## Simplicial Functors and Stable Homotopy Theory (1998)

Citations: | 30 - 0 self |

### BibTeX

@TECHREPORT{Lydakis98simplicialfunctors,

author = {Manos Lydakis},

title = {Simplicial Functors and Stable Homotopy Theory},

institution = {},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper) are well known: They are the FSPs, which were introduced in 1985 by Bokstedt [B]. There are interesting constructions with FSPs, e.g., topological cyclic homology, which can be considerably simplified and conceptualized using the smash product of simplicial functors [LS]. Note however that it is not clear that an E1-FSP has a commutative model, although the analogous statement is true for S-modules and symmetric spectra. This has, e.g., the disadvantage, that it is not clear how to give a model category structure to (or even how to define) MU-algebras using simplicial functors (although this can be done for modules over any any FSP, and algebras over any commutative FSP, by similar methods as in [Sch]). Returning to examining the special features of simplicial functors, in constrast to

### Citations

858 |
Algebraic Topology
- Spanier
- 1966
(Show Context)
Citation Context ...e given by (S n ) K i for some K i in S, and the maps between the E i are induced by maps between the K i . The conclusion follows from the Blakers-Massey theorem (the version given in Spanier's book =-=[S] as theo-=-rem 9.3.5 is sufficient) , which provides a constant c such that, for msk + c, the following two statements are true. First, there are natural isomorphisms between �� k\Gamman E i and �� k+mS ... |

358 |
Homotopical Algebra
- Quillen
- 1967
(Show Context)
Citation Context ...rms, a simplicial functor X from the simplicial category C to the simplicial category D is a family of maps of the form C(K;L) ! D(XK;XL), that preserve composition and units. This is the approach of =-=[Q]-=- (see section II.1), but the definition given there is too restrictive for our purposes. (There it is required that all simplicial categories have functors ? K and ?\Omega K, behaving like the functor... |

200 | Symmetric spectra
- Hovey, Smith
(Show Context)
Citation Context ...etric monoidal product corresponding to the smash product of spectra. Two solutions were found recently, namely the smash product of S-modules of [EKMM], and the smash product of symmetric spectra of =-=[HSS]-=-. Here we present another solution, the smash product of simplicial functors. As a byproduct, we obtain simpler and more descriptive constructions of the model structures on spectra. As another byprod... |

154 |
modules, and algebras in stable homotopy theory
- Elmendorf, Kriz, et al.
- 1997
(Show Context)
Citation Context ...to the model category of spectra, and which has a symmetric monoidal product corresponding to the smash product of spectra. Two solutions were found recently, namely the smash product of S-modules of =-=[EKMM]-=-, and the smash product of symmetric spectra of [HSS]. Here we present another solution, the smash product of simplicial functors. As a byproduct, we obtain simpler and more descriptive constructions ... |

42 | Algebraic K-Theory of Spaces - Waldhausen - 1984 |

33 |
Calculus I: The first derivative of pseudoisotopy theory, K-theory 4
- Goodwillie
- 1990
(Show Context)
Citation Context ...opy functors. As a final byproduct, we obtain a model-category version of the part of Goodwillie's calculus of homotopy functors having to do with "the linear approximation of a homotopy functor =-=at " [G1], without -=-the restriction on the relevant homotopy functors of being "stably excisive". 1. Introduction The problem of constructing a nice smash product of spectra is an old and well-known problem of ... |

29 | Calculus II: analytic functors, K-Theory 5 - Goodwillie - 1992 |

21 |
Calculus III: the Taylor series of a homotopy functor, in preparation
- Goodwillie
(Show Context)
Citation Context ...omotopy functors, cf. remark 4.6 and section 8), but interesting "metastable", as well as a whole tower of "higher stable" theories, which are related to the calculus of homotopy f=-=unctors [G1], [G2], [G3]-=- (this will appear in a future paper). Finally, simplicial functors can be not only smashed, but also composed, with each other. There is a canonical map from the smash product to the composition prod... |

20 | On closed categories of functors, Reports of the Midwest Category Seminar - Day - 1970 |

18 |
Homotopy theory of \Gamma-spaces, spectra, and bisimplicial sets
- Bousfield, Friedlander
- 1978
(Show Context)
Citation Context ...unctors. In section 9, we construct the stable model structure on simplicial functors, which we later show is Quillen-equivalent to the stable model structure on spectra. The latter was introduced in =-=[BF]-=-, but we give a self-contained account of it in section 10, together with all the facts of stable homotopy theory that we need. In section 11, we compare the stable model structures. In section 12, we... |

7 |
Smash products and Gamma-spaces
- Lydakis
- 1999
(Show Context)
Citation Context ...phism in an important special case (proposition 5.13), a fact that provides a very concrete description of the smash product. There is a close connection between simplicial functors and \Gamma-spaces =-=[L2]-=-. In fact, \Gamma-spaces may be identified with a special kind of simplicial functors (see convention 2.11 of [L2]) in such a way that the smash product of \Gamma-spaces corresponds to the smash produ... |

3 | Homotopy limits of categories - Lydakis - 1994 |

1 |
M.: Topological Hochschild homology
- okstedt
- 1985
(Show Context)
Citation Context ... "algebras over the sphere spectrum", after thinking of simplicial functors as spectra by using the results of this paper) are well known: They are the FSPs, which were introduced in 1985 by=-= Bokstedt [B]-=-. There are interesting constructions with FSPs, e.g., topological cyclic homology, which can be considerably simplified and conceptualized using the smash product of simplicial functors [LS]. Note ho... |

1 | S.: Topological Hochschild homology and \Gamma-spaces - Lydakis, Schwede |

1 | Stable homotopical algebra and \Gamma-spaces
- Schwede
(Show Context)
Citation Context ...ory structure to (or even how to define) MU-algebras using simplicial functors (although this can be done for modules over any any FSP, and algebras over any commutative FSP, by similar methods as in =-=[Sch]-=-). Returning to examining the special features of simplicial functors, in constrast to S-modules and symmetric spectra, simplicial functors have not only an interesting unstable theory (modeling certa... |