## Representable Multicategories (2000)

Venue: | Advances in Mathematics |

Citations: | 33 - 6 self |

### BibTeX

@INPROCEEDINGS{Hermida00representablemulticategories,

author = {Claudio Hermida},

title = {Representable Multicategories},

booktitle = {Advances in Mathematics},

year = {2000},

pages = {164--225}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...

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Citation Context ...riangle for the units and the associativity pentagon, follow from the uniqueness of mediating morphisms between (codomains of) universal arrows for the same sequence of objects. In the terminology of =-=[JS93]-=-, such univeral arrows and the mediating morphisms between them form a clique (a category equivalent to the terminal one). We now extend the above construction to morphisms of representable multicateg... |

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Citation Context ...riangle for the units and the associativity pentagon, follow from the uniqueness of mediating morphisms between (codomains of) universal arrows for the same sequence of objects. In the terminology of =-=[JS93]-=-, such univeral arrows and the mediating morphisms between them form a clique (a category equivalent to the terminal one). 38 We now extend the above construction to morphisms of representable multica... |

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Citation Context ...ucture (namely functors and natural transformations) to follow from this identication. This is not quite as straightforward: Street's original formulation of the 2-category of monads in a 2-category [=-=Str72-=-] was designed to account for the category-of-algebras construction, and his denition does not apply to obtain the 2-category of internal categories in a direct way. Street has recently elaborated an ... |

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Citation Context ... topological ones of [Joy97, Tam96] which deal with such n-dimensional gadgets via their nerves, imposing suitableslling conditions on certain cellular structures. Another approach is that of Batanin =-=[Bat98b-=-], based on homotopical algebra, more specically on the theory of operads. He considers contractible operads to organise the `spaces' of coherent operations, so that weak n-categories amount to algebr... |

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Citation Context ...categories with pullbacks, pullback preserving functors and cartesian transformations. Let Bicat denote the tricategory of bicategories, homomorphisms, pseudo-natural transformations and modications [=-=GPS95]-=-. A.3. Proposition. The bicategory of spans construction extends to a trihomomorphism Spn( ) : Pbk ! Bicat Proof. Given a pullback preserving functor F : B ! C , the homomorphism Spn(F) : Spn(B ) ! Sp... |

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Citation Context ...-category we refer to [Kel89]. The reader unfamiliar with this notion at the abstract level might think of it as the 2-categorical generalisation of the familiar category of fractions construction of =-=[G-=-Z67]. We now want to analyse whether M would be an equivalence given the coinverter property of ( M ) 0 . We would also like an explicit description of the coinverter without having to indulge in c... |

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Citation Context ... M // TM M to obtain 0 : TM ! M which is left adjoint to M , i.e. s M (" 1 ) 0 ; id : 0 a M . Notice that : TM ! M preserves universal arrows as it is a left adjoint, cf. Corollary 8.13 and [K=-=oc95]-=-. 2 10.2 Representable multicategories equivalent to strict ones We aim to prove now the main coherence result for representable multicategories, namely that every such is equivalent to a strict one. ... |

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Citation Context ...operads, although the opetopes carry no explicit operations themselves). The approach in ibid. just outlined motivated [HMP98a], which in turn initiated the work presented in this paper. There and in =-=[HMP98b]-=-, the authors develop an alternative (substantially simpler) method to set-up what ought to be the above mentioned n-opetopes, that is, the ambient cellular structures required to specify n-category c... |

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Citation Context ...7 A The functoriality of the Spn( ) construction 50 1 Introduction Among his vast seminal body of work, Grothendieck established a paradigm we might call coherence via universality . He introduced in =-=[Gro71]-=- the notion of fibration, a functor p : E!B characterised by the existence of cartesian liftings for any pair (X 2 E ; u : I!pX) in B . Here we have a strict structure, namely the category E with stri... |

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Citation Context ...iate) lower dimension designated as codomain (n-opetopes are defined in terms of operads, although the opetopes carry no explicit operations themselves). The approach in ibid. just outlined motivated =-=[HMP98a]-=-, which in turn initiated the work presented in this paper. There and in [HMP98b], the authors develop an alternative (substantially simpler) method to set-up what ought to be the above mentioned n-op... |

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Citation Context ...cks Proof. Universal arrows in m Spn(C ) correspond to limit cones. Hence m Spn(C ) i C admits limits of such diagrams. But these diagrams aresnite simply connected , and Pare's characterisation in [P=-=ar90]-=- shows that such diagrams are precisely the ones whose limits can be constructed by pullbacks. 2 It is clear that the bicategory associated to the representable multicategory m Spn(C ) is non other th... |

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Citation Context ...ebra coherence theorem: coherence theorem: fibration split fibration representable multicategory strict representable multicategory 2 Multicategories The notion of multicategory was introduced in [L=-=am69]-=- (see also [Lam89, Lin71]) as a framework for a syntactic calculus. A multicategory can be seen as an algebraic structure which models a (restricted kind of) many-sorted algebraic theory: brie y put, ... |

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Citation Context ...gth. It is convenient then to regard the arrows of CG as labellings of those of C1; the latter give the `shapes' of the former. Evidently, the shape of a sequence is simply its length. As observed in =-=[Str96]-=-, we can characterise a free category C on a graph by the existence of a functor l : C ! C1 with a `unique lifting of decompositions' property. We will reproduce this observation in the context of mul... |

2 |
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Citation Context ...lgebras construction, and his denition does not apply to obtain the 2-category of internal categories in a direct way. Street has recently elaborated an alternative version to sort out this diculty [S=-=tr99]-=- (see also Remark 6.4). We will address this issue in x6.1. 4 Multicategories revisited We now proceed to give a monadic definition of multicategories in the style of x3 above. The crucial point is th... |

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1 |
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Citation Context ... C0 (X) = hXi and codomain X, as required for the identity arrow in a multicategory. 2 The above internal characterisation of multicategories was also found independently from the author by Burroni [=-=Bur71]-=- and Leinster [Lei97]. We came up with it as it was essential to our higher-dimensional generalisation, which we will elaborate in a follow-up article (there have been a number of presentations of thi... |

1 |
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(Show Context)
Citation Context ...7 A The functoriality of the Spn( ) construction 49 1 Introduction Among his vast seminal body of work, Grothendieck established a paradigm we might call coherence via universality . He introduced in =-=[Gro71]-=- the notion ofsbration, a functor p : E ! B characterised by the existence of cartesian liftings for any pair (X 2 E ; u : I ! pX) in B . Here we have a strict structure, namely the category E with st... |

1 |
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Citation Context ...p with it as it was essential to our higher-dimensional generalisation, which we will elaborate in a follow-up article (there have been a number of presentations of this higher-dimensional version cf.=-=[Her97]-=-). 5 Free multicategories and trees In order to aid the visualisation of multicategories and computations in them, we give an explicit account of free multicategories (in Set) in terms of finite trees... |

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1 |
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Citation Context ...odomain X, as required for the identity arrow in a multicategory. 2 The above internal characterisation of multicategories was also found independently from the author by Burroni [Bur71] and Leinster =-=[Lei97]-=-. We came up with it as it was essential to our higher-dimensional generalisation, which we will elaborate in a follow-up article (there have been a number of presentations of this higher-dimensional ... |

1 | The multilinear Yoneda lemmas - Linton - 1971 |

1 |
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Citation Context ...iate) lower dimension designated as codomain (n-opetopes are defined in terms of operads, although the opetopes carry no explicit operations themselves). The approach in ibid. just outlined motivated =-=[HMP98b]-=-, which in turn initiated the work presented in this paper. There and in [HMP98a], the authors develop an alternative (substantially simpler) method to set-up what ought to be the above mentioned n-op... |

1 |
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Citation Context ...domain X , as required for the identity arrow in a multicategory. 2 The above internal characterisation of multicategories was also found independently from the author by Burroni [Bur71] and Leinster =-=[Lei97]-=-. We came up with it as it was essential to our higher-dimensional generalisation, which we will elaborate in a follow-up article (there have been a number of presentations of this higher-dimensional ... |

1 |
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Citation Context ...bras construction, and his definition does not apply to obtain the 2-category of internal categories in a direct way. Street has recently elaborated an alternative version to sort out this difficulty =-=[Str99]-=- (see also Remark 6.4). We will address these issues in x6.1. 4 Multicategories revisited We now proceed to give a monadic definition of multicategories in the style of x3 above. The crucial point is ... |