## Frobenius monads and pseudomonoids (2004)

Venue: | 2-CATEGORIES COMPANION 73 |

Citations: | 20 - 4 self |

### BibTeX

@INPROCEEDINGS{Street04frobeniusmonads,

author = {Ross Street},

title = {Frobenius monads and pseudomonoids},

booktitle = {2-CATEGORIES COMPANION 73},

year = {2004},

pages = {3930--3948}

}

### OpenURL

### Abstract

Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.

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Citation Context ...t it is the same as a comonad d h o is a unit for G J G. G = ( G, , ) §2. Review of enriched categories e d with a natural transformation h :1X æÆ æ G such that References for enriched categories are =-=[Kel]-=- and [Law2]. Let V denote a particularly familiar symmetric monoidal category. The reader really only needs to keep in mind the category Set of sets with cartesian product as tensor product and the ca... |

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Citation Context ...s a little about Morita equivalence. Finally, in Section 6, we discuss wreath products of Frobenius algebras. This is done at the level of generalized distributive laws between monads as developed in =-=[LSt]-=-. §1. Frobenius monads Let T = ( T, h, m) be a monad on a category X . We write X T for the category of Talgebras in the sense of [EM] (although those authors called monads "triples"). We write 7sUT T... |

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Citation Context ...e relevance of categories to physics has become widely acknowledged in at least two particular areas: quantum group theory QGT (see [JS3], [Kas], [Maj]) and topological quantum field theory TQFT (see =-=[Ko]-=-, [KL]). Quantum groups arise from the Yang-Baxter equation of statistical mechanics, while each quantum group has a monoidal (or "tensor") category of representations. A two-dimensional TQFT can be r... |

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Citation Context ...eo r = mo h = 1T e d e m r m e r m h T T T T T T T o = o o = o o = o = 1T do h = Tmo rToh = Tmo T2ho r = r . QED Remark Condition (a) of Lemma 1.2 has occurred in the work of Carboni and Walters (see =-=[CW]-=- and [Cbn]) and of Boyer and Joyal (unfortunately [BJ] is unpublished but see [St4] for some details). The condition relates to separability of algebras and discreteness. Condition (b) expesses that e... |

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Citation Context ...ow that works. Some connection between quantum groups and Frobenius algebras is already apparent from the fact that quantum groups are Hopf algebras and finite-dimensional Hopf algebras are Frobenius =-=[LSw]-=-. W e intend to deepen the connection between Frobenius algebras and quantum group theory. A k-algebra A is called Frobenius when it is equipped with an exact pairing s :AƒAæÆk æ satisfying the condit... |

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Citation Context ...monoidal category. For any category A , the category [ AA , ] of endofunctors on A becomes strict monoidal by taking composition as the tensor product: a monoid in [ AA , ] is called a monadon A (see =-=[ML]-=- for the theory of monads and their algebras). Frobenius structure on a monoid makes sense in any monoidal category. We recall this in Section 1 where we assemble some facts about Frobenius monoids. M... |

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Citation Context ... of generalized distributive laws between monads as developed in [LSt]. §1. Frobenius monads Let T = ( T, h, m) be a monad on a category X . We write X T for the category of Talgebras in the sense of =-=[EM]-=- (although those authors called monads "triples"). We write 7sUT T : X æÆX for a comonad VG G : X æÆX æ for the forgetful functor and G = ( G, e, d ) , we write æ for the forgetful functor, and we wri... |

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Citation Context ...categories are not all monoidally biequivalent to monoidal 2-categories but some degree of strictness can be attained. We do not need more detail than this; however, the interested reader can consult =-=[DS1]-=- and [McC]. In any monoidal bicategory it is possible to define pseudomonoids; these are like monoids except that the associativity and unital conditions only hold up to invertible 2cells that are cal... |

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Citation Context ... in [DS2], the star-autonomy defined there is precisely the higher-dimensional version of Frobenius structure. Section 4 is largely inspired by the discussion of "weak monoidal Morita equivalence" in =-=[Mü1]-=- and [Mü2] where it is shown that monoidal categories that are equivalent in this weak sense still give rise to the same state sum invariants of closed oriented 3-manifolds (see [BW1] and [BW2]). We d... |

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Citation Context ...s algebras are discussed. Introduction Over the last two decades, the relevance of categories to physics has become widely acknowledged in at least two particular areas: quantum group theory QGT (see =-=[JS3]-=-, [Kas], [Maj]) and topological quantum field theory TQFT (see [Ko], [KL]). Quantum groups arise from the Yang-Baxter equation of statistical mechanics, while each quantum group has a monoidal (or "te... |

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Citation Context ... equivalence" in [Mü1] and [Mü2] where it is shown that monoidal categories that are equivalent in this weak sense still give rise to the same state sum invariants of closed oriented 3-manifolds (see =-=[BW1]-=- and [BW2]). We define a notion of projective equivalence between objects in any bicategory. In the same general setting, we define what it means for a Frobenius monad to be strongly separable and rel... |

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Citation Context ...vance of categories to physics has become widely acknowledged in at least two particular areas: quantum group theory QGT (see [JS3], [Kas], [Maj]) and topological quantum field theory TQFT (see [Ko], =-=[KL]-=-). Quantum groups arise from the Yang-Baxter equation of statistical mechanics, while each quantum group has a monoidal (or "tensor") category of representations. A two-dimensional TQFT can be regarde... |

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Citation Context ...the comonad generated by U J C is right adjoint to the monad generated by F J U. We can add the following extra observation on adjoint monads; it is a trivial consequence of Beck's monadicity theorem =-=[Bec]-=-). AM5. if F J U J C and U is conservative (that is, reflects invertibility of morphisms) then the comparison functor, into the category of Eilenberg-Moore algebras for the monad generated by F J U, i... |

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Citation Context ...ositions, is called the centre of the bicategory D. So scalars are endomorphisms of the unit of the centre. If D is the one-object bicategory SC with hom monoidal category C then of C in the sense of =-=[JS2]-=-. ( )( ) Hom DD , 1D, 1D is the centre ZC Definition 4.1 A morphism u: A æÆ æ X in a bicategory D is called a projective equivalence when there is a morphism f: X æÆ æ A adjoint to u on both sides (th... |

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Citation Context ...re a 2dimensional version of natural transformation in which the naturality equations are "broken" by asking them to hold only up to extra invertible 2-cells that satisfy some further conditions (see =-=[KS]-=- for example). There is a bicategory Hom BD , ( ) whose objects are the pseudofunctors from B to D, whose morphisms are the pseudonatural transformations, and whose 2-cells are called modifications. W... |

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Citation Context ... the star-autonomy defined there is precisely the higher-dimensional version of Frobenius structure. Section 4 is largely inspired by the discussion of "weak monoidal Morita equivalence" in [Mü1] and =-=[Mü2]-=- where it is shown that monoidal categories that are equivalent in this weak sense still give rise to the same state sum invariants of closed oriented 3-manifolds (see [BW1] and [BW2]). We define a no... |

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Citation Context ... yet our corollary provides a setting in which even the more general quasi-Hopf algebras of Drinfeld are Frobenius irrespective of dimension. Another example is any autonomous monoidal V-category. In =-=[DS2]-=-, we showed how quantum groups (and more generally "quantum groupoids") and starautonomous monoidal categories are instances of the same mathematical structure. Although the term Frobenius was not use... |

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Citation Context ...potents. It is easy to see that V-categories A and B are Cauchy equivalent if and only if P A and PB are equivalent V-categories (that is, equivalent in the 2-category V -Cat ). It is well known (see =-=[St2]-=- for a proof in a very general context) that V-categories A and B are Cauchy equivalent if and only if QA and QB are equivalent V-categories. The inclusion of QA in QQA is an equivalence, so A is Cauc... |

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Citation Context ...when every object has both a left and right dual. If V is symmetric then every right dual is also a left dual. There is a weaker kind of monoidal duality that was conceived by Barr (see [Ba1], [Ba2], =-=[Ba3]-=-) based on examples in topological algebra yet the notion has received a lot of attention by computer scientists interested in Girard's "linear logic". A monoidal category V is said to be *-autonomous... |

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Citation Context ...mo rToh = Tmo T2ho r = r . QED Remark Condition (a) of Lemma 1.2 has occurred in the work of Carboni and Walters (see [CW] and [Cbn]) and of Boyer and Joyal (unfortunately [BJ] is unpublished but see =-=[St4]-=- for some details). The condition relates to separability of algebras and discreteness. Condition (b) expesses that e is a counit for the comultiplication d . Condition (c) suggests dually introducing... |

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Citation Context ...ed autonomous when every object has both a left and right dual. If V is symmetric then every right dual is also a left dual. There is a weaker kind of monoidal duality that was conceived by Barr (see =-=[Ba1]-=-, [Ba2], [Ba3]) based on examples in topological algebra yet the notion has received a lot of attention by computer scientists interested in Girard's "linear logic". A monoidal category V is said to b... |

9 |
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Citation Context ...m 1.6 that our definition agrees with Lawvere's definition of 12 s m rsFrobenius monad (see pages 151 and 152 of [Law1]). Using the "algebra" terminology, it also agrees for example with Chapter 5 of =-=[Cmd]-=-, Section 6 of [BS], and Definition 3.1 of [Mü]. It follows also that the notion of Frobenius monad is self-dual in the sense that it is the same as a comonad d h o is a unit for G J G. G = ( G, , ) §... |

8 |
group representations
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Citation Context ... h = 1T e d e m r m e r m h T T T T T T T o = o o = o o = o = 1T do h = Tmo rToh = Tmo T2ho r = r . QED Remark Condition (a) of Lemma 1.2 has occurred in the work of Carboni and Walters (see [CW] and =-=[Cbn]-=-) and of Boyer and Joyal (unfortunately [BJ] is unpublished but see [St4] for some details). The condition relates to separability of algebras and discreteness. Condition (b) expesses that e is a coun... |

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Citation Context ...f the V-functor V-category PA = [ A , V ] that contains the representable V-functors ( ) A -,A and is closed under absolute V-colimits. (Absolute V-colimits are those preserved by all V-functors; see =-=[St3]-=-.) For example, if V = Set then QA is the completion of the category A under splitting of idempotents; and if V = Vect k then QA is the completion of the additive category A under direct sums and spli... |

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5 |
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Citation Context ...nomous when every object has both a left and right dual. If V is symmetric then every right dual is also a left dual. There is a weaker kind of monoidal duality that was conceived by Barr (see [Ba1], =-=[Ba2]-=-, [Ba3]) based on examples in topological algebra yet the notion has received a lot of attention by computer scientists interested in Girard's "linear logic". A monoidal category V is said to be *-aut... |

4 |
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Citation Context ... are not all monoidally biequivalent to monoidal 2-categories but some degree of strictness can be attained. We do not need more detail than this; however, the interested reader can consult [DS1] and =-=[McC]-=-. In any monoidal bicategory it is possible to define pseudomonoids; these are like monoids except that the associativity and unital conditions only hold up to invertible 2cells that are called associ... |

2 |
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Citation Context ...is is an immediate consequence of Propositions 1.4 and 1.5. QED It is clear from Theorem 1.6 that our definition agrees with Lawvere's definition of 12 s m rsFrobenius monad (see pages 151 and 152 of =-=[Law1]-=-). Using the "algebra" terminology, it also agrees for example with Chapter 5 of [Cmd], Section 6 of [BS], and Definition 3.1 of [Mü]. It follows also that the notion of Frobenius monad is self-dual i... |

1 |
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Citation Context ... o o = o o = o = 1T do h = Tmo rToh = Tmo T2ho r = r . QED Remark Condition (a) of Lemma 1.2 has occurred in the work of Carboni and Walters (see [CW] and [Cbn]) and of Boyer and Joyal (unfortunately =-=[BJ]-=- is unpublished but see [St4] for some details). The condition relates to separability of algebras and discreteness. Condition (b) expesses that e is a counit for the comultiplication d . Condition (c... |

1 |
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Citation Context ... A and B are defined to be Cauchy equivalent when PA and PB are equivalent monoidal V-categories (that is, equivalent in the 2category of monoidal V-categories and monoidal V-functors). Scott Johnson =-=[Jn1]-=- showed that the convolution tensor product on PA restricts to QA and that monoidal V23scategories A and B are Cauchy equivalent if and only if QA and QB are equivalent monoidal V-categories. Moreover... |