## Polynomial functors and polynomial monads (2009)

Citations: | 3 - 0 self |

### BibTeX

@TECHREPORT{Gambino09polynomialfunctors,

author = {Nicola Gambino and Joachim Kock},

title = {Polynomial functors and polynomial monads},

institution = {},

year = {2009}

}

### OpenURL

### Abstract

Abstract. We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.

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Citation Context ... and Carboni-Johnstone [13] (see also [40, App. C]), and local right adjoints of Lamarche [36], Taylor [55], and Weber [56, 57]. We also comment on the relationship with species and analytic functors =-=[27, 9]-=-, and with Girard’s normal functors [17]. Although the category of topological spaces is not locally cartesian closed, the notion of polynomial functor makes sense if one separately requires the ‘midd... |

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Citation Context ...nomial monad is the free-monoid monad of Example 1.9.�� � �� 24 GAMBINO AND KOCK 4.2. We are interested in the construction of the free monad on a polynomial endofunctor, and start by recalling from =-=[28, 7]-=- some general facts about free monads. Let C be a category and P : C → C an endofunctor. The free monad on P is a monad (T,η,µ) on C together with a natural transformation α : P ⇒ T enjoying the follo... |

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Citation Context ...nd streamlined proofs. These results mainly concern the diagram representation of strong natural transformations between polynomial functors, and some of these results can be found in Abbott’s thesis =-=[1]-=-. Having laid the groundwork, our first main result is to assemble polynomial functors into a double category, in fact a framed bicategory in the sense of Shulman [52], hence providing a convenient an... |

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Citation Context ...’ (multiplicative transfer). In4 GAMBINO AND KOCK Paragraph 1.22, we give an algebraic-theory interpretation of one of his discoveries. Further study of Tambara functors has been carried out by Brun =-=[11]-=-, with applications to Witt vectors. The name polynomial functor is often given to endofunctors of the category of vector spaces involving actions of the symmetric groups, cf. Appendix A of Macdonald’... |

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Citation Context ...ms. The objects of D0 are called objects of D, the morphisms of D0 are called vertical arrows, the objects of D1 are called horizontal arrows, and the morphisms of D1 are called squares. As is custom =-=[18]-=-, we allow the possibility for the horizontal composition to be associative and unital only up to specified coherent isomorphisms. Precisely, a double category is a pseudo-category [44] in the 2-categ... |

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Citation Context ...ied. In Paragraph 1.17 we list six equivalent characterisations of polynomial functors over Set, and briefly comment on the contexts of the related notions: familially representable functors of Diers =-=[14]-=- and Carboni-Johnstone [13] (see also [40, App. C]), and local right adjoints of Lamarche [36], Taylor [55], and Weber [56, 57]. We also comment on the relationship with species and analytic functors ... |

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Citation Context ...und. Notions of polynomial functor have proved useful in many areas of mathematics, ranging from algebra [41, 34] and topology [10, 50] to mathematical logic [17, 45] and theoretical computer science =-=[24, 2, 20]-=-. The present paper deals with the notion of polynomial functor over locally cartesian closed categories. Before outlining our results, let us briefly motivate this level of abstraction. Among the dev... |

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Citation Context ... with operads and other related notions is explored. Introduction Background. Notions of polynomial functor have proved useful in many areas of mathematics, ranging from algebra [41, 34] and topology =-=[10, 50]-=- to mathematical logic [17, 45] and theoretical computer science [24, 2, 20]. The present paper deals with the notion of polynomial functor over locally cartesian closed categories. Before outlining o... |

2 |
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Citation Context ...t to Manes and Arbib [43], and was recently explored from a different viewpoint under the name ‘interaction systems’ in the setting of dependent type theory by Hancock and Setzer [20] and by Hyvernat =-=[22]-=-, where polynomials are also given a game-theoretic interpretation. The morphisms there are certain bisimulations, more general than the strong natural transformations used in the present work. Within... |

2 |
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Citation Context ... and Carboni-Johnstone [13] (see also [40, App. C]), and local right adjoints of Lamarche [36], Taylor [55], and Weber [56, 57]. We also comment on the relationship with species and analytic functors =-=[27, 9]-=-, and with Girard’s normal functors [17]. Although the category of topological spaces is not locally cartesian closed, the notion of polynomial functor makes sense if one separately requires the ‘midd... |

1 |
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Citation Context ...r notions of polynomial and polynomial functor are almost exactly the same as the notions of container and container functor introduced in theoretical computer science by Abbott, Altenkirch and Ghani =-=[1, 2, 3, 4]-=- to provide semantics for recursive data types, and studied further in [5]. The differences, mostly stylistic, are explained in Paragraph 2.16. A predecessor to containers were the shapely types of Ja... |

1 |
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Citation Context ...abelled by an operation of the same arity. 3.16. We finish this section with a digression on the relationship between polynomial functors and the shapely functors and shapely types of Jay and Cockett =-=[24, 23]-=-, since the double-category setting provides some conceptual simplification of the latter notion. A shapely functor [24] is a pullback-preserving functor F : E m → E n equipped with a strength. Since,... |

1 |
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Citation Context ...und. Notions of polynomial functor have proved useful in many areas of mathematics, ranging from algebra [41, 34] and topology [10, 50] to mathematical logic [17, 45] and theoretical computer science =-=[24, 2, 20]-=-. The present paper deals with the notion of polynomial functor over locally cartesian closed categories. Before outlining our results, let us briefly motivate this level of abstraction. Among the dev... |