## 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.

### Citations

389 | Basic concepts of enriched category theory
- Kelly
- 1982
(Show Context)
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... |

164 |
Quantum groups, Graduate Texts
- Kassel
- 1995
(Show Context)
Citation Context ...ras 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 "tensor") ... |

154 |
The geometry of tensor calculus
- Joyal, Street
- 1991
(Show Context)
Citation Context ...T with counit s = e m æ Æ æ o :T2 1 and unit comonad for the monad T . 2 r :1æÆ æ T . Moreover, 2 2 G = ( T, e, d) is a right adjoint Proof We shall do this using the string calculus (as justified by =-=[JS1]-=-). We use Lemma 1.2. One of the counit/unit identities is proved by the following calculation; look in a mirror for the proof of the other. s r For the second sentence we need to show that = = = m e h... |

144 |
Introduction to bicategories
- Bénabou
(Show Context)
Citation Context ...onoidal 2category. Composition of functors is strictly associative so Cat itself is stricter than it might be in the 2-dimensional setting. This leads to a weaker version of 2-category due to Bénabou =-=[Bu]-=-. A bicategory B has objects, and, for objects A and B, we have a category ( ) B AB , (called a hom-category) whose objects are called morphisms f: A æÆ æ B of B, whose morphisms are called 2-cells q ... |

130 |
The Formal Theory of Monads
- Street
- 1972
(Show Context)
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... |

115 |
Metric spaces, generalized logic, and closed categories, Rendiconti del seminario matématico e fisico di
- Lawvere
- 1973
(Show Context)
Citation Context ...e 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 category Vect... |

110 |
Foundations of quantum group theory.” Cambridge
- Majid
- 1995
(Show Context)
Citation Context ... 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 "tensor") categor... |

98 |
On closed categories of functors
- Day
- 1970
(Show Context)
Citation Context ...dal V-category (that is, a pseudomonoid in becomes a cocomplete monoidal V-category via the convolution tensor product XY , ( ) = ( ƒ ) ƒ ƒ Ú M * N ( A) A A, X Y MX NY V -Cat ). Then P A of Brian Day =-=[Dy]-=-. Monoidal V-categories 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-fu... |

83 |
Frobenius algebras and 2D topological quantum field theories
- Kock
- 2004
(Show Context)
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... |

83 |
An associative orthogonal bilinear form for Hopf algebras
- Larson, Sweedler
- 1969
(Show Context)
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... |

81 |
Cartesian bicategories
- Carboni, Walters
- 1987
(Show Context)
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... |

80 |
Categories for the Working
- Lane
- 1998
(Show Context)
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... |

72 |
Adjoint functors and triples
- EILENBERG, MOORE
(Show Context)
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... |

61 |
Monoidal bicategories and Hopf algebroids
- Day, Street
(Show Context)
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... |

54 | subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
- Müger, From
(Show Context)
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... |

49 |
An introduction to Tannaka duality and quantum groups
- Joyal, Street
- 1991
(Show Context)
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... |

40 |
Invariants of Piecewise-Linear 3-manifolds
- Barrett, Westbury
- 1994
(Show Context)
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... |

40 |
Lyubashenko V V, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with
- Kerler
- 2001
(Show Context)
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... |

38 | algebras and cohomology
- Beck
(Show Context)
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... |

31 |
Tortile Yang-Baxter operators in tensor categories
- Joyal, Street
- 1991
(Show Context)
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... |

31 |
Review of the elements of
- Kelly, Street
- 1974
(Show Context)
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... |

28 | From subfactors to categories and topology III, in preparation
- Muger
(Show Context)
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... |

21 | Quasi-Hopf algebras, (Russian) Algebra i Analiz 1 - Drinfeld - 1989 |

19 | Quantum categories, star autonomy, and quantum groupoids
- Day, Street
(Show Context)
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... |

19 | Enriched categories and cohomology
- Street
- 1981
(Show Context)
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... |

13 | The Chu construction
- Barr
- 1996
(Show Context)
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... |

12 |
Higher categories, strings, cubes and simplex equations’, Applied Categorical Structures 3
- Street
- 1995
(Show Context)
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... |

10 |
Cobordism Categories
- Carmody
- 1996
(Show Context)
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, , ) §... |

9 |
Autonomous categories, with an appendix by Po Hsiang Chu
- Barr
- 1979
(Show Context)
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... |

8 |
group representations
- Matrices
- 1991
(Show Context)
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... |

7 | The Dedekind completion - Mack, Johnson - 1967 |

7 |
Absolute colimits in enriched categories, Cahiers de Top. et Géom
- Street
- 1983
(Show Context)
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... |

6 |
Non-symmetric -autonomous categories. Theoret
- Barr
- 1995
(Show Context)
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 |
Categories of representations of balanced coalgebroids
- McCrudden
- 1999
(Show Context)
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 |
Ordinal sums and equational doctrines, in: "Seminar on Triples and Categorical Homology Theory
- Lawvere
- 1969
(Show Context)
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 |
Separable algebras and Seifert surfaces
- Boyer, Joyal
- 1994
(Show Context)
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 |
Monoidal Morita equivalence
- Johnson
- 1989
(Show Context)
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... |