### Abstract

For a coaugmented functor J on spaces, we consider J-modules and finite J-limits. The former are spaces X which are retracts of JX via the natural map. The latter are homotopy limits of J-modules arranged in diagrams whose shape is finite dimensional. Familiar examples are generalised Eilenberg MacLane spaces, which are the SPinfinity-modules. Finite SPinfinity-limits are nilpotent spaces with a very strong finiteness property. We show that the cofacial Bousfield-Kan construction of the functors J(n) is universal for finite J-limits in the sense that every map X --> Y where Y is a finite J-limit, factors through such natural map X --> J(n)X, for some n < infinity. (C) 2002 Elsevier Science Ltd. All rights reserved.

Original language | English |
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Pages (from-to) | 555-568 |

Number of pages | 13 |

Journal | Topology |

Volume | 42 |

DOIs | |

Publication status | Published - 2003 |

### Keywords

- homotopy limits
- coaugmented functors
- Bousfield-Kan spectral sequence

### Cite this

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

Research output: Contribution to journal › Article

*Topology*, vol. 42, pp. 555-568. https://doi.org/10.1016/S0040-9383(02)00019-8

}

TY - JOUR

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

AU - Libman, Assaf

PY - 2003

Y1 - 2003

N2 - For a coaugmented functor J on spaces, we consider J-modules and finite J-limits. The former are spaces X which are retracts of JX via the natural map. The latter are homotopy limits of J-modules arranged in diagrams whose shape is finite dimensional. Familiar examples are generalised Eilenberg MacLane spaces, which are the SPinfinity-modules. Finite SPinfinity-limits are nilpotent spaces with a very strong finiteness property. We show that the cofacial Bousfield-Kan construction of the functors J(n) is universal for finite J-limits in the sense that every map X --> Y where Y is a finite J-limit, factors through such natural map X --> J(n)X, for some n < infinity. (C) 2002 Elsevier Science Ltd. All rights reserved.

AB - For a coaugmented functor J on spaces, we consider J-modules and finite J-limits. The former are spaces X which are retracts of JX via the natural map. The latter are homotopy limits of J-modules arranged in diagrams whose shape is finite dimensional. Familiar examples are generalised Eilenberg MacLane spaces, which are the SPinfinity-modules. Finite SPinfinity-limits are nilpotent spaces with a very strong finiteness property. We show that the cofacial Bousfield-Kan construction of the functors J(n) is universal for finite J-limits in the sense that every map X --> Y where Y is a finite J-limit, factors through such natural map X --> J(n)X, for some n < infinity. (C) 2002 Elsevier Science Ltd. All rights reserved.

KW - homotopy limits

KW - coaugmented functors

KW - Bousfield-Kan spectral sequence

U2 - 10.1016/S0040-9383(02)00019-8

DO - 10.1016/S0040-9383(02)00019-8

M3 - Article

VL - 42

SP - 555

EP - 568

JO - Topology

JF - Topology

SN - 0040-9383

ER -