## On the homotopy theory of n-types

Venue: | Homology, Homotopy Appl |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Biedermann_onthe,

author = {Georg Biedermann},

title = {On the homotopy theory of n-types},

journal = {Homology, Homotopy Appl},

year = {},

pages = {2008}

}

### OpenURL

### Abstract

Abstract. An n-truncated model structure on simplicial (pre-)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine’s intermediate model structures we construct such an n-type model structure via Bousfield-Friedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called n-hyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analogous n-truncations on simplicial groupoids and prove a Quillen equivalence between these settings. We achieve a classification of n-types of simplicial presheaves in terms of (n −1)-types of presheaves of simplicial groupoids. Our classification holds for general n. Therefore this can also be viewed as the homotopy theory of (pre-)sheaves of (weak) higher groupoids. Contents

### Citations

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(Show Context)
Citation Context ...jective and the injective ones and denote a choice of one of them by I. Remark 3.5. The local or projective model structure on sPreC with respect to the chaotic topology on C was first constructed in =-=[BK72]-=- and generalized to other topologies in [Bla01]. The global or injective structure was constructed in [Jar87]. Its predecessor in the sheaf case was found in [Joy]. The S-model structures on sPreC wer... |

146 |
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Citation Context ... e.g. we identify sets of generating trivial cofibrations relative to the underlying model structures on sPreC or sShvC. I begin in section 2 with a very brief overview of a localization process from =-=[BF78]-=- and improved on in [Bou00], which I will use later on to obtain the truncations. Then in section 3, I quickly recast the necessary homotopy theory on sPreC and sShvC. There are several choices for su... |

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Citation Context ...ries on the groupoid side are developed in section 5. It is an important and quite intricate point to not neglect set-theoretic difficulties. We will not be concerned with it though, since Jardine in =-=[Jar87]-=- has dealt with this problem by considering the concept of small sites and I simply adopt his point of view. See also [Jar05]. In section 4 the Bousfield-Friedlander localization from section 2 is emp... |

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Citation Context ...approximation is given by taking n-th Postnikov stages. The existence of such a model structure was certainly folklore for a long time and itON THE HOMOTOPY THEORY OF n-TYPES 3 has been described in =-=[Hir03]-=- or in [Lur] and [TV] as the left Bousfield localization along the map ∂∆ n+2 → ∆ n+2 . In this article the structure is obtained by Bousfield-Friedlander localization, [BF78] and [Bou00], along the n... |

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Citation Context ...ucted in [Jar87]. Its predecessor in the sheaf case was found in [Joy]. The S-model structures on sPreC were constructed in [Jar03]. Their existence can also be derived from a more general context in =-=[Bek00]-=-. All of these structures serve equally well as a starting point for a theory of (pre-)sheaves of n-types in the next section. Of course, these model structures are all Quillen equivalent to each othe... |

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Citation Context ...ace” in our context means simplicial set or, more generally, (pre-)sheaf of simplicial sets on a small Grothendieck site C. A good way to describe homotopy theory is via Quillen model structures, see =-=[DS95]-=-. The right notion of equivalence of this structure is a Quillen equivalence. The goal of this article is to give sense to the following diagram of Quillen equivalences: (1.1) sPre n C i L 2 sShv n C ... |

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Citation Context ...oice of one of them by I. Remark 3.5. The local or projective model structure on sPreC with respect to the chaotic topology on C was first constructed in [BK72] and generalized to other topologies in =-=[Bla01]-=-. The global or injective structure was constructed in [Jar87]. Its predecessor in the sheaf case was found in [Joy]. The S-model structures on sPreC were constructed in [Jar03]. Their existence can a... |

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Citation Context ...d groupoids enriched in simplicial sets. These injectively fibrant models of n-types of simplicial (pre-)sheaves have received some attention in recent publications as n-(hyper-)stacks, see [Lur] and =-=[TV]-=-. So the above diagram gives different ways of describing the homotopy theory of higher hyperstacks, of which the right hand side seems to be completely new. Also on the left hand side this technique ... |

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Citation Context ...generating trivial cofibrations relative to the underlying model structures on sPreC or sShvC. I begin in section 2 with a very brief overview of a localization process from [BF78] and improved on in =-=[Bou00]-=-, which I will use later on to obtain the truncations. Then in section 3, I quickly recast the necessary homotopy theory on sPreC and sShvC. There are several choices for such underlying model structu... |

14 |
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Citation Context ...e later on to obtain the truncations. Then in section 3, I quickly recast the necessary homotopy theory on sPreC and sShvC. There are several choices for such underlying model structures exhibited in =-=[Jar03]-=- called intermediate model structures, because they sit in between the well known projective and injective model structures. Each of these structures serves equally well as a starting point for our th... |

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Citation Context ...f truncated groupoids enriched in simplicial sets. These injectively fibrant models of n-types of simplicial (pre-)sheaves have received some attention in recent publications as n-(hyper-)stacks, see =-=[Lur]-=- and [TV]. So the above diagram gives different ways of describing the homotopy theory of higher hyperstacks, of which the right hand side seems to be completely new. Also on the left hand side this t... |

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Citation Context ...C, s ≥0 } ∪ {∅ → LU∆ 0 | U ∈ C} Jproj := { LUΛ s+1 k → LU∆ s+1 | U ∈ C, s ≥0, s + 1 ≥k ≥0} The projective model structure exists more generally for small presheaves on arbitrary sites as described in =-=[CD05]-=- and [BCR06]. The real challenge lies in the injective structure. Then the case of the other intermediate model structures is done by juggling with what we already have [Jar03]. Remark 3.9. It is in t... |

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Citation Context ... G: sPreC → sPreC(Gd) is the loop groupoid functor and W is the universal cocycle functor from [DK84], also discussed in [GJ99, V.]. The first pair was used in [JT93]. The second pair was taken up by =-=[Luo]-=-. I follow the second approach. Let I denote an intermediate model structure on sPreC as in 3.5. Definition 5.1. We call the following classes of maps the transfered I-model structure or simply I-stru... |

3 |
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(Show Context)
Citation Context .... The functors i: sShvC ⇆ sPreC :L 2 form a Quillen equivalence for the respective structures on both sides. The proof of the lifting axioms in this theorem relies on the useful observation proved in =-=[Jar05]-=-, that a map is a fibration in one of the model structures from 3.11 in sShvC if and only if it is a fibration in the corresponding structure on sPreC. We will use the analogous argument in the proof ... |

2 |
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Citation Context ... {∅ → LU∆ 0 | U ∈ C} Jproj := { LUΛ s+1 k → LU∆ s+1 | U ∈ C, s ≥0, s + 1 ≥k ≥0} The projective model structure exists more generally for small presheaves on arbitrary sites as described in [CD05] and =-=[BCR06]-=-. The real challenge lies in the injective structure. Then the case of the other intermediate model structures is done by juggling with what we already have [Jar03]. Remark 3.9. It is in the proof of ... |

2 |
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Citation Context ...d pair is G: sPreC ⇆ sPreC(Gd) :W, where G: sPreC → sPreC(Gd) is the loop groupoid functor and W is the universal cocycle functor from [DK84], also discussed in [GJ99, V.]. The first pair was used in =-=[JT93]-=-. The second pair was taken up by [Luo]. I follow the second approach. Let I denote an intermediate model structure on sPreC as in 3.5. Definition 5.1. We call the following classes of maps the transf... |

1 | Truncated resolution model structures. arXiv:math.AT/0602564, to appear - Biedermann |

1 |
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Citation Context ...ertical maps in (6.1) are clearly projective fibrations. Note that for this reason we do not need to apply a fibrant replacement to give Pn homotopy meaning. The following observation can be found in =-=[Jar06]-=-. For a simplicial groupoid G and an object y in it let G/y denote the slice category of G over y, whose objects are the morphisms x → y and whose morphisms are the obvious commuting triangles. This i... |