## Polycategories via pseudo-distributive laws

Citations: | 7 - 1 self |

### BibTeX

@MISC{Garner_polycategoriesvia,

author = {Richard Garner},

title = {Polycategories via pseudo-distributive laws},

year = {}

}

### OpenURL

### Abstract

In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘two-sided Kleisli bicategory’ of this pseudo-distributive law are precisely symmetric polycategories. 1

### Citations

133 |
Review of the elements of 2-categories
- Kelly, Street
(Show Context)
Citation Context ... Proof. We take ˆ T = ˆ S, whilst ˆǫ and ˆ ∆ have respective components at C given by ˆǫC = (ηC) ∗ and ˆ ∆C = (µC) ∗ . We obtain the remaining data for the pseudocomonad ˆ T via the calculus of mates =-=[12]-=-, making use of the adjunctions ˆηC ⊣ ˆǫC and ˆµC ⊣ ˆ ∆C. The key idea is to produce a pseudo-distributive law (δ, η, ǫ, µ, ∆) of the pseudocomonad ˆ T over the pseudomonad ˆ S; that is, there should ... |

110 | Introduction to distributive categories
- Cockett
- 1993
(Show Context)
Citation Context ...zabo’s theory of polycategories [19] has been the target of renewed interest over recent years. Polycategories are the ‘not-necessarily-representable’ cousins of the weakly distributive categories of =-=[5]-=-; their relationship mirrors that of multicategories to monoidal categories. Though it is possible, as Szabo did, to give a ‘hands on’ description of a polycategory, such a description leaves a lot to... |

74 | Higher-dimensional algebra III: n-categories and the algebra of opetopes
- Baez, Dolan
- 1998
(Show Context)
Citation Context ...ng like a ‘distributive law’. Where we deviate from Koslowski is in the description of multicategories that we build upon. In Section 1, we recount this alternative description: it is the approach of =-=[1]-=- and [4], based on profunctors rather than spans. We go on to describe how we may generalise this description to one for polycategories; to do this we invoke a pseudodistributive law (in the sense of ... |

61 |
Monoidal bicategories and Hopf algebroids
- Day, Street
(Show Context)
Citation Context ...y [Mod,Mod]ψ. However, it makes sense in any bicategory equipped with well-behaved notions of ‘whiskering’ (well-behaved in the sense that they obey axioms formally similar to those for a Gray-monoid =-=[6]-=-). We show in the Appendix of [8] that for any double club, Coll(S) is not only a monoidal pseudo double category, but is equipped with a notion of ‘whiskering’, and it follows from this that B ( Coll... |

57 |
Deductive Systems and Categories II: Standard Constructions and Closed Categories
- Lambek
- 1969
(Show Context)
Citation Context ... Γ, ∆2; z), This data satisfies axioms expressing the fact that exchange isomorphisms compose as expected, and that composition is associative, unital, and compatible with exchange isomorphisms. (See =-=[14]-=- for the full details of this definition.) Now, this data expresses composition as a binary operation performed between two multimaps; however, there is another view, where we ‘multicompose’ a family ... |

33 | Representable multicategories
- Hermida
- 2000
(Show Context)
Citation Context ...lone. Thus far, the paper [13] has provided the only attempt to rectify this situation. Koslowski provides an abstract description of polycategories that generalises the elegant work of [2] and later =-=[10]-=- and [15] on ‘T-multicategories’. However, whilst this latter theory uses only some rather simple and obvious constructions on categories with finite limits, the structures that Koslowski uses to buil... |

32 | Operads in higher-dimensional category theory
- Leinster
- 2000
(Show Context)
Citation Context ...tions of pseudomonad, pseudocomonad and pseudodistributive law. 2 Multicategories and polycategories We begin by re-examining the theory of multicategories: the material here summarises [1], [11] and =-=[16]-=-, amongst others. Note that throughout, we shall only be interested in the theory of symmetric multicategories, and, later, of symmetric polycategories; that is, we allow ourselves to reorder freely t... |

19 | Frobenius monads and pseudomonoids
- Street
(Show Context)
Citation Context ...uston: from the paper [7], multicategories with object set X can be viewed as lax monoids on the discrete object X in Mod. We might hope to extend this to a notion of lax Frobenius algebra, following =-=[18]-=-; then a polycategory would be such a lax Frobenius algebra on a discrete object of Mod. However, we shall not pursue this further here.] 3 Deriving the pseudo-distributive law δ We intend to construc... |

16 |
Higher Operads, Higher Categories, volume 298 of London Mathematical Society Lecture Notes
- Leinster
- 2004
(Show Context)
Citation Context ...s far, the paper [13] has provided the only attempt to rectify this situation. Koslowski provides an abstract description of polycategories that generalises the elegant work of [2] and later [10] and =-=[15]-=- on ‘T-multicategories’. However, whilst this latter theory uses only some rather simple and obvious constructions on categories with finite limits, the structures that Koslowski uses to build his des... |

12 |
On clubs and data-type constructors
- Kelly
- 1992
(Show Context)
Citation Context ...tend to construct the pseudo-distributive law δ by exploiting the theory of double clubs, as developed in the companion paper [8]. A double club is a generalisation of Kelly’s abstract notion of club =-=[11]-=- from the level of categories to that of pseudo (or weak) double categories. Let us recap briefly the details we shall need here. A pseudo double category K is a ‘pseudo-category’ object in Cat. Expli... |

11 | Pseudo-distributive laws
- Cheng, Hyland, et al.
(Show Context)
Citation Context ... = SC(Γ, ⊗ ∆i). Now, just as each monad on a category gives rises to a Kleisli category, so each pseudomonad on a bicategory gives rise to a ‘Kleisli bicategory’. This construction was first given in =-=[3]-=- for the special case of a pseudomonad on a 2-category; and the following is the obvious generalisation to the bicategorical case: Definition 3. Let B be a bicategory and let (S,η,µ,λ,ρ,τ) be a pseudo... |

10 | Distributive laws for pseudo-monads
- Marmolejo
- 1999
(Show Context)
Citation Context ...and [3], based on profunctors rather than spans. We go on to describe how we may generalise this description to one for polycategories; to do this we invoke a pseudo-distributive law (in the sense of =-=[17]-=-, [20]) of a pseudocomonad (the ‘target arity’) over a pseudomonad (the ‘source arity’). Polycategories now arise as monads in the ‘two-sided Kleisli bicategory’ of this pseudo-distributive law. There... |

8 |
Lax monoids, pseudo-operads, and convolution
- Day, Street
- 2003
(Show Context)
Citation Context ...pseudo-distributive law δ. It is the task of the remainder of this paper to do this. [The following alternative approach to the theory of polycategories was suggested by Robin Houston: from the paper =-=[7]-=-, multicategories with object set X can be viewed as lax monoids on the discrete object X in Mod. We might hope to extend this to a notion of lax Frobenius algebra, following [18]; then a polycategory... |

7 |
Proof theory in the abstract
- Hyland
- 2002
(Show Context)
Citation Context ... definitions of pseudomonad, pseudocomonad and pseudodistributive law. 22 Multicategories and polycategories We begin by re-examining the theory of multicategories: the material here summarises [1], =-=[10]-=- and [16], amongst others. Note that throughout, we shall only be interested in the theory of symmetric multicategories, and, later, of symmetric polycategories: that is, we allow ourselves to reorder... |

7 |
Metric, topology and multicategory—a common approach
- Clementino, Tholen
- 2003
(Show Context)
Citation Context ...a ‘distributive law’. Where we deviate from Koslowski is in the description of multicategories that we build upon. In Section 1, we recount this alternative description: it is the approach of [1] and =-=[4]-=-, based on profunctors rather than spans. We go on to describe how we may generalise this description to one for polycategories; to do this we invoke a pseudodistributive law (in the sense of [17], [2... |

7 | Monoidal bicategories and Hopf algebroids, Advance in maths - Day, Street - 1997 |

5 |
T-catégories (catégories dans un triple
- Burroni
- 1971
(Show Context)
Citation Context ...on intuition alone. Thus far, the paper [13] has provided the only attempt to rectify this situation. Koslowski provides an abstract description of polycategories that generalises the elegant work of =-=[2]-=- and later [9] and [15] on ‘T-multicategories’. However, whilst this latter theory uses only some rather simple and obvious constructions on categories with finite limits, the structures that Koslowsk... |

2 | A monadic approach to polycategories
- Koslowski
- 2005
(Show Context)
Citation Context ...? What is a polycategorical limit? In attempting to answer such questions without a formal framework, one is forced into the unsatisfactory position of relying on intuition alone. Thus far, the paper =-=[13]-=- has provided the only attempt to rectify this situation. Koslowski provides an abstract description of polycategories that generalises the elegant work of [2] and later [10] and [15] on ‘T-multicateg... |

1 |
Pseudo-distributive laws
- Tanaka
- 2004
(Show Context)
Citation Context ...], based on profunctors rather than spans. We go on to describe how we may generalise this description to one for polycategories; to do this we invoke a pseudo-distributive law (in the sense of [17], =-=[20]-=-) of a pseudocomonad (the ‘target arity’) over a pseudomonad (the ‘source arity’). Polycategories now arise as monads in the ‘two-sided Kleisli bicategory’ of this pseudo-distributive law. There are s... |