## Classification of Stable Model Categories (0)

Citations: | 7 - 5 self |

### BibTeX

@TECHREPORT{Schwede_classificationof,

author = {Stefan Schwede and Brooke Shipley},

title = {Classification of Stable Model Categories},

institution = {},

year = {}

}

### OpenURL

### Abstract

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. 1.

### Citations

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327 |
Homotopical algebra
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- 1967
(Show Context)
Citation Context ...(over symmetric spectra) category theory with the language of closed model categories. We give specic references throughout; for general background on model categories see Quillen's original article [=-=45-=-], a modern introduction [12], or [21] for a more complete overview. We want to point out the conceptual similarities between the present paper and the work of Keller [31]. Keller uses dierential grad... |

265 | Simplicial objects in algebraic topology - May - 1967 |

197 | Locally Presentable and Accessible Categories - Adámek, Rosický - 1994 |

191 | Symmetric spectra
- Hovey, Shipley, et al.
(Show Context)
Citation Context ...ic monoidal and homotopically well behaved smash product before passing to the homotopy category. Thesrst examples of such categories were the S-modules of [15] and the symmetric spectra of Je Smith [=-=25]-=-; by now several more such categories have been constructed [35, 38]. We work with symmetric spectra because we can replace stable model categories by Quillen equivalent ones which are enriched over s... |

154 |
Des catégories dérivées des catégories abéliennes. Astérisque
- VERDIER
- 1996
(Show Context)
Citation Context ...KE SHIPLEY The homotopy category of a stable model category has a large amount of extra structure, some of which plays a role in this paper. First of all, it is naturally a triangulated category (cf. =-=[64-=-] or [24, A.1]). A complete reference for this fact can be found in [21, 7.1.6]; we sketch the constructions: by denition of `stable' the suspension functor is a self-equivalence of the homotopy categ... |

148 |
Homotopy theory of Γ-spaces, spectra, and bisimplicial sets
- Bousfield, Friedlander
- 1977
(Show Context)
Citation Context ...ategory of spectra in the sense of stable homotopy theory. The sphere spectrum is a compact generator. Many model categories of spectra have been constructed, for example by Bousfield and Friedlander =-=[BF78]; Robinson [Rob87a, -=-‘spectral sheaves’]; Jardine [Jar97, ‘n-fold spectra’]; Elmendorf, Kriz, Mandell and May [EKMM, ‘coordinate free spectra’, ‘L-spectra’, ‘S-modules’]; Hovey, Shipley and Smith [HSS,... |

145 | Algebras and modules in monoidal model categories - Schwede, Shipley |

144 |
modules, and algebras in stable homotopy theory
- Elmendorf, Kriz, et al.
- 1997
(Show Context)
Citation Context ...e category of spectra which admits a symmetric monoidal and homotopically well behaved smash product before passing to the homotopy category. Thesrst examples of such categories were the S-modules of =-=[15-=-] and the symmetric spectra of Je Smith [25]; by now several more such categories have been constructed [35, 38]. We work with symmetric spectra because we can replace stable model categories by Quill... |

143 |
Model categories, Mathematical surveys and monographs
- Hovey
- 1998
(Show Context)
Citation Context ...ory with the language of closed model categories. We give specic references throughout; for general background on model categories see Quillen's original article [45], a modern introduction [12], or [=-=21-=-] for a more complete overview. We want to point out the conceptual similarities between the present paper and the work of Keller [31]. Keller uses dierential graded categories to give an elegant refo... |

136 |
Morita theory for derived categories
- Rickard
- 1989
(Show Context)
Citation Context ...ete abelian categories with a single small projective generator; the classical Morita theory for equivalences between module categories (see for example [1, x21, 22]) follows from this. Later Rickard =-=[47, 48]-=- developed a Morita theory for derived categories based on the notion of a tilting complex. In this paper we carry this line of thought one step further. Spectra are the homotopy theoretical generaliz... |

133 |
Categories and cohomology theories, Topology 13
- Segal
- 1974
(Show Context)
Citation Context ... give a symmetric spectrum, which we denote F (S). In the situation at hand we thus have DX = QX(S). If we restrict the simplicial functor QX to the category op ofsnite pointed sets we obtain a-space =-=[59,-=- 4] denoted QX . Every-space can be prolonged to a simplicial functor dened on the category of pointed simplicial sets [4, x4]. Prolongation is left adjoint to the restriction functor , and we denote ... |

123 |
Homotopy Theories and Model Categories, Handbook of Algebraic Topology
- Dwyer, Spalinski
- 1995
(Show Context)
Citation Context ...egory theory with the language of closed model categories. We give specic references throughout; for general background on model categories see Quillen's original article [45], a modern introduction [=-=12-=-], or [21] for a more complete overview. We want to point out the conceptual similarities between the present paper and the work of Keller [31]. Keller uses dierential graded categories to give an ele... |

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

114 | Model categories of diagram spectra
- Mandell, May, et al.
(Show Context)
Citation Context ...passing to the homotopy category. Thesrst examples of such categories were the S-modules of [15] and the symmetric spectra of Je Smith [25]; by now several more such categories have been constructed [=-=35, 38]-=-. We work with symmetric spectra because we can replace stable model categories by Quillen equivalent ones which are enriched over symmetric spectra (Section 3.6). Also, symmetric spectra are reasonab... |

111 | The Grothendieck duality theorem via Bousfield's techniques and Brown .representability - Neeman - 1996 |

106 |
Equivariant Stable Homotopy Theory
- Jr, May, et al.
- 1986
(Show Context)
Citation Context ...ts', or spectral categories, see Denition 3.3.1 and Theorem A.1.1. (iii) Equivariant stable homotopy theory. If G is a compact Lie group, there is a category of G-equivariant coordinate free spectra [=-=34]-=- which is a stable model category. Modern versions of this model category are the G-equivariant orthogonal spectra of [37] and Gequivariant S-modules of [15]. In this case the equivariant suspension s... |

79 |
Derived equivalences as derived functors
- Rickard
(Show Context)
Citation Context ...ete abelian categories with a single small projective generator; the classical Morita theory for equivalences between module categories (see for example [1, x21, 22]) follows from this. Later Rickard =-=[47, 48]-=- developed a Morita theory for derived categories based on the notion of a tilting complex. In this paper we carry this line of thought one step further. Spectra are the homotopy theoretical generaliz... |

78 | On the groups of H(Π, n - Eilenberg, Lane - 1953 |

76 | Motivic symmetric spectra
- Jardine
(Show Context)
Citation Context ...s that theories like algebraic K-theory or motivic cohomology are represented by objects in this stable homotopy category [Voe98, Sec. 6], at least when the base scheme is the spectrum of a field. In =-=[Jar00b]-=-, Jardine provides the details of the construction of model categories of Tspectra over the spectrum of a field k. He constructs two Quillen equivalent proper, simplicial model categories of Bousfield... |

70 |
Deriving DG categories, Ann
- Keller
- 1994
(Show Context)
Citation Context ...ee Quillen's original article [45], a modern introduction [12], or [21] for a more complete overview. We want to point out the conceptual similarities between the present paper and the work of Keller =-=[31-=-]. Keller uses dierential graded categories to give an elegant reformulation (and generalization) of Rickard's results on derived equivalences for rings. Our approach is similar to Keller's, but where... |

69 | Homotopy limits in triangulated categories - Boekstedt, Neeman - 1993 |

64 | Morava K-theories and localisation - Hovey, Strickland - 1999 |

64 | Triangulated categories, Annals of Mathematics Studies - Neeman - 2001 |

63 | Equivariant orthogonal spectra and S-modules
- Mandell, May
- 2002
(Show Context)
Citation Context ...mpact Lie group, there is a category of G-equivariant coordinate free spectra [34] which is a stable model category. Modern versions of this model category are the G-equivariant orthogonal spectra of =-=[37-=-] and Gequivariant S-modules of [15]. In this case the equivariant suspension spectra of the coset spaces G=H+ for all closed subgroups H G form a set of compact generators. This equivariant model ca... |

55 | Homology of symmetric products and other functors of complexes - Dold - 1958 |

55 | Spectra and symmetric spectra in general model categories
- Hovey
(Show Context)
Citation Context ... rise to an associated stable model category by `inverting' the suspension functor, i.e., by passage to internal spectra. This has been carried out for certain simplicial model categories in [52] and =-=[2-=-3]. The construction of symmetric spectra over a model category (see Section 3.6) is another approach to stabilization. (vi) Bouseld localization. Following Bouseld [3], localized model structures for... |

53 |
Spectra in model categories and applications to the algebraic cotangent complex
- Schwede
- 1997
(Show Context)
Citation Context ...ould give rise to an associated stable model category by `inverting' the suspension functor, i.e., by passage to internal spectra. This has been carried out for certain simplicial model categories in =-=[5-=-2] and [23]. The construction of symmetric spectra over a model category (see Section 3.6) is another approach to stabilization. (vi) Bouseld localization. Following Bouseld [3], localized model struc... |

49 | The Milnor conjecture
- Voevodsky
- 1996
(Show Context)
Citation Context ...r schemes over a base. An associated stable homotopy category of A 1 -local T -spectra (where T = A 1 =(A 1 0) is the `Tate-sphere') is an important tool in Voevodsky's proof of the Milnor conjecture =-=[65]-=-. This stable homotopy category arises from a stable model category with a set of compact generators, see Example 3.4 (ii) for more details. 2.4. Examples: abelian stable model categories. Some exampl... |

42 |
Topological Hochschild homology
- Bokstedt
- 1985
(Show Context)
Citation Context ...e category of Gamma-rings (where it is easy to prove) to the category of symmetric ring spectra and extend it to the `multiple object case'. We use in a crucial way Bokstedt's hocolim I construction [=-=6]-=-. The functors M;E as well as an intermediate functor D all arise as lax monoidal functors from the category of symmetric spectra to itself, and the natural maps between them are monoidal transformati... |

39 |
Stable homotopy theory of simplicial presheaves
- Jardine
- 1987
(Show Context)
Citation Context ...osed subgroups H G form a set of compact generators. This equivariant model category is taken up again in Examples 3.4 (i) and 5.1.2. (iv) Presheaves of spectra. For every Grothendieck site Jardine [=-=27-=-] constructs a proper, simplicial, stable model category of presheaves of Bouseld-Friedlander type spectra; the weak equivalences are the maps which induce isomorphisms of the associated sheaves of st... |

30 |
Bousfield, The localization of spaces with respect to homology
- K
(Show Context)
Citation Context ...l model categories in [52] and [23]. The construction of symmetric spectra over a model category (see Section 3.6) is another approach to stabilization. (vi) Bouseld localization. Following Bouseld [3=-=]-=-, localized model structures for modules over an S-algebra are constructed in [15, VIII 1.1]. Hirschhorn [20] shows that under quite general hypotheses the localization of a model category is again a ... |

29 | Simplicial functors and stable homotopy theory
- Lydakis
- 1998
(Show Context)
Citation Context ...passing to the homotopy category. Thesrst examples of such categories were the S-modules of [15] and the symmetric spectra of Je Smith [25]; by now several more such categories have been constructed [=-=35, 38]-=-. We work with symmetric spectra because we can replace stable model categories by Quillen equivalent ones which are enriched over symmetric spectra (Section 3.6). Also, symmetric spectra are reasonab... |

29 | Smash products and Γ-spaces - Lydakis - 1999 |

28 |
Localization of model categories
- Hirschhorn
- 1999
(Show Context)
Citation Context ...n 3.6) is another approach to stabilization. (vi) Bouseld localization. Following Bouseld [3], localized model structures for modules over an S-algebra are constructed in [15, VIII 1.1]. Hirschhorn [2=-=0]-=- shows that under quite general hypotheses the localization of a model category is again a model category. The localization of a stable model category is stable and localization preserves generators. ... |

28 |
Obstruction theory and the strict associativity of Morava K-theories
- Robinson
- 1988
(Show Context)
Citation Context ...ucture. The following example illustrates this point. Consider the n-th Morava K-theory spectrum K(n) for asxed prime and some number n > 0. This spectrum admits the structure of an A1 -ring spectrum =-=[51]-=-. Hence it also has a model as an S-algebra or a symmetric ring spectrum and the category of its module spectra is a stable model category. The ring of homotopy groups of K(n) is the gradedseld F p [v... |

27 | Model category structures on chain complexes of sheaves - Hovey |

26 | Generalized Etale Cohomology Theories - Jardine - 1997 |

25 |
Stable homotopy of algebraic theories, Topology 40
- Schwede
- 2001
(Show Context)
Citation Context ... Stable homotopy of algebraic theories. Another motivation for this paper and an early instance of Theorem 3.1.1 came from the stabilization of the model category of algebras over an algebraic theory =-=[54-=-]. For every pointed algebraic theory T , the category of simplicial T -algebras is a simplicial model category so that one has a category Sp(T ) of (Bouseld-Friedlander type) spectra of T -algebras, ... |

24 | Equivalences of monoidal model categories
- Schwede, Shipley
(Show Context)
Citation Context ...eness results are then developed further in [62, 55]. Moreover, the results in this paper form a basis for developing an algebraic model for any rational stable model category. This is carried out in =-=[58]-=- and applied in [61, 19]. In order to carry out our program it is essential to have available a highly structured model for the category of spectra which admits a symmetric monoidal and homotopically ... |

24 | On the groups of H(Π,n - Eilenberg, Lane - 1953 |

22 |
Rings and Categories of Modules. Second edition
- Anderson, Fuller
- 1992
(Show Context)
Citation Context ... simplicial model category we use the notation X and X for the simplicial suspension and loop functors (i.e., the pointed tensor and cotensor of an object X with the pointed simplicial circle S 1 = [1]=@[1]); one should keep in mind that these objects may have the `wrong' homotopy type if X is not cobrant orsbrant respectively. Our notation for various kinds of morphism objects is as follows: the... |

21 |
A 1 -homotopy theory of schemes, Inst
- Morel, Voevodsky
- 1999
(Show Context)
Citation Context ...odel category. The localization of a stable model category is stable and localization preserves generators. Compactness need not be preserved, see Example 3.2 (iii). (vii) Motivic stable homotopy. In =-=[42, 66]-=- Morel and Voevodsky introduced the A 1 -local model category structure for schemes over a base. An associated stable homotopy category of A 1 -local T -spectra (where T = A 1 =(A 1 0) is the `Tate-sp... |

19 |
Replacing model categories with simplicial ones
- Dugger
(Show Context)
Citation Context ... [45, II.2]). This is not a big loss of generality; it is shown in [46] that every cobrantly generated, proper, stable model category is in fact Quillen equivalent to a simplicial model category. In [=-=11-=-], Dugger obtains the same conclusion under somewhat dierent hypotheses. In both cases the candidate is the category of simplicial objects over the given model category endowed with a suitable localiz... |

18 |
Quillen model structures for relative homological algebra
- Christensen, Hovey
(Show Context)
Citation Context ...d one can consider model categories of dierential graded modules over a dierential graded algebra, or even a `DGA with many objects', alias DG-categories [31]. (ii) Relative homological algebra. In [8=-=]-=-, Christensen and Hovey introduce model category structures for chain complexes over an abelian category based on a projective class. In the special case where the abelian category is modules over som... |

17 | Simplicial structures on model categories and functors
- Rezk, Schwede, et al.
(Show Context)
Citation Context ...variant under Quillen equivalences of model categories. For convenience we restrict our attention to simplicial model categories (see [45, II.2]). This is not a big loss of generality; it is shown in =-=[4-=-6] that every cobrantly generated, proper, stable model category is in fact Quillen equivalent to a simplicial model category. In [11], Dugger obtains the same conclusion under somewhat dierent hypoth... |

17 | Symmetric spectra and topological Hochschild homology, K-Theory 19(2 - Shipley - 2000 |

17 | Stable homotopical algebra and Γ-spaces - Schwede |

16 |
Homotopy theory of -spaces, spectra, and bisimplicial sets, in
- eld, Friedlander
- 1977
(Show Context)
Citation Context ...category of spectra in the sense of stable homotopy theory. The sphere spectrum is a compact generator. Many model categories of spectra have been constructed, for example by Bouseld and Friedlander [=-=4]-=-; Robinson [49, `spectral sheaves']; Jardine [28, `n-fold spectra']; Elmendorf, Kriz, Mandell and May [15, `coordinate free spectra', `L-spectra', `S-modules']; Hovey, Shipley and Smith [25, `symmetri... |

15 | A uniqueness theorem for stable homotopy theory
- Schwede, Shipley
(Show Context)
Citation Context ...ategories which are Quillen (or derived) equivalent to the derived category of a ring. Another result which is very closely related to this characterization of stable model categories can be found in =-=[57]-=- where we give necessary and sucient conditions for when a stable model category is Quillen equivalent to spectra, see also Example 3.2 (i). These uniqueness results are then developed further in [62,... |