structure on algebras and their module categories - table

**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 |

Last revised on March 31, 2015 at 17:01:04. See the history of this page for a list of all contributions to it.