Closed model category structures on the category of chain functors (Q1398194)

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.