# nLab associative unital algebra

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definitions

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

• a monoid internal to $R$Mod equipped with the tensor product of modules $\otimes$;

• a pointed one-object category enriched over $(R Mod, \otimes)$;

• a pointed $R$-algebroid with one object;

• an $R$-module $V$ equipped with linear maps $p : V \otimes V \to V$ and $i : R \to V$ satisfying the associative and unit laws;

• a ring $A$ under $R$ such that the corresponding map $R \to A$ lands in the center of $A$.

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 $i: R \to V$; 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 $Vect$.

## Properties

### Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
$A$$Mod_A$
$R$-algebra$Mod_R$-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
$A$$Mod_A$
$R$-2-algebra$Mod_R$-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
$A$$Mod_A$
$R$-3-algebra$Mod_R$-4-module

Revised on September 12, 2014 12:01:42 by Urs Schreiber (185.26.182.37)