## G G GGG (2003)

### BibTeX

@MISC{Ggg03gg,

author = {G G G Ggg and T T Tttt and Ak Bousfield},

title = {G G GGG},

year = {2003}

}

### OpenURL

### Abstract

Cosimplicial resolutions and homotopy spectral sequences in model categories

### Citations

266 | Categories for the Working Mathematician, Graduate Texts - Lane - 1971 |

222 | Model Categories
- Hovey
- 1999
(Show Context)
Citation Context ...egory with three classes of maps called weak equivalences, cofibrations, and fibrations, satisfying the usual axioms labeled MC1–MC5 in [25, pages 83–84]. We refer the reader to [25], [29], [30], and =-=[31]-=- for good recent treatments of model categories. A model category is called bicomplete when it is closed under all small limits and colimits. It is called factored when the factorizations provided by ... |

164 |
Homotopy theory of Γ-spaces, spectra, and bisimplicial sets
- Bousfield, Friedlander
- 1978
(Show Context)
Citation Context ... if C and D are factored and H is functorial, then G is also functorial and hence c C G and c D H are factored. 4.9 Another homotopical example Let Sp be the model category of spectra in the sense of =-=[17]-=- (see also [32]), and let Ho s = Ho(Sp) be the stable homotopy category. The infinite suspension and 0-space functors S∗ ⇆ Sp are Quillen adjoints, and their total derived functors are the usual infin... |

135 |
Homotopy theories and model categories, in Handbook of Algebraic Topology
- Dwyer, Spalinski
- 1995
(Show Context)
Citation Context ...This consists of a category with three classes of maps called weak equivalences, cofibrations, and fibrations, satisfying the usual axioms labeled MC1–MC5 in [25, pages 83–84]. We refer the reader to =-=[25]-=-, [29], [30], and [31] for good recent treatments of model categories. A model category is called bicomplete when it is closed under all small limits and colimits. It is called factored when the facto... |

69 |
The localization of spectra with respect to homology
- Bousfield
- 1979
(Show Context)
Citation Context ...to the S/p∗–localization and, when A is nilpotent, is equivalent to the p–completion (Z/p)∞A of [18]. For a spectrum E, we likewise let Ê = E S/p be the p–completion given by the S/p∗–localization of =-=[10]-=-. Thus, when the groups π∗E are finitely generated, we have π∗Ê = π∗E ⊗ ˆ Zp using the p–adic integers ˆ Zp. We now introduce the following: 11.2 The p–adic K–completion The triple on Ho∗ carrying a s... |

46 |
Homologie nicht-additiver Funktoren
- Dold, Puppe
- 1961
(Show Context)
Citation Context ...general, a cosimplicial G–injective resolution, or G–resolution, of an object A ∈ C consists of a trivial cofibration A → Ā• to a fibrant target Ā• in c CG . By applying the constructions of [18] and =-=[21]-=- to G–resolutions, we obtain right derived functors Rs GT(A) = πsT( Ā• ), G–completions ˆ LGA = Tot Ā• , and G– homotopy spectral sequences {E s,t r (A;M)G}r≥2 = {E s,t r ( Ā• ;M)}r≥2 abutting to [M, ... |

41 | Stable homotopy and generalized homology - Adams - 1974 |

36 |
The localization of spaces with respect to homology. Topology 14
- Bousfield
- 1975
(Show Context)
Citation Context ...ike U(M) structures as in [13]. We first recall the following: 11.1 The p–completion of a space or spectrum For a space A ∈ S∗, we let Â = A H/p be the p–completion given by the H/p∗– localization of =-=[9]-=-. This is equivalent to the S/p∗–localization and, when A is nilpotent, is equivalent to the p–completion (Z/p)∞A of [18]. For a spectrum E, we likewise let Ê = E S/p be the p–completion given by the ... |

34 |
Homotopy spectral sequences and obstructions
- Bousfield
- 1989
(Show Context)
Citation Context ...rning the natural inclusions E s,t ∞+ (X•;M) ⊂ E s,t ∞ (X•;M) where E s,t ∞+ (X•;M) denotes the kernel of Qsπt−s(Tot X• ;M) → Qs−1πt−s(Tot X• ;M) and where E s,t ∞ (X•;M) = ⋂ r>s Es,t r (X•;M). As in =-=[11]-=-, the spectral sequence may be partially extended beyond the t ≥ s ≥ 0 sector, and there is an associated obstruction theory. Finally, in preparation for our work on resolution model categories, we co... |

33 |
Foundations of relative homological algebra
- Eilenberg, Moore
- 1965
(Show Context)
Citation Context ...om( Ã• ,G) → Hom(A,G) is acyclic for each G ∈ G. When T is additive, we have Rs GT(A) = HsT Ã• for s ≥ 0, and we recover the usual right derived functors Rs GT : C → M of relative homological algebra =-=[26]-=-. In general, we obtain relative versions of the Dold–Puppe [21] derived functors. Now suppose that the model category C is simplicial and bicomplete. Geometry & Topology, Volume 7 (2003)1024 A K Bou... |

29 |
An E2 model category structure for pointed simplicial spaces
- Dwyer, Kan, et al.
- 1993
(Show Context)
Citation Context ...eeded because the original chain level constructions of pairings and products [20] do not readily extend to that setting. We rely heavily on a generalized cosimplicial version of the Dwyer–Kan–Stover =-=[24]-=- theory of resolution model categories (or E2 model categories in their parlance). This provides a simplicial model category structure c C G on the category c C of cosimplicial objects over a left pro... |

28 |
Steenrod operations in the Eilenberg-Moore spectral sequence
- Rector
- 1970
(Show Context)
Citation Context ...Bousfield 10.8 The geometric cobar construction Let B(A,Λ,B) • ∈ c C be the usual geometric cobar construction with B(A,Λ,B) n = A × Λ × · · · × Λ × B for n ≥ 0 where the factor Λ occurs n times (see =-=[40]-=-). It is straightforward to show that B(A,Λ,B) • is Reedy fibrant with Tot B(A,Λ,B) • ∼ = P(A,Λ,B) where P(A,Λ,B) is the double mapping path object defined by the pullback P(A,Λ,B) −−−−→ hom(∆1 ,Λ) ⏐ ... |

21 | Homotopy theory of model categories
- Reedy
- 1974
(Show Context)
Citation Context ...omotopy spectral sequences of cosimplicial objects in model categories, thereby generalizing the constructions of Bousfield–Kan [18] for cosimplicial spaces. This generalization is mainly due to Reedy=-=[41]-=-, but we offer some details to establish notation and terminology. We first consider the following: Geometry & Topology, Volume 7 (2003)Cosimplicial resolutions and homotopy spectral sequences 1005 2... |

18 |
J.Beck, Homology and standard constructions
- Barr
- 1969
(Show Context)
Citation Context ... α ∗ : [Γ • A,Ω t G] → [A,Ω t G] in S for each G ∈ G and t ≥ 0. This follows by Lemma 7.2 since 〈Γ,η,µ〉 gives a triple on Ho C such that each Ω t G is Γ–injective. Various authors including Barr–Beck =-=[2]-=-, Bousfield–Kan [18], and Bendersky– Thompson [7] have used triple resolutions to define right derived functors, completions, or homotopy spectral sequences, and we can now fit these constructions int... |

14 | On the telescopic homotopy theory of spaces - Bousfield |

12 |
The unstable Adams spectral sequence for generalized homology, Topology 17
- Bendersky, Curtis, et al.
- 1978
(Show Context)
Citation Context ...ield 1 Introduction In [18] and [19], Bousfield–Kan developed unstable Adams spectral sequences and completions of spaces with respect to a ring, and this work was extended by Bendersky–Curtis–Miller =-=[3]-=- and Bendersky–Thompson [7] to allow a ring spectrum in place of a ring. In the present work, we develop a much more general theory of cosimplicial resolutions, homotopy spectral sequences, and comple... |

11 |
The Bous spectral sequence for periodic homology theories
- Bendersky, Thompson
(Show Context)
Citation Context ... and [19], Bousfield–Kan developed unstable Adams spectral sequences and completions of spaces with respect to a ring, and this work was extended by Bendersky–Curtis–Miller [3] and Bendersky–Thompson =-=[7]-=- to allow a ring spectrum in place of a ring. In the present work, we develop a much more general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model ... |

11 |
An arithmetic square for virtually nilpotent spaces
- Dror, Dwyer, et al.
(Show Context)
Citation Context ...f Theorem 11.4 For 0 ≤ s ≤ ∞, we obtain a homotopy fiber square Tot s (K • A) −−−−→ Tot s (̂K • A) ⏐ ↓ ⏐ ↓ Tot s(K • A) (0) −−−−→ Tot s(̂K • A) (0) by applying Tot s to the termwise arithmetic square =-=[23]-=- of K • A. Since the lower spaces of the square are HQ∗–local [9, page 192], the upper map has an HQ∗–local homotopy fiber and induces an equivalence map ∗(M,Tot s(K • A)) ≃ map ∗(M,Tot s(̂K • A)) Thu... |

9 | The K-theory localizations and v1-periodic homotopy groups of H-spaces, Topology 38 - Bousfield - 1999 |

8 | The homotopy spectral sequence of a space with coefficients in a ring, Topology 11 - BOUSFIELD, KAN - 1972 |

8 | Localization with respect to K-theory - Mislin - 1977 |

7 |
Localization of model categories. Preprint, available at http://www-math.mit.edu/ psh
- Hirschhorn
- 1997
(Show Context)
Citation Context ...s of a category with three classes of maps called weak equivalences, cofibrations, and fibrations, satisfying the usual axioms labeled MC1–MC5 in [25, pages 83–84]. We refer the reader to [25], [29], =-=[30]-=-, and [31] for good recent treatments of model categories. A model category is called bicomplete when it is closed under all small limits and colimits. It is called factored when the factorizations pr... |

6 |
Pairings and products in the homotopy spectral sequence
- Bous, Kan
- 1973
(Show Context)
Citation Context ...ngs, this provides a flexible approach to the Bendersky–Thompson spectral sequences and completions, which is especially needed because the original chain level constructions of pairings and products =-=[20]-=- do not readily extend to that setting. We rely heavily on a generalized cosimplicial version of the Dwyer–Kan–Stover [24] theory of resolution model categories (or E2 model categories in their parlan... |

5 |
On the coalgebraic ring and Bousfield–Kan spectral sequence for a Landweber exact spectrum
- Bendersky, Hunton
(Show Context)
Citation Context ...spectral sequence {E s,t r (A;M)E} = {E s,t r (A;M)G} over an arbitrary ring spectrum E which need not be an S–algebra. As pointed out by Dror Farjoun [22, page 36], Libman [34], and Bendersky–Hunton =-=[6]-=-, this generality can also be achieved by using restricted cosimplicial E–resolutions without codegeneracies. However, we believe that codegeneracies remain valuable; for instance, they are essential ... |

5 |
On p-adic λ-rings and the K-theory of H-spaces
- Bousfield
- 1996
(Show Context)
Citation Context ... properties. For instance, the p–adic K– homotopy spectral sequences seem especially applicable to spaces whose p–adic K–cohomologies are torsion-free with Steenrod–Epstein-like U(M) structures as in =-=[13]-=-. In much of this work, for simplicity, we assume that our model categories are pointed. However, as in [28], this assumption can usually be eliminated, and we offer a brief account of the unpointed t... |

4 | A stable approach to an unstable homotopy spectral sequence
- Bendersky, Davis
(Show Context)
Citation Context ...in the present topics, we were prompted to formulate this theory by Martin Bendersky and Don Davis who are using Geometry & Topology, Volume 7 (2003)1004 A K Bousfield some of our results in [4] and =-=[5]-=-, and we thank them for their questions and comments. We also thank Assaf Libman for his suggestions and thank the organizers of BCAT 2002 for the opportunity to present this work. Throughout, we assu... |

3 | Compositions in the v1 {periodic homotopy groups of spheres, Forum Math
- Bendersky, Davis
(Show Context)
Citation Context ...erested in the present topics, we were prompted to formulate this theory by Martin Bendersky and Don Davis who are using Geometry & Topology, Volume 7 (2003)1004 A K Bousfield some of our results in =-=[4]-=- and [5], and we thank them for their questions and comments. We also thank Assaf Libman for his suggestions and thank the organizers of BCAT 2002 for the opportunity to present this work. Throughout,... |

3 | E2 model structures for presheaf categories
- Jardine
(Show Context)
Citation Context ...eorems 3.3 and 12.4). Of course, our cosimplicial statements have immediate simplicial duals. Other more specialized versions of the simplicial theory are developed by Goerss–Hopkins [28] and Jardine =-=[33]-=- using small object arguments which are not applicable in the duals of many familiar model categories. When C is discrete, our version reduces to a variant of Quillen’s model category structure [39, I... |

3 |
Universal spaces for homotopy limits of modules over coaugmented functors
- Libman
(Show Context)
Citation Context ... the accompanying homotopy spectral sequence {E s,t r (A;M)E} = {E s,t r (A;M)G} over an arbitrary ring spectrum E which need not be an S–algebra. As pointed out by Dror Farjoun [22, page 36], Libman =-=[34]-=-, and Bendersky–Hunton [6], this generality can also be achieved by using restricted cosimplicial E–resolutions without codegeneracies. However, we believe that codegeneracies remain valuable; for ins... |

2 |
The core of a ring
- Bousfield, Kan
- 1972
(Show Context)
Citation Context ..., and suppose the ring π0E is commutative. Let R = core(π0E) be the subring R = {r ∈ π0E | r ⊗ 1 = 1 ⊗ r ∈ π0E ⊗ π0E}, and recall that R is solid (ie, the multiplication R⊗R → R is an isomorphism) by =-=[8]-=-. Geometry & Topology, Volume 7 (2003)Cosimplicial resolutions and homotopy spectral sequences 1039 Theorem 9.7 If E is a connective ring spectrum with commutative π0E, then there are natural equival... |

2 | Two completion towers for generalized homology - Farjoun |

2 |
Homotopy limits of triples, preprint
- Libman
(Show Context)
Citation Context ...tual summetric group actions on the n-fold composites R · · · R for n ≥ 3. The partial failure of our triple lemma in [18] does not seem to invalidate any of our other results, and new work of Libman =-=[35]-=- on homotopy limits for coaugmented functors shows that the functors Rs must all still belong to triples on the homotopy category Ho∗. Geometry & Topology, Volume 7 (2003)Cosimplicial resolutions and... |

1 | On λ–rings and the K–theory of infinite loop spaces, K– Theory 10 - Bousfield - 1996 |

1 |
Pairings of homotopy spectral sequences in model categories
- Bousfield
(Show Context)
Citation Context ...ever, as in [28], this assumption can usually be eliminated, and we offer a brief account of the unpointed theory in Section 12. We thank Paul Goerss for suggesting such a generalization. In a sequel =-=[16]-=-, we develop composition pairings for our homotopy spectral sequences and discuss the E2–terms from the standpoint of homological algebra. This extends the work of [20], replacing the original chain-l... |