nLab (infinity,n)-module

Idea

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

  $$0 Vect_k = k \,.$$

• 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 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 $(\infty,n-1)Mod$.

Definition

Following the above idea we have the following definition.

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

$$(\infty,1)Vect_k := k Mod \,,$$

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.

Examples

$(\infty,1)$-vector spaces

See (∞,1)-vector space for more.

2-Modules

Remark

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

  $$Mod_A \stackrel{(-) \otimes_A N }{\to} Mod_B$$

• 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

$$Mod_A \simeq Vect Cat(B A, Vect) \,.$$

Comparing this for the formula

$$V \simeq Set(S,k)$$

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.

3-Modules

A 3-vector space according to def. 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.

4-Modules

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).

$n$-Representations

See infinity-representation.

Related concepts

• vector space,

• (∞,1)-module, (∞,1)-module bundle, (∞,1)-category of (∞,1)-modules

• 2-ring, 2-module

• 2-vector space

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

• $n$-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

References

The notion of $n$-vector spaces is (defined for $n = 2$ and sketched recursively for greater $n$) in

appendix A of

• Urs Schreiber, AQFT from $n$-functorial QFT Communications in Mathematical Physics, Volume 291, Issue 2, pp.357-401 (2008) (pdf)

section 7 of

• Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups (2009)

Full details are in

• Rune Haugseng, The higher Morita category of $E_n$-algebras (arXiv:1412.8459)

Review of work on 4-modules (implicitly) as trialgebras/Hopf monoidal categories is around p. 98 of

• John Baez, Aaron Lauda, A prehistory of $n$-categorical physics, in Deep beauty, 13-128, Cambridge Univ. Press, Cambridge, 2011 (arXiv:0908.2469)