are [[fields]]) in direct analogy of how ordinary vector spaces form the basis of ordinary [[algebra]]. For more on this see \hyperlink{StructuresInternalToSuperVectorSpaces}{below}, and for yet more see at \emph{[[geometry of physics -- superalgebra]]}.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\begin{defn}
\label{VectorSpaces}\hypertarget{VectorSpaces}{}
For $k$ a [[field]], we write $Vect_k$ for the [[category]] whose

\begin{itemize}%
\item [[object|objects]] are $k$-[[vector spaces]];


\item [[morphism|morphisms]] are $k$-[[linear functions]] between these.



\end{itemize}
When the [[ground field]] $k$ is understood or when its precise nature is irrelevant, we will often notationally suppress it and speak of just the category [[Vect]] of vector spaces.

This is the category inside which [[linear algebra]] takes place.

\end{defn}
Of course the category [[Vect]] has some special properties. Not only are its objects ``linear spaces'', but the whole category inherits linear structure of sorts. This is traditionally captured by the following terminology for \textbf{[[additive and abelian categories]]}. Notice that there are several different but equivalent ways to state the following properties (discussed behind the relevant links).

\begin{defn}
\label{AdditiveAndAbelianCategories}\hypertarget{AdditiveAndAbelianCategories}{}
Let $\mathcal{C}$ be a [[category]].

\begin{enumerate}%
\item Say that $\mathcal{C}$ has \textbf{[[direct sums]]} if it has [[finite products]] and [[finite coproducts]] and if the canonical comparison morphism between these is an [[isomorphism]]. We write $V \oplus W$ for the direct sum of two objects of $\mathcal{C}$.


\item Say that $\mathcal{C}$ is an \textbf{[[additive category]]} if it has [[direct sums]] and in addition it is [[Ab-enriched category|enriched in abelian groups]], meaning that every [[hom-set]] is equipped with the structure of an [[abelian group]] such that [[composition]] of morphisms is a [[bilinear map]].


\item Say that $\mathcal{C}$ is an \textbf{[[abelian category]]} if it is an [[additive category]] and has property that its [[monomorphisms]] are precisely the inclusions of [[kernels]] and its [[epimorphisms]] are precisely the projections onto [[cokernels]].



\end{enumerate}
\end{defn}
We also make the following definition of $k$-linear category, but notice that conventions differ as to which extra properties beyond [[Vect]]-[[enriched category|enrichment]] to require on a linear category:

\begin{defn}
\label{LinearCategory}\hypertarget{LinearCategory}{}
For $k$ a [[field]] (or more generally just a [[commutative ring]]), call a [[category]] $\mathcal{C}$ a \textbf{$k$-[[linear category]]} if

\begin{enumerate}%
\item it is an [[abelian category]] (def. \ref{AdditiveAndAbelianCategories});


\item its [[hom-sets]] have the structure of $k$-[[vector spaces]] (generally $k$-[[modules]]) such that [[composition]] of morphisms in $\mathcal{C}$ is a [[bilinear map]]



\end{enumerate}
and the underlying additive [[abelian group]] structure of these [[hom-spaces]] is that of the underlying [[abelian category]].

In other words, a $k$-linear category is an [[abelian category]] with the additional structure of a [[Vect]]-[[enriched category]] (generally $k$[[Mod]]-enriched) such that the underlying [[Ab-enriched category|Ab-enrichment]] according to def. \ref{AdditiveAndAbelianCategories} is obtained from the $Vect$-enrichment under the [[forgetful functor]] $Vect \to Ab$.

A [[functor]] between $k$-linear categories is called a \textbf{$k$-[[linear functor]]} if its component functions on [[hom-sets]] are [[linear maps]] with respect to the given $k$-linear structure, hence if it is a [[Vect]]-[[enriched functor]].

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The category [[Vect]]${}_k$ of [[vector spaces]] (def. \ref{VectorSpaces}) is a $k$-[[linear category]] according to def. \ref{LinearCategory}.

Here the abstract [[direct sum]] is the usual direct sum of [[vector spaces]], whence the name of the general concept.

For $V,W$ two $k$-vector spaces, the vector space structure on the [[hom-set]] $Hom_{Vect}(V,W)$ of [[linear maps]] $\phi \colon V \to W$ is given by ``pointwise'' multiplication and addition of functions:

\begin{displaymath}
(c_1 \phi_1 + c_2 \phi_2)
  \;\colon\,
   v \;\mapsto\;
   c_1 \phi_1(v) + c_2 \phi_2(v)
\end{displaymath}
for all $c_1, c_2 \in k$ and $\phi_1, \phi_2 \in Hom_{Vect}(V,W)$.

\end{example}
Recall the basic construction of the [[tensor product of vector spaces]]:

\begin{defn}
\label{TensorProductOfVectorSpaces}\hypertarget{TensorProductOfVectorSpaces}{}
Given two [[vector spaces]] over some [[field]] $k$, $V_1, V_2 \in Vect_k$, their [[tensor product of vector spaces]] is the vector space denoted

\begin{displaymath}
V_1 \otimes_k V_2 \;\in\; Vect
\end{displaymath}
whose elements are [[equivalence classes]] of [[tuples]] of elements $(v_1,v_2)$ with $v_i \in V_i$, for the [[equivalence relation]] given by

\begin{displaymath}
(c v_1 , v_2) \;\sim\; (v_1, c v_2)
\end{displaymath}
\begin{displaymath}
(v_1 + v'_1 , v_2) \; \sim \; (v_1,v_2) + (v'_1, v_2)
\end{displaymath}
\begin{displaymath}
(v_1 , v_2 + v'_2) \; \sim \; (v_1,v_2) + (v_1, v'_2)
\end{displaymath}
More abstractly this means that the [[tensor product of vector spaces]] is the vector space characterized by the fact that

\begin{enumerate}%
\item it receives a [[bilinear map]]

\begin{displaymath}
V_1 \times V_2 \longrightarrow V_1 \otimes V_2
\end{displaymath}
(out of the [[Cartesian product]] of the underlying sets)


\item any other [[bilinear map]] of the form

\begin{displaymath}
V_1 \times V_2 \longrightarrow V_3
\end{displaymath}
factors through the above bilinear map via a unique [[linear map]]

\begin{displaymath}
\itexarray{
     V_1 \times V_2 &\overset{bilinear}{\longrightarrow}& V_3
     \\
     \downarrow & \nearrow_{\mathrlap{\exists ! \, linear}}
      \\
      V_1 \otimes_k V_2
  }
\end{displaymath}


\end{enumerate}
\end{defn}
The existence of the [[tensor product of vector spaces]], def. \ref{TensorProductOfVectorSpaces}, equips the category [[Vect]] of vector spaces with extra structure, which is a ``[[categorification]]'' of the familiar structure of a [[semi-group]]. One also says ``[[monoid]]'' for [[semi-group]] and therefore [[categories]] equipped with a [[tensor product]] operation are also called \emph{[[monoidal categories]]}:

\begin{defn}
\label{MonoidalCategory}\hypertarget{MonoidalCategory}{}
A \textbf{[[monoidal category]]} is a [[category]] $\mathcal{C}$ equipped with

\begin{enumerate}%
\item a [[functor]]

\begin{displaymath}
\otimes
     \;\colon\;
   \mathcal{C} \times \mathcal{C}
    \longrightarrow
   \mathcal{C}
\end{displaymath}
out of the [[product category]] of $\mathcal{C}$ with itself, called the \textbf{[[tensor product]]},


\item an object

\begin{displaymath}
1 \in \mathcal{C}
\end{displaymath}
called the \textbf{[[unit object]]} or \textbf{[[tensor unit]]},


\item a [[natural isomorphism]]

\begin{displaymath}
a
    \;\colon\;
  ((-)\otimes (-)) \otimes (-)
    \overset{\simeq}{\longrightarrow}
  (-) \otimes ((-)\otimes(-))
\end{displaymath}
called the \textbf{[[associator]]},


\item a [[natural isomorphism]]

\begin{displaymath}
\ell
\;\colon\;
(1 \otimes (-))
\overset{\simeq}{\longrightarrow}
(-)
\end{displaymath}
called the \textbf{[[left unitor]]}, and a natural isomorphism

\begin{displaymath}
r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)
\end{displaymath}
called the \textbf{[[right unitor]]},



\end{enumerate}
such that the following two kinds of [[commuting diagram|diagrams commute]], for all objects involved:

\begin{enumerate}%
\item \textbf{triangle identity}:

\begin{displaymath}
\itexarray{
     & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y)
      \\
      & {}_{\rho_x \otimes 1_y}\searrow
      && \swarrow_{1_x \otimes \lambda_y}
      &
     \\
     &&
     x \otimes y
     &&
  }
\end{displaymath}

\item the \textbf{[[pentagon identity]]}:

\begin{displaymath}
\itexarray{
    && (w \otimes x) \otimes (y \otimes z)
    \\
    & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow
     &&
    \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}}
    \\
    ((w \otimes x ) \otimes y) \otimes z
     && &&
    (w \otimes (x \otimes (y \otimes z)))
    \\
    {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow
     && &&
    \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }}
    \\
    (w \otimes (x \otimes y)) \otimes z
      && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} &&
    w \otimes ( (x \otimes y) \otimes z )
  }
\end{displaymath}


\end{enumerate}
\end{defn}
As expected, we have the following basic example:

\begin{example}
\label{VectAsAMonoidalCategory}\hypertarget{VectAsAMonoidalCategory}{}
For $k$ a [[field]], the category [[Vect]]${}_k$ of $k$-[[vector spaces]] becomes a [[monoidal category]] (def. \ref{MonoidalCategory}) as follows

\begin{itemize}%
\item the abstract [[tensor product]] is the [[tensor product of vector spaces]] $\otimes_k$ from def. \ref{TensorProductOfVectorSpaces};


\item the [[tensor unit]] is the [[field]] $k$ itself, regarded as a 1-dimensional vector space over itself;


\item the [[associator]] is the map that on representing [[tuples]] acts as

\begin{displaymath}
\alpha_{V_{1}, V_2, V_3}
     \;\colon\;
 ((v_1, v_2), v_3) \mapsto (v_1, (v_2,v_3))
\end{displaymath}

\item the left [[unitor]] is the map that on representing [[tuples]] is given by

\begin{displaymath}
\ell_{V} \colon (k,v) \mapsto k v
\end{displaymath}
and the right unitor is similarly given by

\begin{displaymath}
r_V \colon (v,k) \mapsto k v
  \,.
\end{displaymath}


\end{itemize}
That this satisifes the [[pentagon identity]] (def. \ref{MonoidalCategory}) and the left and right unit identities is immediate on representing tuples.

\end{example}
But the point of the abstract definition of [[monoidal categories]] is that there are also more exotic examples. The followig one is just a minimal enrichment of example \ref{VectAsAMonoidalCategory}, and yet it will be important.

\begin{example}
\label{GradedVectorSpacesAsAMonoidaCategory}\hypertarget{GradedVectorSpacesAsAMonoidaCategory}{}
Let $G$ be a [[group]] (or in fact just a [[monoid]]/[[semi-group]]). A \textbf{$G$-[[graded vector space]]} $V$ is a [[direct sum]] of vector spaces labeled by the elements in $G$:

\begin{displaymath}
V = \underset{g \in G}{\oplus} V_g
  \,.
\end{displaymath}
A [[homomorphism]]

\begin{displaymath}
\phi \;\colon\; V \longrightarrow W
\end{displaymath}
of $G$-graded vector spaces is a [[linear map]] that respects this direct sum structure, hence equivalently a [[direct sum]] of [[linear maps]]

\begin{displaymath}
\phi_g \;\colon\; V_g \longrightarrow W_g
\end{displaymath}
for all $g \in G$, such that

\begin{displaymath}
\phi = \underset{g \in G}{\oplus} \phi_g
  \,.
\end{displaymath}
This defines a [[category]], denoted $Vect^G$. Equip this category with a [[tensor product]] which on the underlying vector spaces is just the [[tensor product of vector spaces]] from def. \ref{TensorProductOfVectorSpaces}, equipped with the $G$-grading which is obtained by multiplying degree labels in $G$:

\begin{displaymath}
(V \otimes W)_g
    \;\coloneqq\;
  \underset{{g_1, g_2 \in G} \atop {g_1 g_2 = g}}{\oplus}
   V_{g_1} \otimes_k V_{g_2}
  \,.
\end{displaymath}
The [[tensor unit]] for the tensor product is the ground field $k$, regarded as being in the degree of the [[neutral element]] $e \in G$

\begin{displaymath}
1_g
  \;=\;
  \left\{
    \itexarray{
      k & | g = e
      \\
      0 & | otherwise
    }
  \right.
  \,.
\end{displaymath}
The [[associator]] and [[unitors]] are just those of the monoidal structure on plain vector spaces, from example \ref{VectAsAMonoidalCategory}.

\end{example}
One advantage of abstracting the concept of a [[monoidal category]] is that it allows to prove general statements uniformly for all kinds of tensor products, familiar ones and more exotic ones. The following lemma \ref{kel1} and remark \ref{CoherenceForMonoidalCategories} are two important such statements.

\begin{lemma}
\label{kel1}\hypertarget{kel1}{}
\textbf{(\hyperlink{Kelly64}{Kelly 64})}

Let $(\mathcal{C}, \otimes, 1)$ be a [[monoidal category]], def. \ref{MonoidalCategory}. Then the left and right [[unitors]] $\ell$ and $r$ satisfy the following conditions:

\begin{enumerate}%
\item $\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1$;


\item for all objects $x,y \in \mathcal{C}$ the following [[commuting diagram|diagrams commutes]]:

\begin{displaymath}
\itexarray{
    (1 \otimes x) \otimes y  & &
    \\
    {}^\mathllap{\alpha_{1, x, y}} \downarrow
    & \searrow^\mathrlap{\ell_x \otimes id_y} &
    \\
    1 \otimes (x \otimes y)
    & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y
  }
  \,;
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    x \otimes (y \otimes 1) & &
    \\
    {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow
    & \searrow^\mathrlap{id_x \otimes r_y} &
    \\
    (x \otimes y) \otimes 1
      &
    \underset{r_{x \otimes y}}{\longrightarrow}
      &
    x \otimes y
  }
  \,;
\end{displaymath}


\end{enumerate}
\end{lemma}
For \textbf{proof} see at \emph{[[monoidal category]]} \href{monoidal+category#kel1}{this lemma} and \href{monoidal+category#kel2}{this lemma}.

\begin{remark}
\label{CoherenceForMonoidalCategories}\hypertarget{CoherenceForMonoidalCategories}{}
Just as for an [[associative algebra]] it is sufficient to demand $1 a = a$ and $a 1 = a$ and $(a b) c = a (b c)$ in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a \emph{[[coherence theorem for monoidal categories]]} which states that all ways of freely composing the [[unitors]] and [[associators]] in a [[monoidal category]] (def. \ref{MonoidalCategory}) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an [[associative algebra]] altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.

(Here the qualifier ``freely'' means informally that we must not use any non-formal identification between objects, and formally it means that the diagram in question must be in the image of a [[strong monoidal functor]] from a \emph{free} monoidal category. For example if in a particular monoidal category it so happens that the object $X \otimes (Y \otimes Z)$ is actually \emph{equal} to $(X \otimes Y)\otimes Z$, then the various ways of going from one expression to another using only associators \emph{and} this ``accidental'' equality no longer need to coincide.)

\end{remark}
The above discussion makes it clear that a [[monoidal category]] is like a [[monoid]]/[[semi-group]], but ``[[categorified]]''. Accordingly we may consider additional properties of [[monoids]]/[[semi-groups]] and correspondingly lift them to monoidal categories. A key such property is \emph{[[commutative ring|commutativity]]}. But while for a monoid commutativity is just an extra [[property]], for a [[monoidal category]] it involves choices of commutativity-[[isomorphisms]] and hence is [[stuff, structure and property|extra structure]]. We will see \hyperlink{SuperGroupsAsSuperHopfAlgebras}{below} that this is the very source of [[superalgebra]].

The [[categorification]] of ``commutativity'' comes in two stages: [[braiding]] and [[symmetric monoidal category|symmetric braiding]].

\begin{defn}
\label{BraidedMonoidalCategory}\hypertarget{BraidedMonoidalCategory}{}
A \textbf{[[braided monoidal category]]}, is a [[monoidal category]] $\mathcal{C}$ (def. \ref{MonoidalCategory}) equipped with a [[natural isomorphism]]

\begin{displaymath}
\tau_{x,y} \;\colon\; x \otimes y \to y \otimes x
\end{displaymath}
(for all [[objects]] $x,y in \mathcal{C}$) called the \textbf{[[braiding]]}, such that the following two kinds of [[commuting diagram|diagrams commute]] for all [[objects]] involved (``hexagon identities''):

\begin{displaymath}
\itexarray{
   (x \otimes y) \otimes z
   &\stackrel{a_{x,y,z}}{\to}&
   x \otimes (y \otimes z)
   &\stackrel{\tau_{x,y \otimes z}}{\to}&
   (y \otimes z) \otimes x
   \\
   \downarrow^{\tau_{x,y}\otimes Id}
   &&&&
   \downarrow^{a_{y,z,x}}
   \\
   (y \otimes x) \otimes z
   &\stackrel{a_{y,x,z}}{\to}&
   y \otimes (x \otimes z)
   &\stackrel{Id \otimes \tau_{x,z}}{\to}&
   y \otimes (z \otimes x)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
   x \otimes (y \otimes z)
   &\stackrel{a^{-1}_{x,y,z}}{\to}&
   (x \otimes y) \otimes z
   &\stackrel{\tau_{x \otimes y, z}}{\to}&
   z \otimes (x \otimes y)
   \\
   \downarrow^{Id \otimes \tau_{y,z}}
   &&&&
   \downarrow^{a^{-1}_{z,x,y}}
   \\
   x \otimes (z \otimes y)
   &\stackrel{a^{-1}_{x,z,y}}{\to}&
   (x \otimes z) \otimes y
   &\stackrel{\tau_{x,z} \otimes Id}{\to}&
   (z \otimes x) \otimes y
  }
  \,,
\end{displaymath}
where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the [[associator]] of $\mathcal{C}^\otimes$.

\end{defn}
\begin{defn}
\label{SymmetricMonoidalCategory}\hypertarget{SymmetricMonoidalCategory}{}
A \textbf{[[symmetric monoidal category]]} is a [[braided monoidal category]] (def. \ref{BraidedMonoidalCategory}) for which the [[braiding]]

\begin{displaymath}
\tau_{x,y} \colon x \otimes y \to y \otimes x
\end{displaymath}
satisfies the condition:

\begin{displaymath}
\tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}
\end{displaymath}
for all objects $x, y$

\end{defn}
\begin{remark}
\label{SymmetricMonoidalCategoriesCoherenceTheorem}\hypertarget{SymmetricMonoidalCategoriesCoherenceTheorem}{}
In analogy to the [[coherence theorem for monoidal categories]] (remark \ref{CoherenceForMonoidalCategories}) there is a [[coherence theorem for symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}), saying that every diagram built freely (see remark \ref{SymmetricMonoidalCategoriesCoherenceTheorem}) from [[associators]], [[unitors]] and [[braidings]] such that both sides of the diagram correspond to the same [[permutation]] of objects, coincide.

\end{remark}
Consider the simplest non-trivial special case of $G$-[[graded vector spaces]] from example \ref{GradedVectorSpacesAsAMonoidaCategory}, the case where $G = \mathbb{Z}/2$ is the [[cyclic group of order two]].

\begin{example}
\label{Z2Zgradedvectorspaces}\hypertarget{Z2Zgradedvectorspaces}{}
A \textbf{$\mathbb{Z}/2$-[[graded vector space]]} is a [[direct sum]] of two vector spaces

\begin{displaymath}
V = V_{even} \oplus V_{odd}
  \,,
\end{displaymath}
where we think of $V_{even}$ as the summand that is graded by the [[neutral element]] in $\mathbb{Z}/2$, and of $V_{odd}$ as being the summand that is graded by the single non-trivial element.

A [[homomorphism]] of $\mathbb{Z}/2$-graded vector spaces

\begin{displaymath}
f \;\colon\; V_1 \longrightarrow V_2
\end{displaymath}
is a [[linear map]] of the underlying vector spaces that respects the grading, hence equivalently a pair of linear maps

\begin{displaymath}
f_{even} \;\colon\; (V_1)_{even} \longrightarrow (V_1)_{even}
\end{displaymath}
\begin{displaymath}
f_{odd} \;\colon\; (V_1)_{odd} \longrightarrow (V_1)_{odd}
\end{displaymath}
between then summands in even degree and in odd degree, respectively:

\begin{displaymath}
f = f_{even} \oplus f_{odd}
  \,.
\end{displaymath}
The [[tensor product]] of $\mathbb{Z}/2$-graded vector space is the [[tensor product of vector spaces]] of the underlying vector spaces, but with the grading obtained from multiplying the original gradings in $\mathbb{Z}/2$. Hence

\begin{displaymath}
(V_1 \otimes V_2)_{even}
    \;\coloneqq\;
  \left((V_1)_{even} \otimes (V_2)_{even}\right)
    \oplus
  \left((V_1)_{odd} \otimes (V_2)_{odd}\right)
\end{displaymath}
and

\begin{displaymath}
(V_1 \otimes V_2)_{odd}
    \;\coloneqq\;
  \left((V_1)_{even} \otimes (V_2)_{odd}\right)
    \oplus
  \left((V_1)_{odd} \otimes (V_2)_{even}\right)
  \,.
\end{displaymath}
As in example \ref{GradedVectorSpacesAsAMonoidaCategory}, this definition makes $\mathbb{Z}/2$ a [[monoidal category]] def. \ref{MonoidalCategory}.

\end{example}
\begin{prop}
\label{TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces}\hypertarget{TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces}{}
There are, up to [[braided monoidal functor|braided monoidal]] [[equivalence of categories]], precisely two choices for a [[symmetric monoidal category|symmetric]] [[braiding]] (def. \ref{SymmetricMonoidalCategory})

\begin{displaymath}
V_1 \otimes V_2
     \stackrel{\tau_{V_1,V_2}}{\longrightarrow}
  V_2 \otimes V_1
\end{displaymath}
on the [[monoidal category]] $(Vect_k^{\mathbb{Z}/2}, \otimes_k)$ of $\mathbb{Z}/2$-[[graded vector spaces]] from def. \ref{Z2Zgradedvectorspaces}:

\begin{enumerate}%
\item the \textbf{trivial braiding} which is the [[natural transformation|natural]] [[linear map]] given on tuples $(v_1,v_2)$ representing an element in $V_1 \otimes V_2$ (according to def. \ref{TensorProductOfVectorSpaces}) by

\begin{displaymath}
\tau^{triv}_{V_1, V_2} \;\colon\; (v_1,v_2) \mapsto (v_2, v_1)
\end{displaymath}

\item the \textbf{super-braiding} which is the [[natural transformation|natural]] [[linear function]] given on tuples $(v_1,v_2)$ of \emph{homogeneous degree} (i.e. $v_i \in (V_i)_{\sigma_i} \hookrightarrow V_i$, for $\sigma_i \in \mathbb{Z}/2$) by

\begin{displaymath}
\tau^{super}_{V_1, V_2}
     \;\colon\;
  (v_1, v_2)
     \mapsto
  (-1)^{deg(v_1) deg(v_2)}
  \,
  (v_2,v_1)
  \,.
\end{displaymath}


\end{enumerate}
\end{prop}
\begin{proof}
For $(\mathcal{C}, \otimes, 1)$ a [[monoidal category]], write

\begin{displaymath}
(Line(\mathcal{C}), \otimes, 1)
    \hookrightarrow
  (\mathcal{C}, \otimes, 1)
\end{displaymath}
for the [[full subcategory]] on those $L \in \mathcal{C}$ which are [[invertible objects]] under the [[tensor product]], i.e. such that there is an object $L^{-1} \in \mathcal{C}$ with $L \otimes L^{-1} \simeq 1$ and $L^{-1} \otimes L \simeq 1$. Since the [[tensor unit]] is clearly in $Line(L)$ (with $1^{-1} \simeq 1$) and since with $L_1, L_2 \in Line(\mathcal{C}) \hookrightarrow \mathcal{C}$ also $L_1 \otimes L_2 \in Line(\mathcal{C})$ (with $(L_1 \otimes L_2)^{-1} \simeq L_2^{-1} \otimes L_1^{-1}$) the [[monoidal category]] structure on $\mathcal{C}$ restricts to $Line(\mathcal{C})$.

Accordingly any [[braiding]] on $(\mathcal{C}, \otimes,1)$ restricts to a braiding on $(Line(\mathcal{C}), \otimes, 1)$. Hence it is sufficient to show that there is an essentially unique non-trivial symmetric braiding on $(Line(\mathcal{C}), \otimes, 1)$, and that this is the restriction of a braiding on $(\mathcal{C}, \otimes, 1)$.

Now $(Line(\mathcal{C}, \otimes , 1))$ is necessarily a [[groupoid]] (the ``[[Picard groupoid of a monoidal category|Picard groupoid]]'' of $\mathcal{C}$) and in fact is what is called a \emph{[[2-group]]}. As such we may regard it equivalently as a [[homotopy 1-type]] with group structure, and as such it it is equivalent to its [[delooping]]

\begin{displaymath}
B_\otimes Line(\mathcal{C})
\end{displaymath}
regarded as a [[pointed homotopy type]]. (See at \emph{[[looping and delooping]]}).

The [[Grothendieck group]] of $(\mathcal{C}, \otimes, 1)$ is

\begin{displaymath}
\pi_0(Line(\mathcal{C}))
    \simeq
  \pi_1(B Line(\mathcal{C}))
\end{displaymath}
the [[fundamental group]] of the delooping space.

Now a symmetric braiding on $Line(\mathcal{C})$ is precisely the structure that makes it a [[symmetric 2-group]] which is equivalently the structure of a second [[delooping]] $B^2 Line(\mathcal{C})$ (for the braiding) and then a third delooping $B^3 Line(\mathcal{C})$ (for the symmetry), regarded as a [[pointed homotopy type]].

This way we have rephrased the question equivalently as a question about the possible [[k-invariants]] of spaces of this form.

Now in the case at hand, $Line(Vect^{\mathbb{Z}/2})$ has precisely two [[isomorphism classes]] of objects, namely the [[ground field]] $k$ itself, regarded as being in even degree and regarded as being in odd degree. We write $k^{1\vert 0}$ and $k^{0 \vert 1}$ for these, respectively. By the rules of the tensor product of [[graded vector spaces]] we have

\begin{displaymath}
k^{1\vert 0} \otimes_k k^{1\vert 0} \simeq k^{1\vert 0}
\end{displaymath}
\begin{displaymath}
k^{1\vert 0} \otimes_k k^{0\vert 1} \simeq k^{0\vert 1}
\end{displaymath}
and

\begin{displaymath}
k^{0 \vert 1} \otimes_k k^{0 \vert 1} \simeq k^{1 \vert 0}
  \,.
\end{displaymath}
In other words

\begin{displaymath}
\pi_0(Line(Vect^{\mathbb{Z}/2})) \simeq \mathbb{Z}/2
  \,.
\end{displaymath}
Now under the above homotopical identification the non-trivial braiding is identified with the elements

\begin{displaymath}
1 =
  k^{1 \vert 0}
     \simeq
  k^{0\vert 1} \otimes_k k^{0 \vert 1}
    \stackrel{\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}}}{\longrightarrow}
  k^{0\vert 1} \otimes_k k^{0\vert 1}
    \simeq
  k^{1 \vert 0}
  = 1
\end{displaymath}
Due to the [[symmetric monoidal category|symmetry]] condition (def. \ref{SymmetricMonoidalCategory}) we have

\begin{displaymath}
(\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}})^2 = id
\end{displaymath}
which implies that

\begin{displaymath}
\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}}
  \in
  \{+ id, -id\}
  \,.
\end{displaymath}
Therefore for classifying just the symmetric braidings, it is sufficient to restrict the [[hom-spaces]] in $Line(Vect^{\mathbb{Z}/2})$ from being either $k$ or empty, to [[hom-sets]] being $\mathbb{Z}/2 = \{+1-1\} \hookrightarrow k$ or empty. Write $\widetilde{Line}(sVect)$ for the resulting [[2-group]].

In conclusion then the [[equivalence classes]] of possible [[k-invariants]] of $B^3 Line(sVect)$, hence the possible symmetric braiding on $Line(Vect^{\mathbb{Z}/2})$ are in the degree-4 [[ordinary cohomology]] of the [[Eilenberg-MacLane space]] $K(\mathbb{Z}/2,3)$ with [[coefficients]] in $\mathbb{Z}/2$. One finds (\ldots{})

\begin{displaymath}
H^4(K(\mathbb{Z}/2, 3), \mathbb{Z}/2)
  \;\simeq\;
  \mathbb{Z}/2
  \,.
\end{displaymath}
\end{proof}
\begin{defn}
\label{CategoryOfSuperVectorSpaces}\hypertarget{CategoryOfSuperVectorSpaces}{}
The [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory})

\begin{itemize}%
\item whose underlying [[monoidal category]] is that of $\mathbb{Z}/2$-[[graded vector spaces]] (example \ref{Z2Zgradedvectorspaces});


\item whose [[braiding]] (def. \ref{BraidedMonoidalCategory}) is the unique non-trivial symmetric grading $\tau^{super}$ from prop. \ref{TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces} is called the \textbf{[[category of super vector spaces]]}



\end{itemize}
\begin{displaymath}
sVect_k
    \;\coloneqq\;
  (Vect_k^{\mathbb{Z}/2}, \otimes = \otimes_k, 1 = k, \tau = \tau^{super} )
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The non-full symmetric monoidal subcategory

\begin{displaymath}
(\widetilde{Line}(sVect), \otimes_k, k, \tau^{super})
\end{displaymath}
of

\begin{displaymath}
(Line(sVect) , \otimes_k, k, \tau^{super})
    \hookrightarrow
  (sVect, \otimes_k, k, \tau^{super})
\end{displaymath}
(on the two objects $k^{1\vert 0}$ and $k^{0\vert 1}$ and with [[hom-sets]] restricted to $\{+1,-1\} \subset k$, as in the proof of prop. \ref{TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces}) happens to be the [[truncated object of an (infinity,1)-category|1-truncation]] of the [[looping]] of the [[sphere spectrum]] $\mathbb{S}$, regarded as a group-like [[E-infinity space]] (``[[abelian infinity-group]]'')

\begin{displaymath}
(\widetilde{Line}(sVect), \otimes, k, \tau^{super})
  \;\simeq\;
  \trunc_1 \Omega \mathbb{S}
  \,.
\end{displaymath}
It has been suggested (in \href{super+algebra#Kapranov15}{Kapranov 15}) that this and other phenomena are evidence that in the wider context of [[homotopy theory]]/[[stable homotopy theory]] super-grading (and hence [[superalgebra]]) is to be regarded as but a shadow of grading in [[higher algebra]] over the [[sphere spectrum]]. Notice that the [[sphere spectrum]] is just the analog of the group of [[integers]] in [[stable homotopy theory]].

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

\hypertarget{StructuresInternalToSuperVectorSpaces}{}\subsubsection*{{Structures internal to super-vector spaces}}\label{StructuresInternalToSuperVectorSpaces}

By [[internalization|internalizing]] [[algebra]] and [[geometry]] in the category $sVect$ of super vector spaces, one obtains the corresponding [[superalgebra]] and [[supergeometry]].

For example

\begin{itemize}%
\item [[monoids]] in $sVect$ are [[super algebras]] and, more interestingly, [[commutative monoids]] in $sVect$ are [[supercommutative superalgebra]];


\item hence for example [[commutative Hopf algebras]] in $sVect$ are equivalently [[supercommutative Hopf algebras]], which are the [[formal duals]] of affine algebraic [[supergroups]];


\item [[Lie algebras]] in $sVect$ are [[super Lie algebras]];


\item [[manifolds]] modeled on objects of $sVect$ are [[supermanifolds]]


\item etc.



\end{itemize}
By the above definition, any structure in $sVect$ works just like the corresponding structure in [[Vect]], but with a sign inserted whenever two odd-graded symbols are interchanged. For more on this see also at \emph{[[signs in supergeometry]]}.

\hypertarget{delignes_theorem_on_tensor_categories}{}\subsubsection*{{Deligne's theorem on tensor categories}}\label{delignes_theorem_on_tensor_categories}

[[Deligne's theorem on tensor categories]] says that all suitable [[tensor categories]] of subexponential growth have a [[fiber functor]] to $sVect$ and are equivalent to [[categories of representations]] of affine algebraic [[supergroups]].

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

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


\item [[superdeterminant]]


\item [[super vector bundle]]


\item [[chain complex of super vector spaces]]


\item [[model structure on chain complexes of super vector spaces]]


\item [[superalgebra]], [[supercommutative superalgebra]], [[differential graded-commutative superalgebra]],

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


\item [[geometry of physics -- supergeometry]]



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

\begin{itemize}%
\item [[Veeravalli Varadarajan]], section 3.1 of \emph{[[Supersymmetry for mathematicians]]: An introduction}, Courant lecture notes in mathematics, American Mathematical Society Providence, R.I 2004


\item [[Dennis Westra]], section 3 of \emph{Superrings and supergroups}, 2009 (\href{http://www.mat.univie.ac.at/~michor/westra_diss.pdf}{pdf})



\end{itemize}
[[!redirects SVect]] [[!redirects sVect]] [[!redirects SuperVect]]

[[!redirects super vector space]] [[!redirects super vector spaces]] [[!redirects supervector space]] [[!redirects supervector spaces]] [[!redirects super-vector space]] [[!redirects super-vector spaces]]

[[!redirects category of super vector spaces]] [[!redirects category of super-vector spaces]] [[!redirects category of supervector spaces]]



\end{document}