Closed model category structures on the category of chain functors
type: Item
instance of: scholarly article
title: Closed model category structures on the category of chain functors
zbMATH Open document ID: 1027.55018
DOI: 10.1016/S0166-8641(02)00295-X
author: Tamar Datuashvili
author: Friedrich Wilhelm Bauer
published in: Topology and its Applications
publication date: 2003-07-29T00:00:00Z
review text: A chain functor is a pair of functors from the topological or simplicial category into the category of chain complexes, together with some additional data (see [\textit{F. W. Bauer}, Homology Homotopy Appl. 3, 37-53 (2001; Zbl 0971.55007)]. They are used to calculate homology groups of a given homology theory by means of chains, cycles and boundaries as in the case of ordinary singular, simplicial or cellular homology. In the category \(\mathfrak C h\) of chain functors, the authors define fibrations, cofibrations and weak equivalences verifying all the properties of a close model category [\textit{D. G. Quillen}, Homotopical algebra, Lect. Notes Math. 43 Springer, Berlin-N.Y. (1967; Zbl 0168.20903)], except for the existence of finite limits and colimits: there exist finite products and sums in \(\mathfrak C h\), however, if \(f\) is a morphism in \(\mathfrak C h\), then ker \(f\) is not necessarily a chain functor. The same occurs with cokernels. Nevertheless they show that there are significant cases where kernels and cokernels exist: there exist suspension and loop functors, which are invertible and, up to isomorphism, inverses to each other, turning the homotopy category \(\mathfrak C h_h\) into a stable category.
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 18C35
Mathematics Subject Classification ID: 55U15
Mathematics Subject Classification ID: 55U40
zbMATH DE Number: 1955988
zbMATH Keywords: chain functors
zbMATH Keywords: Kan and Hurewicz fibrations
zbMATH Keywords: cofibrations
reviewed by: J. Remedios
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1955988