symmetric monoidal (∞,1)-category of spectra
The concept of a -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.
There are at least three distinct conceptual roles which vectors and vector spaces play in mathematics:
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.
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.
The first of these may be thought of as motivating the notion of
the second that of of
the third the notion of
These were introduced in:
The idea here is that just as a vector space can be regarded as a module over the ground field , a -vector space 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 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 -vector space as an abelian -module category equivalent to for some .
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 .
Because Kapranov–Voevodsky -vector spaces categorify the idea of a vector space as a ‘state-space’ of a system, they are the notion of -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 -vector space is , the category of representations of a finite group .
These were explicitly described in:
A Baez–Crans -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 are equivalent to chain complexes in non-negative degree and can be regarded as strict --modules. This allows to conceive much of homological algebra and many of the structures appearing in higher Lie theory – for instance the definition of -algebras, as being about -vector spaces. Regarding a chain complex as an -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 -vector spaces which they are equivalent to. (See also the remark about different notions of 2-vector spaces further below.)
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 -module category is a category whose space of objects and space of morphisms are both -modules – ordinary vector spaces! – such that all structure morphisms (source, target, identity, composition) respect the -action – hence are linear maps. This are categories internal to which are equivalent to chain complexes of vector spaces concentrated in degree 0 and 1.
In other words, a Baez–Crans -vector space can be thought of as a Kapranov–Voevodsky -vector space, if one ‘categorifies’ the ground field by simply regarding it as a discrete monoidal category.
For a general symmetric closed monoidal category the full bicategory of all monoidal category modules over 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 -modules with basis namely the category in the sense of enriched category theory with
morphisms are bimodules ;
-morphisms are natural transformations.
Notice that all -categories of modules over a -category are naturally tensored over and hence are monoidal category modules over . In analogy to how a vector space (a -module) is equipped with a basis by finding a set such that , we can think of a general monoidal category module over to be equipped with a basis by providing an equivalence , for some -category . In this sense is the category of 2-vector spaces with basis.
All of the examples on this page are special cases of this one.
According to the above a -enriched category can be regarded as a basis for the -module . A -enriched category is just an algebroid. If it has a single object it is an algebra and 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 -vector space, iff they are Morita equivalent.
-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)
-vector spaces in the sub-bicategory of algebras (-enriched categories with a single object), bimodules and intertwiners are discussed in
Some blog discussion of this point is at 2-Vectors in Trodheim.
More generally one can replace vector spaces by complexes of vector spaces and consider as a model for the -category of -vector spaces (with basis): its objects are dg-categories.
It is argued in
that the generalization from to is necessary to have a good notion of higher sheaves of sections of 2-vector bundles, i.e. of higher coherent sheaves.
Upon further restriction of to 2-vector spaces whose basis is a discrete category, namely a set (or the -enriched category over which has just the ground field object sitting over each element of ) one arrives at -modules of the form
(where denotes the algebra of diagonal -matrices). These are precisely Kapranov–Voevodsky -vector spaces.
Another notion of 2-vector space which also includes Kaparanov–Voevodsky as particular instances is given in
The idea is to categorify the construction of a vector space as the space of finite linear combinations of elements in any set . Instead of , we start now with any category , and take first the free -linear category generated by , and next the additive completion of this. Kapranov–Voevodsky -vector spaces are recovered when 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 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 -modules, which is nonKaroubian.
We can regard the objects of the -dimensional Kapranov–Voevodsky -vector space – which are -tuples of vector spaces – as vector bundles over the finite set of 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
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 in question is a modular tensor category. A result by Victor Ostrik showed that all -module categories are equivalent to for some one-object -enriched category (i.e., an algebra internal to ) in
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.
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 -module categories are crucial for higher Lie theory but 2-bundles with fibers -module categories are comparatively boring as far as general 2-bundles go, as they are essentially complexes of ordinary vector bundles. See
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 -enriched categories with some extra properties were discribed in
In applications one often assumes these 2-Hilbert spaces to be semisimple in which case such a 2-Hilbert space is a Kapranov–Voevodsky -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
|monoid/associative algebra||category of modules|
|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|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
2-vector space, 2-representation
A systematic definition of 2-modules over 2-rings (see there for more) is in
See also at
the section on 2-modules.