The Hurewicz model structure on simplicial \(R\)-modules
For a nice bicomplete abelian category \(\mathcal{A}\), two of the well-known standard model structures on the category \(\mbox{Ch}(\mathcal{A})\) of unbounded chain complexes of objects of \(\mathcal{A}\) are the projective model structure, for which weak equivalences are quasi-isomorphisms and fibrations are degreewise epimorphisms, and the Hurewicz model structure, for which weak equivalences are chain homotopy equivalences and fibrations are degreewise split epimorphisms. \N\NThe Hurewicz model structure on \(\mbox{Ch}(\mathcal{A})\) was introduced by \textit{M. Golasiński} and \textit{G. Gromadzki} [Colloq. Math. 47, 173--178 (1982; Zbl 0532.55025)]. It was later also constructed using different methods by e.g., [\textit{J. D. Christensen} and \textit{M. Hovey}, Math. Proc. Camb. Philos. Soc. 133, No. 2, 261--293 (2002; Zbl 1016.18008)]. \textit{D. G. Quillen} [Homotopical algebra. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0168.20903)] introduced the projective model structure on the category \(\mbox{Ch}_{\ge 0}(\mathcal{A})\) of non-negatively graded chain complexes of objects of \(\mathcal{A}\), i.e., with quasi-isomorphisms as weak equivalences and degreewise monomorphisms with degreewise projective cokernel as cofibrations. \N\NThe Dold-Kan correspondence induces model structures on the category of simplicial \(R\)-modules \(s\mbox{Mod}_R\). The author presents a description of the two induced model categories on \(s\mbox{Mod}_R\) and some of their properties, notably the fact that both are monoidal. Furthermore, fibrations and cofibrations are characterized in terms of homotopy lifting and homotopy extension, respectively.
scientific article; zbMATH DE number 7944385
scientific article