## Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids (2006)

Citations: | 6 - 3 self |

### BibTeX

@MISC{Paoli06semistrictmodels,

author = {Simona Paoli},

title = {Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids },

year = {2006}

}

### Years of Citing Articles

### OpenURL

### Abstract

Homotopy 3-types can be modelled algebraically by Tamsamani’s weak 3-groupoids as well as, in the path connected case, by cat 2-groups. This paper gives a comparison between the two models in the path-connected case. This leads to two different semistrict algebraic models of connected 3-types using Tamsamani’s model. Both are then related to Gray groupoids.

### Citations

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Citation Context ...by [·] : Cat C → Cat C/∼ the localization functor. The above notion of weak equivalence in C∆op is part of the Quillen model category structure on simplicial objects in an algebraic category given in =-=[19]-=-, [21]; in this model structure, if X∗ ∈ C∆op is cofibrant, then each Xn is projective in C with respect to the class of regular epimorphisms (see [19]). Lemma 3.1. Let C be a category of groups with ... |

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Citation Context ...nt classifying spaces. On the other hand, being the underlying simplicial n n12 SIMONA PAOLI sets of a simplicial group, they are both fibrant and cofibrant. By a general theorem of model categories =-=[9]-=-, they are also simplicially homotopy equivalent, so the corresponding categories are equivalent. We also have n TUψn∗ ∼ =Uπ0ψn∗ ∼ =Uπ0(ψ1∗×ψ0∗ · · ·×ψ0∗ψ1∗) ∼ = U(π0ψ1∗×π0ψ0∗ · · ·×π0ψ0∗π0ψ1∗). Hence... |

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Citation Context ...as in Section 2. Let Bigpd be the category of bigroupoids and their homomorphisms. Then Bic restricts to a functor Bic : T2 → Bigpd. Let st : Bicat → 2-cat be the strictification functor described in =-=[7]-=-. Then st restricts to a functor st : Bigpd → 2-gpd. Let Bic : N2 → Bicat and ν : 2-gpd → T st 2 Definition 6.1. Let St : T2 → T st 2 be the composite functor T2 Bic → Bigpd st → 2-gpd ν → T st 214 S... |

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Citation Context ...her category theory, homotopy models serve as a “test” for a good definition of weak higher category, which should give a model of n-types in the weak ngroupoid case. This property has been proved in =-=[22]-=-,[23] for the Tamsamani’s model of weak n-categories. Tamsamani’s model and cat n -groups are both multi-simplicial models but they have distinctly different features: the first is a strict but cubica... |

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Citation Context ...se: the cat n -group model, introduced by Loday [14] and later developed in [4],[18] generalized the earlier work of Whitehead on crossed modules [24]; another model was built by Carrasco and Cegarra =-=[6]-=-. In higher category theory, homotopy models serve as a “test” for a good definition of weak higher category, which should give a model of n-types in the weak ngroupoid case. This property has been pr... |

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Citation Context ...here the equivalence relation is induced by the cells of the next higher dimension.18 SIMONA PAOLI When G = Gφ, recalling the algebraic expression of the homotopy groups of a strict 2-groupoid as in =-=[15]-=-, we therefore find: π1(Gφ) = AutGφ (∗)/ ∼ = π0(st Bic φ1) π2(Gφ) = AutGφ (1∗)/ ∼ ∼ = π0Homst Bicφ1(∗, ∗) = π1 st Bic φ1 π3(Gφ) = AutGφ (11∗) ∼ = Autst Bic φ1(1∗) = π2 st Bic φ1 By Proposition 6.2, πi... |

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Citation Context ...t to both homotopy theory and higher category theory. In homotopy theory, various models exist for the path-connected case: the cat n -group model, introduced by Loday [14] and later developed in [4],=-=[18]-=- generalized the earlier work of Whitehead on crossed modules [24]; another model was built by Carrasco and Cegarra [6]. In higher category theory, homotopy models serve as a “test” for a good definit... |

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Citation Context ...ons: ✷SEMISTRICT MODELS OF CONNECTED 3-TYPES AND TAMSAMANI’S... 11 a) The constant functor ψ([0],-,-) : ∆2op → Set takes values in the oneelement set. b) For each m ≥ 2 the Segal maps ψ([m],-,-) → ψ(=-=[1]-=-,-,-) × m · · · × ψ([1],-,-) are bijections. Note that objects of H are not strict 3-groupoids because, in general, ψ([m],-,-) are weak, not strict, 2-nerves. We say a morphism f : ψ → ψ ′ in H is an ... |

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Citation Context ...F0F1). Let Cat C be the category of internal categories in C and internal functors. In this paper we use the category Cat C in the case where C is a category of groups with operations in the sense of =-=[16]-=-, [17]. Recall that this consists of a category of groups with a set of additional operations Ω = Ω0 ∪ Ω1 ∪ Ω2, where Ωi is the set of i-ary operations in Ωi such that the group operations of identity... |

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Citation Context ...evant to both homotopy theory and higher category theory. In homotopy theory, various models exist for the path-connected case: the cat n -group model, introduced by Loday [14] and later developed in =-=[4]-=-,[18] generalized the earlier work of Whitehead on crossed modules [24]; another model was built by Carrasco and Cegarra [6]. In higher category theory, homotopy models serve as a “test” for a good de... |

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Citation Context ... Let Cat C be the category of internal categories in C and internal functors. In this paper we use the category Cat C in the case where C is a category of groups with operations in the sense of [16], =-=[17]-=-. Recall that this consists of a category of groups with a set of additional operations Ω = Ω0 ∪ Ω1 ∪ Ω2, where Ωi is the set of i-ary operations in Ωi such that the group operations of identity, inve... |

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Citation Context ...[φ]. Therefore (i St , j) is an equivalence of categories. ✷ 7. The comparison with Gray groupoids It is well known that Gray groupoids are semi-strict algebraic models of homotopy 3-types; see [10], =-=[13]-=-, [1], [11]. Recall that Gray is the category of 2-categories with monoidal structure given by the Gray tensor product. A Gray category is a category enriched in Gray . A Gray groupoid is a Gray categ... |

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Citation Context ...ore (i St , j) is an equivalence of categories. ✷ 7. The comparison with Gray groupoids It is well known that Gray groupoids are semi-strict algebraic models of homotopy 3-types; see [10], [13], [1], =-=[11]-=-. Recall that Gray is the category of 2-categories with monoidal structure given by the Gray tensor product. A Gray category is a category enriched in Gray . A Gray groupoid is a Gray category whose c... |

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2-nerves for bicategories
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Citation Context ...ictification functor from T2 to T st 2 . In the following theorem, N Bicat denotes the 2-category of bicategories, normal homomorphisms and oplax natural transformations with identity components (see =-=[12]-=-) Theorem 2.2. [12, Th. 7.2] There is a fully faithful 2-functor N : NBicat → N2 with a left 2-adjoint Bic. The counit Bic N → 1 is invertible and each component u : X → N BicX of the unit is a pointw... |

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Citation Context ...tφ] = [φ]. Therefore (i St , j) is an equivalence of categories. ✷ 7. The comparison with Gray groupoids It is well known that Gray groupoids are semi-strict algebraic models of homotopy 3-types; see =-=[10]-=-, [13], [1], [11]. Recall that Gray is the category of 2-categories with monoidal structure given by the Gray tensor product. A Gray category is a category enriched in Gray . A Gray groupoid is a Gray... |

3 | Model categories and their localizations - Hirschorn |