Homology, Homotopy and Applications

Weak (co)fibrations in categories of (co)fibrant objects

R. W. Kieboom, G. Sonck, T. Van der Linden, and P. J. Witbooi

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Abstract

We introduce a fibre homotopy relation for maps in a category of cofibrant objects equipped with a choice of cylinder objects. Weak fibrations are defined to be those morphisms having the weak right lifting property with respect to weak equivalences. We prove a version of Dold's fibre homotopy equivalence theorem and give a number of examples of weak fibrations. If the category of cofibrant objects comes from a model category, we compare fibrations and weak fibrations, and we compare our fibre homotopy relation, which is defined in terms of left homotopies and cylinders, with the fibre homotopy relation defined in terms of right homotopies and path objects. We also dualize our notion of weak fibration in a category of cofibrant objects to a notion of weak cofibration in a category of fibrant objects, and give examples of these weak cofibrations. A section is devoted to the case of chain complexes in an abelian category.

Article information

Source
Homology Homotopy Appl., Volume 5, Number 1 (2003), 345-386.

Dates
First available in Project Euclid: 13 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.hha/1139839938

Mathematical Reviews number (MathSciNet)
MR2006405

Zentralblatt MATH identifier
1071.18008

Subjects
Primary: 18D30: Fibered categories
Secondary: 18G55: Homotopical algebra 55U40: Topological categories, foundations of homotopy theory

Citation

Kieboom, R. W.; Sonck, G.; Van der Linden, T.; Witbooi, P. J. Weak (co)fibrations in categories of (co)fibrant objects. Homology Homotopy Appl. 5 (2003), no. 1, 345--386. https://projecteuclid.org/euclid.hha/1139839938


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