## Interpolation categories for homology theories (2004)

### BibTeX

@MISC{Biedermann04interpolationcategories,

author = {Georg Biedermann},

title = {Interpolation categories for homology theories},

year = {2004}

}

### OpenURL

### Abstract

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by proving the existence of truncated versions of resolution or E2-model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As application we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.

### Citations

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Citation Context ...t Therefore we get by adjointness (X • ⊗ int K) n := X n ⊗M K map int (X • , Y • )n := HomcM(X • ⊗ int ∆ n , Y • ) as well as hom int (K, X • ) n := homM(K, X n ). The following theorem was proved in =-=[Ree74]-=-, see also [GJ99, VII.2.12.] and [Hir03, 16.3.4.]. Theorem 1.3.3 The category cM together with the Reedy structure becomes a model category. It becomes a simplicial model structure if we provide it wi... |

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

144 | Stable homotopy and generalised homology - Adams - 1974 |

89 |
Model categories and their localizations, volume 99
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Citation Context ...nd not the full result on the existence of interpolation categories. 2. Resolution model structures Let M be a model category. Let cM be the category of cosimplicial objects over M. We refer to [18], =-=[19]-=- or [21] for the necessary background, in particular forINTERPOLATION CATEGORIES 3 the internal simplicial structure, which is compatible with the Reedy structure, and for latching- and matching obje... |

87 | Simplicial homotopy theory, volume 174 - Goerss, Jardine - 1999 |

69 |
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Citation Context ...ies The notion of a model category was introduced in [Qui67]. Today there are several axiom systems, but they differ just in minor technical details. In general we refer the reader to [DS95], [GJ99], =-=[Hov99]-=- and [Hir03] for the theory of model categories and for simplicial techniques in homotopy theory. Definition 1.1.1 When we speak of a model category we mean a category together with three subcategorie... |

54 |
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(Show Context)
Citation Context ...en them, that induces an isomorphism via F, it makes no sense to keep them separate. In homotopy theory this process is known as Bousfield localization and is available for all homology theories, see =-=[Bou79]-=- and [Hir03]. This process supplies a new stable model structure on M whose weak equivalences are exactly those morphisms that induce isomorphisms via F. We replace T by the homotopy category with res... |

53 |
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(Show Context)
Citation Context ....1 Model categories The notion of a model category was introduced in [Qui67]. Today there are several axiom systems, but they differ just in minor technical details. In general we refer the reader to =-=[DS95]-=-, [GJ99], [Hov99] and [Hir03] for the theory of model categories and for simplicial techniques in homotopy theory. Definition 1.1.1 When we speak of a model category we mean a category together with t... |

50 |
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Citation Context ... (Fibs+1X • ) → G ∗ (FibsX • ) coincides up to sign with the boundary of the normalized cochain complex N • (G ∗ X • ). Hence: E s 2 = πs[X • , G] =⇒ colim [ Tots X s • , G] (3.3) Theorem 6.1(a) from =-=[Boa99]-=- states that the convergence of this spectral se= 0. Problems arise now when we try quences is strong if and only if lim1 r E∗ r to relate the target with the term HomA(π0FX • , FG). Let us formulate ... |

35 |
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(Show Context)
Citation Context .... In the case of F = E∗ given by a suitable ring spectrum E this will be the E-nilpotent completion, which is proved in [Bie04]. The answer we can offer is that, whenever the convergence results from =-=[Bou87]-=- apply or the nilpotent completion is known to be the localization, we can derive the existence of X in T . Later, when it comes to realization questions, we will simply assume that the injective dime... |

35 |
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(Show Context)
Citation Context ...omotopy class. A realization along this functor is a strictification of homotopy commutative diagram. There is the Dwyer-Kan-Smith obstruction calculus for realizing diagrams described in [DK84b] and =-=[DKS89]-=-. We think, that one can get back this calculus from our interpolation categories at least in the stable case. Since the abstract theory on (n-)G-structures is so pure, everything also works for the s... |

29 | Invertible spectra in the E(n)-local stable homotopy category
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(Show Context)
Citation Context ... case where we have a class of projective models instead of injective ones. In fact all the other articles considering resolution model structures except of [Bou03] were written simplicially. Compare =-=[Jar04]-=-. Examples that qualify as a homological functor for our setting include the homology theories induced by every ring spectrum satisfying an Adams-Atiyah condition given in [Ada74, 15.1.], see 3.1.7. A... |

28 | Uniqueness theorems for certain triangulated categories with an Adams spectral sequence
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(Show Context)
Citation Context ...ssical one uses relative injective resolutions. We explain the connection with our constructions in 3.2.30. Accounts of the modified Adams spectral sequence are given in [Bri68], [Bou85], [Dev97] and =-=[Fra96]-=-. Returning to our realization process it is conceptually and technically easier to start with already strict cosimplicial objects over M and search for those that look like a constant cosimplicial ob... |

23 |
An E2 model category structure for pointed simplicial spaces
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Citation Context ...e for our purpose. They depend on the additional data of a class of injective models in M, which in our case is supplied by the class of F-injectives. The first resolution model structure appeared in =-=[DKS93]-=- and was used to study the realization problem for the homotopy group functor on spaces. This was pursued further in [DKS95] and [BDG01]. Another resolution model structure was considered in [GH04] to... |

22 |
Cosimplicial resolutions and homotopy spectral sequences in model categories
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(Show Context)
Citation Context ...are looking for a model structure on cM with precisely these maps as weak equivalences. Luckily a very general formulation of such a model structure was found just in time for us in the elegant paper =-=[Bou03]-=-. The model structures constructed there have the very suggestive name “resolution model structures” and, indeed, they perfectly serve for our purpose. They depend on the additional data of a class of... |

18 |
Combinatorial foundation of homology and homotopy
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(Show Context)
Citation Context ...eveals more information about homological functors, at least when the injective dimension of the target category is finite and probably small. There is a similar theory of interpolation categories in =-=[Bau99]-=-, but at the moment we do not understand the relationship between the different approaches. We remark that our results dualize. This means that the whole theory of truncated model structures also work... |

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(Show Context)
Citation Context ... F-injectives. The first resolution model structure appeared in [DKS93] and was used to study the realization problem for the homotopy group functor on spaces. This was pursued further in [DKS95] and =-=[BDG01]-=-. Another resolution model structure was considered in [GH04] to study the existence of A∞- or E∞-structures on ring spectra. All these model structures are exhibited as a special case of the general ... |

18 | Realizability of modules over Tate cohomology
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(Show Context)
Citation Context ... compared to [5] result from the use of truncated resolution model structures. An obstruction calculus for realizing objects using only the triangulated structure is described, among other things, in =-=[2]-=-. We apologize in advance for using so much notation, but it seems unavoidable. Our task was to look out for realizations in Ho(M) = T of objects in A. To motivate our next definition, let X be an obj... |

17 | M.J.Hopkins. Moduli spaces of commutative ring spectra
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(Show Context)
Citation Context ...nd the algebraic target category. The obstruction groups will lie in the E2-term of the F-based Adams spectral sequence. The way, we set up the obstruction calculus, follows the philosophy of [5] and =-=[17]-=-. We use injective resolutions in A and try to realize them step by step as cosimplicial objects over M, where M is a stable model category having T as its homotopy category. As we go along by gluing ... |

17 |
Homotopical algebra”, Lecture Notes in Mathematics No. 43, Springer-Verlag 1967. 203 1] ”Revêtements étales et groupes fondamental
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(Show Context)
Citation Context ... I-structure, where the class I of injective objects in A is taken as a class of injective models. It follows from [7, 4.4] that this model structure corresponds to the classical model structure from =-=[24]-=- for the nonnegative cochain complexes CoCh ≥0 (A) via the Dold-Kan correspondence. So in CoCh ≥0 (A) we have: The I-equivalences are the cohomology equivalences, the I-cofibrations are the maps that ... |

14 |
On the telescopic homotopy theory of spaces
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(Show Context)
Citation Context ...ral homotopy groups just up to degree n. We will obtain these structures by employing co-Q-structures, the dual concept of Q-structures which were invented in [BF78] and were considerably improved in =-=[Bou00]-=-. We also would like to point out, that this truncation process does not work for the other type of homotopy groups which the author had to learn painfully while struggling for the right constructions... |

13 | Homotopy theory of comodules over a Hopf algebroid. In Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, volume
- Hovey
- 2004
(Show Context)
Citation Context ...duces a homological functor E∗ : Ho(Spectra)E → E∗E − comod from the E-local stable homotopy category to the category of E∗E-comodules This functor possesses enough injectives. This is proved e.g. in =-=[Hov04]-=-. It also detects isomorphism, since we localized at E. From this data we can construct a spectral sequence, which is known as the E-based modified Adams-spectral sequence. Remark 3.1.8 For our functo... |

12 |
On the homotopy theory of K-local spectra at an odd
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- 1985
(Show Context)
Citation Context ...lutions while the classical one uses relative injective resolutions. We explain the connection with our constructions in 3.2.30. Accounts of the modified Adams spectral sequence are given in [Bri68], =-=[Bou85]-=-, [Dev97] and [Fra96]. Returning to our realization process it is conceptually and technically easier to start with already strict cosimplicial objects over M and search for those that look like a con... |

12 |
The bigraded homotopy groups πi,jX of a pointed simplicial space X
- Dwyer, Kan, et al.
(Show Context)
Citation Context ...ucture on M is simplicial. We review in subsection 2.1 the relevant definitions from [7] on resolution model structures. In subsection 2.2 we give a dualized account of the spiral exact sequence from =-=[13]-=-. In the last subsection 2.3 we explain the truncated model structures from [3]. 2.1. The G-structure on cM. The following definitions are taken from [7] who gave the definitive treatment on resolutio... |

9 |
An obstruction theory for diagrams of simplicial sets
- Dwyer, Kan
- 1984
(Show Context)
Citation Context ...M to their homotopy class. A realization along this functor is a strictification of homotopy commutative diagram. There is the Dwyer-Kan-Smith obstruction calculus for realizing diagrams described in =-=[DK84b]-=- and [DKS89]. We think, that one can get back this calculus from our interpolation categories at least in the stable case. Since the abstract theory on (n-)G-structures is so pure, everything also wor... |

8 |
Morava modules and Brown-Comenetz duality
- Devinatz
- 1997
(Show Context)
Citation Context ...hile the classical one uses relative injective resolutions. We explain the connection with our constructions in 3.2.30. Accounts of the modified Adams spectral sequence are given in [Bri68], [Bou85], =-=[Dev97]-=- and [Fra96]. Returning to our realization process it is conceptually and technically easier to start with already strict cosimplicial objects over M and search for those that look like a constant cos... |

7 |
Constructions of elements in Picard groups
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- 1992
(Show Context)
Citation Context ...invertible objects inherit an abelian group structure which we will call the Picard group Pic(C). It is an abelian group, but sometimes in a higher universe! It was defined by Hopkins and we refer to =-=[HMS94]-=-, where it is proved, that the Picard group of the whole stable homotopy category of spectra is�. There are also computations involving the Picard group of the K(n)-local category, where K(n) denotes ... |

4 |
Relative homological algebra and the Adams spectral sequence
- Brinkmann
- 1968
(Show Context)
Citation Context ...d Adams spectral sequence is constructed in the same way as the original Adams spectral sequence, but it uses absolute injective resolutions instead of relative ones. It was 36�� �� �� introduced in =-=[Bri68]-=-. Other accounts are given in [Bou85], [Dev97] and [Fra96]. It can be considered as the Bousfield-Kan spectral sequence of the simplicial space map pro (X, Y • ). The E1-term is given by E s,t 1 = π0m... |

1 | Injective completion with respect to homology. arXiv:math.AT/0412387
- Biedermann
- 2004
(Show Context)
Citation Context ...letion of the initial object, which may not coincide with the (localization of the) object. In the case of F = E∗ given by a suitable ring spectrum E this will be the E-completion, which is proved in =-=[4]-=-. The answer we can offer is that, whenever the convergence results from [9] apply or the completion is known to be isomorphic to the localization, we can derive the existence of X in T . Later, when ... |

1 |
Realizability of modules over Tate cohomology. to appear
- Benson, Krause, et al.
- 2004
(Show Context)
Citation Context ... this are referred to as Massey products or Toda brackets. There is an easy account of such an obstruction calculus for realizing objects using just the triangulated structure of T in the appendix of =-=[BKS04]-=-. Carrying out our realization process means, that we start to strictify our up-to-homotopy-cosimplicial objects. Here we view cosimplicial objects as resolutions of objects in the underlying category... |

1 | A classification for diagrams of simplicial sets - Dwyer, Kan - 1984 |

1 |
Resolutions in Model Categories. unpublished. P.G. Goerss and M.J. Hopkins. Moduli spaces of commutative ring spectra. http://www.math.nwu.edu∼pgoerss, to appear in
- Goerss, Hopkins
- 2004
(Show Context)
Citation Context ...t induces an isomorphism via F, it makes no sense to keep them separate. In homotopy theory this process is known as Bousfield localization and is available for all homology theories, see [Bou79] and =-=[Hir03]-=-. This process supplies a new stable model structure on M whose weak equivalences are exactly those morphisms that induce isomorphisms via F. We replace T by the homotopy category with respect to this... |

1 |
Truncated resolution model structures. to appear in
- Biedermann
(Show Context)
Citation Context ... and used for exactly this purpose. Recently Bousfield in [7] has given a very general and elegant treatment of resolution model structures, which exhibits all previous instances as special cases. In =-=[3]-=- I truncate these resolution model structures and this article is based on the results there. Weak equivalences are now given by maps, which induce isomorphisms on cohomology just up to degree n, 2.7.... |