## Categorical homotopy theory

Venue: | Homology, Homotopy Appl |

Citations: | 185 - 8 self |

### BibTeX

@ARTICLE{Jardine_categoricalhomotopy,

author = {J. F. Jardine},

title = {Categorical homotopy theory},

journal = {Homology, Homotopy Appl},

year = {},

pages = {2006}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small

### Citations

375 |
Homotopy limits, completions and localizations
- Bous, Kan
- 1972
(Show Context)
Citation Context ...n. Now suppose that π: Z → X is a projective cofibrant model for the I-diagram X. Then I claim that there is a weak equivalence lim Z ≃ holim −→ −−−→ X, where holim −−−→ X has the standard definition =-=[2]-=-,[5]. In effect, the map π induces a weak equivalence π∗: holim −−−→ Z → holim −−−→ X by standard results about bisimplicial sets. If Y is a Kan complex, then the function complex hom(Z, Y ) is a glob... |

368 |
Higher algebraic K-theory. I
- Quillen
- 1972
(Show Context)
Citation Context ... more effectively proved with a comparison of homotopy colimits — part of the appeal of the result lies in the simplicity of that proof. All of this has been known since the early 1970s, when Quillen =-=[17]-=- introduced these concepts and results as a foundation for his description of higher algebraic K-theory. This set of techniques is still fundamental for algebraic K-theory, and Theorem B is now one of... |

121 |
Simplicial presheaves
- Jardine
(Show Context)
Citation Context ...ves on a Grothendieck site and derive much of their power in applications from holding at that level of generality. The development of these theories and their applications was initiated in the 1980s =-=[7]-=-, [12], [19] and continues to the present [8], [16]. The homotopy theory of stacks [9], [13] is a vital and important subindustry of this work.�� Homology, Homotopy and Applications, vol. 8(1), 2006 ... |

80 | Presheaves of symmetric spectra
- Jardine
(Show Context)
Citation Context ... their power in applications from holding at that level of generality. The development of these theories and their applications was initiated in the 1980s [7], [12], [19] and continues to the present =-=[8]-=-, [16]. The homotopy theory of stacks [9], [13] is a vital and important subindustry of this work.�� Homology, Homotopy and Applications, vol. 8(1), 2006 73 Thomason’s work on the model structure for... |

37 | Local projective model structures on simplicial presheaves
- Blander
(Show Context)
Citation Context ...i) → Y (i) of A-sets have the property P. In particular, a map π: Z → W is a pointwise trivial fibration if and only if all maps π: Z(i) → Y (i) are trivial fibrations of M. Recall (see, for example, =-=[1]-=-) that, since M is cofibrantly generated, there is a model structure on the category of I-diagrams I → M for which the weak equivalences and fibrations are defined pointwise. The cofibrations for the ... |

25 |
préfaisceaux comme modèles des types d’homotopie, Astérisque 308
- Cisinski, Les
- 2006
(Show Context)
Citation Context ...so part of a history of the subject, but its impact has so far been rather muted. It is strongly related to but not necessary for the ideas exposed in this paper. The thesis of Denis-Charles Cisinski =-=[3]-=- represents the next leap forward for the subject. Cisinski’s thesis is primarily concerned with the proof of some conjectures of Grothendieck [6], [15] concerning diagram categories that model homoto... |

25 | Localization theories for simplicial presheaves
- Goerss, Jardine
- 1998
(Show Context)
Citation Context ...of the form f: ∗ → I for some simplicial presheaf I. Example 4.22. Suppose that X is a scheme of finite dimension, and let C be the site (Sm|X)Nis of smooth schemes over X with the Nisnevich topology =-=[4]-=-, [8], [16]. The motivic model structure for the category of simplicial presheaves on the smooth Nisnevich site (Sm|X)Nis is the f-local theory, where f: ∗ → A 1 is some choice of rational point. It f... |

22 |
Letter to A. Grothendieck
- Joyal
- 1984
(Show Context)
Citation Context ...n a Grothendieck site and derive much of their power in applications from holding at that level of generality. The development of these theories and their applications was initiated in the 1980s [7], =-=[12]-=-, [19] and continues to the present [8], [16]. The homotopy theory of stacks [9], [13] is a vital and important subindustry of this work.�� Homology, Homotopy and Applications, vol. 8(1), 2006 73 Tho... |

14 |
La théorie de l’homotopie de Grothendieck, Astérisque 301 (2005), 1–140. ha l-0 0, v er sio
- Maltsiniotis
(Show Context)
Citation Context ...this paper. The thesis of Denis-Charles Cisinski [3] represents the next leap forward for the subject. Cisinski’s thesis is primarily concerned with the proof of some conjectures of Grothendieck [6], =-=[15]-=- concerning diagram categories that model homotopy theory, but the techniques that he has developed are arguably more important than the conjectures themselves. The theory begins with Grothendieck’s c... |

12 |
Abstract homotopy I
- Kan
- 1955
(Show Context)
Citation Context ...omotopy of morphisms. The main examples of such theories arise either from taking iterated products X × I ×n with objects I having two distinct global sections, or from Kan’s tensor product operation =-=[14]-=- for cubical sets. The affine line A 1 and the global sections 0, 1: ∗ → A 1 generate an interval theory in the motivic context. The construction of the resulting (⊗, S)-model structure on the categor... |

7 |
A1-homotopy theory of schemes, Publ
- Morel, Voevodsky
- 1999
(Show Context)
Citation Context ...r power in applications from holding at that level of generality. The development of these theories and their applications was initiated in the 1980s [7], [12], [19] and continues to the present [8], =-=[16]-=-. The homotopy theory of stacks [9], [13] is a vital and important subindustry of this work.�� Homology, Homotopy and Applications, vol. 8(1), 2006 73 Thomason’s work on the model structure for the c... |

6 |
and the homotopy theory of simplicial sheaves
- Jardine, Stacks
(Show Context)
Citation Context ... at that level of generality. The development of these theories and their applications was initiated in the 1980s [7], [12], [19] and continues to the present [8], [16]. The homotopy theory of stacks =-=[9]-=-, [13] is a vital and important subindustry of this work.�� Homology, Homotopy and Applications, vol. 8(1), 2006 73 Thomason’s work on the model structure for the category of small categories [18] is... |

5 |
On the homotopy theory of sheaves of simplicial groupoids
- Joyal, Tierney
- 1996
(Show Context)
Citation Context ...hat level of generality. The development of these theories and their applications was initiated in the 1980s [7], [12], [19] and continues to the present [8], [16]. The homotopy theory of stacks [9], =-=[13]-=- is a vital and important subindustry of this work.�� Homology, Homotopy and Applications, vol. 8(1), 2006 73 Thomason’s work on the model structure for the category of small categories [18] is also ... |

4 | Cubical homotopy theory: a beginning
- Jardine
- 2002
(Show Context)
Citation Context ...py theoretic language and notation, and to begin to describe their applications. I was initially attracted to Cisinksi’s thesis as a result of my own work on the homotopy theory of cubical sets — see =-=[10]-=-. This paper displays a model structure on the category of cubical sets whose associated homotopy category was equivalent to that of simplicial sets. The cofibrations are the monomorphisms and the wea... |

3 |
Pursuing stacks. Letter to D
- GROTHENDIECK
- 1983
(Show Context)
Citation Context ...d in this paper. The thesis of Denis-Charles Cisinski [3] represents the next leap forward for the subject. Cisinski’s thesis is primarily concerned with the proof of some conjectures of Grothendieck =-=[6]-=-, [15] concerning diagram categories that model homotopy theory, but the techniques that he has developed are arguably more important than the conjectures themselves. The theory begins with Grothendie... |

3 |
Cat as a closed model category
- Thomason
- 1980
(Show Context)
Citation Context ...cks [9], [13] is a vital and important subindustry of this work.�� Homology, Homotopy and Applications, vol. 8(1), 2006 73 Thomason’s work on the model structure for the category of small categories =-=[18]-=- is also part of a history of the subject, but its impact has so far been rather muted. It is strongly related to but not necessary for the ideas exposed in this paper. The thesis of Denis-Charles Cis... |

1 |
Simplicial approximation, Theory
- Jardine
- 2004
(Show Context)
Citation Context ...al set fibrations. After the fact, cubical set excision (Theorem 8.9) turns out to be a direct consequence of the formal techniques displayed here, along with the excision theorem for simplicial sets =-=[11]-=-. Much of the rest of the paper is an exposition of the basic theory. I have chosen to emphasize presheaves of A-sets (here called A-presheaves) on an arbitrary small Grothendieck site C because I bel... |

1 |
N6A 5B7 Canada This article is available at http://intlpress.com/HHA/v8/n1/a3
- Sup
- 1985
(Show Context)
Citation Context ...othendieck site and derive much of their power in applications from holding at that level of generality. The development of these theories and their applications was initiated in the 1980s [7], [12], =-=[19]-=- and continues to the present [8], [16]. The homotopy theory of stacks [9], [13] is a vital and important subindustry of this work.�� Homology, Homotopy and Applications, vol. 8(1), 2006 73 Thomason’... |