symmetric monoidal (∞,1)-category of spectra
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 $Vect$.
A cosimplicial algebra is a cosimplicial object in the category of algebras.
A dg-algebra is a monoid not in Vect but in the category of (co)chain complexes.
A smooth algebra is an associative $\mathbb{R}$-algebra that has not only the usual binary product induced from the product $\mathbb{R}\times \mathbb{R} \to \mathbb{R}$, but has a $n$-ary product operation for every smooth function $\mathbb{R}^n \to \mathbb{R}$.
This may be understood as a special case of an algebra over a Lawvere theory, here the Lawvere theory CartSp.
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 |