nLab 2-vector space

Idea

The concept of a $2$-vector space is supposed to be a 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.

What is a vector space?

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

1. A vector is a column of numbers. This is the way vector spaces appear in quantum mechanics, sections of line bundles, elementary linear algebra, etc.

2. A vector is a 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.

3. A vector is an element of a module over the base ring/field.

The first of these may be thought of as motivating the notion of

• Kapranov-Voevodsky $2$-vector space

the second the notion of

• Baez-Crans $2$-vector spaces

the third the notion of

• $2$-Modules.

Kapranov–Voevodsky $2$-vector spaces

These were introduced in Kapranov & Voevodsky 1991.

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 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 $G$.

Baez–Crans $2$-vector spaces

These were explicitly described in Baez & Crans 2004.

A Baez–Crans $2$-vector space is defined as a category internal to Vect. They categorify the idea of a vector as a ‘direction in space’, and crop up when considering the infinitesimal directions of a structure, such as in higher Lie theory. In fact, (following for instance from an extension of the Dold-Kan theorem by Brown and Higgins), 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 $L_\infty$-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 Forrester-Barker (2004).

$2$-modules and $2$-linear maps as algebras and bimodules

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 with basis namely the category $V-Mod$ in the sense of enriched category theory with

• objects are categories $C$ enriched over $V$, to be thought of as placeholders for their categories of modules, $Mod_C := [C,V]$

• morphisms $C \to D$ are bimodules $C^{op}\otimes D \to V$;

• $2$-morphisms are natural transformations.

Notice that all $V$-categories $Mod_C$ of modules over a $V$-category $C$ are naturally 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.

$\Vect$-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

• Jacob Lurie, On the classification of topological field theories (pdf) (see example 1.2.4)

• B. Toën, G. Vezzosi, A note on Chern character, loop spaces and derived algebraic geometry, (arXiv, p. 6)

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

• U. Schreiber, AQFT from $n$-functorial QFT (arXiv) (appendix A)

and

• U. Schreiber and K. Waldorf, Connections on non-abelian gerbes and their holonomy (arXiv)

Some blog discussion of this point is at 2-Vectors in Trodheim.

$Ch(Vect)$-enriched categories

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-categories.

It is argued in

• B. Toën, G. Vezzosi, A note on Chern character, loop spaces and derived algebraic geometry, (arXiv, p. 6)

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.

Revisiting Kapranov–Voevodsky 2-vector spaces

Upon further restriction of $\Vect\Mod$ to 2-vector spaces whose basis is a 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

$$[S, Vect] = Mod_{k^n} \simeq (Vect)^n$$

(where $k^n$ denotes the algebra of diagonal $n\times n$-matrices). These are precisely Kapranov–Voevodsky $2$-vector spaces.

Elgueta $2$-vector spaces

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

• Josep Elgueta, Generalized 2-vector spaces and general linear 2-groups (arXiv)

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 (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.

Infinite-dimensional K-V 2-vector 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 measure space such infinite dimensional K-V 2-vector spaces have been studied in

• John C. Baez, Aristide Baratin, Laurent Freidel, Derek K. Wise, Infinite-dimensional representations of 2-groups (arXiv)

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 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

• V. Ostrik, Module Categories, weak Hopf Algebras and Modular Invariants (arXiv, blog)

2-Modules as modules over a 2-ring

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 2-ring – Compatibly monoidal presentable categories for more details.

Remark on the different notions of $2$-vector spaces

As the above list shows, there are 2-vector spaces of very different kind. There is not 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

• Nils. A. Baas?, Marcel Bökstedt, Tore August Kro, Two-Categorical Bundles and Their Classifying Spaces, J. K-Theory, 10 (2012) 299 - 369, with a preliminary version at (arXiv)

$2$-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 categories with some extra properties were discribed in

• John Baez, Higher-Dimensional Algebra II: 2-Hilbert Spaces (arXiv) .

In applications one often assumes these 2-Hilbert spaces to be 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

• Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory (arXiv)

Properties

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-4-module

Related concepts

• vector space

• module category, (“actegory”)

• 2-vector space, 2-representation

  • 2-ring

  • 2Mod

  • 2-vector bundle

  • TwoVect is a Mathematica software package for computer algebra with 2-vector spaces

• n-vector space

References

As $n$-tuples of vector spaces

The notion of 2-vector spaces as $n$-tuples of vector spaces is due to

• Mikhail Kapranov, Vladimir Voevodsky, $2$-categories and Zamolodchikov tetrahedra equations in 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., Providence RI 1994), pp. 177-259 [pdf]

As 2-term chain complexes

The notion of 2-vector spaces as 2-term chain complexes is due to

• John C. Baez, Alissa S. Crans, §3 of: Higher-Dimensional Algebra VI: Lie 2-Algebras, Theor. Appl. Categor. 12 (2004) 492-528 [arXiv:math/0307263]

and used in

• Magnus Forrester-Barker, Representations of Crossed Modules and $Cat^1$-Groups, PhD thesis U. Wales Bangor (2004) [pdf]

As algebras with bimodules between them

The notion of 2-vector spaces with 2-linear maps between them as algebras with bimodules between them (subsuming the definition in Kapranov & Voevodsky 1991 as the special case of algebras that are direct sums of the ground field) is due to

• Urs Schreiber, §A of: AQFT from n-functorial QFT, Commun. Math. Phys. 291 (2009) 357-401 [arXiv:0806.1079, doi:10.1007/s00220-009-0840-2]

following earlier discussion in

• Urs Schreiber, 2-vectors in Trondheim (2006)

• Urs Schreiber, Topology in Trondheim and Kro, Baas & Bökstedt on 2-vector bundles (2007)

which is picked up in

• Urs Schreiber, Konrad Waldorf, §4.4 of: Connections on non-abelian Gerbes and their Holonomy, Theory Appl. Categ., 28 17 (2013) 476-540 [arXiv:0808.1923, tac:28-17]

and further developed into a theory of 2-vector bundles (via algebra bundles with bundles of bimodules between them) in:

• Peter Kristel, Matthias Ludewig, Konrad Waldorf, The insidious bicategory of algebra bundles [arXiv:2204.03900]

Essentially the same notion also appears, apparently independently, in:

• Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, Constantin Teleman, §7.1 with Ex. 2.13 in: Topological Quantum Field Theories from Compact Lie Groups [arXiv:0905.0731]

The notion is reviewed in a list of “standard” definitions in BDSPV15, without however referencing it.

See also at

• geometry of physics: geometry of physics - modules

the section on 2-modules.

Further discussion

Review includes

• Bruce Bartlett, Christopher L. Douglas, Chris Schommer-Pries, Jamie Vicary, A bestiary of 2–vector spaces, Appendix A in: Modular categories as representations of the 3-dimensional bordism 2-category [arXiv:1509.06811]

Another definition of 2-modules over 2-rings (see there for more) is in

• Alexandru Chirvasitu, Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. (DOI, arXiv:1105.3104)

A treatment of 2-representations of Lie 2-groups is in

• Zhen Huan, 2-Representations of Lie 2-groups and 2-Vector Bundles (arXiv:2208.10042)