## JOURNAL OF PURE AND APPLIED ALGEBRA Comparison of the geometric bar and W-constructions (1995)

### BibTeX

@MISC{Berger95journalof,

author = {Clemens Berger and Johannes Huebschmann},

title = {JOURNAL OF PURE AND APPLIED ALGEBRA Comparison of the geometric bar and W-constructions},

year = {1995}

}

### OpenURL

### Abstract

Dedicated to the memory of V.K.A.M. Gugenheim For a simplicial group K, the realization of the W-construction WK + WK of K is shown to be naturally homeomorphic to the universal bundle E]K]--t BIK of its geometric realization]Kl. The argument involves certain recursive descriptions of the W-construction and classifying bundle and relies on the facts that the realization functor carries an action of a simplicial group to a geometric action of its realization and preserves reduced cones and colimits. @ 1998 Elsevier Science B.V. All rights reserved. AMS Cluss$ccrrior~: 55405; 55P35; I8G30

### Citations

146 |
Categories and cohomology theories, Topology 13
- Segal
- 1974
(Show Context)
Citation Context ...m the corresponding total space to IINGII is always locally trivial whether or not (G, e) is an NDR. Below (G, e) will always be a CW-pair and hence an NDR, cf. e.g. the discussion in the appendix to =-=[22]-=-, and we shall deal exclusively with the lean realization BG = INGI. To reproduce a description thereof, and to introduce notation, write d for the category of finite ordered sets [q] = (0, 1,. . . q)... |

28 | On the Chern-Weil homomorphism and the continuous cohomology of Lie groups - Bott - 1973 |

28 | On the de Rham theory of certain classifying spaces - Bott, Shulman, et al. - 1976 |

20 |
Homotopy Theory of Simplicial Groupoids
- Dwyer, Kan
- 1984
(Show Context)
Citation Context ... the present paper to simplicial groupoids, so that a result of the kind given in the corollary would follow for an arbitrary connected simplicial set, with the Kan group replaced by the Kan groupoid =-=[5]-=-. Such an extension would have to rely on correct descriptions of the requisite monads for groupoid actions and conical contractions in the general non-reduced setting. We hope to return to this issue... |

4 | On the groups H(#,n - Eilenberg, Lane - 1954 |

1 |
Une bewon cH’ective du thtor&nc de Hurewcr. These de doctorat. Unibersiti- de Cirenoble
- Bcrgcr
(Show Context)
Citation Context ... appropriate sense. See [2.5] for details. This kind of universal construction is available whenever one is given an algebra structure over a monad preserving push out diagrams. The unit interval I = =-=[0, I]-=- is a topological monoid under ordinary multiplication having 1 as its unit, and hence we can talk about an l-action X x I +X on a space X. Such an l-action is plainly a special kind of homotopy which... |

1 | Simplicial homotopy theory - Curtw - 1971 |

1 | On the groups H(rr.u - Lane - 1953 |

1 | On the chain complex of a fibration - Ciuyenhcim - 1972 |

1 | On the theory and appllcatlons of ditTerential torsion products - May - 1974 |

1 | C‘omparison de la bar construction h la construction It’ et aux complexes K( rc.,r). ExposC1 13, Si-minalre Il. Cartan ( 1954:1955) 242 250. 1191 D. Puppe, Ilomotopie und Iiomologie in abelschcn Gruppen und Monoldkomplexen - Moore - 1958 |

1 |
Classifying spaces and spectral sequences
- Sepal
- 1968
(Show Context)
Citation Context ...on, Mac Lane observed in [ 131 that ING] coincides with the classifying space for G constructed by Stasheff [23] and Milgram [15]; see also Section 1 of Stashefl’s survey paper [24] and Segal’s paper =-=[21]-=-. Mac Lane actually worked with a variant of the category d which enabled him to handle simultaneously the total space EC and the base BG. Steenrod [25] gave a recursive description of ING] which we s... |

1 |
H-spaces and classifying spaces: foundations and recent developments, in
- Stashell
- 1971
(Show Context)
Citation Context ...ploiting this observation, Mac Lane observed in [ 131 that ING] coincides with the classifying space for G constructed by Stasheff [23] and Milgram [15]; see also Section 1 of Stashefl’s survey paper =-=[24]-=- and Segal’s paper [21]. Mac Lane actually worked with a variant of the category d which enabled him to handle simultaneously the total space EC and the base BG. Steenrod [25] gave a recursive descrip... |

1 |
Comparison between the reduced bar construction and the reduced W-construction, Dlplomarbelt. Math. lnstitut der Universitiit
- Wang
(Show Context)
Citation Context ...ction BlAl of the normalized chain algebra IAl of A and raised the question whether this reduction is in fact part of a contraction. By means of homological perturbation theory, in his “Diplomarbeit” =-=[27]-=- supervised by the second named author, Wong answered this question by establishing such a contraction. Wong’s basic tool is the “perturbation lemma” exploited in [8]; see [IO] for details and history... |