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\begin{document}

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\section*{(infinity,n)-module}

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

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{higher_linear_algebra}{}\paragraph*{{Higher linear algebra}}\label{higher_linear_algebra}

[[!include homotopy - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{vector_spaces}{$(\infty,1)$-vector spaces}\dotfill \pageref*{vector_spaces} \linebreak
\noindent\hyperlink{2modules}{2-Modules}\dotfill \pageref*{2modules} \linebreak
\noindent\hyperlink{3modules}{3-Modules}\dotfill \pageref*{3modules} \linebreak
\noindent\hyperlink{4modules}{4-Modules}\dotfill \pageref*{4modules} \linebreak
\noindent\hyperlink{representations}{$n$-Representations}\dotfill \pageref*{representations} \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 notion of \emph{$n$-module} ($n$-vector space) is a [[categorification]] of the notion of [[module]] ([[vector space]]).

There are various different notions of $n$-vector spaces.

One notion is: an $n$-vector space is a [[chain complex]] of [[vector space]]s in degrees 0 to $n$. For $n=2$ this is a [[Baez-Crans 2-vector space]]. This is useful for lots of things, but tends to be too restrictive in other contexts.

Another is, recursively: an $(n-1)$-algebra object (or its $(n-1)$-category of modules) in the $n$-category of $(n-1)$-bimodules. For higher $n$ this is envisioned in (\href{FHLT}{FHLT, section 7}), details are in spring. It includes the previous concept as a special case.

For $n=2$ this subsumes various other definitions of [[2-vector space]] that are in the literature, such as notably the notion of [[Kapranov-Voevodsky 2-vector space]].

We sketch the iterative definition of $n$-vector spaces. More details are below.

Assume that a notion of [[n-category]] is chosen for each $n$ (for instance [[(n,1)-category]]), that a notion of [[symmetric monoidal category|symmetric monoidal]] $n$-category is fixed (for instance [[symmetric monoidal (∞,1)-category]]) and that a notion of (weak) commutative [[monoid]] objects and [[module]] and [[bimodule]] object in a symmetric monoidal $n$-category is fixed (for instance the notion of [[algebra in an (∞,1)-category]]).

Then we have the following recursive (rough) definition:

fix a ground [[field]] $k$.

\begin{itemize}%
\item a 0-vector space over $k$ is an elemment of $k$. The [[0-category]] of 0-vector spaces is the set

\begin{displaymath}
0 Vect_k = k
  \,.
\end{displaymath}

\item The category $1 Vect_k$ is just [[Vect]].


\item For $n \gt 1$, the [[n-category]] $n Vect$ of \textbf{$n$-vector spaces} over $k$ is the $n$-category with objects algebra objects in $(n-1)Vect$ and morphisms bimodule objects in $(n-1)Vect$.



\end{itemize}
Here we think of an algebra object $A \in (n-1)Vect$ as a basis for the $n$-vector space which is the $(n-1)$-category $A Mod$.

With this definition we have that $2 Vect$ is the [[2-category]] of $k$-[[algebra]]s, [[bimodule]]s and bimodule homomorphisms.

More generally, let $k$ here be a [[ring spectrum]]. Set

\begin{itemize}%
\item $(\infty,0)Vect_k := k$ -- a [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[∞-groupoid]];


\item $(\infty,1)Vect_k := k Mod$ the [[symmetric monoidal (∞,1)-category]] of modules over that ring spectrum;


\item $(\infty,n)Vect_k := (\infty,n-1) Mod$ the [[symmetric monoidal (∞,n)-category]] of modules over $(\infty,n-1)Mod$.



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

Following the \href{IteratedModules}{above idea} we have the following definition.

\begin{defn}
\label{nVectViaALgObjects}\hypertarget{nVectViaALgObjects}{}
Fix a [[ring]] $k$ (usually taken to be a [[field]] if one speaks of ``vector spaces'' instead of just [[module]]s, but this is not actually essential for the construction). This may be an [[ring spectrum|∞-ring]].

For $n \in \mathbb{N}$, define an [[symmetric monoidal (∞,n)-category]] $n Vect_k$ of \textbf{$(\infty,n)$-vector spaces} as follows (the bi-counting follows the pattern of [[(n,r)-categories]]).

An \textbf{$(\infty,0)$-vector space} is an element of $k$. If $k$ is an ordinary ring, then the [[0-category]] $0 Vect$ is the underlying set of $k$, regarded as a [[symmetric monoidal category]] using the product structure on $k$. If $k$ is more generally an [[ring spectrum|∞-ring]], then the ``stabilized [[(∞,0)-category]]'' (= [[spectrum]]) of $(\infty,0)$-vector spaces is $k$ itself: $(\infty,0)Vect_k \simeq k$.

An \textbf{[[(∞,1)-vector space]]} is an [[module spectrum|∞-module]] over $k$. The [[(∞,1)-category]] of $(\infty,1)$-vector spaces is

\begin{displaymath}
(\infty,1)Vect_k := k Mod
  \,,
\end{displaymath}
the $(\infty,1)$-category of $k$-[[module spectra]].

For $k$ a field ordinary [[vector space]]s over $k$ are a [[full sub-(∞,1)-category]] of this: $1Vect_k \hookrightarrow (\infty,1)Vect_k$ .

For $n \geq 2$, an \textbf{$(\infty,n)$-vector space} is an [[algebra in an (infinity,1)-category|algebra object in the symmetric monoidal (∞,1)-category]] $(\infty,n-1)Vect$. A [[morphism]] is a [[bimodule object]]. [[k-morphism|Higher morphisms]] are defined recursively.

\end{defn}
For $\infty$ replaced by $n$ this appears as (\hyperlink{Schreiber}{Schreiber, appendix A}) and then with allusion to more sophisticated [[higher category theory|higher categorical]] tools in (\hyperlink{FHLT}{FHLT, def. 7.1}).

Notice that FHLT say ``$(n-1)$-algebra'' instead of ``$n$-vector space'', but only for the reason (p. 29) that

\begin{quote}%
The discrepancy between $m$ (the algebra level) and $n$ the algebra level -- for which we apologize -- is caused by the fact that the term ``$n$-vector space'' has been used for a much more restrictive notion than our $(n-1)$-algebras.


\end{quote}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{vector_spaces}{}\subsubsection*{{$(\infty,1)$-vector spaces}}\label{vector_spaces}

See [[(∞,1)-vector space]] for more.

\hypertarget{2modules}{}\subsubsection*{{2-Modules}}\label{2modules}

\begin{remark}
\label{}\hypertarget{}{}
The symmetric monoidal 3-category $Alg_k^b = 2 Mod_k$ of [[2-modules]] over $k$ is:

\begin{itemize}%
\item [[objects]] are [[associative algebras]] over $k$;


\item [[morphisms]] are [[bimodules]] of associative algebras; [[composition]] is the [[tensor product]] of bimodules;


\item [[2-morphisms]] are bimodule homomorphisms.



\end{itemize}
We think of this equivalently as its essential image in $Vect_k Mod$, where

\begin{itemize}%
\item an algebra $A$ is a placeholder for its [[module category]] $Mod_A$;


\item an $A$-$B$ [[bimodule]] $N$ is a placeholder for the [[functor]]

\begin{displaymath}
Mod_A \stackrel{(-) \otimes_A N }{\to} Mod_B
\end{displaymath}

\item a bimodule homomorphism is a placeholder for a [[natural transformation]] of two such functors.



\end{itemize}
If we think of an algebra $A$ in terms of its [[delooping]] [[Vect]]-[[enriched category]] $B A$, then we have an [[equivalence of categories]]

\begin{displaymath}
Mod_A \simeq Vect Cat(B A, Vect)
  \,.
\end{displaymath}
Comparing this for the formula

\begin{displaymath}
V \simeq Set(S,k)
\end{displaymath}
for a $k$-vector space $V$ with [[basis]] $S$, we see that we may

\begin{itemize}%
\item think of the algebra objects appearing in the above as being \emph{bases} for a higher vector space;


\item think of the bimodules as being higher [[matrix|matrices]].



\end{itemize}
\end{remark}
\hypertarget{3modules}{}\subsubsection*{{3-Modules}}\label{3modules}

A \emph{3-vector space} according to def. \ref{nVectViaALgObjects} is

\begin{itemize}%
\item a $k$-algebra $A$;


\item equipped with an $A$-$A\otimes A$-[[bimodule]] defining the 2-multiplication, and a left $A$-[[module]] defining the unit.



\end{itemize}
Equivalently this is a [[sesquiunital sesquialgebra]].

Classes of examples come from the following construction:

\begin{itemize}%
\item Every \emph{commutative} [[associative algebra]] $A$ becomes a 3-vector space.


\item Every [[Hopf algebra]] canonically becomes a 3-vector space (amplified in \hyperlink{FHLT}{FHLT, p. 27}).

More generally: every [[hopfish algebra]].



\end{itemize}
\hypertarget{4modules}{}\subsubsection*{{4-Modules}}\label{4modules}

Next, an algebra object internal to $2 Alg_k^b = 3Mod_3$, is an algebra equipped with three compatible algebra structures, a [[trialgebra]].

Its [[category of modules]] is a [[monoidal category]] equipped with two compatible product structures a [[Hopf category]].

The 2-category of 2-modules of that is a [[monoidal 2-category]].

For a review see (\href{BaezLauda}{Baez-Lauda 09, p. 98}).

\hypertarget{representations}{}\subsection*{{$n$-Representations}}\label{representations}

See [[infinity-representation]].

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

\begin{itemize}%
\item [[vector space]],


\item [[(∞,1)-module]], [[(∞,1)-module bundle]], [[(∞,1)-category of (∞,1)-modules]]


\item [[2-ring]], [[2-module]]


\item [[2-vector space]]

\begin{itemize}%
\item \emph{[[TwoVect]]} is a Mathematica software package for computer algebra with 2-vector spaces

\end{itemize}

\item \textbf{$n$-vector space}, [[n-vector bundle]],



\end{itemize}
[[!include structure on algebras and their module categories - table]]

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

The notion of $n$-vector spaces is (defined for $n = 2$ and sketched recursively for greater $n$) in

appendix A of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{AQFT from $n$-functorial QFT} Communications in Mathematical Physics, Volume 291, Issue 2, pp.357-401 (2008) (\href{http://ncatlab.org/schreiber/files/AQFTfromFQFT.pdf}{pdf})

\end{itemize}
section 7 of

\begin{itemize}%
\item [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]} (2009)

\end{itemize}
Full details are in

\begin{itemize}%
\item [[Rune Haugseng]], \emph{The higher Morita category of $E_n$-algebras} (\href{http://arxiv.org/abs/1412.8459}{arXiv:1412.8459})

\end{itemize}
Review of work on 4-modules (implicitly) as [[trialgebras]]/[[Hopf monoidal categories]] is around p. 98 of

\begin{itemize}%
\item [[John Baez]], [[Aaron Lauda]], \emph{A prehistory of $n$-categorical physics}, in \emph{Deep beauty}, 13-128, Cambridge Univ. Press, Cambridge, 2011 (\href{http://arxiv.org/abs/0908.2469}{arXiv:0908.2469})

\end{itemize}
[[!redirects n-vector spaces]] [[!redirects n-vector spaces]]

[[!redirects nMod]]

[[!redirects (∞,n)-vector space]] [[!redirects (infinity,n)-vector space]] [[!redirects (∞,n)-vector spaces]] [[!redirects (infinity,n)-vector spaces]]

[[!redirects (infinity,n)-modules]] [[!redirects (∞,n)-module]] [[!redirects (∞,n)-modules]]

[[!redirects (infinity,2)-module]] [[!redirects (infinity,2)-modules]] [[!redirects (∞,2)-module]] [[!redirects (∞,2)-modules]]

[[!redirects (∞, n)-vector space]] [[!redirects (infinity, n)-vector space]] [[!redirects (∞, n)-vector spaces]] [[!redirects (infinity, n)-vector spaces]]

[[!redirects (infinity, n)-modules]] [[!redirects (∞, n)-module]] [[!redirects (∞, n)-modules]]

[[!redirects (infinity, 2)-module]] [[!redirects (infinity, 2)-modules]] [[!redirects (∞, 2)-module]] [[!redirects (∞, 2)-modules]]

[[!redirects 3-vector space]] [[!redirects 3-vector spaces]]

[[!redirects 3-module]] [[!redirects 3-modules]]

[[!redirects 4-module]] [[!redirects 4-modules]] [[!redirects 4-vector space]] [[!redirects 4-vector spaces]]

[[!redirects 2-algebra]] [[!redirects 2-algebras]]

[[!redirects 3-algebra]] [[!redirects 3-algebras]]

[[!redirects n-module]] [[!redirects n-modules]]

[[!redirects (infinity,n)-vector space]] [[!redirects (infinity, n)-vector space]] [[!redirects n-vector space]]



\end{document}