## Modern foundations of stable homotopy theory. Handbook of Algebraic Topology, edited by (1995)

Citations: | 22 - 7 self |

### BibTeX

@MISC{Elmendorf95modernfoundations,

author = {A. D. Elmendorf and I. Kriz and M. A. Mandell and J. P. May},

title = {Modern foundations of stable homotopy theory. Handbook of Algebraic Topology, edited by},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

2. Smash products and twisted half-smash products 11

### Citations

921 | Categories for the working mathematician - Lane - 1998 |

474 | Categories for the working mathematician - MacLane - 1998 |

325 | Homotopical algebra - Quillen - 1967 |

258 | The geometry of iterated loop spaces - May - 1972 |

118 |
modules, and algebras in stable homotopy theory
- Rings
- 1997
(Show Context)
Citation Context ...sight that these up to homotopy notions are the only ones possible. It is a recent discovery that there is a category MS of S-modules that has an associative, commutative, and unital smash product ∧S =-=[11]-=-. Its objects are spectra with additional structure, and we say that a map of S-modules is a weak equivalence if it is a weak equivalence as a map of spectra. The derived category DS is obtained from ... |

68 |
Equivariant Homotopy and Cohomology Theory. Volume 91
- May
- 1996
(Show Context)
Citation Context ... MANDELL, AND J. P.MAY 1. Spectra and the stable homotopy category We here give a bare bones summary of the construction of the stable homotopy category, referring to [16] and [11] for details and to =-=[22]-=- for a more leisurely exposition. We aim to give just enough of the basic definitional framework that the reader can feel comfortable with the ideas. By Brown’s representability theorem [6], if E ∗ is... |

49 |
Conditionally convergent spectral sequences
- Boardman
(Show Context)
Citation Context ...l sequence is of standard cohomological type, with dr : E p,q r → E p+r,q−r+1 r .32 A. D. ELMENDORF, I. KRIZ, M. A. MANDELL, AND J. P.MAY It lies in the right half plane. In the language of Boardman =-=[5]-=- (see also [12, App B]), it is conditionally convergent. It therefore converges strongly if, for each fixed (p, q), there are only finitely many r such that dr is non-zero on E p,q r . Setting M = FRX... |

46 | On the formal group laws of unoriented and complex cobordism theory
- Quillen
- 1969
(Show Context)
Citation Context ...rate by explaining how BP appears in this context. Fix a prime p and write (?)p for localization at p. Let BP be the Brown-Peterson spectrum at p. We are thinking of Quillen’s idempotent construction =-=[24]-=-, and we have the splitting maps i : BP −→ MUp and e : MUp −→ BP. These are maps of commutative and associative ring spectra such that e ◦ i = id. Let I be the kernel of the composite MU∗ −→ MUp∗ −→ B... |

44 | Generalized Tate cohomology - Greenlees, May - 1995 |

39 |
with contributions by
- May
- 1977
(Show Context)
Citation Context ...he point-set level; we say that R is commutative if the standard commutativity diagram also commutes. There were earlier notions with a similar flavor, namely the A∞ and E∞ ring spectra introduced in =-=[19, 20]-=-. Here “A∞” stands historically for “associative up to an infinite sequence of higher homotopies”; similarly, “E∞” stands for “homotopy everything”, meaning that the product is associative and commuta... |

35 | H∞ ring spectra and their applications - Bruner, May, et al. - 1986 |

31 |
Stable Homotopy and Generalized Homology
- Adams
- 1974
(Show Context)
Citation Context ...n ad hoc calculational hypothesis that requires case-by-case verification. It covers some cases that are not covered by the results above, and conversely. The cited paper of Adams, and his later book =-=[2]-=-, are prime sources for the first flowering of stable homotopy theory. While some of their foundational parts may be obsolete, their applications and calculational parts certainly are not. [11]. The f... |

30 |
Abstract homotopy theory
- Brown
- 1965
(Show Context)
Citation Context ...ls and to [22] for a more leisurely exposition. We aim to give just enough of the basic definitional framework that the reader can feel comfortable with the ideas. By Brown’s representability theorem =-=[6]-=-, if E ∗ is a reduced cohomology theory on based spaces, then there are CW complexes En such that, for CW complexes X, E n (X) is naturally isomorphic to the set [X, En] of homotopy classes of based m... |

30 |
On the theory and applications of differential torsion products
- Gugenheim, May
- 1974
(Show Context)
Citation Context ...R, we find spectral sequences for the calculation of our Tor and Ext groups that are analogous to the EilenbergMoore (or hyperhomology) spectral sequences in differential homological algebra. Compare =-=[9, 13, 15]-=-. They may be viewed as giving universal coefficient and Künneth spectral sequences for homology and cohomology theories on R-modules, and they specialize to give such spectral sequences for homology ... |

26 | Multiplicative infinite loop space theory
- May
(Show Context)
Citation Context ...he point-set level; we say that R is commutative if the standard commutativity diagram also commutes. There were earlier notions with a similar flavor, namely the A∞ and E∞ ring spectra introduced in =-=[19, 20]-=-. Here “A∞” stands historically for “associative up to an infinite sequence of higher homotopies”; similarly, “E∞” stands for “homotopy everything”, meaning that the product is associative and commuta... |

22 |
Catégories dérivées
- Verdier
- 1977
(Show Context)
Citation Context ...HN) coincides with the algebraic Yoneda product. The proof is clear enough: we just check the axioms for Tor and Ext. We can elaborate this result to an equivalence of derived categories. Recall from =-=[28]-=- or [15, Ch.III] that the derived category DR is obtained from the homotopy category of chain complexes over R by formally inverting the quasi-isomorphisms, exactly as we obtained the category DHR fro... |

18 |
Elements of homotopy theory, Springer-Verlag, Graduate Texts in Mathematics, 61
- Whitehead
- 1978
(Show Context)
Citation Context ...category of spectra is a map E ∧ I+ −→ E ′ . We have cofibration and fibration □8 A. D. ELMENDORF, I. KRIZ, M. A. MANDELL, AND J. P.MAY sequences that are defined exactly as on the space level (e.g. =-=[29]-=-) and enjoy the same homotopical properties. Let [E, E ′] denote the set of homotopy classes of maps E −→ E ′; we shall later understand that, when using this notation, E must be of the homotopy type ... |

15 |
Lectures on generalised cohomology
- Adams
- 1968
(Show Context)
Citation Context ... sequences. Theorem 9.3 (Künneth). For any spectra X and Y , there are spectral sequences of the form and Tor R∗ ∗,∗ (R∗(X), R∗(Y )) =⇒ R∗(X ∧ Y ) Ext ∗,∗ R ∗(R−∗(X), R ∗ (Y )) =⇒ R ∗ (X ∧ Y ). Adams =-=[1]-=- first observed that one can derive Künneth spectral sequences from universal coefficient spectral sequences, and he observed that, by duality, the four spectral sequences of Theorems 9.2 and 9.3 impl... |

13 | Commutative algebra in stable homotopy theory and a completion theorem - Elmendorf, Greenlees, et al. - 1994 |

13 |
with contributions by J.E. McClure). Equivariant stable homotopy theory
- Lewis, May, et al.
- 1986
(Show Context)
Citation Context ...6 A. D. ELMENDORF, I. KRIZ, M. A. MANDELL, AND J. P.MAY 1. Spectra and the stable homotopy category We here give a bare bones summary of the construction of the stable homotopy category, referring to =-=[16]-=- and [11] for details and to [22] for a more leisurely exposition. We aim to give just enough of the basic definitional framework that the reader can feel comfortable with the ideas. By Brown’s repres... |

12 | Derived tensor products in stable homotopy theory, Topology 22 - Robinson - 1983 |

11 |
A variant of E.H. Brown’s representability theorem. Topology 10
- Adams
- 1971
(Show Context)
Citation Context ...grams of spectra; the latter require application of the functor S ∧L (?). Thus we have enough information to quote the categorical form of Brown’s representability theorem given in [6]. Adams’ analog =-=[3]-=- for functors defined only on finite CW spectra also applies in our context, with the same proof. Theorem 6.6 (Brown). A contravariant functor k : DR → Sets is representable in the form k(M) ∼ = DR(M,... |

10 | The extraordinary derived category - Robinson - 1987 |

9 | Spectra of derived module homomorphisms - Robinson - 1987 |

2 | Towers of E1 -ring spectra with an application to BP , preprint - Kriz - 1993 |

2 |
Derived Categories in Algebra and Topology
- May
(Show Context)
Citation Context ...ules is a chain homotopy equivalence (Whitehead theorem) and every chain complex is quasi-isomorphic to a cell R-module. Then DR is equivalent to the ordinary homotopy category of cell R-modules. See =-=[15, 21]-=-. This is a topologist’s way of thinking about the appropriate generalization to chain complexes of projective resolutions of modules. We think of the sphere spectrum S as the analog of R. We think of... |

1 | Stable homotopy theory. Thesis, Warwick 1964; mimeographed notes from Warwick and Johns Hopkins Universities - Boardman |

1 |
Homotopy theories and model categories. This volume
- Dwyer, Spalinski
(Show Context)
Citation Context ... converts homotopy pushouts to weak pullbacks of underlying sets. In fact, Brown’s theorem is the kind of formal result that can be derived in any (closed) model category in the sense of Quillen (see =-=[8]-=- for a good exposition), and we have the following result. Serre fibrations of spectra are maps that satisfy the covering homotopy property with respect to the set of cone spectra {Σ ∞ q CS n |q ≥ 0 a... |

1 |
Towers of E∞ ring spectra with an application to BP
- Kriz
- 1993
(Show Context)
Citation Context ...unnecessary to appeal to Baas-Sullivan theory or to Landweber’s exact functor theorem for the construction and analysis of spectra such as these. With more sophisticated techniques, the second author =-=[14]-=- has proven that BP can be constructed as a commutative S-algebra, and in fact admits uncountably many distinct such structures. There is much other ongoing work on the construction and application of... |