## FCEyN, Universidad de Buenos Aires (810)

### BibTeX

@MISC{810fceyn,universidad,

author = {},

title = {FCEyN, Universidad de Buenos Aires},

year = {810}

}

### OpenURL

### Abstract

Grothendieck fibrations have played an important role in homotopy theory. Among others, they were used by Thomason to describe homotopy colimits of small categories and by Quillen to derive long exact sequences of higher K-theory groups. We construct simplicial objects, namely the fibred and the cleaved nerve, to characterize the homotopy type of a Grothendieck fibration by using the additional structure. From these, we derive long exact sequences of homotopy groups and spectral sequences for homology groups, establishing new results and placing those of Thomason and Quillen into our framework.

### Citations

331 |
Algebraic K-Theory I
- Quillen
- 1973
(Show Context)
Citation Context ... an important role in homotopy theory. Thomason described homotopy colimits of small categories as the Grothendieck construction of the involved functor in [Th], and famous Quillen’s Theorems A and B =-=[Qu]-=- might be understood as consequences of the behaviour of classifying space functor with respect to fibrations. Other applications can be found, for example, in [dH, He1, He2]. Lastly, we also want to ... |

164 | Simplicial homotopy theory
- Goerss, Jardine
- 1999
(Show Context)
Citation Context ...nd recall some results about the classifying space functor and a key proposition on simplicial sets (1.0.1). The reader is referred to [Qu] for an introduction to homotopy of small categories, and to =-=[GoJa]-=- for a comprehensive treatment of bisimplical objects. The principal reference on Grothendieck fibrations is [SGA1], however our viewpoint is slightly different from that. In section 2 we recall from ... |

83 | Classifying spaces and spectral sequences - Segal - 1968 |

34 |
Homotopy colimits in the category of small categories
- Thomason
- 1979
(Show Context)
Citation Context ...Ti, Mi]. Grothendieck fibrations have played an important role in homotopy theory. Thomason described homotopy colimits of small categories as the Grothendieck construction of the involved functor in =-=[Th]-=-, and famous Quillen’s Theorems A and B [Qu] might be understood as consequences of the behaviour of classifying space functor with respect to fibrations. Other applications can be found, for example,... |

27 |
Revêtements étales et groupe fondamental (SGA1
- Grothendieck
- 1963
(Show Context)
Citation Context ...ader is referred to [Qu] for an introduction to homotopy of small categories, and to [GoJa] for a comprehensive treatment of bisimplical objects. The principal reference on Grothendieck fibrations is =-=[SGA1]-=-, however our viewpoint is slightly different from that. In section 2 we recall from loc. cit. some main facts about fibrations, adapt some others and develop some technical results which will be need... |

23 | préfaisceaux comme modèles des types d’homotopie - Cisinski, Les - 2006 |

19 | Strong stacks and classifying spaces, Category theory - Joyal, Tierney - 1990 |

13 | La théorie de l’homotopie de Grothendieck - Maltsiniotis - 2005 |

11 |
Cat as a closed model category. Cahiers Topologie Géom. Différentielle
- Thomason
- 1980
(Show Context)
Citation Context ...on, with normal closed cleavageΣ. 3.3.1 Theorem. The composition ki : d(NcE)→NE is a homotopy equivalence.�� � �� M. del Hoyo - Grothendieck fibrations and classifying spaces 11 Proof. (Compare with =-=[Th2]-=-, 1.2) Since 2.2.3 we know that the cleavageΣgives a very good map s : E(p)→Ewhich induces a commutative square d(Nc(E(p))) s d(NcE) k k �� N(E(p)) s � NE whose vertical arrows are homotopy equivalenc... |

2 | Homotopy colimits in presheaf categories. Cahiers Topologie Geom Differentielle - Heggie - 1993 |

1 |
On the subdivision of small categories. Topology and its Applications 155
- Hoyo
- 2008
(Show Context)
Citation Context ...Se]). The study of homotopy in categories might be motivated by the following fact. For any topological space X there is a small category C such that X and BC are weakly homotopy equivalent (see e.g. =-=[dH]-=-). An interesting question that arises naturally is how to compute the discrete invariants of X, such as homotopy and homology groups, directly from C. The nerve of a category is a simplicial set. We ... |

1 |
Pursuing stacks. Unpublished letter
- Grothendieck
(Show Context)
Citation Context ... in [dH, He1, He2]. Lastly, we also want to mention that Grothendieck fibrations are basal in the theory of tests categories, the axiomatic homotopy theory initiated by Grothendieck’s Pursuing Stacks =-=[PS]-=- and followed by Maltsiniotis and Cisinski [Ma, Ci]. The paper is organized as follows. Section 1 deals with preliminaries. We fix some notations and recall some results about the classifying space fu... |

1 | Homotopy cofibrations in (cat). Cahiers Topologie Geom Differentielle 33 - Heggie - 1992 |

1 | Numerably Contractible Categories. K-Theory 36 - Minian - 2005 |