A notion of $n$-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 spaces 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 (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 $n$-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 $k$.
a 0-vector space over $k$ is an elemment of $k$. The 0-category of 0-vector spaces is the set
The category $1 Vect_k$ is just Vect.
For $n \gt 1$, the n-category $n Vect$ of $n$-vector spaces over $k$ is the $n$-category with objects algebra objects in $(n-1)Vect$ and morphisms bimodule objects in $(n-1)Vect$.
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 $k$-algebras, bimodules and bimodule homomorphisms.
More generally, let $k$ here be a ring spectrum. Set
$(\infty,0)Vect_k := k$ – a symmetric monoidal ∞-groupoid;
$(\infty,1)Vect_k := k Mod$ the (∞,1)-category of modules over that ring spectrum;
$(\infty,1)Vect_k := k Mod$ the symmetric monoidal (∞,1)-category of modules over that ring spectrum;
$(\infty,n)Vect_k := (\infty,n-1) Mod$ the symmetric monoidal (∞,n)-category of modules over $(\infty,n-1)Mod$.
Following the above idea we have the following definition.
Fix a ring $k$ (usually taken to be a field if one speaks of “vector spaces” instead of just modules, but this is not actually essential for the construction). This may be an ∞-ring.
For $n \in \mathbb{N}$, define an symmetric monoidal (∞,n)-category $n Vect_k$ of $(\infty,n)$-vector spaces as follows (the bi-counting follows the pattern of (n,r)-categories).
An $(\infty,0)$-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, then the “stabilized (∞,0)-category” (= spectrum) of $(\infty,0)$-vector spaces is $k$ itself: $(\infty,0)Vect_k \simeq k$.
An (∞,1)-vector space is an ∞-module over $k$. The (∞,1)-category of $(\infty,1)$-vector spaces is
the $(\infty,1)$-category of $k$-module spectra.
For $k$ a field ordinary vector spaces over $k$ are a full sub-(∞,1)-category of this: $1Vect_k \hookrightarrow (\infty,1)Vect_k$ .
For $n \geq 2$, an $(\infty,n)$-vector space is an algebra object in the symmetric monoidal (∞,1)-category $(\infty,n-1)Vect$. A morphism is a bimodule object. Higher morphisms are defined recursively.
For $\infty$ replaced by $n$ this appears as (Schreiber, appendix A) and then with allusion to more sophisticated higher categorical tools in (FHLT, def. 7.1).
Notice that FHLT say “$(n-1)$-algebra” instead of “$n$-vector space”, but only for the reason (p. 29) that
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.
See (∞,1)-vector space for more.
The symmetric monoidal 3-category $Alg_k^b = 2 Mod_k$ of 2-modules over $k$ is:
objects are associative algebras over $k$;
morphisms are bimodules of associative algebras; composition is the tensor product of bimodules;
2-morphisms are bimodule homomorphisms.
We think of this equivalently as its essential image in $Vect_k Mod$, where
an algebra $A$ is a placeholder for its module category $Mod_A$;
an $A$-$B$ bimodule $N$ is a placeholder for the functor
a bimodule homomorphism is a placeholder for a natural transformation of two such functors.
If we think of an algebra $A$ in terms of its delooping Vect-enriched category $B A$, then we have an equivalence of categories
Comparing this for the formula
for a $k$-vector space $V$ with basis $S$, 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 $k$-algebra $A$;
equipped with an $A$-$A\otimes A$-bimodule defining the 2-multiplication, and a left $A$-module defining the unit.
Equivalently this is a sesquiunital sesquialgebra.
Classes of examples come from the following construction:
Every commutative associative algebra $A$ becomes a 3-vector space.
Every Hopf algebra canonically becomes a 3-vector space (amplified in FHLT, p. 27).
More generally: every hopfish algebra.
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 (Baez-Lauda 09, p. 98).
(∞,1)-module, (∞,1)-module bundle, (∞,1)-category of (∞,1)-modules
$n$-vector space, n-vector bundle,
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 |
The notion of $n$-vector spaces is (defined for $n = 2$ and sketched recursively for greater $n$) in
appendix A of
section 7 of
Full details are in
Review of work on 4-modules (implicitly) as trialgebras/Hopf monoidal categories is around p. 98 of