## Computads and slices of operads. (2002)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Batanin02computadsand,

author = {M. A. Batanin},

title = {Computads and slices of operads.},

year = {2002}

}

### OpenURL

### Abstract

For a given ω-operad A on globular sets we introduce a sequence of symmetric operads on Set called slices of A and show how the connected limit preserving properties of slices are related to the property of the

### Citations

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Monoidal globular categories as a natural environment for the theory of weak n-categories
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(Show Context)
Citation Context ...eory of computads for finitary monads on globular sets. An important class of such monads consists of so called analytic monads [5] which can be identified with higher operads in Span in the sense of =-=[1]-=-. The examples in the previous paragraph all belong to this class of monads. In [3] some properties of computads for analytic monads were established. In particular, it was claimed that computads form... |

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Citation Context ...strict one; • the notion of ‘semistrict’ n-category is ‘minimal’ with the above property. In dimension 2 this is just the notion of strict 2-category. In dimension 3 it is the notion of Gray-category =-=[8]-=-. Crans has a candidate for dimension 4 and some ideas about higher dimensions [7]. Here we risk to suggesting a conjecture. Conjecture 3.2 There is a unique contractible n-operad Gn with the property... |

83 |
Foncteurs analytiques et especes de structures. Combinatoire Enumerative
- Joyal
- 1986
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Citation Context ...analytic and unit and multiplication are cartesian natural transformation. The category of analytic monads is equivalent to the category of ω-operads in Span. The following definition is due to Joyal =-=[9]-=-. An endofunctor a on Set is called analytic if it can be represented as a ‘Taylor series’ a(X) = ∑ A[n] × X Σn n , n≥0 where A[n], n ≥ 0, is a symmetric collection, i.e. a family of sets equipped wit... |

51 |
A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so
- Kelly
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(Show Context)
Citation Context ...composite of V and Γ which is left adjoint to the restriction functor l ⋆ : Algn −→ AlgIA induced by an obvious morphism of monads l : IA → An. This left adjoint exists due to the finitary assumption =-=[10]-=-. We also can talk about ω-computads. Recall [3] that the n-truncation of an (n + 1)-computad (C, φ, C) is the n-computad C. 4Definition 2.2 Let A be a finitary monad on Glob. An ω-computad for A is ... |

34 |
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- 1976
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Citation Context ...ing properties of slices are related to the property of the category of n-computads of A being a presheaf topos. 1991 Math. Subj. Class. 18C20, 18D05 1 Introduction. Computads were invented by Street =-=[13]-=- as a tool for the presentation of strict n-categories. They attracted a new wave of interest in recent years due to the development of the theory of weak higher categories. It also became evident tha... |

33 | The Eckmann-Hilton argument and higher operads - Batanin |

29 |
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- Street
(Show Context)
Citation Context .... 5We will also denote by (D, µ, ǫ) the monad generated by this adjunction (notice, that in [1] this monad was denoted by Ds). In [1] a description of D in terms of plain trees was presented. Recall =-=[14]-=- that a natural transformation p : R → Q between two functors is called cartesian if for every morphism f : X → Y the naturality square R(f) R(X) ✲ R(Y ) p ❄ Q(X) Q(f) ✲ p ❄ Q(Y ) is a pullback. Recal... |

19 |
The universal property of the multitude of trees
- Batanin, Street
(Show Context)
Citation Context ... globular theories were used. In our paper [3] we construct a general theory of computads for finitary monads on globular sets. An important class of such monads consists of so called analytic monads =-=[5]-=- which can be identified with higher operads in Span in the sense of [1]. The examples in the previous paragraph all belong to this class of monads. In [3] some properties of computads for analytic mo... |

11 | Computads for finitary monads on globular sets; in: Higher category theory
- Batanin
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(Show Context)
Citation Context ...For example, the theory of surface diagrams in 3D-space naturally leads to the use of so called Gray-computads [12]. In our paper [4] computads for magmatype globular theories were used. In our paper =-=[3]-=- we construct a general theory of computads for finitary monads on globular sets. An important class of such monads consists of so called analytic monads [5] which can be identified with higher operad... |

9 |
On the Penon method of weakening algebraic structures
- Batanin
(Show Context)
Citation Context ...en need some more general types of computads than Street’s computads. For example, the theory of surface diagrams in 3D-space naturally leads to the use of so called Gray-computads [12]. In our paper =-=[4]-=- computads for magmatype globular theories were used. In our paper [3] we construct a general theory of computads for finitary monads on globular sets. An important class of such monads consists of so... |

6 | A tensor product for Gray-categories
- Crans
- 1999
(Show Context)
Citation Context ...operty. In dimension 2 this is just the notion of strict 2-category. In dimension 3 it is the notion of Gray-category [8]. Crans has a candidate for dimension 4 and some ideas about higher dimensions =-=[7]-=-. Here we risk to suggesting a conjecture. Conjecture 3.2 There is a unique contractible n-operad Gn with the property that Pk(Gn), 0 ≤ k ≤ n − 1, is the free k-fold monoid operad. A semistrict n-cate... |

5 |
Symmetric Operads for Globular Sets
- Weber
- 2001
(Show Context)
Citation Context ...minal algebras of An. We have a restriction of the forgetful functor W W (k) : Alg (k−1) n → Glob (k−1) n , k ≥ 1. It is not hard to prove that this functor is monadic at least for a finitary monad A =-=[15]-=-. Hence, we have a monad S k An on Globn−k such that its category of algebras is equivalent to Alg (k−1) n . We also put S 0 A = A. Definition 3.2 [15] S k An is called the k-fold suspension of An Now... |

3 |
3-computads do not form a presheaf category
- Makkai, Zawadowski
- 2001
(Show Context)
Citation Context ...tunately, the proof we gave in [3] and [4] ∗ The author holds the Scott Russell Johnson Fellowship in the Centre of Australian Category Theory at Macquarie University 1turned out to be incorrect. In =-=[11]-=- Makkai and Zawadowski observed that the category of Street’s 3-computads can not be a presheaf topos. In this paper we study this question more carefully. We find a sufficient condition when computad... |

2 |
Trimble T., Surface diagrams for Gray-categories,(submitted
- McIntyre
- 1997
(Show Context)
Citation Context ...evident that we often need some more general types of computads than Street’s computads. For example, the theory of surface diagrams in 3D-space naturally leads to the use of so called Gray-computads =-=[12]-=-. In our paper [4] computads for magmatype globular theories were used. In our paper [3] we construct a general theory of computads for finitary monads on globular sets. An important class of such mon... |