### Abstract

Let M be a simplicial model category and J : M --> M a simplicial coaugmented functor. Given an object X, the assignment n bar right arrow J(n+1)X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J(s)X = tot(s)([n] bar right arrow J(n+1)X). An object X is called J-injective if it is a retract of JX in Ho(M) via the natural map. We show that certain homotopy limits of J-injective objects are J(s)-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) --> X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}(sgreater than or equal to0)-->{tot(S)X}(s greater than or equal to 0).

Original language | English |
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Pages (from-to) | 385-407 |

Number of pages | 22 |

Journal | Mathematische Zeitschrift |

Volume | 247 |

DOIs | |

Publication status | Published - Apr 2004 |

### Cite this

*Mathematische Zeitschrift*,

*247*, 385-407. https://doi.org/10.1007/s00209-003-0621-0

**Tower Techniques for Cosimplicial Resolutions.** / Chacholski, W.; Libman, Assaf.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 247, pp. 385-407. https://doi.org/10.1007/s00209-003-0621-0

}

TY - JOUR

T1 - Tower Techniques for Cosimplicial Resolutions

AU - Chacholski, W.

AU - Libman, Assaf

PY - 2004/4

Y1 - 2004/4

N2 - Let M be a simplicial model category and J : M --> M a simplicial coaugmented functor. Given an object X, the assignment n bar right arrow J(n+1)X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J(s)X = tot(s)([n] bar right arrow J(n+1)X). An object X is called J-injective if it is a retract of JX in Ho(M) via the natural map. We show that certain homotopy limits of J-injective objects are J(s)-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) --> X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}(sgreater than or equal to0)-->{tot(S)X}(s greater than or equal to 0).

AB - Let M be a simplicial model category and J : M --> M a simplicial coaugmented functor. Given an object X, the assignment n bar right arrow J(n+1)X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J(s)X = tot(s)([n] bar right arrow J(n+1)X). An object X is called J-injective if it is a retract of JX in Ho(M) via the natural map. We show that certain homotopy limits of J-injective objects are J(s)-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) --> X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}(sgreater than or equal to0)-->{tot(S)X}(s greater than or equal to 0).

U2 - 10.1007/s00209-003-0621-0

DO - 10.1007/s00209-003-0621-0

M3 - Article

VL - 247

SP - 385

EP - 407

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

ER -