## Monads on Tensor Categories (2002)

Venue: | J. Pure Appl. Algebra |

Citations: | 25 - 1 self |

### BibTeX

@ARTICLE{Moerdijk02monadson,

author = {I. Moerdijk},

title = {Monads on Tensor Categories},

journal = {J. Pure Appl. Algebra},

year = {2002},

volume = {168},

pages = {189--208}

}

### OpenURL

### Abstract

this paper we will discuss the combination of two classical notions of category theory, both treated extensively in [CWM]. One of these is the notion of a monad or triple on a category, which goes back to Godement [G] and was rst developed by Eilenberg, Moore, Beck and others. The other is that of a monoidal category or tensor category, which originates with Benabou [Be] and with Mac Lane's famous coherence theorem [MacL], and which pervades much of present day mathematics. For a monad S on a tensor category, there is a natural additional structure that one can impose, namely that of a comparison map S(X

### Citations

921 |
Categories for the working mathematician
- Lane
- 1998
(Show Context)
Citation Context ...erdijk To Saunders Mac Lane, on the occasion of his 90 th birthday Introduction. In this paper we will discuss the combination of two classical notions of category theory, both treated extensively in =-=[CWM]-=-. One of these is the notion of a monad or triple on a category, which goes back to Godement [G] and wassrst developed by Eilenberg, Moore, Beck and others. The other is that of a monoidal category or... |

260 | The geometry of iterated loop spaces
- May
- 1972
(Show Context)
Citation Context ...S(Y ) c S(X);S(Y ) ## S(Y X) ## S(Y ) S(X) commutes. 3.1 Example (Operads) Suppose the symmetric tensor category C has countable sums and quotiens of actions bysnite permutation groups. Recall from [=-=-=-Ma], [KM] that any operad Pon C denes a monad SP on C, by SP (X) = a n0 P(n) n X n : The algebras for this monad SP are exactly the P -algebras for the operad P. This monad SP has the structure of a H... |

196 | Locally Presentable and Accessible Categories - Adámek, Rosicky - 1994 |

155 |
Topologie Algébrique et Théorie des Faisceaux
- Godement
- 1958
(Show Context)
Citation Context ...will discuss the combination of two classical notions of category theory, both treated extensively in [CWM]. One of these is the notion of a monad or triple on a category, which goes back to Godement =-=[-=-G] and wassrst developed by Eilenberg, Moore, Beck and others. The other is that of a monoidal category or tensor category, which originates with Benabou [Be] and with Mac Lane's famous coherence theo... |

155 |
On the Hopf algebra structure of perturbative quantum field theories
- Kreimer
- 1998
(Show Context)
Citation Context ...ures. The original motivation for this paper came from my attempt to understand the nature of the Hopf algebra structure on the polynomial algebra ofsnite rooted trees, originally due to Kreimer (see =-=[K]-=-, [CK]). As it turns out, this algebra is the initial S[t]-algebra for the particular case where S 1 is the symmetric algebra monad on the category of vector spaces. It thus follows that this Hopf alg... |

145 |
The geometry of tensor calculus
- JOYAL, STREET
- 1991
(Show Context)
Citation Context ...er k in the usual sense (but, without antipode, so far.) 5 3 Symmetry and cocommutativity. Suppose the tensor category C is equipped with a symmetry (or braiding) c, c = c X;Y : X Y ! Y X (cf. [CWM], [JS]). Let S be a Hopf monad on C, with S = (S; ; ; ; ) as before. We will call S cocommutative if cs = sS(c); (9) i.e. for any two objects X and Y the square S(X Y ) S(cX;Y ) ## ## S(X) S(Y ) c S... |

125 |
The formal theory of monads
- Street
- 1972
(Show Context)
Citation Context ...noidal comonad" used e.g. by Boardman [Bo]. Following an earlier version of this paper, McCrudden [McC] has explained how Hopf monadsst into the general framework of monads on objects in a 2-cate=-=gory [S]-=-, and has proved general versions of Proposition 2.2 and Theorem 7.1 in this context. Acknowledgements. This paper is a faithful write-up of my talk at the Category Meeting in Coimbra (July 1999), and... |

124 |
homotopy algebra and iterated integrals for double loop spaces arXiv:hep-th/9403055v1
- Getzler, Jones, et al.
(Show Context)
Citation Context ...fying natural conditions. I will call a monad on a tensor category equipped with this additional structure a Hopf monad, thus extending the terminology already existing in the special case of operads =-=[GJ]. The-=- basic property of such Hopf monads S is that the tensor product lifts to the category Alg(S) of algebras for the monad. In fact, there is a bijective correspondence between \Hopf structures" on ... |

118 | Hopf algebras, renormalization and noncommutative geometry
- Connes, Kreimer
- 1998
(Show Context)
Citation Context ... The original motivation for this paper came from my attempt to understand the nature of the Hopf algebra structure on the polynomial algebra ofsnite rooted trees, originally due to Kreimer (see [K], =-=[CK]-=-). As it turns out, this algebra is the initial S[t]-algebra for the particular case where S 1 is the symmetric algebra monad on the category of vector spaces. It thus follows that this Hopf algebra s... |

58 |
Natural associativity and commutativity
- Lane, S
- 1963
(Show Context)
Citation Context ... wassrst developed by Eilenberg, Moore, Beck and others. The other is that of a monoidal category or tensor category, which originates with Benabou [Be] and with Mac Lane's famous coherence theorem [M=-=a-=-cL], and which pervades much of present day mathematics. For a monad S on a tensor category, there is a natural additional structure that one can impose, namely that of a comparison map S(X 1 Xn ) ... |

22 |
J.P.May “Operads, algebras, modules and motives.” Astrisque No
- Kriz
- 1995
(Show Context)
Citation Context ...c S(X);S(Y ) ## S(Y X) ## S(Y ) S(X) commutes. 3.1 Example (Operads) Suppose the symmetric tensor category C has countable sums and quotiens of actions bysnite permutation groups. Recall from [Ma], [=-=-=-KM] that any operad Pon C denes a monad SP on C, by SP (X) = a n0 P(n) n X n : The algebras for this monad SP are exactly the P -algebras for the operad P. This monad SP has the structure of a Hopf mo... |

16 | On the Connes–Kreimer construction of Hopf Algebras, Homotopy methods in algebraic topology
- Moerdijk
- 1999
(Show Context)
Citation Context ...a monad on the category of vector spaces. It thus follows that this Hopf algebra structure is one of an entire family of such structures. This is discussed in more detail in the context of operads in =-=[Mo]. The mere den-=-ition of \Hopf monad" is an obvious variation on that of a \monoidal monad", and is strictly dual to that of \monoidal comonad" used e.g. by Boardman [Bo]. Following an earlier version ... |

12 | Catégories avec multiplication - BÉNABOU - 1963 |

9 | Unstable operations in generalized cohomology
- Boardman, Johnson, et al.
- 1995
(Show Context)
Citation Context ...l in the context of operads in [Mo]. The mere denition of \Hopf monad" is an obvious variation on that of a \monoidal monad", and is strictly dual to that of \monoidal comonad" used e.g=-=. by Boardman [Bo]-=-. Following an earlier version of this paper, McCrudden [McC] has explained how Hopf monadsst into the general framework of monads on objects in a 2-category [S], and has proved general versions of Pr... |

1 |
Opmonoidal monads, unpublished manuscript
- McCrudden
- 1999
(Show Context)
Citation Context ...Hopf monad" is an obvious variation on that of a \monoidal monad", and is strictly dual to that of \monoidal comonad" used e.g. by Boardman [Bo]. Following an earlier version of this pa=-=per, McCrudden [McC]-=- has explained how Hopf monadsst into the general framework of monads on objects in a 2-category [S], and has proved general versions of Proposition 2.2 and Theorem 7.1 in this context. Acknowledgemen... |