There are various different notions of -vector spaces.
One notion is: an -vector space is a chain complex of vector spaces in degrees 0 to . For 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 -algebra object (or its -category of modules) in the -category of -bimodules. For higher this is in (FHLT, section 7). It includes the previous concept as a special case.
We sketch the iterative definition of -vector spaces. More details are below.
Assume that a notion of n-category is chosen for each (for instance (n,1)-category), that a notion of symmetric monoidal -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 -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 .
a 0-vector space over is an elemment of . The 0-category of 0-vector spaces is the set
The category is just Vect.
For , the n-category of -vector spaces over is the -category with objects algebra objects in and morphisms bimodule objects in .
Here we think of an algebra object as a basis for the -vector space which is the -category .
More generally, let here be a ring spectrum. Set
the (∞,1)-category of modules over that ring spectrum;
the symmetric monoidal (∞,1)-category of modules over that ring spectrum;
the symmetric monoidal (∞,n)-category of modules over .
Following the above idea we have the following definition.
An -vector space is an element of . If is an ordinary ring, then the 0-category is the underlying set of , regarded as a symmetric monoidal category using the product structure on . If is more generally an ∞-ring, then the “stabilized (∞,0)-category” (= spectrum) of -vector spaces is itself: .
the -category of -module spectra.
For , an -vector space is an algebra object in the symmetric monoidal (∞,1)-category . A morphism is a bimodule object. Higher morphisms are defined recursively.
Notice that FHLT say “-algebra” instead of “-vector space”, but only for the reason (p. 29) that
The discrepancy between (the algebra level) and [the algebra level] – for which we apologize – is caused by the fact that the term “-vector space” has been used for a much more restrictive notion than our -algebras.
See (∞,1)-vector space for more.
The symmetric monoidal 3-category of 2-modules over is:
2-morphisms are bimodule homomorphisms.
We think of this equivalently as its essential image in , where
an algebra is a placeholder for its module category ;
a bimodule homomorphism is a placeholder for a natural transformation of two such functors.
Comparing this for the formula
for a -vector space with basis , we see that we may
think of the algebra objects appearing in the above as being bases for a higher vector space;
think of the bimodules as being higher matrices.
A 3-vector space according to def. 1 is
a -algebra ;
Equivalently this is a sesquiunital sesquialgebra.
Classes of examples come from the following construction:
Every commutative associative algebra becomes a 3-vector space.
More generally: every hopfish algebra.
Next, an algebra object internal to , is an algebra equipped with three compatible algebra structures, a trialgebra.
The 2-category of 2-modules of that is a monoidal 2-category.
For a review see (Baez-Lauda 09, p. 98).
-vector space, n-vector bundle,
|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|
The notion of -vector spaces is (defined for and sketched recursively for greater ) in
appendix A of
section 7 of