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(infinity,n)-module

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Idea

A notion of nn-module (nn-vector space) is a categorification of the notion of module (vector space).

There are various different notions of nn-vector spaces.

One notion is: an nn-vector space is a chain complex of vector spaces in degrees 0 to nn. For n=2n=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 (n1)(n-1)-algebra object (or its (n1)(n-1)-category of modules) in the nn-category of (n1)(n-1)-bimodules. For higher nn this is in (FHLT, section 7). It includes the previous concept as a special case.

For n=2n=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 nn-vector spaces. More details are below.

Assume that a notion of n-category is chosen for each nn (for instance (n,1)-category), that a notion of symmetric monoidal nn-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 nn-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 kk.

  • a 0-vector space over kk is an elemment of kk. The 0-category of 0-vector spaces is the set

    0Vect k=k. 0 Vect_k = k \,.
  • The category 1Vect k1 Vect_k is just Vect.

  • For n>1n \gt 1, the n-category nVectn Vect of nn-vector spaces over kk is the nn-category with objects algebra objects in (n1)Vect(n-1)Vect and morphisms bimodule objects in (n1)Vect(n-1)Vect.

Here we think of an algebra object A(n1)VectA \in (n-1)Vect as a basis for the nn-vector space which is the (n1)(n-1)-category AModA Mod.

With this definition we have that 2Vect2 Vect is the 2-category of kk-algebras, bimodules and bimodule homomorphisms.

More generally, let kk here be a ring spectrum. Set

Definition

Following the above idea we have the following definition.

Definition

Fix a ring kk (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 nn \in \mathbb{N}, define an symmetric monoidal (∞,n)-category nVect kn Vect_k of (,n)(\infty,n)-vector spaces as follows (the bi-counting follows the pattern of (n,r)-categories).

An (,0)(\infty,0)-vector space is an element of kk. If kk is an ordinary ring, then the 0-category 0Vect0 Vect is the underlying set of kk, regarded as a symmetric monoidal category using the product structure on kk. If kk is more generally an ∞-ring, then the “stabilized (∞,0)-category” (= spectrum) of (,0)(\infty,0)-vector spaces is kk itself: (,0)Vect kk(\infty,0)Vect_k \simeq k.

An (∞,1)-vector space is an ∞-module over kk. The (∞,1)-category of (,1)(\infty,1)-vector spaces is

(,1)Vect k:=kMod, (\infty,1)Vect_k := k Mod \,,

the (,1)(\infty,1)-category of kk-module spectra.

For kk a field ordinary vector spaces over kk are a full sub-(∞,1)-category of this: 1Vect k(,1)Vect k1Vect_k \hookrightarrow (\infty,1)Vect_k .

For n2n \geq 2, an (,n)(\infty,n)-vector space is an algebra object in the symmetric monoidal (∞,1)-category (,n1)Vect(\infty,n-1)Vect. A morphism is a bimodule object. Higher morphisms are defined recursively.

For \infty replaced by nn 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 ”(n1)(n-1)-algebra” instead of ”nn-vector space”, but only for the reason (p. 29) that

The discrepancy between mm (the algebra level) and nn [the algebra level] – for which we apologize – is caused by the fact that the term ”nn-vector space” has been used for a much more restrictive notion than our (n1)(n-1)-algebras.

Examples

(,1)(\infty,1)-vector spaces

See (∞,1)-vector space for more.

2-Modules

Remark

The symmetric monoidal 3-category Alg k b=2Mod kAlg_k^b = 2 Mod_k of 2-modules over kk is:

We think of this equivalently as its essential image in Vect kModVect_k Mod, where

  • an algebra AA is a placeholder for its module category Mod AMod_A;

  • an AA-BB bimodule NN is a placeholder for the functor

    Mod A() ANMod B 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 AA in terms of its delooping Vect-enriched category BAB A, then we have an equivalence of categories

Mod AVectCat(BA,Vect). Mod_A \simeq Vect Cat(B A, Vect) \,.

Comparing this for the formula

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

for a kk-vector space VV with basis SS, 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. 1 is

  • a kk-algebra AA;

  • equipped with an AA-AAA\otimes A-bimodule defining the 2-multiplication, and a left AA-module defining the unit.

Equivalently this is a sesquiunital sesquialgebra.

Classes of examples come from the following construction:

4-Modules

Next, an algebra object internal to 2Alg k b=3Mod 32 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).

nn-Representations

See infinity-representation.

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

References

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

appendix A of

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

section 7 of

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

Revised on August 29, 2013 22:51:31 by Urs Schreiber (89.204.139.41)