Axiomatic stable homotopy theory

Hovey, Mark; Palmieri, John H.; Strickland, Neil P.

doi:10.1090/memo/0610

Cited by 279 publications

(474 citation statements)

References 44 publications

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“…On the other hand, in either Ho Sp Σ or Ho Sp N , X ∧Y is determined by the collection of F ∧Y for all finite spectra F mapping to X. To be precise, X∧Y is the minimal weak colimit [HPS97] of the F ∧Y . As an equivalence of categories, RU preserves minimal weak colimits, so there is an isomorphism RU (X ∧ Y ) ∼ = (RU )(X) ∧ (RU )(Y ).…”

Section: Proof We Inductively Define a New Sequence C I And Mapsmentioning

confidence: 99%

Symmetric spectra

Hovey

Shipley

Smith³

1999

J. Amer. Math. Soc.

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476

569

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The stable homotopy category, much studied by algebraic topologists, is a closed symmetric monoidal category. For many years, however, there has been no well-behaved closed symmetric monoidal category of spectra whose homotopy category is the stable homotopy category. In this paper, we present such a category of spectra; the category of symmetric spectra. Our method can be used more generally to invert a monoidal functor, up to homotopy, in a way that preserves monoidal structure. Symmetric spectra were discovered at about the same time as the category of S S -modules of Elmendorf, Kriz, Mandell, and May, a completely different symmetric monoidal category of spectra.

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Section: Proof We Inductively Define a New Sequence C I And Mapsmentioning

confidence: 99%

Symmetric spectra

Hovey

Shipley

Smith³

1999

J. Amer. Math. Soc.

Self Cite

476

569

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“…In this section, we recall some axiomatic stable homotopy theory from [HPS97] and apply it to our situation.…”

Section: Noetherian Stable Homotopy Theorymentioning

confidence: 99%

“…For a general discussion of localization, see [HPS97,Chapter 3]. In this paper, we will use one kind of Bousfield localization, called "finite localization".…”

Section: Noetherian Stable Homotopy Theorymentioning

confidence: 99%

“…In this paper, we will use one kind of Bousfield localization, called "finite localization". Theorem 2.3 (Theorem 3.3.3 in [HPS97]). Let C be a bigraded Noetherian stable homotopy category, and let A be a tensor-closed thick subcategory of finite objects of C. There is a functor L A : C → C with the following properties:…”

Section: Noetherian Stable Homotopy Theorymentioning

confidence: 99%

“…In any bigraded Noetherian stable homotopy category C, we have the following equality of Bousfield classes, from Theorem 6.1.9 of [HPS97]:…”

Section: The Tensor Product Propertymentioning

confidence: 99%

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Stably thick subcategories of modules over Hopf algebras

Hovey¹,

Palmieri²

2001

Math. Proc. Camb. Phil. Soc.

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Abstract. We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional cocommutative Hopf algebra B. Our method is based on that of , who classify such subcategories when B = kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of F 2 . Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

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Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta

Christensen¹

1998

Advances in Mathematics

121

135

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We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case of A modules over an A ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps from X to Y can always be described as a 1 group. The last two sections focus on algebraic examples. In the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of Kelly on the vanishing of composites of such maps. We also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non-trivially. Academic Press

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Axiomatic stable homotopy theory

Cited by 279 publications

References 44 publications

Symmetric spectra

Symmetric spectra

Stably thick subcategories of modules over Hopf algebras

Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta

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