"Strøm-type" model structure on chain complexes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:53:25Z http://mathoverflow.net/feeds/question/52508 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52508/strom-type-model-structure-on-chain-complexes "Strøm-type" model structure on chain complexes? John Klein 2011-01-19T12:34:59Z 2011-02-04T02:29:55Z <p><b> Background </b></p> <p>The <i> Quillen model structure </i> on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by the lifting property, but in the end they are those inclusions which are built up by cell attachments (or are retracts of such things).</p> <p>The <i> Strøm model structure </i> on spaces has weak equivalences = the homotopy equivalences and fibrations the Hurewicz fibrations. Cofibrations the closed inclusions which satisfy the homotopy extension property. </p> <p>The <i> projective model structure </i> on non-negatively graded chain complexes over \$\Bbb Z\$ has fibrations given by the degreewise surjections and weak equivalences given by the quasi-isomorphisms. Cofibrations are given by the degree-wise split inclusions such that the quotient complex is degree-wise free. </p> <p>From the above, it would appear to me that the projective model structure on chain complexes is analogous to the Quillen model structure on spaces.</p> <p><b> Question </b></p> <p><i> Is there a model structure on (non-negatively graded) chain complexes over \$\Bbb Z\$ for which the weak equivalences are the chain homotopy equivalences? </i></p> <p>(Extra wish: I want cofibrations in the projective model structure to be a sub-class of the cofibrations in the model structure answering the question. Conjecturally, they should be the inclusions satisfying the chain homotopy extension property.) </p> http://mathoverflow.net/questions/52508/strom-type-model-structure-on-chain-complexes/54279#54279 Answer by Peter May for "Strøm-type" model structure on chain complexes? Peter May 2011-02-04T02:29:55Z 2011-02-04T02:29:55Z <p>There are several other papers, I think earlier ones, that cover this. </p> <p>[32] M. Cole. The homotopy category of chain complexes is a homotopy category. Preprint (1990's)</p> <p>[29] J. Daniel Christensen and Mark Hovey. Quillen model structures for relative homological algebra. Math. Proc. Cambridge Philos. Soc., 133(2):261–293, 2002.</p> <p>[121] R. Schw¨anzl and R. M. Vogt. Strong coﬁbrations and ﬁbrations in enriched categories. Arch. Math. (Basel), 79(6):449–462, 2002. </p> <p>The references are from the book ``More concise algebraic topology: localizations, completions, and model categories'', by Kate Ponto and myself. It will be published this year by the University of Chicago Press. It includes and compares three natural model structures on spaces and chain complexes, the Strom model structure, the Quillen model structure, and Cole's mixed model structures:</p> <p>33] Michael Cole. Mixing model structures. Topology Appl., 153(7):1016–1032, 2006.</p>