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\section*{2-vector space}

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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

[[!include higher category theory - contents]]

\hypertarget{content}{}\section*{{Content}}\label{content}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{what_is_a_vector_space}{What is a vector space?}\dotfill \pageref*{what_is_a_vector_space} \linebreak
\noindent\hyperlink{KV2VectorSpace}{Kapranov--Voevodsky $2$-vector spaces}\dotfill \pageref*{KV2VectorSpace} \linebreak
\noindent\hyperlink{BaezCrans2VectorSpaces}{Baez--Crans $2$-vector spaces}\dotfill \pageref*{BaezCrans2VectorSpaces} \linebreak
\noindent\hyperlink{AbstractApproach}{$2$-modules and $2$-linear maps as algebras and bimodules}\dotfill \pageref*{AbstractApproach} \linebreak
\noindent\hyperlink{enriched_categories}{$\Vect$-enriched categories}\dotfill \pageref*{enriched_categories} \linebreak
\noindent\hyperlink{enriched_categories_2}{$Ch(Vect)$-enriched categories}\dotfill \pageref*{enriched_categories_2} \linebreak
\noindent\hyperlink{revisiting_kapranovvoevodsky_2vector_spaces}{Revisiting Kapranov--Voevodsky 2-vector spaces}\dotfill \pageref*{revisiting_kapranovvoevodsky_2vector_spaces} \linebreak
\noindent\hyperlink{elgueta_vector_spaces}{Elgueta $2$-vector spaces}\dotfill \pageref*{elgueta_vector_spaces} \linebreak
\noindent\hyperlink{infinitedimensional_kv_2vector_spaces}{Infinite-dimensional K-V 2-vector spaces}\dotfill \pageref*{infinitedimensional_kv_2vector_spaces} \linebreak
\noindent\hyperlink{using_a_modular_tensor_category}{Using a modular tensor category}\dotfill \pageref*{using_a_modular_tensor_category} \linebreak
\noindent\hyperlink{2modules_as_modules_over_a_2ring}{2-Modules as modules over a 2-ring}\dotfill \pageref*{2modules_as_modules_over_a_2ring} \linebreak
\noindent\hyperlink{remark_on_the_different_notions_of_vector_spaces}{Remark on the different notions of $2$-vector spaces}\dotfill \pageref*{remark_on_the_different_notions_of_vector_spaces} \linebreak
\noindent\hyperlink{hilbert_spaces}{$2$-Hilbert spaces}\dotfill \pageref*{hilbert_spaces} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{tannaka_duality}{Tannaka duality}\dotfill \pageref*{tannaka_duality} \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}

The concept of a \emph{$2$-vector space} is supposed to be a [[vertical categorification|categorification]] of the concept of a [[vector space]]. As usual in the game of `categorification', this requires us to think deeply about what an ordinary vector space really is, and then attempt to categorify that idea.

\hypertarget{what_is_a_vector_space}{}\subsubsection*{{What is a vector space?}}\label{what_is_a_vector_space}

There are at least three distinct conceptual roles which vectors and vector spaces play in mathematics:

\begin{enumerate}%
\item A vector is a \emph{column of numbers}. This is the way vector spaces appear in quantum mechanics, sections of line bundles, elementary linear algebra, etc.


\item A vector is a \emph{direction in space}. Vector spaces of this kind are often the infinitesimal data of some global structure, such as tangent spaces to manifolds, Lie algebras of Lie groups, and so on.


\item A vector is an element of a [[module]] over the base [[ring]]/[[field]].



\end{enumerate}
The first of these may be thought of as motivating the notion of

\begin{itemize}%
\item \hyperlink{KV2VectorSpace}{Kapranov-Voevodsky $2$-vector space}

\end{itemize}
the second the notion of

\begin{itemize}%
\item \hyperlink{BaezCrans2VectorSpaces}{Baez-Crans $2$-vector spaces}

\end{itemize}
the third the notion of

\begin{itemize}%
\item \hyperlink{AbstractApproach}{$2$-Modules}.

\end{itemize}
\hypertarget{KV2VectorSpace}{}\subsubsection*{{Kapranov--Voevodsky $2$-vector spaces}}\label{KV2VectorSpace}

These were introduced in:

\begin{itemize}%
\item M. Kapranov and V. Voevodsky, \emph{$2$-categories and Zamolodchikov tetrahedra equations} in \emph{Algebraic groups and their generalization: quantum and infinite-dimensional methods, University Park, PA (1991) (eds: W. J. Haboush and B. J. Parshall), Proc. Sympos. Pure Math. 56 (Amer. Math. Soc., Providencem RIm 1994), pp. 177-259}

\end{itemize}
The idea here is that just as a vector space can be regarded as a [[module]] over the [[ground field]] $k$, a $2$-vector space $W$ should be a [[category]] which is a [[monoidal category module]] with some nice properties (such as being an abelian category) over a suitable [[monoidal category]] $V$ which plays the role of the categorified ground field. There is then an obvious [[bicategory]] of such module categories.

In fact, Kapranov and Voevodsky defined a \textbf{Kapranov--Voevodsky $2$-vector space} as an abelian $\Vect$-module category equivalent to $\Vect^n$ for some $n$.

While this definition makes a lot of sense it does not give an abstract characterization of 2-vector spaces. That is, it is hardly different to simply defining a 2-vector space as a category equivalent to $Vect^n$.

Because Kapranov--Voevodsky $2$-vector spaces categorify the idea of a vector space as a `state-space' of a system, they are the notion of $2$-vector space which feature on the right hand side of extended TQFTs (functors from higher [[cobordism]] categories to higher vector spaces).

An example of a Kapranov--Voevodsky $2$-vector space is $Rep(G)$, the category of [[representations]] of a finite group $G$.

\hypertarget{BaezCrans2VectorSpaces}{}\subsubsection*{{Baez--Crans $2$-vector spaces}}\label{BaezCrans2VectorSpaces}

These were explicitly described in:

\begin{itemize}%
\item John C. Baez and Alissa S. Crans, \emph{Higher-Dimensional Algebra VI: Lie $2$-Algebras} (\href{http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html}{tac}, \href{http://math.ucr.edu/home/baez/hda6.pdf}{pdf}).

\end{itemize}
A \textbf{Baez--Crans $2$-vector space} is defined as a [[internal category|category internal]] to [[Vect]]. They categorify the idea of a vector as a `direction in space', and crop up when considering the \emph{infinitesimal directions} of a structure, such as in higher [[Lie theory]]. In fact, (following for instance from an extension of the [[Dold-Kan correspondence|Dold-Kan theorem]] by Brown and Higgins), [[strict omega-category|strict omega-categories]] internal to $\Vect$ are equivalent to chain complexes in non-negative degree and can be regarded as strict $Disc(k)$-$\infty$-modules. This allows to conceive much of [[homological algebra]] and many of the structures appearing in higher [[Lie theory]] -- for instance the definition of $L_\infty$-[[L-infinity-algebra|algebras]], as being about $\infty$-vector spaces. Regarding a chain complex as an $\infty$-vector space is useful conceptually for understanding the meaning of some constructions on chain complexes, while of course chain complexes themselves are well suited for direct computation with the $\infty$-vector spaces which they are equivalent to. (See also the remark about different notions of 2-vector spaces further below.)

They were also independently introduced and studied by [[Magnus Forrester-Barker]] in his thesis (\hyperlink{Forrester-Barker}{Forrester-Barker 2004}).

\hypertarget{AbstractApproach}{}\subsection*{{$2$-modules and $2$-linear maps as algebras and bimodules}}\label{AbstractApproach}

It is possible to conceive of 2-vector spaces of the Kapranov--Voevodsky and Baez--Crans type from a single unified perspective. Namely, by regarding the [[ground field]] itself as a [[discrete category]] we can think of it as a [[monoidal category]]. A $Disc(k)$-module category is a category whose space of objects and space of morphisms are both $k$-modules -- ordinary vector spaces! -- such that all structure morphisms (source, target, identity, composition) respect the $k$-action -- hence are linear maps. These are categories internal to $\Vect_k$ which are equivalent to chain complexes of vector spaces concentrated in degree 0 and 1.

In other words, a Baez--Crans $2$-vector space can be thought of as a Kapranov--Voevodsky $2$-vector space, if one `categorifies' the ground field by simply regarding it as a discrete monoidal category.

For $V$ a general symmetric [[closed monoidal category]] the full bicategory of all [[monoidal category modules]] over $V$ is in general hard to get under control, but what is more tractable is the sub-bicategory which may be addressed as the bicategory of $V$-modules \emph{with basis} namely the category $V-Mod$ in the sense of [[enriched category theory]] with

\begin{itemize}%
\item objects are categories $C$ [[enriched category|enriched over]] $V$, to be thought of as placeholders for their categories of [[modules]], $Mod_C := [C,V]$


\item morphisms $C \to D$ are [[bimodules]] $C^{op}\otimes D \to V$;


\item $2$-morphisms are natural transformations.



\end{itemize}
Notice that all $V$-categories $Mod_C$ of modules over a $V$-category $C$ are naturally [[copower|tensored]] over $V$ and hence are [[monoidal category modules]] over $V$. In analogy to how a vector space $W$ (a $k$-module) is equipped with a basis by finding a set $S$ such that $W \simeq [S,k]$, we can think of a general [[monoidal category module]] $W$ over $V$ to be equipped with a basis by providing an [[equivalence]] $W \simeq [C,V]$, for some $V$-category $C$. In this sense $V-Mod$ is the category of $V$ 2-vector spaces with basis.

All of the examples on this page are special cases of this one.

\hypertarget{enriched_categories}{}\subsubsection*{{$\Vect$-enriched categories}}\label{enriched_categories}

According to the above a $Vect$-[[enriched category]] $C$ can be regarded as a basis for the $Vect$-module $Mod_C = [C,Vect]$. A $Vect$-enriched category is just an [[algebroid]]. If it has a single object it is an [[algebra]] and $Mod_C$ is the familiar category of modules over an algebra.

Notice that, by the very definition of [[Morita equivalence]], two algebras (algebroids) have equivalent module categories, and hence can be regarded as different bases for the same $\Vect$ $2$-vector space, iff they are Morita equivalent.

$Vect$-enriched categories as models for 2-vector spaces appear in

\begin{itemize}%
\item Jacob Lurie, \emph{On the classification of topological field theories} (\href{http://www.math.harvard.edu/~lurie/papers/cobordism.pdf}{pdf}) (see example 1.2.4)


\item B. To\"e{}n, G. Vezzosi, \emph{A note on Chern character, loop spaces and derived algebraic geometry}, (\href{http://arxiv.org/abs/0804.1274}{arXiv}, p. 6)



\end{itemize}
$2$-vector spaces in the sub-bicategory of algebras ($Vect$-enriched categories with a single object), bimodules and intertwiners are discussed in

\begin{itemize}%
\item U. Schreiber, \emph{AQFT from $n$-functorial QFT} (\href{http://arxiv.org/abs/0806.1079}{arXiv}) (appendix A)

\end{itemize}
and

\begin{itemize}%
\item U. Schreiber and K. Waldorf, \emph{Connections on non-abelian gerbes and their holonomy} (\href{http://arxiv.org/abs/0808.1923}{arXiv})

\end{itemize}
Some blog discussion of this point is at \href{http://golem.ph.utexas.edu/category/2007/11/2vectors_in_trondheim.html}{2-Vectors in Trodheim}.

\hypertarget{enriched_categories_2}{}\subsubsection*{{$Ch(Vect)$-enriched categories}}\label{enriched_categories_2}

More generally one can replace vector spaces by complexes of vector spaces and consider $Ch(Vect)\Mod$ as a model for the $2$-category of $2$-vector spaces (with basis): its objects are [[dg-category|dg-categories]].

It is argued in

\begin{itemize}%
\item B. To\"e{}n, G. Vezzosi, \emph{A note on Chern character, loop spaces and derived algebraic geometry}, (\href{http://arxiv.org/abs/0804.1274}{arXiv}, p. 6)

\end{itemize}
that the generalization from $Vect\Mod$ to $Ch(Vect)\Mod$ is necessary to have a good notion of higher sheaves of sections of 2-vector bundles, i.e. of higher coherent sheaves.

\hypertarget{revisiting_kapranovvoevodsky_2vector_spaces}{}\subsubsection*{{Revisiting Kapranov--Voevodsky 2-vector spaces}}\label{revisiting_kapranovvoevodsky_2vector_spaces}

Upon further restriction of $\Vect\Mod$ to 2-vector spaces whose basis is a \emph{discrete category}, namely a set $S$ (or the $Vect$-enriched category over $S$ which has just the [[ground field]] object sitting over each element of $S$) one arrives at $Vect$-modules of the form

\begin{displaymath}
[S, Vect] = Mod_{k^n} \simeq (Vect)^n
\end{displaymath}
(where $k^n$ denotes the algebra of diagonal $n\times n$-matrices). These are precisely Kapranov--Voevodsky $2$-vector spaces.

\hypertarget{elgueta_vector_spaces}{}\subsubsection*{{Elgueta $2$-vector spaces}}\label{elgueta_vector_spaces}

Another notion of 2-vector space which also includes Kaparanov--Voevodsky as particular instances is given in

\begin{itemize}%
\item Josep Elgueta, \emph{Generalized 2-vector spaces and general linear 2-groups} (\href{http://arxiv.org/abs/math/0606472}{arXiv})

\end{itemize}
The idea is to categorify the construction of a vector space as the space of finite linear combinations of elements in any set $S$. Instead of $S$, we start now with any category $C$, and take first the free $k$-linear category generated by $C$, and next the additive completion of this. Kapranov--Voevodsky $2$-vector spaces are recovered when $C$ is discrete. In some cases this gives nonabelian and even non-[[Karoubian category|Karoubian]] (i.e., nonidempotent complete) categories. This is the case, for instance, when we take as $C$ the one-object category defined by the additive monoid of natural numbers. The 2-vector space this category generates can be identified with the category of free $k[T]$-modules, which is nonKaroubian.

\hypertarget{infinitedimensional_kv_2vector_spaces}{}\subsubsection*{{Infinite-dimensional K-V 2-vector spaces}}\label{infinitedimensional_kv_2vector_spaces}

We can regard the objects of the $n$-dimensional Kapranov--Voevodsky $2$-vector space $Vect^n$ -- which are $n$-tuples of vector spaces -- as vector bundles over the finite set of $n$ elements. This has an obvious generalization to vector bundles over any topological space -- in terms of modules these are the finitely generated projective modules of the algebra of continuous functions on this space. So categories of vector bundles can be regarded as infinite-dimensional 2-vector spaces. For the case that the underlying topological space is a \emph{measure space} such infinite dimensional K-V 2-vector spaces have been studied in

\begin{itemize}%
\item John C. Baez, Aristide Baratin, Laurent Freidel, Derek K. Wise, \emph{Infinite-Dimensional Representations of 2-Groups} (\href{http://arxiv.org/abs/0812.4969}{arXiv})

\end{itemize}
\hypertarget{using_a_modular_tensor_category}{}\subsubsection*{{Using a modular tensor category}}\label{using_a_modular_tensor_category}

The relevance of module categories as models for 2-vector spaces was apparently first realized in the context of [[conformal field theory]], where the monoidal category $V$ in question is a [[modular tensor category]]. A result by Victor Ostrik showed that \emph{all} $V$-module categories are equivalent to $Mod_A$ for $A$ some one-object $V$-enriched category (i.e., an algebra internal to $V$) in

\begin{itemize}%
\item V. Ostrik, \emph{Module Categories, weak Hopf Algebras and Modular Invariants} (\href{http://arxiv.org/abs/math.QA/0111139}{arXiv}, \href{http://golem.ph.utexas.edu/string/archives/000717.html}{blog})

\end{itemize}
\hypertarget{2modules_as_modules_over_a_2ring}{}\subsection*{{2-Modules as modules over a 2-ring}}\label{2modules_as_modules_over_a_2ring}

One can go further and derive the identification of 2-modules and 2-linear maps with algebras and bimodules from a more fundamental notion of modules over [[2-rings]]. For the moment see there at \emph{\href{2-rig#CompatiblyMonoidalPresentableCategories}{2-ring -- Compatibly monoidal presentable categories}} for more details.

\hypertarget{remark_on_the_different_notions_of_vector_spaces}{}\subsection*{{Remark on the different notions of $2$-vector spaces}}\label{remark_on_the_different_notions_of_vector_spaces}

As the above list shows, there are 2-vector spaces of very different kind. There is not \emph{the} notion of 2-vector space which is the universal right answer. Different notions of vector spaces are applicable and useful in different situations. This can be regarded as nothing but a more pronounced incarnation of the fact that already ordinary vector space appear in different flavors which are useful in different situations (real vector spaces, complex vector spaces, vector spaces over a [[finite field]], etc.)

For instance $Disc(k)$-module categories are crucial for higher [[Lie theory]] but 2-bundles with fibers $Disc(k)$-module categories are comparatively boring as far as general 2-bundles go, as they are essentially complexes of ordinary vector bundles. See

\begin{itemize}%
\item [[Nils. A. Baas]], Marcel B\"o{}kstedt, Tore August Kro, \emph{Two-Categorical Bundles and Their Classifying Spaces}, J. K-Theory, 10 (2012) 299 - 369, with a preliminary version at (\href{http://arxiv.org/abs/math/0612549}{arXiv})

\end{itemize}
\hypertarget{hilbert_spaces}{}\subsection*{{$2$-Hilbert spaces}}\label{hilbert_spaces}

2-vector spaces have to a large extent been motivated by and applied in (2-dimensional) [[quantum field theory]]. In that context it is usually not the concept of a plain vector space which needs to be categorified, but that of a Hilbert space.

2-Hilbert spaces as a $\Hilb$-[[enriched category|enriched categories]] with some extra properties were discribed in

\begin{itemize}%
\item John Baez, \emph{Higher-Dimensional Algebra II: 2-Hilbert Spaces} (\href{http://arxiv.org/abs/q-alg/9609018}{arXiv}) .

\end{itemize}
In applications one often assumes these 2-Hilbert spaces to be [[semisimple algebra|semisimple]] in which case such a 2-Hilbert space is a Kapranov--Voevodsky $2$-vector space equipped with extra structure.

A review of these ideas of 2-Hilbert spaces as well as applications of 2-Hilbert spaces to finite group representation theory are in

\begin{itemize}%
\item Bruce Bartlett, \emph{On unitary 2-representations of finite groups and topological quantum field theory} (\href{http://arxiv.org/abs/0901.3975}{arXiv})

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

\hypertarget{tannaka_duality}{}\subsubsection*{{Tannaka duality}}\label{tannaka_duality}

[[!include structure on algebras and their module categories - table]]

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

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


\item \textbf{2-vector space}, [[2-representation]]

\begin{itemize}%
\item [[2-ring]]


\item [[2Mod]]


\item [[2-vector bundle]]


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



\end{itemize}

\item [[n-vector space]]



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

\begin{itemize}%
\item [[Magnus Forrester-Barker]], \emph{Representations of Crossed Modules and $Cat^1$-Groups}, PhD thesis U. Wales Bangor (2004) (\href{http://www.maths.bangor.ac.uk/research/ftp/theses/forrester-barker.pdf}{pdf})

\end{itemize}
A systematic definition of 2-modules over [[2-rings]] (see there for more) is in

\begin{itemize}%
\item [[Alexandru Chirvasitu]], [[Theo Johnson-Freyd]], \emph{The fundamental pro-groupoid of an affine 2-scheme} (\href{http://arxiv.org/abs/1105.3104}{arXiv:1105.3104})

\end{itemize}
See also at

\begin{itemize}%
\item \emph{[[geometry of physics]]: [[geometry of physics - modules]]}

\end{itemize}
the section on 2-modules.

[[!redirects 2-vector spaces]]

[[!redirects 2-module]] [[!redirects 2-modules]]



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