## DOI: 10.1007/s00222-007-0061-2 Triangulated categories without models (2007)

### BibTeX

@MISC{Muro07doi:10.1007/s00222-007-0061-2,

author = {O Muro and Stefan Schwede and Neil Strickl},

title = {DOI: 10.1007/s00222-007-0061-2 Triangulated categories without models},

year = {2007}

}

### OpenURL

### Abstract

Abstract. We exhibit examples of triangulated categories which are neither

### Citations

155 | Homotopy theory of Γ-spaces, spectra, and bisimplicial sets - Bousfield, Friedlander - 1977 |

80 | On triangulated orbit categories
- Keller
(Show Context)
Citation Context ...τ : k[ε]/ε 2 → k[ε]/ε 2 with τ(ε) =−ε, and (9) k[ε]/ε 2 ε −→ k[ε]/ε 2 ε −→ k[ε]/ε 2 ε −→ τ ∗ k[ε]/ε 2 is an exact triangle. An algebraic model for this triangulated category was obtained by Keller in =-=[Kel05]-=-. Keller’s model is a differerential graded (dg) k-category. Here we exhibit an alternative model, which is a dg k-algebra A such that F (k[ε]/ε 2 ) is exact equivalent to the category of compact obje... |

64 | On differential graded categories - Keller |

51 | on Modules and Rings - Lam - 1999 |

28 | Uniqueness theorems for certain triangulated categories with an Adams spectral sequence
- Franke
- 1996
(Show Context)
Citation Context ...r, due to Grothendieck [Gro90], and the closely related notions of a stable homotopy theory in the sense of Heller [Hel88, Hel97] or a system of triangulated diagram categories in the sense of Franke =-=[Fra96]-=-. In each of these settings, the stable homotopy category is the underlying category of the free example on one generator (the sphere spectrum). We do not know a precise reference of this fact for tri... |

24 | On the structure and ideal theory of complete local rings - Cohen - 1946 |

22 |
Stable homotopy categories
- Heller
- 1968
(Show Context)
Citation Context ...l to Bernhard Keller for helpful conversations on the results of this paper, to Amnon Neeman, who suggested the possibility of constructing a triangulated structure on F(Z/4) by using Heller’s theory =-=[Hel68]-=-, and to the referee for valuable suggestions. In the original version of this note the first author alone constructed the triangulation of the category F(Z/4) and proved that it does not admit any mo... |

21 | Homotopy theories - Heller - 1988 |

15 | A uniqueness theorem for stable homotopy theory - Schwede, Shipley |

9 | Stable homotopy theories and stabilization - Heller - 1997 |

8 |
Algebraic versus topological triangulated categories. Extended notes of a talk given at
- Schwede
- 2006
(Show Context)
Citation Context ...he triangulated categories which belong, up to exact equivalence, to one of these two families will be termed algebraic and topological, respectively. We borrow this terminology from [Kel06, 3.6] and =-=[Sch06]-=-. Algebraic triangulated categories are generally also topological, but there are many well-known examples of topological triangulated categories which are not algebraic. In the present paper we exhib... |

7 | Propriétés universelles et extensions de Kan dérivées - Cisinski |

3 |
Derived algebraic geometry I. Stable infinity categories. Available from http://www.math.harvard.edu/∼lurie
- Lurie
(Show Context)
Citation Context ...lassical Hopf map for the sphere spectrum to a Hopf map for A. Another kind of structure which underlies many triangulated categories is that of a stable infinity category as investigated by Lurie in =-=[Lur06]-=-. The appropriate universal property of the infinity category of spectra is established in [Lur06, Cor. 17.6], so again every object of the homotopy category of any stable, presentable infinity catego... |

1 | Y.I.: Methods of Homological Algebra, 2nd edn - Gelfand, Manin - 2003 |

1 |
Dérivateurs, manuscript, around 1990, partially available from http://www.math.jussieu.fr/~maltsin/groth/Derivateurs.html
- Grothendieck
(Show Context)
Citation Context ...then for some (hence any) exact triangle (10) we have 2 · 1C = 0; so the zero map is a Hopf map. Another example of such extra structure is the notion of a triangulated derivator, due to Grothendieck =-=[Gro90]-=-, and the closely related notions of a stable homotopy theory in the sense of Heller [Hel88, Hel97] or a system of triangulated diagram categories in the sense of Franke [Fra96]. In each of these sett... |