## Monoidal uniqueness of stable homotopy theory (2001)

Venue: | Adv. in Math. 160 |

Citations: | 11 - 7 self |

### BibTeX

@INPROCEEDINGS{Shipley01monoidaluniqueness,

author = {Brooke Shipley},

title = {Monoidal uniqueness of stable homotopy theory},

booktitle = {Adv. in Math. 160},

year = {2001},

pages = {217--240}

}

### OpenURL

### Abstract

Abstract. We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlying model structures Margolis ’ axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, W-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. The equivalences of modules, algebras and commutative algebras in these categories are also considered. 1.

### Citations

327 | Homotopical algebra - Quillen - 1967 |

307 | Homotopy limits, completions and localizations - Bousfield, Kan - 1972 |

236 | Categories for the working mathematician. Graduate Texts - Lane - 1998 |

202 | Model Categories - Hovey - 1999 |

191 | Symmetric spectra
- Hovey, Shipley, et al.
(Show Context)
Citation Context ...roduct) induced by the smash product [1, 26]. Recently, several categories of spectra have been constructed which have symmetric monoidal smash products even before the weak equivalences are inverted =-=[6, 10, 11, 14]-=-. Such categories are of interest because they facilitate the development of algebraic constructions such as ring spectra and module spectra. In each of these examples, inverting the weak equivalences... |

152 |
Des Catégories Dérivées des Categories Abéliennes, Asterisque 239
- Verdier
- 1996
(Show Context)
Citation Context ...n the homotopy category are inverse equivalences. Certain extra structures on the homotopy category of a stable model category are key here. The homotopy category is naturally a triangulated category =-=[25]-=-. The suspension functor defines the shift functor and the cofiber sequences of [16, Chapter I §3] define the distinguished triangles (the fiber sequences agree up to sign [8, Theorem 7.1.11]); see [8... |

145 | Algebras and modules in monoidal model categories
- Schwede, Shipley
(Show Context)
Citation Context ... Chapter II §2]. In particular, the product on a monoidal model category induces a derived product on the homotopy category which is symmetric monoidal. Monoidal model categories have been studied in =-=[20]-=- and [8]. Here, instead of requiring a closed monoidal structure, we use the weaker hypotheses that the product commutes with colimits. Definition 2.1. A model category C is a monoidal model category ... |

145 |
modules, and algebras in stable homotopy theory
- Elmendorf, Kriz, et al.
- 1997
(Show Context)
Citation Context ...roduct) induced by the smash product [1, 26]. Recently, several categories of spectra have been constructed which have symmetric monoidal smash products even before the weak equivalences are inverted =-=[6, 10, 11, 14]-=-. Such categories are of interest because they facilitate the development of algebraic constructions such as ring spectra and module spectra. In each of these examples, inverting the weak equivalences... |

143 |
Model categories, Mathematical surveys and monographs
- Hovey
- 1998
(Show Context)
Citation Context ...II §2]. In particular, the product on a monoidal model category induces a derived product on the homotopy category which is symmetric monoidal. Monoidal model categories have been studied in [20] and =-=[8]-=-. Here, instead of requiring a closed monoidal structure, we use the weaker hypotheses that the product commutes with colimits. Definition 2.1. A model category C is a monoidal model category if it is... |

114 | Model categories of diagram spectra
- Mandell, May, et al.
(Show Context)
Citation Context ...roduct) induced by the smash product [1, 26]. Recently, several categories of spectra have been constructed which have symmetric monoidal smash products even before the weak equivalences are inverted =-=[6, 10, 11, 14]-=-. Such categories are of interest because they facilitate the development of algebraic constructions such as ring spectra and module spectra. In each of these examples, inverting the weak equivalences... |

112 | Axiomatic stable homotopy theory - Hovey, Palmieri, et al. - 1997 |

61 | Equivariant orthogonal spectra and S-modules - Mandell, May |

58 |
Function complexes in homotopical algebra, Topology 19
- Dwyer, Kan
- 1980
(Show Context)
Citation Context ...een Ho(C) and Ho(SpΣ ). 2. I, is a small weak generator and [I, I] Ho(C) ∗ To construct the right adjoint Hom(I, −) we use cosimplicial resolutions since C is not simplicial. These were first used in =-=[5]-=- to construct function complexes on homotopy categories, but in [4] this theory has been extended to provide function complexes on model categories. Our main reference here is [8, Chapter 5], see also... |

29 | Simplicial functors and stable homotopy theory, preprint
- Lydakis
- 1998
(Show Context)
Citation Context ...in C. A cofibrant desuspension of the unit exists in every known symmetric monoidal model category of spectra. In the diagram categories of spectra investigated in [14] and their simplicial analogues =-=[10, 11]-=-, (orthogonal spectra, symmetric spectra, and simplicial functors or W -spaces) the cofibrant desuspension can be chosen as the object denoted F1S 0 , with the weak equivalence η: F1S 1 −→ F0S 0 ; see... |

18 | Replacing model categories with simplicial ones - Dugger |

18 | Homotopy theory of model categories
- Reedy
- 1973
(Show Context)
Citation Context ...(X · ⊗ K) ⊗ L since they both represent the same functor. In particular, Σ m (X · ) is the mth iterated suspension of X · . We consider the Reedy model category on C ∆ , the cosimplicial objects on C =-=[17]-=-, [8, Theorem 5.2.5]. An object A · is Reedy cofibrant if the map A · ⊗ ∂∆[k]+ −→ A · ⊗ ∆[k]+ ∼ = A k is a cofibration for each k. A cosimplicial resolution is then a Reedy cofibrant object of C ∆ suc... |

16 | Simplicial structures on model categories and functors
- Rezk, Schwede, et al.
(Show Context)
Citation Context ...ngation of the product commutes with the simplicial action. Using [7, Proposition 16.11.1, Theorem 16.4.2], one can show that if C is a monoidal model category then the simplicial model category from =-=[18]-=- is also monoidal. Hence, under these conditions, one can apply the constructions in this section to the stable simplicial monoidal model category on C∆op. This remark can also be applied if C is simp... |

16 | Symmetric spectra and topological Hochschild homology, K-Theory 19 - Shipley |

14 | A uniqueness theorem for stable homotopy theory
- Schwede, Shipley
(Show Context)
Citation Context ...odules, orthogonal spectra, W -spaces, simplicial functors and symmetric spectra and the associated categories of modules and algebras. The conditions on the unit in Theorem 1.2 were first studied in =-=[21]-=-. There we considered the uniqueness of model categories of spectra but ignored the monoidal product structure. Theorems 1.1 and 1.2 give the most highly structured uniqueness properties of the monoid... |

13 |
S-modules and symmetric spectra
- Schwede
(Show Context)
Citation Context ...xtended to modules, algebras and commutative algebras. Remark 4.9 shows that the monoidal Quillen equivalences constructed in Theorems 4.7, 5.2 and 5.6 recover and unify those constructed in [14] and =-=[19]-=- between S-modules, orthogonal spectra, W -spaces, simplicial functors and symmetric spectra and the associated categories of modules and algebras. The conditions on the unit in Theorem 1.2 were first... |

12 | Spectra and the Steenrod Algebra. Modules over the Steenrod algebra and the stable homotopy category - Margolis - 1983 |

11 |
Stable homotopy theory” Mimeographed notes 1966–1970
- Boardman
(Show Context)
Citation Context ...ion The homotopy category of spectra, obtained by inverting the weak equivalences of spectra, has long been known to have a symmetric monoidal product (or tensor product) induced by the smash product =-=[1, 26]-=-. Recently, several categories of spectra have been constructed which have symmetric monoidal smash products even before the weak equivalences are inverted [6, 10, 11, 14]. Such categories are of inte... |

11 |
Model categories and general abstract homotopy theory, in preparation
- Dwyer, Hirschhorn, et al.
(Show Context)
Citation Context ...Ho(C) ∗ To construct the right adjoint Hom(I, −) we use cosimplicial resolutions since C is not simplicial. These were first used in [5] to construct function complexes on homotopy categories, but in =-=[4]-=- this theory has been extended to provide function complexes on model categories. Our main reference here is [8, Chapter 5], see also [21]. Given a cosimplicial object X · in C ∆ and a pointed simplic... |

11 | M.A.Mandell and J.P.May “Rings, modules, and algebras in stable homotopy theory.” With an appendix by M - Elmendorf - 1997 |

10 |
A first approximation to homotopy theory
- Spanier, Whitehead
- 1953
(Show Context)
Citation Context ...monogenic, monoidal, triangulated category S with an exact and strong symmetric monoidal equivalence6 BROOKE SHIPLEY R : SWf −→ S small between the Spanier-Whitehead category of finite CW-complexes (=-=[24]-=-, [15, Chapter 1, §2]) and the full subcategory of small objects in S . As shown in [21, Section 3], such an equivalence induces a π s ∗-linear structure on the triangulated category S . In fact, we c... |

6 | Classification of stable model categories - Schwede, Shipley |

5 |
Boardman’s stable homotopy category
- Vogt
- 1970
(Show Context)
Citation Context ...ion The homotopy category of spectra, obtained by inverting the weak equivalences of spectra, has long been known to have a symmetric monoidal product (or tensor product) induced by the smash product =-=[1, 26]-=-. Recently, several categories of spectra have been constructed which have symmetric monoidal smash products even before the weak equivalences are inverted [6, 10, 11, 14]. Such categories are of inte... |

4 | Localization, cellularization, and homotopy colimits, preprint - Hirschhorn |

3 | Classification of stable model categories, preprint - Schwede, Shipley - 2000 |

1 | Localization, cellularization, and homotopy colimits - Hirschhorn - 1999 |