nLab
associative unital algebra

Contents

Definitions

For RR a commutative ring, an associative unital RR-algebra is equivalently

If there is no danger for confusion, one often says simply ‘associative algebra’, or even only ‘algebra’.

More generally, a (merely) associative algebra need not have i:RVi: R \to V; that is, it is a semigroup instead of a monoid.

Less generally, a commutative algebra (where associative and unital are usually assumed) is an abelian monoid in VectVect.

Variants

Examples

Properties

Tannaka duality

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

Revised on August 20, 2014 07:52:03 by Urs Schreiber (89.204.139.117)