### Abstract

A contraction for a cosimplicial resolution X-1 --> X-. is an "extra codegeneracy map", and the existence of such, is well known to induce a homotopy equivalence between the augmentation and the total space of the resolution. We generalise and strengthen this result by considering cofacial cosimplicial resolutions of length n of diagrams of spaces. We show that if X-1 is a P-diagram and dim P less than or equal to n, and the cofacial resolution X-. admits termwise contractions, then holim X-1 is a retract of tot, holim(p)X(.), and that the tower map {holimX(-1)} --> {tot(n)holim(p)X(.)}(n) is a pro-equivalence in the homotopy category of spaces. (C) 2002 Elsevier Science Ltd. All rights reserved.

Original language | English |
---|---|

Pages (from-to) | 569-602 |

Number of pages | 33 |

Journal | Topology |

Volume | 42 |

DOIs | |

Publication status | Published - 2003 |

### Keywords

- homotopy limits
- cosimplicial resolutions
- contractions
- SPECTRAL SEQUENCE
- HOMOLOGY

### Cite this

**Universal Spaces for Homotopy Limits of Modules over Coaugmented Functions II.** / Libman, Assaf.

Research output: Contribution to journal › Article

*Topology*, vol. 42, pp. 569-602. https://doi.org/10.1016/S0040-9383(02)00020-4

}

TY - JOUR

T1 - Universal Spaces for Homotopy Limits of Modules over Coaugmented Functions II

AU - Libman, Assaf

PY - 2003

Y1 - 2003

N2 - A contraction for a cosimplicial resolution X-1 --> X-. is an "extra codegeneracy map", and the existence of such, is well known to induce a homotopy equivalence between the augmentation and the total space of the resolution. We generalise and strengthen this result by considering cofacial cosimplicial resolutions of length n of diagrams of spaces. We show that if X-1 is a P-diagram and dim P less than or equal to n, and the cofacial resolution X-. admits termwise contractions, then holim X-1 is a retract of tot, holim(p)X(.), and that the tower map {holimX(-1)} --> {tot(n)holim(p)X(.)}(n) is a pro-equivalence in the homotopy category of spaces. (C) 2002 Elsevier Science Ltd. All rights reserved.

AB - A contraction for a cosimplicial resolution X-1 --> X-. is an "extra codegeneracy map", and the existence of such, is well known to induce a homotopy equivalence between the augmentation and the total space of the resolution. We generalise and strengthen this result by considering cofacial cosimplicial resolutions of length n of diagrams of spaces. We show that if X-1 is a P-diagram and dim P less than or equal to n, and the cofacial resolution X-. admits termwise contractions, then holim X-1 is a retract of tot, holim(p)X(.), and that the tower map {holimX(-1)} --> {tot(n)holim(p)X(.)}(n) is a pro-equivalence in the homotopy category of spaces. (C) 2002 Elsevier Science Ltd. All rights reserved.

KW - homotopy limits

KW - cosimplicial resolutions

KW - contractions

KW - SPECTRAL SEQUENCE

KW - HOMOLOGY

U2 - 10.1016/S0040-9383(02)00020-4

DO - 10.1016/S0040-9383(02)00020-4

M3 - Article

VL - 42

SP - 569

EP - 602

JO - Topology

JF - Topology

SN - 0040-9383

ER -