nLab 2Mod

Idea

The 2-category of 2-modules/2-vector spaces.

Fix some commutative ring or more generally an E-infinity ring $R$. Then a basis for a 2-module/2-vector space over $R$ may be taken to be an $R$-associative algebra or more generally an $R$Mod-enriched category $\mathcal{A}$ (an algebras): the correspponding 2-vector space is the category of modules $Hom_{R Mod}(\mathcal{A}, R Mod)$. Then $2Mod_R$ is equivalently the 2-category whose

• objects are $R$-algebras;

• 1-morphisms are bimodules,

• 2-morphisms are intertwiners.

In enriched category theory this is equivalently the 2-category of Mod${}_R$-enriched categories and profunctors between them. In this context one can write

$$2Mod_R = Mod(Mod_R)$$

or $2 Mod_R =$ Prof$(Mod_R)$ for this 2-category.

Related constructions

Algebra objects in $2Mod$

An algebra object in 2Mod${}_3$ is equivalently a sesquiunital sesquialgebra over $R$. This may be taken to be a basis for a 3-module/3-vector space over $R$.

Line 2-bundles

(…) line 2-bundle (…)

Related concepts

• module, vector space

• 2-module/2-vector space, 2-representation

  • 2-vector bundle

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

• (∞,1)-module, (∞,1)Mod

• n-vector space

• nMod