## Operads, clones, and distributive laws (2008)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Curien08operads,clones,,

author = {Pierre-louis Curien},

title = {Operads, clones, and distributive laws},

year = {2008}

}

### OpenURL

### Abstract

Abstract We show how non-symmetric operads (or multicategories), sym-metric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory of generalized species of structures,but, for the multicategory case, the general idea goes back to Burroni's T-categories (1971). We show how other previous categorical analysesof operad (via Day's tensor products, or via analytical functor) fit with the profunctor approach.

### Citations

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Citation Context ...)(c, c ′′ )= � c ′ Ψ(c ′ ,c ′′ ) × Φ(c, c ′ ) Therefore, profunctors compose only up to isomorphism. Categories, profunctors, and natural transformations form thus, not a 2-category, but a bicategory =-=[3]-=-. 8sThe (pseudo-)category Prof of profunctors is autodual, via the isomorphism op : Prof → Prof op which maps C to C op and Φ : C1 + C2 to Φ op = λ(c2,c1).Φ(c1,c2) :C2 op + C1 op The composition of pr... |

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Citation Context ...tion operation • is via a different Kan extension: ?1 ⊆ Set X Lan ⊆X Set where ⊆ is the faithful (and non full) functor described at the end of section 2. This is the approach taken by Joyal (for !s) =-=[13]-=-. In Joyal’s language, X is called a species of structure, andLan ⊆X is the associated analytic functor, whose explicit formula is (for any set z): � m Lan ⊆Xz = z m × Xm Joyal’s approach requires mor... |

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Citation Context ...wedge is λ(α, u).αc ′′ u, and for any wedge from the diagram to some set z,wegetthemapfromYc ′′ to z by considering its component at C ′′ [ ,c ′′ ]. 8 Distributive laws Recall that a distributive law =-=[2]-=- (see also [1]) is a natural transformation λ : TS → ST, where(S, ηS,µS) and(T,ηT ,µT ) are two monads over the same category C, satisfying the following laws (expressed using self-explanatory picture... |

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Citation Context ...aches have led to different types of successes. The current state of the art seems to be that: 1. using spans, an impressive variety of shapes in the non-symmetric case have been covered. We refer to =-=[16]-=- for a book-length accont; 2. using profunctors one may cover the two other kinds of variations mentioned in this introduction. For the rest of these notes, we use (and introduce) the profunctor road.... |

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Citation Context ... ◦ η. Hereweshall only need to know that the solution of this universal problem is given by the following formula for Lan K(T ): Lan KTc = � m C[Km,c] · Tm where we use Mac Lane’s notation for coends =-=[17]-=-. Coends are sorts of colimits (or inductive limits), adapted to the case of diagrams which vary both covariantly and contravariantly over some parameter: here, m appears contravariantly in C[Km,c] an... |

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Citation Context ...sc1 c2 σ c2 σ c1 c1 c2 = c1 c2 c c δ σ c The other equations are the familiar ones for comonoids, and (mutatis mutandis) for distributive laws (see below). For a complete list, we refer the reader to =-=[7, 15]-=-. Take any combination X of these three combinators σ, δ, ɛ. We build the category !XC as follows: objects are sequences (c1,...,cn) of objects of C. Morphisms are diagrams built out of the combinator... |

21 | A Koszul duality for props
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Citation Context ...logue of operads for bioperations, and for X = {oe} and * as described above, what we obtain (replacing Set by Vect) is exactly the notion of properad introduced by Vallette in his Th`ese de Doctorat =-=[21, 22]-=-. Acknowledgements. I collected the material presented here for an invited talk at the conference Operads 2006. When preparing this talk, I benefited a lot from discussions with Marcelo Fiore and Mart... |

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Citation Context ... in a monad, and one then extends the monad to the category of spans, or to the category of profunctors. Spans are pairs of morphisms with the same domain, in a suitable category C. Burroni has shown =-=[6]-=- that every cartesian monad T on C (which means that C has pullbacks, that T preserves pullbacks, and that the naturality squares of the unit and multiplication of the monad are pullbacks) 1sextends t... |

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Citation Context ...?1 Lan Y(X ⊗ )= •X In this diagram, the functor X ⊗ associates with n the nth iterated tensor of X, with respect to the following monoidal structure on Set ?1 , due (in a more general setting) to Day =-=[8]-=-: (X ⊗ Y )p = � m,n Xm × Yn×?1[m + n, p] In =?1[0,n] The n-ary tensor product is described by the following formula: (X1 ⊗ X2 ⊗ ...⊗ Xm)p = � n1,...,nm Hence the formula for Y • X = Lan Y(X ⊗ )Y is X1... |

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Citation Context ...sc1 c2 σ c2 σ c1 c1 c2 = c1 c2 c c δ σ c The other equations are the familiar ones for comonoids, and (mutatis mutandis) for distributive laws (see below). For a complete list, we refer the reader to =-=[7, 15]-=-. Take any combination X of these three combinators σ, δ, ɛ. We build the category !XC as follows: objects are sequences (c1,...,cn) of objects of C. Morphisms are diagrams built out of the combinator... |

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Citation Context ...amples, ≈ will not always be termwise (it will be in the operad cases, but not in the clone case). 4 Kelly’s account of operads In 1972, Kelly gave the following description of operads [14] (see also =-=[9]-=-): 1 X !1 X ⊗ Set ?1 Y Set ?1 Lan Y(X ⊗ )= •X In this diagram, the functor X ⊗ associates with n the nth iterated tensor of X, with respect to the following monoidal structure on Set ?1 , due (in a mo... |

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Citation Context ...iso. The qualifiers ”(pseudo-)” or ”(bi)” will be the only traces of this concern here. Size issues arise from the fact that we shall consider the presheaf construction as a monad on Cat. We refer to =-=[18, 11]-=- for indications on how these issues can be handled rigorously. 7 Profunctors as a Kleisli category As we have already done when defining composition, we can alternatively write a profunctor Φ : C1 + ... |

5 |
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(Show Context)
Citation Context ...Tmd). In our examples, ≈ will not always be termwise (it will be in the operad cases, but not in the clone case). 4 Kelly’s account of operads In 1972, Kelly gave the following description of operads =-=[14]-=- (see also [9]): 1 X !1 X ⊗ Set ?1 Y Set ?1 Lan Y(X ⊗ )= •X In this diagram, the functor X ⊗ associates with n the nth iterated tensor of X, with respect to the following monoidal structure on Set ?1 ... |

4 |
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(Show Context)
Citation Context ...]) = Lan ⊆(Y • X)z (the summation over p is superflluous since from f ∈?1[n1 + ...+ nm,p we get z ⊆(f) from z p to z n1+...+nm ). 6 Profunctors Recall that a profunctor (or distributor) [4] (see also =-=[5]-=-) Φ from C to C ′ , notation Φ : C + C ′ , is a functor Φ:C × C ′op → Set Composition of profunctors is given by the following formula: (Ψ ◦ Φ)(c, c ′′ )= � c ′ Ψ(c ′ ,c ′′ ) × Φ(c, c ′ ) Therefore, p... |

4 |
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(Show Context)
Citation Context ...logue of operads for bioperations, and for X = {oe} and * as described above, what we obtain (replacing Set by Vect) is exactly the notion of properad introduced by Vallette in his Th`ese de Doctorat =-=[21, 22]-=-. Acknowledgements. I collected the material presented here for an invited talk at the conference Operads 2006. When preparing this talk, I benefited a lot from discussions with Marcelo Fiore and Mart... |

3 |
Les distributeurs, Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Rapport 33
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(Show Context)
Citation Context ...n1 + ...+ nm,p]) = Lan ⊆(Y • X)z (the summation over p is superflluous since from f ∈?1[n1 + ...+ nm,p we get z ⊆(f) from z p to z n1+...+nm ). 6 Profunctors Recall that a profunctor (or distributor) =-=[4]-=- (see also [5]) Φ from C to C ′ , notation Φ : C + C ′ , is a functor Φ:C × C ′op → Set Composition of profunctors is given by the following formula: (Ψ ◦ Φ)(c, c ′′ )= � c ′ Ψ(c ′ ,c ′′ ) × Φ(c, c ′ ... |

3 | On the coherence conditions for pseudo-distributive laws
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(Show Context)
Citation Context ...iso. The qualifiers ”(pseudo-)” or ”(bi)” will be the only traces of this concern here. Size issues arise from the fact that we shall consider the presheaf construction as a monad on Cat. We refer to =-=[18, 11]-=- for indications on how these issues can be handled rigorously. 7 Profunctors as a Kleisli category As we have already done when defining composition, we can alternatively write a profunctor Φ : C1 + ... |

3 |
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(Show Context)
Citation Context ...the cocomplete categories (and hence that there can be at most one Pshalgebra structure on a given category). This can be proved using Beck's characterization of monadic adjunctions (Theorem 4.4.4 of =-=[6]-=-). 18sConsider a slightly more general version of Day's tensor product than that given in section 4, with now some monoidal category C in place of !1 (X \OmegasY )C = Z C1,C2 XC1 * XC2 * C[C, C1 \Omeg... |

2 |
N.Gambino, M.Hyland, and G.Winskel, The cartesian closed bicategory of generalised species of structures
- Fiore
- 2006
(Show Context)
Citation Context ...of operad litterature) arise as endomorphisms endowed with a monoid structure in the (co)Kleisli (bi)category of the category of spans Profunctors are functors Φ : C × C op → Set. As observed, say in =-=[10]-=-, every monad in Cat (the category of categories) satisfying a distributive law (explicited in these notes) can be extended to the category of profunctors (whose objects are categories and whose morph... |

1 |
A universal property of the convolution monoidal structure
- Garner, Polycategories
- 2005
(Show Context)
Citation Context ...?, ?\Phi , *, !\Psi , and the multiplication of !. The identity is the composition ot the counit of ? and of the unit of !. This idea has been carried out in detail by Garner, in the case of X = {oe} =-=[13]-=-. We set *((m1, . . . , mp), (n1, . . . , nq)) 6= ; iff m1 + . . . + mp = n1 + . . . + nq and when the equality holds, *((m1, . . . , mp), (n1, . . . , nq)) is the set of permutations s from m1 + . . ... |